Reputational Bargaining with Minimal Knowledge of
Rationality
Alexander Wolitzky
Stanford University and Microsoft Research
August 23, 2011
Abstract
Two players announce bargaining postures to which they may become committed
and then bargain over the division of a surplus. The share of the surplus that a player
can guarantee herself under …rst-order knowledge of rationality is determined (as a
function of her probability of becoming committed), as is the bargaining posture that
she must announce in order to guarantee herself this much. This “maxmin” share of
the surplus is large relative to the probability of becoming committed (e.g., it equals
30% if the commitment probability is 1 in 10, and equals 13% if the commitment
probability is 1 in 1000), and the corresponding bargaining posture simply demands
this share plus compensation for any delay in reaching agreement.
Keywords: bargaining, knowledge of rationality, posturing, reputation
I thank the editor, Philippe Jehiel, and three anonymous referees for helpful comments.
I thank my
advisors, Daron Acemoglu, Glenn Ellison, and Muhamet Yildiz, for extensive discussions and comments as
well as continual advice and support.
For additional helpful comments, I thank Sandeep Baliga, Abhijit
Banerjee, Alessandro Bonatti, Je¤ Ely, Drew Fudenberg, Bob Gibbons, Michael Grubb, Peter Klibano¤,
Anton Kolotilin, Mihai Manea, Parag Pathak, Jeroen Swinkels, Juuso Toikka, Iván Werning, and (especially)
Gabriel Carroll, as well as many seminar audiences. I thank the NSF for …nancial support.
1
1
Introduction
Economists have long been interested in how individuals split gains from trade. The division
of surplus often determines not only equity, but also e¢ ciency, as it a¤ects individuals’ex
ante incentives to make investments; this e¤ect of surplus division on e¢ ciency is a major
theme of, for example, property rights theories of the …rm (Grossman and Hart, 1986),
industrial organization models of cumulative innovation (Green and Scotchmer, 1995), and
search-and-matching models of the labor market (Hosios, 1990). Recently, “reputational”
models of bargaining have been developed that make sharp prediction about the division
of the surplus independently of many details of the bargaining procedure (Myerson, 1991;
Abreu and Gul, 2000; Kambe, 1999; Compte and Jehiel, 2002). In these models, players
may be committed to a range of possible bargaining strategies, or “postures,”before the start
of bargaining, and bargaining consists of each player attempting to convince her opponent
that she is committed to a strong posture.
These models assume that the probabilities
with which the players are committed to various bargaining postures (either ex ante or after
a stage where players strategically announce bargaining postures) are common knowledge,
and that play constitutes a (sequential) equilibrium.
In this paper, I study reputational
bargaining while assuming only that the players know that each other is rational, and show
that each player can guarantee herself a relatively large share of the surplus— even if her
probability of being committed is small— by announcing the posture that simply demands
this share plus compensation for any delay in reaching agreement. Furthermore, announcing
any other posture does not guarantee as much.
A key feature of my model is the existence of a positive number " such that, if a player
announces any bargaining posture (i.e., any in…nite path of demands) at the beginning of
the game, she then becomes committed to that posture with probability at least " (or, equivalently, she convinces her opponent that she is committed to that posture with probability
at least "). I derive the highest payo¤ that a player can guarantee herself by announcing
any posture, regardless of her opponent’s beliefs about her bargaining strategy, so long as
her opponent is rational and believes that she is committed to her announced posture with
probability at least ". More precisely, player 1’s “highest guaranteed,”or “maxmin,”payo¤
2
is the highest payo¤ u1 with the property that there exists a corresponding posture (the
“maxmin posture”) and bargaining strategy such that player 1 receives at least u1 whenever
she announces this posture and follows this strategy and player 2 plays any best-response to
any belief about player 1’s strategy that assigns probability at least " to player 1 following
her announced posture.
The main result of this paper characterizes the maxmin payo¤ and posture when only one
player may become committed to her announced posture; as discussed below, a very similar
characterization applies when both players may become committed. While the maxmin
payo¤ may be very small when " is small in general two-person games, it is relatively large
in my model: in particular, it equals 1= (1
log "). This equals 1 when " = 1 (i.e., when
the player makes a take-it-or-leave-it o¤er) and, more interestingly, goes to 0 very slowly as
" goes to 0 (more precisely, it goes to 0 at a logarithmic rate). For example, a bargainer can
guarantee herself approximately 30% of the surplus if her commitment probability is 1 in
ten; 13% if it is 1 in 1 thousand; and 7% if it is 1 in 1 million. The second part of the main
result is that the unique bargaining posture that guarantees this share of the surplus simply
demands this share in addition to compensation for any delay; that is, it demands a share of
the surplus that increases at rate equal to the discount rate, r. This compensation amounts
to the entire surplus after a long enough delay, so the unique maxmin posture demands
min ert = (1
log ") ; 1
at every time t. This posture is depicted in Figure 1, for commitment probability " = 1=1000
and discount rate r = 1.
The intuition for why the maxmin payo¤ is large relative to " is that, when player 1’s
demand is small, player 2 must accept unless he believes that he will be rewarded with
high probability for rejecting. In the latter case, if player 1 does not reward player 2 for
rejecting, then player 2 quickly updates his belief toward player 1’s being committed to her
announced posture (i.e., player 1 builds reputation quickly), and player 2 accepts when he
becomes convinced that she is committed.1 Thus, player 1 is able to compensate for a small
1
This is related to the argument in the existing reputational bargaining literature that player 1 builds
reputation more quickly in equilibrium when her demand is small, though my analysis is not based on
equilibrium.
3
Demand
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Time
Figure 1: The Unique Maxmin Bargaining Posture for " = 1=1000 and r = 1
commitment probability by demanding (somewhat) less, thus avoiding a long delay before
her demand is accepted.
The intuition for why the unique maxmin posture demands compensation for delay involves two key ideas. First, as I have argued, player 1’s demand is accepted sooner when
it is lower (when player 2’s beliefs are those that lead him to reject for as long as possible).
Second, the maxmin posture can never make demands that would give player 1 less than
her maxmin payo¤ if they were accepted, because otherwise player 2 could simply accept
some such demand and give player 1 a payo¤ below her maxmin payo¤, which was supposed
to be guaranteed to player 1 (though it must be veri…ed that such behavior by player 2 is
rational).
Combining these ideas implies that player 1 must always demand at least her
maxmin level of utility (hence, compensation for delay), but no more.
I also characterize the maxmin payo¤s and postures when both players may become
committed to their announced postures. Each player’s maxmin posture is exactly the same as
in the one-sided commitment model, and each player’s maxmin payo¤ is close to her maxmin
payo¤ in the one-sided commitment model as long as her opponent’s commitment probability
is small. Thus, the one-sided commitment analysis applies to each player separately.
Three distinctive features of my approach are the timing of commitment (players freely
choose which bargaining postures to announce, but may become bound by their announce4
ments), the range of bargaining postures players may announce (all possible paths of demands
for the duration of bargaining), and the solution concept (…rst-order knowledge of rationality).
The timing of commitment is appropriate if players are rational but may credibly
announce bargaining postures.
This assumption has many precedents in the literature,
starting with Schelling (1956), who discusses observable factors that make announced postures more credible, corresponding to a higher value of " in my model.2 The assumption that
a player may announce any path of demands and that all such announcements are equally
credible seems unappealing a priori, because announcing a “simple” posture may be more
credible than announcing a “complicated”posture. Fortunately, the unique maxmin posture
is simply announcing, “I want a certain share of the surplus, and if you make me wait to get
it then you must compensate me for the delay.”3 Thus, allowing players to credibly announce
complicated postures ensures that my results do not depend on ad hoc restrictions on the
range of credible postures, but my results would still apply if only “simple” postures were
credible. I also provide a less explicit characterization of the maxmin payo¤ and posture in
a more general model where credibility does vary across announcements (cf Section 5.5).
A player’s maxmin payo¤ is her lowest payo¤ consistent with (…rst-order) knowledge of
rationality (at the start of the game). In bargaining, a player cannot guarantee herself any
positive payo¤ without knowledge of her opponent’s rationality, as, for example, she receives
payo¤ 0 if her opponent always rejects her o¤er and demands the entire surplus.
Thus,
knowledge of rationality is the weakest solution concept consistent with positive guaranteed
payo¤s.
Furthermore, I show that any feasible payo¤ greater than the maxmin payo¤ is
consistent with knowledge of rationality, which implies that the maxmin payo¤ and posture
2
For example, an announcement is more credible if the stakes in the current negotiation are small relative
to the stakes in potential future negotiations; if the announcement is observable to a large number of third
parties; if the bargainer can side-contract with third parties to bind herself to her announcement; if the
bargainer may be acting as an agent for a third party and does not have independent authority to change
her posture; or if the bargainer displays emotions that suggest an unwillingness to modify her posture.
3
To my knowledge, this is the …rst bargaining model that predicts that such a posture will be adopted,
though it seems like a reasonable bargaining position to stake out. For example, in most U.S. states defendants must pay “prejudgment interest” on damages in torts cases, which amounts to plainti¤s demanding
the initial damages in addition to compensation for any delay (e.g., Knoll, 1996); similarly, unions sometimes
include payment for strike days among their demands.
5
are the key objects of interest under knowledge of rationality.
Imposing only knowledge of rationality rather than a stronger solution concept, such as
rationalizability or equilibrium, leads to more robust predictions. In particular, predictions
under knowledge of rationality (such as the prediction that each player receives at least her
maxmin payo¤) do not depend on each player’s beliefs about her opponent’s strategies— so
long as each player believes that her opponent is playing a best-response to some belief
about her own play— and also do not depend on unmodelled strategic considerations that
do not a¤ect a player’s payo¤s or her beliefs about her opponent’s payo¤s, but may a¤ect
higher-order beliefs about payo¤s.
Such considerations arise naturally in bargaining: for
example, a local union that strategically announces that it will strike until o¤ered a wage of
at least $25 an hour may be concerned that the …rm’s management may believe that it is
actually required by the national union to strike until o¤ered a wage of at least $20 an hour.
The maxmin payo¤ and posture may also be viewed as the predicted payo¤ and posture of
a positive theory of bargaining in which each player is either maximally pessimistic (in a
Bayesian sense) about her opponent’s strategy or expects her opponent to play her “worstcase”strategy (in a maxmin sense), given her knowledge of her opponent’s rationality. These
two approaches are equivalent in my model, though they di¤er in general games.
The paper proceeds as follows: Section 2 relates the paper to the literature. Section
3 presents the model and de…nes maxmin payo¤s and postures.
Section 4 analyzes the
baseline case with one-sided commitment and presents the main characterization of maxmin
payo¤s and postures. Section 5 presents …ve brief extensions. Section 6 considers two-sided
commitment. Section 7 concludes. Omitted proofs are in the appendix. A supplementary
appendix shows that the main characterization is robust to details of the bargaining procedure such as the order and relative frequency of o¤ers (so long as o¤ers are frequent), as
well as to strengthening the solution concept from knowledge of normal-form rationality to
iterated conditional dominance.4
4
Rubinstein (1982) implies that the main result is not robust to simultaneously specifying a discrete-time
bargaining procedure and strengthening the solution concept to iterated conditional dominance.
supplementary appendix for details.
6
See the
2
Related Literature
The seminal paper on reputational bargaining is Abreu and Gul (2000), which generalizes
Myerson (1991).5
In their model, there is a vector of probabilities for each player corre-
sponding to the probability that she is committed to each of a variety of behavioral types
(which are analogous to bargaining postures in my model), and these vectors are common
knowledge. In the (e¤ectively unique) sequential equilibrium, players randomize over mimicking di¤erent behavioral types, with mixing probabilities determined by the prior, and
play proceeds according to a war of attrition, where each player hopes that her opponent
will concede. A player’s equilibrium payo¤ is higher when she is more likely to be committed
to strong behavioral types and when she is more patient. Thus, Abreu and Gul present a
complete and elegant bargaining theory in which the bargaining procedure is unimportant
and sharp predictions are driven by the vector of prior commitment probabilities.
The main di¤erence between my analysis and Abreu and Gul’s is that I characterize
maxmin payo¤s and postures rather than sequential equilibria. My approach entails weaker
assumptions on knowledge of commitment probabilities (i.e., second-order knowledge that
each player is committed to her announced posture with probability at least ", rather than
common knowledge of a vector of commitment probabilities) and on behavior (i.e., …rstorder knowledge of rationality, rather than sequential equilibrium), and does not yield unique
predictions about the division of surplus or about the details of how bargaining will proceed.
One motivation for this complementary approach is that behavioral types are sometimes
viewed as “perturbations”re‡ecting the fact that a player (or an outside observer) cannot be
sure that the model captures all of the other player’s strategic considerations. Thus, it seems
reasonable to assume that players realize that their opponent’s type may be perturbed in
some manner (e.g., that a rich set of types have positive prior weight), but assuming that the
distribution over perturbations is common knowledge goes against the spirit of introducing
perturbations.
The paper most closely related to mine is Kambe (1999). In Kambe’s model, each player
5
Other important antecedents of Abreu and Gul include Kreps and Wilson (1982) and Milgrom and
Roberts (1982), who pioneered the incomplete information approach to reputation-formation, and Chatterjee
and Samuelson (1987, 1988), who study somewhat simpler reputational bargaining models.
7
…rst strategically announces a posture and then becomes committed to her announced posture with probability ", as in my model.
of Abreu and Gul.
Thus, Kambe endogenizes the behavioral types
The structure of equilibrium and the determinants of the division of
the surplus are similar to those in Abreu and Gul’s model. There are two di¤erences between Kambe’s model and mine.
First, Kambe requires that players announce postures
that demand a constant share of the surplus (as do Abreu and Gul), while I allow players to
announce non-constant postures (and players do bene…t from announcing non-constant postures in my model). Second, and more fundamentally, Kambe studies sequential equilibria,
while I study maxmin payo¤s and postures. These di¤erences lead my analysis and results
to be quite di¤erent from Kambe’s, with the exception that Kambe’s calculation of bounds
on the set of sequential equilibrium payo¤s resembles my calculation of the maxmin payo¤
in the special case where players can only announce constant postures (Section 5.1).
There are also a number of earlier bargaining models in which players try to commit
themselves to advantageous postures. Crawford (1982) studies a two-stage model in which
players …rst announce demands and then learn their private costs of changing these demands,
and shows that such a model can lead to impasse. Fershtman and Seidmann (1993) show
that agreement is delayed until an exogenous deadline if each player is unable to accept an
o¤er that she has previously rejected. Muthoo (1996) studies a two-stage model related to
Crawford (1982), with the feature that making a larger change to one’s initial demand is
more costly, and shows that a player’s equilibrium payo¤ is increasing in her marginal cost
of changing her demand. Ellingsen and Miettinen (2008) point out that if commitment is
costly in Crawford’s model then impasse not only can result, but must. These papers study
equilibrium and do not involve reputation formation.
This paper is also related to the literatures on bargaining with incomplete information
either without common priors (Yildiz 2003, 2004; Feinberg and Skrzypacz, 2005) or with
rationalizability rather than equilibrium (Cho, 1994; Watson, 1998), in that players may
disagree about the distribution over outcomes of bargaining.
Finally, this paper weakens the solution concept from equilibrium to knowledge of rationality in reputational bargaining models in the same way that Watson (1993) and Battigalli
and Watson (1996) weaken the solution concept from equilibrium to knowledge of rationality
8
in Fudenberg and Levine’s (1989) model of reputation in repeated games. They …nd that
Fudenberg and Levine’s equilibrium predictions also apply under knowledge of rationality,
whereas my predictions di¤er dramatically from the existing reputational bargaining literature. The reason for this di¤erence is that in repeated games with one long-run player and
one short-run player, or in bargaining where one player is in…nitely more patient than her
opponent, the long-run or patient player receives close to her Stackelberg payo¤ (which in
bargaining equals the entire surplus) under either knowledge of rationality or equilibrium.
Thus, the predictions of Watson and Battigalli and Watson coincide with those of Fudenberg
and Levine, just as my predictions coincide with those of Abreu and Gul (and others) in
the special case where one player is in…nitely more patient than the other (Section 5.4).6
However, the main focus of this paper is on the case of equally (or at least comparably)
patient players, where predictions under equilibrium and knowledge of rationality di¤er for
both repeated games and bargaining.7
3
Model and Key De…nitions
This section describes the model and de…nes maxmin payo¤s and postures, which are the
main objects of analysis.
3.1
Model
Two players (“she,” “he”) bargain over one unit of surplus in two phases: a “commitment
phase”followed by a “bargaining phase.” I describe the bargaining phase …rst. It is intended
to capture a continuous bargaining process where players can change their demands and
accept their opponents’ demands at any time, but in order to avoid well-known technical
issues that emerge when players can condition their play on “instantaneous” actions of
6
A potentially confusing point is that if only one player has commitment power in reputational bargaining
then that player obtains the entire surplus in equilibrium regardless of the discount rates (Myerson, 1991).
This result relies on equilibrium as well as on a contraction argument that does not apply in Fudenberg and
Levine, Watson, Battigalli and Watson, or my paper.
7
Little is known about the role of reputation in repeated games with equally patient players under
knowledge of rationality. I brie‡y discuss this in the conclusion.
9
their opponents (Simon and Stinchcombe, 1989; Bergin and MacLeod, 1993) I assume that
players can revise their paths of demands only at integer times (while letting them accept
their opponents’demands at any time).8
Time runs continuously from t = 0 to 1. At every integer time t 2 N (where N is the
natural numbers), each player i 2 f1; 2g chooses a path of demands for the next length-1
period of time, uti : [t; t + 1) ! [0; 1], which is required to be the restriction to [t; t + 1) of a
continuous function on [t; t + 1]. Let U t be the set of all such functions, and let
(U t ) be
the space of probability measures on the Borel -algebra of U t endowed with the product
topology.
The interpretation is that uti ( ) is the demand that player i makes at time
(this is simply denoted by ui ( ) when t is understood; note that ui : R+ ! R+ can be
discontinuous at integer times but is right-continuous everywhere9 ). Even though player i’s
path of demands for [t; t + 1) is decided at t, player j only observes demands as they are made.
Intuitively, each player i may accept her opponent’s demand uj (t) at any time t, which ends
the game with payo¤s (e
(throughout, j =
rt
(1
uj (t)) ; e
rt
uj (t)), where r > 0 is the common discount rate
i). Formally, every instant of time t is divided into three dates, (t; 1),
(t; 0), and (t; 1) (except for time 0, which is divided only into dates (0; 0) and (0; 1)), with
the following timing:10 First, at date (t; 1), each player i announces accept or reject. If
both players reject, the game continues; if only player i accepts, the game ends with payo¤s
(e
rt
(1
lim
"t
uj ( )) ; e
rt
lim
"t
uj ( )); and if both players accept, the games ends with
payo¤s determined by the average of the two demands, lim
"t
u1 ( ) and lim
"t
u2 ( ). Next,
at date (t; 0), both players simultaneously announce their time-t demands (u1 (t) ; u2 (t))
(which were determined at the most recent integer time); if t is an integer, this is also the
date where each player i chooses a path of demands for the next length-1 period, uti . Finally,
8
In this paper, working in continuous time not only allows the players more ‡exibility than does discrete
time but also yields simpler results.
In particular, the continuous-time maxmin posture demands exact
compensation for delay and thus changes over time in a simple way, while I conjecture that the discrete-time
maxmin posture demands approximate compensation for delay but that the details of the approximation are
complicated and depend on the exact timing of o¤ers.
9
A function f : R ! R is right-continuous if, for every x 2 R and every
0
0
> 0, there exists
that jf (x) f (x )j < for all x 2 (x; x + ).
10
This is similar to the notion of date introduced by Abreu and Pearce (2007).
10
> 0 such
at date (t; 1), each player i again announces accept or reject. If both players reject, the game
continues; if only player i accepts, the game ends with payo¤s (e
rt
(1
uj (t)) ; e
rt
uj (t));
and if both players accept, the game ends and the demands u1 (t) and u2 (t) are averaged.
This timing ensures that there is a …rst and last date at which each player can accept each
of her opponent’s demands. In particular, at integer time t, player i may accept either her
opponent’s “left” demand, lim
"t
uj ( ), or her time-t demand, uj (t). I say that agreement
is reached at time t if the game ends at time t (i.e., at date (t; 1) or (t; 1)). Both players
receive payo¤ 0 if agreement is never reached.
The public history up to time t excluding the time-t demands is denoted by ht
<t ,
(u1 ( ) ; u2 ( ))
=
and the public history up to time t including the time-t demands is de-
noted by ht+ = (u1 ( ) ; u2 ( ))
t
(with the convention that this corresponds to all o¤ers
having been rejected, as otherwise the game would have ended). A generic time-t history is
denoted by ht . Since lim
"t
ui (t) = ui (t) for non-integer t, I generally distinguish between
ht and ht+ only for integer t. Formally, a bargaining phase (behavior) strategy for player i
is a pair
Fi (ht )
i
= (Fi ; Gi ), where Fi is a map from histories ht into [0; 1] with the properties that
0
Fi ht
0
whenever ht is a successor of ht and Fi (ht+ ) is a right-continuous function
of t; and Gi is a map from histories ht with t 2 N into
(U t ). Let
i
be the set of player i’s
bargaining phase strategies. The interpretation is that Fi (ht ) is the probability that player
i accepts player j’s demand at or before date (t; 1), Fi (ht+ ) is the probability that player i
accepts player j’s demand at or before date (t; 1), and Gi (ht ) is the probability distribution
over paths of demands uti : [t; t + 1) ! [0; 1] chosen by player i at date (t; 0). This formalism
implies that player i’s hazard rate of acceptance at history ht , fi (ht ) = (1
Fi (ht )), is well-
de…ned at any time t at which the realized distribution function Fi admits a density fi ; and
in addition player i’s probability of acceptance at history ht+ (resp., ht ), Fi (ht+ )
(resp., Fi (ht )
lim
"t
Fi (h )), is well-de…ned for all times t.
Fi (ht )
However, so long as one
bears in mind these formal de…nitions, it su¢ ces for the remainder of the paper to omit
the notation (Fi ; Gi ) and instead simply view a (bargaining phase) strategy
i
2
i
as a
function that maps every history ht to a hazard rate of acceptance, a discrete probability
of acceptance, and (if ht = ht for t 2 N) a probability distribution over paths of demands
uti . A pure bargaining phase strategy is a strategy
11
i
such that Fi (ht ) 2 f0; 1g for all ht and
Gi (ht ) is a degenerate distribution for all ht .11
At the beginning of the bargaining phase, player i has an initial belief
behavior of her opponent. Formally,
over
j,
so
i
i
2
Let supp ( i )
be the support of
j
strategy pro…le ( i ;
i,
j ),
and let
responses to belief
i
j ),
( i)
i
let ui ( i ;
i)
j
i,
about the
the set of …nite-dimensional distributions
is a …nite-dimensional distribution over behavior strategies
alternatively be viewed as an element of
belief
(
i
j;
note that
can
by reducing lotteries over behavior strategies.12
let ui ( i ;
j)
be player i’s expected utility given
be player i’s expected utility given strategy
argmax i ui ( i ;
i
i)
i
and
be the set of player i’s (normal-form) best-
(which may be empty; see discussion below).
I say that an action
(accepting, rejecting, or choosing a demand path uti ) is optimal at history ht under belief
if there exists a pure strategy
i
that prescribes that action at ht such that
i
2
i
i
( i ).13
At the beginning of the game (prior to time 0), player 1 (but not player 2) publicly
announces a bargaining posture
: [0; 1) ! [0; 1], which must be continuous at non-
integer times t, be right-continuous everywhere, and have well-de…ned left limits everywhere.
Slightly abusing notation, a posture
mands
is identi…ed with the strategy of player 1’s that de-
(t) for all t 2 R+ and always rejects player 2’s demand; with this notation,
2
1.
In other words, a posture is a pure bargaining phase strategy that does not condition on
player 2’s play or accept player 2’s demand. After announcing posture , player 1 becomes
committed to
with some probability " > 0, meaning that she must play strategy
bargaining phase. With probability 1
", she is free to play any strategy in the bargaining
phase. Whether or not player 1 becomes committed to
3.2
in the
is observed only by player 1.
De…ning the Maxmin Payo¤ and Posture
This subsection de…nes player 1’s maxmin payo¤ and posture. Intuitively, player 1’s maxmin
payo¤ is the highest payo¤ she can guarantee herself when all she knows about player 2 is that
he is rational (i.e., maximizes his expected payo¤ given his belief about player 1’s behavior,
11
12
Thus, behavior strategies are primitives of the model and pure strategies are special behavior strategies.
The assumption that i puts weight on only …nitely many behavior strategies is not very restrictive since
the de…nition of behavior strategies allows for continuous mixing.
13
This is optimality under initial beliefs. For iterated conditional dominance, see the supplementary
appendix.
12
and updates his belief according to Bayes’ rule when possible) and that he believes that
player 1 follows her announced posture
with probability at least ".
Formally, that player 2 is rational and assigns probability at least " to player 1 following
her announced posture
means that his strategy satis…es the following condition:
De…nition 1 A strategy
2
of player 2’s is rational given posture
of player 2’s such that
2
( )
2
A belief
1
f
2
:
2
1
2
2
2
( 2 ).
of player 1’s is consistent with knowledge of rationality given posture
if every strategy
of beliefs
" and
if there exists a belief
2
2 supp ( 1 ) is rational given posture
.
In other words, the set
that are consistent with knowledge of rationality given posture
is
1
is rational given posture g.
I assume that player 1’s belief is consistent with knowledge of rationality.
here is that the set
2
( 2 ) may be empty for some beliefs
2.
A subtlety
It is inevitable in bargaining
models that players do not have best responses to all beliefs; for example, a player has no
best response to the belief that her opponent will accept any strictly positive o¤er but will
refuse an o¤er of 0. Thus, the assumption that player 2 plays a strategy
some belief
2
2
2
2
( 2 ) for
is in fact a joint assumption on his belief and strategy. While the implied
assumption on beliefs is certainly not without loss of generality, it is extremely natural and
indeed fundamental, as it says precisely that player 2’s choice set is non-empty. It is also
weaker than any equilibrium assumption, as players play best responses in equilibrium.
Given that her belief is consistent with knowledge of rationality, the highest payo¤ that
player 1 can guarantee herself after announcing posture
is the following:
De…nition 2 Player 1’s maxmin payo¤ given posture
u1 ( )
sup inf u1 ( 1 ;
1
A strategy
1
12
is
1) .
1
( ) of player 1’s is a maxmin strategy given posture
1
( ) 2 argmax inf u1 ( 1 ;
1
12
13
1
1) .
if
Equivalently, u1 ( ) is the highest payo¤ player 1 can receive when she chooses a strategy
1
and then player 2 chooses a rational strategy
u1 ( ) = sup
1
2: 2
2
that minimizes u1 ( 1 ;
inf
is rational given posture
u1 ( 1 ;
2 );
that is,
2) .
In particular, to guarantee herself a high payo¤, player 1 must play a strategy that does well
against any rational strategy of player 2’s.14
Finally, I de…ne player 1’s maxmin payo¤, the highest payo¤ that player 1 can guarantee
herself before announcing a posture, as well as the corresponding maxmin posture.
De…nition 3 Player 1’s maxmin payo¤ is
u1
A posture
n
(t) !
sup u1 ( ) .
is a maxmin posture if there exists a sequence of postures f
(t) for all t 2 R+ and u1 (
n)
ng
such that
! u1 .
on " by writing u1 (") and
I sometimes emphasize the dependence of u1 and
Both the set of maxmin strategies given any posture
15
".
and the set of maxmin postures are
non-empty, though at this point this is not obvious.
The reason why
is de…ned as a limit of postures f
ng
such that u1 (
n)
! u1 , rather
than as an element of argmax u1 ( ), is that the latter set may be empty because of an
openness problem that is standard in bargaining models. To see the problem, consider the
ultimatum bargaining game, where player 1 makes a take-it-or-leave-it demand in [0; 1] to
player 2.
By knowledge of rationality, player 1 knows that any demand strictly below 1
will be accepted, but demanding 1 does not guarantee her a positive payo¤ because it is a
best-response for player 2 to reject. De…nition 3 is analogous to specifying that in this game
player 1’s maxmin payo¤ is 1 and her maxmin strategy is demanding 1.
14
A potential criticism of the concept of the maxmin payo¤ given posture
is that it appears to neglect
the fact that, in the event that player 1 does become committed to posture
inf
12
1
u1 ( ;
1)
in the bargaining phase, rather than sup
1
inf
12
Section 4.3 that these two numbers are actually identical in my model.
15
The notation ( ) is already taken by the time-t demand of posture
1
u1 (
1;
, she is guaranteed only
1 ).
However, I show in
. I apologize for abusing notation
in writing u1 ( ) and u1 (") for di¤erent objects and hope that this will not cause confusion.
14
Note that De…nitions 2 and 3 are “non-Bayesian”in the sense that they characterize the
largest payo¤ that player 1 can guarantee herself, rather than the maximum payo¤ that she
can obtain given some belief. The following is the “Bayesian”version of these de…nitions:
De…nition 4 Player 1’s pessimistic payo¤ given posture
upess
( )
1
Player 1’s pessimistic payo¤ is upess
1
inf sup u1 ( 1 ;
12
1
(
and upess
1
n)
.
! upess
1
1) .
1
sup upess
( ).
1
posture if there exists a sequence of postures f
is
ng
A posture
such that
n
(t) !
pess
is a pessimistic
pess
(t) for all t 2 R+
Player 1’s pessimistic payo¤ is the worst payo¤ she can receive by best-responding to
a …xed rational strategy of player 2’s. Player 1’s maxmin payo¤ is weakly lower than her
pessimistic payo¤, because in the de…nition of the pessimistic payo¤ player 1 “knows” the
distribution over player 2’s strategies when she chooses her strategy, while in the de…nition
of the maxmin payo¤ her strategy is evaluated with respect to the worst-case response of
player 2’s.
However, I show in Section 4.3 that these payo¤s are in fact identical in my
model. For expositional consistency, I focus on maxmin payo¤s and strategies.
Another reason for studying player 1’s maxmin payo¤ is that it determines the entire
range of payo¤s that are consistent with knowledge of rationality, as shown by the following
proposition.
Proposition 1 For any posture
1
4
such that max 1 u1 ( 1 ;
1)
and any payo¤ u1 2 [u1 ( ) ; 1), there exists a belief
1
2
= u1 .
Characterization of the Maxmin Payo¤ and Posture
This section states and proves Theorem 1, the main result of the paper, which solves for
player 1’s maxmin payo¤ and posture. Section 4.1 states Theorem 1 and provides intuition,
and Sections 4.2 through 4.4 provide the proof. The approach is as follows: In Section 4.2,
I …x a posture
best-response,
and …nd a belief of player 2’s,
2
2
2
(
2 ),
2
that minimize u1 ( ;
15
(satisfying
2 ),
2
( )
"), and corresponding
player 1’s payo¤ from mimicking
in
the bargaining phase;
2
and
2
are called the -o¤setting belief and -o¤setting strategy,
respectively, and play a key role in the analysis. Section 4.3 shows that
itself is a maxmin
strategy given posture , for any , which implies that u1 ( ) = u1 ( ;
for any posture .
That is, player 1’s maxmin payo¤ given posture
2)
is the payo¤ she receives from mimicking
when her opponent follows his -o¤setting strategy. Section 4.4 maximizes u1 ( ;
2)
over
to prove Theorem 1.
4.1
Main Result
The main result is the following:
Theorem 1 Player 1’s maxmin payo¤ is
u1 (") = 1= (1
and the unique maxmin posture
"
"
log ") ,
is given by
(t) = min ert = (1
log ") ; 1
for all t 2 R+ .
A priori, one might have expected player 1’s maxmin payo¤ to be very small when " is
small (because player 2’s beliefs and strategy may be chosen quite freely in the de…nition of
the maxmin payo¤), and might have expected player 1’s maxmin posture to be complicated
(as player 1 is not restricted to announcing monotone, continuous, or otherwise well-behaved
postures). Theorem 1 shows that, on the contrary, player 1’s maxmin payo¤ is relatively
high for even very small commitment probabilities ", as shown by Table 1, and that player
1’s unique maxmin posture is simply demanding the maxmin payo¤ plus compensation for
any delay in reaching agreement.16
16
The importance of non-constant postures is a di¤erence between this paper and existing reputational
bargaining models, where it is usually assumed that players may only be committed to strategies that demand
a constant share of the surplus (as in Abreu and Gul (2000), Compte and Jehiel (2002), and Kambe (1999)).
A notable exception is Abreu and Pearce (2007), where players may be committed to non-constant postures
that can also condition their play on their opponents’behavior. However, Abreu and Pearce’s main result
is that a particular posture that demands a constant share of the surplus is approximately optimal in their
model, when commitment probabilities are small.
16
"
u1 (")
:25
:42
:1
:30
10
3
:13
10
6
:07
10
9
:05
Table 1: The Maxmin Payo¤ u1 (") for Di¤erent Commitment Probabilities "
I …rst give intuition for why the maxmin posture demands the maxmin payo¤ plus compensation for delay, and then give intuition for why the maxmin payo¤ equals 1= (1
log ").
The …rst step is determining the worst belief that player 2 can have after player 1 announces an arbitrary posture
(the “ -o¤setting belief”). A preliminary observation is that
itself is a maxmin strategy give posture
(cf Section 4.3). This is because player 1 is not
guaranteed a positive payo¤ at histories following a deviation from her announced posture,
because at such histories player 2’s beliefs and strategy are unrestricted.
It follows from
this observation that the -o¤setting belief is whatever belief leads player 2 to rejects player
1’s demand for as long as possible, when player 1 mimics .17
What belief leads player 2 to reject for as long as possible? It is the belief that player 1
is committed to
with the smallest possible probability (i.e., "), and that she concedes the
entire surplus to him at the rate that makes him indi¤erent between accepting and rejecting.
For if player 1 conceded more slowly, he would accept, and if player 1 conceded more quickly,
she would build reputation more quickly when mimicking
and would thus eventually have
her demand accepted sooner.18 And what is this concession rate (call it (t))? For player 2,
accepting yields ‡ow payo¤ r (1
17
This is not true if
(t)), while rejecting yields ‡ow payo¤ (t) (t)
0
(t),
ever increases so quickly that delay bene…ts player 1. For this intuitive discussion,
consider instead the “typical” case where delay hurts player 1.
18
This point is clearest in the extreme case where player 2 thinks that player 1 will surely concede in one
second. Then player 1 needs only mimic
for two seconds to convince player 2 that she is committed.
17
and equalizing these ‡ow payo¤s gives
(t) =
r (1
(t)) +
(t)
0
(t)
.19
Thus, the -o¤setting belief is that player 1 is committed to
the entire surplus at rate
with probability " and concedes
(t) (which depends on ). Note that
(t) is greater when
(t)
is smaller; this formalizes the intuition from the introduction that player 1 builds reputation
more quickly when her demand is smaller.
Given this characterization of the -o¤setting belief, what posture
1’s payo¤?
maximizes player
Since player 1 builds reputation more quickly when her demand is smaller,
she bene…ts from demanding as little as possible, subject to the constraint that she always
demands at least her maxmin payo¤ plus compensation for delay (as otherwise player 2 might
rationally accept at a time where she demands less than this, leaving her with less than her
maxmin payo¤). This implies that the maxmin posture demands exactly the maxmin payo¤
plus compensation for delay, until player 1’s reputation reaches 1 (i.e., until player 2 becomes
certain that she is committed to ).20 Furthermore, this time at which player 1’s reputation
reaches 1 is exactly the time at which her demand reaches 1. To see this, note that the fact
that player 1 demands compensation for delay until her reputation reaches 1 implies that her
reputation reaches 1 weakly before her demand reaches 1 (as she cannot demand more than
1). And if player 1’s reputation reaches 1 strictly before her demand reaches 1, she could
instead announce a posture that demands slightly more at every time and still be sure to
receive her initial demand plus compensation for delay (as her reputation would still reach 1
strictly before her demand reaches 1, at which point player 2 would accept). Finally, player
1 must demand 1 forever once her demand reaches 1, as otherwise player 2 would wait and
19
Two remarks:
assumes that
First, the formal de…nition of
is di¤erentiable and that r (1
rigorous derivation of
(t) =
order expansion in dt.
20
A subtlety is that some postures
(t) >
0
(t) for some t and thus
in the derivative term
(t)) +
0
(t)
0, for example).
Second, a slightly more
(t) comes from considering the equation for player 2 to be indi¤erent between
accepting at t and t + dt, 1
0
(t) is provided in Section 4.2 (the current de…nition
0
(t) dt + (1
(t) dt) (1
rdt) (1
(t + dt)), and taking a …rst-
that always demand more than the maxmin posture
(t) >
may have
(t) for some t. However, it can be shown that this advantage
is always more than o¤set by the disadvantage in the level term
over an interval, so that player 1’s reputation is always greater with posture
18
when integrating
than with .
accept a lower demand rather than accept when player 1’s demand is close to 1.
It remains to describe why the maxmin payo¤ equals 1= (1
given by
log "). Consider a posture
(t) = min fert u1 ; 1g for all t 2 R+ , for arbitrary u1 2 R+ . As I have argued,
the -o¤setting belief is that player 1 is committed to
with probability " and concedes at
rate
(t) =
until
(t) reaches 1.
r (1
(t)) +
(t)
0
(t)
=
r
ert u
(1)
1
Now player 1’s reputation reaches 1 at the time T such that the
probability that player 1 has not conceded by T equals ", since if player 1 does not concede
by this time then she must be committed to
by
Z
exp
(as
never concedes). This time T is given
T
(t) dt
= ".21
0
Substituting for
(t) using (1), this becomes
T =
log (1 + u1 log ") =r.
1
On the other hand, player 1’s demand reaches 1 at the time T 1 given by erT u1 = 1, or
T1 =
log (u1 ) =r.
Now, so long as T < T 1 , player 1’s maxmin payo¤ given posture
equals her initial demand,
u1 , as in this case player 2 must accept at T , the time at which he becomes convinced that
player 1 is committed to
(which would not be the case if player 1 were demanding 1 when
he becomes convinced, i.e., if T
T 1 ). Clearly, T < T 1 if and only if u1 < 1= (1
Therefore, player 1 can guarantee herself any payo¤ less than 1= (1
that the maxmin payo¤ is at least 1= (1
more than 1= (1
log ").
log ").
log "), which implies
And player 1 cannot guarantee herself
log "), because it can be shown that any posture that guarantees close to
1= (1
log ") must be close to the maxmin posture
4.2
O¤setting Beliefs and Strategies
.
In this subsection, I …x an arbitrary posture for player 1, , and …nd a rational strategy
of player 2’s,
21
2
(the “ -o¤setting strategy”), that minimizes player 1’s payo¤ when she
A more general de…nition of T is provided in Section 4.2
19
announces posture
and then mimics
in the bargaining phase.
That is, I solve the
problem
inf
2) .
u1 ( ;
(2)
( 2 ; 2 ): 2 ( ) "; 2 2 2 ( 2 )
The resulting strategy
2
always demands the entire surplus and rejects player 1’s demand
until some time t , and accepts player 1’s smaller time-t demand, min flim
(t )), if player 1 follows
(henceforth denoted by
from , then
( ) ; (t )g
"t
until time t . If player 1 ever deviates
2
(the “ -
with probability ", and with probability 1
" plays
rejects player 1’s demand forever. The corresponding belief
2
o¤setting belief”) is that player 1 plays
a particular strategy ~ that mixes between mimicking
and accepting player 2’s demand up
until time t , and subsequently mimics .
The key step in solving (2) is computing the smallest time T by which agreement must
be reached under strategy pro…le ( ;
mint
T
e
rt
2 ).
I then show that the value of (2) is simply
(t), and that the time t at which the strategy
T that minimizes e
rt
2
accepts
(t) is a time before
(t) :
Toward computing T , let v (t; 1) be the continuation value of player 2 from bestresponding to
starting from date (t; 1) (for integer t), and let v (t) be the corresponding
continuation value starting from date (t; 1) (for any t 2 R+ ):
v (t; 1)
max e
r(
t
v (t)
max 1
t)
1
( ) ,
(t) ; sup e
r(
t)
1
,
( )
(3)
>t
where
min flims"
( )
(s) ; ( )g. Thus, the di¤erence between v (t; 1) and v (t) is
that only v (t; 1) gives player 2 the opportunity to accept the demand 1
particular, v (t; 1) = v (t) if (or v) is continuous at t. Note that max
is well-de…ned because
is lower semi-continuous and lim
!1
e
r(
that v is continuous at all non-integer times t; let fs1 ; s2 ; : : :g
t)
S
t
e
1
lim
r(
t)
"t
( ). In
1
( )
( ) = 0, and
N be the set of
discontinuity points of v. Finally, note that v can increase at rate no faster than r. That
is, v (t)
then v (t)
e
r(t0 t)
e
r(
v (t0 ) for all t0
t)
1
t, because if v (t0 ) = e
( ) =e
r(t0 t)
r(
t0 )
1
( ) for some
t0 ,
v (t0 ). This implies that v is continuous but for
20
downward jumps,22 and that v is di¤erentiable almost everywhere.23 These are but two of
the useful properties of the function v (which are not shared by ) that reward working with
v rather than
in the subsequent analysis.
Next, I introduce two functions
player 1 mixes between mimicking
: R+ ! R+ and p : R+ ! R+ with the property that if
and conceding the entire surplus to player 2, then
(t)
(resp., p (t)) is the smallest non-negative hazard rate (resp., discrete probability) at which
player 1 must concede in order for player 2 to be willing to reject player 1’s time-t demand,
(t). Let
rv (t) v 0 (t)
1 v (t)
if v is di¤erentiable at t and v (t) < 1, and let (t) = 0 otherwise; note that
(4)
(t) =
(t)
0 for all
t, because v cannot increase at rate faster than r. Also, let
p (t) =
v (t; 1) v (t)
1 v (t)
(5)
if v (t) < 1, and let p (t) = p (0) = 0 otherwise. To see intuitively why the aforementioned
property holds, note that accepting player 1’s time-t demand gives player 2 ‡ow payo¤ rv (t),
while rejecting gives player 2 ‡ow payo¤
(t) (1
v (t)) + v 0 (t), and equalizing these ‡ow
payo¤s yields (4);24;25 similarly, accepting player 1’s demand at date (t; 1) gives player 2
payo¤ v (t; 1), while delaying acceptance until date (t; 1) gives player 2 payo¤ p (t) (1) +
(1
p (t)) v (t), and equalizing these payo¤s yields (5).
When player 2 expects player 1 to accept his demand at rate (resp., probability)
p), he becomes convinced that player 1 is committed to posture
22
at the time T~ de…ned
A function f : R !R is continuous but for downward jumps if lim inf fx"x (x)
lim supx#x f (x) for all x 2 R.
23
Proof: Let f (t) = e rt v (t).
(resp.,
f (x )
Then f is non-increasing, which implies that f is di¤erentiable almost
everywhere (e.g., Royden, 1988, p. 100). Hence, v is di¤erentiable almost everywhere.
24
This intuition is correct when v (t) = 1
(t). When v (t) > 1
(t), player 2 prefers to reject player
1’s time-t demand even when player 1 concedes at rate 0. At these times, rv (t) = v 0 (t), which implies that
(t) = 0. Hence,
(t) is always the smallest non-negative hazard rate at which player 1 must concede in
order for player 2 to be willing to reject (t). A similar caveat applies to the intuition for p.
25
If v 0 (t) = 0, then (t) becomes the concession rate that makes player 2 indi¤erent between accepting and
rejecting the constant o¤er v (t), which is familiar from the literatures on wars of attrition and reputational
bargaining. However, in these literatures
(t) is the rate at which player 1 concedes in equilibrium, while
here is the rate at which player 1 concedes according to player 2’s -o¤setting belief, as will become clear.
21
in the following lemma, which leads him to accept player 1’s demand no later than the
time T de…ned in the lemma. In the lemma, and throughout the paper, maximization or
minimization over times t should be read as taking place over t 2 R+ [ f1g (i.e., as allowing
t = 1, with the convention that e
r1
0 for all postures ).
(1)
Lemma 1 Let
T~
sup
and let
8
<
:
t : exp
T
Then, for any
2
Z
such that
Y
t
(s) ds
0
rt
(1
(t)) if t = T~
rt
1
(t) if t > T~
" and any
( )
than time T under strategy pro…le ( ;
(1
s2S\[0;t)
8
< e
max argmax
: e
t T~
2
9
=
p (s)) > " ,
;
2
2
( 2 ), agreement is reached no later
In particular,
2 ).
inf
2
.
u1 ( ;
2)
min e
t T
( 2 ; 2 ): 2 ( ) "; 2 2 2 ( 2 )
rt
(t) .
(6)
Lemma 1 amounts to the statement that T~ is the latest time at which it is possible that
agreement has not yet been reached and player 2 is not certain that player 1 is playing ,
when player 2’s initial belief is some
for some
2
2
2
2
such that
2
( )
", under strategy pro…le ( ;
2)
( 2 ). The …rst step of the proof of Lemma 1 shows that, in computing this
time, one can restrict attention to beliefs
2
that assign probability 1 to player 1’s accepting
player 2’s demand whenever player 1 deviates from strategy , and to strategies
2
that
always demand the entire surplus. This is because giving more surplus to player 2 in the
event that player 1 deviates from
makes player 2 more willing to reject player 1’s demand,
without changing player 2’s beliefs about the probability that player 1 is playing
at any
history. The second step expresses the condition that agreement is not reached by a given
time under strategy pro…le ( ;
2)
in terms of player 1’s concession rate and probability. The
…nal step shows that agreement is delayed for as long as possible when player 1’s concession
rate and probability are given by
and p.
The remainder of this subsection is devoted to showing that (6) holds with equality,
which proves that (2) equals mint
T
e
rt
(t). The idea is that player 2 may hold a belief
22
that leads him to demand the entire surplus until time t
min argmint
T
e
rt
(t) and then
accept player 1’s o¤er; this is the -o¤setting belief.26 I …rst de…ne the -o¤setting belief,
and then show that (6) holds with equality.
I begin by introducing a strategy, ~ , which is used in de…ning the -o¤setting belief.27
Let
(t) = max
let
8
< exp
Rt
0
: exp
9
=
(1
p
(s))
"
s2S\[0;t)
;
0
;
Q
;
(s) ds
(1
p
(s))
s2S\[0;t)
(s) ds
Rt
0
^ (t) =
Q
(7)
(t)
(t)
(8)
(t) > 0, and let ^ (t) = 0 otherwise; and let
if
p^ (t) = min
if
p (t)
;1
(t)
(t) > 0, and let p^ (t) = 0 otherwise. Intuitively,
(9)
(t) is the posterior probability that
player 2 assigns to player 1’s playing a strategy other than
unconditional concession rate and probability are
at time t when player 1’s
(t) and p (t), and ^ (t) and p^ (t) are the
conditional (on not playing ) concession rate and probability needed for the unconditional
concession rate and probability to equal
(t) and p (t).
De…nition 5 ~ is the strategy that demands
(t) at all t, accepts with hazard rate ^ (t) at
all t < t , accepts with probability p^ (t) at date (t; 1) for all t < t , and rejects for all t
t,
for all histories ht .
I now de…ne the -o¤setting belief. Throughout, a history ht (resp., ht+ ) is consistent
with posture
if u1 ( ) =
( ) for all
< t (resp.,
De…nition 6 The -o¤setting belief, denoted
The -o¤setting strategy, denoted
2,
2,
t).
is given by
2
( ) = " and
2
(~ ) = 1
".
is the strategy that always demands 1 and accepts or
rejects player 1’s demand as follows:
26
Note that min argmint
T
e
rt
(t) is well-de…ned, because
equal 1, if T = 1). Note also that
(t) is lower semi-continuous (though it may
(t) > 0 for all t < t . This property of t makes it more convenient
to de…ne the -o¤setting belief with reference to t , rather than some other element of argmint
27
This approach is related to a construction in Wolitzky (2011).
23
T
e
rt
(t).
1. If ht is consistent with , then reject if t < t ; accept at date (t ; 1) if and only if
lim
(t ); accept at date (t ; 1) if and only if lim
( )
"t
"t
(t ); and
( ) >
reject if t > t .
2. If ht is inconsistent with , then reject.
Finally, I show that (6) holds with equality, and also that the -o¤setting (belief, strategy)
pair (
2;
2)
is a solution to (2). If t = 1, then the following statement that agreement is
reached at time t means that agreement is never reached.
Lemma 2 Agreement is reached at time t under strategy pro…le ( ;
In particular, the pair (
2;
2)
is a solution to (2), and u1 ( ;
2)
2 ),
= mint
and
T
e
2
rt
2
2
(
2 ).
(t).
Proof. It is immediate from De…nition 6 that agreement is reached at t under strategy
pro…le ( ;
2 ),
(6). Since
2
which implies that u1 ( ;
2)
equals mint
T
e
rt
(t), the right-hand side of
( ) ", it remains only to show that 2 2 2 ( 2 ).
o
n
~
and ht is consistent with , then, by construction of ~ , player 1
If t < min T ; t
accepts player 2’s demand of 1 with unconditional hazard rate (t) and unconditional discrete
probability p (t) under
It is established in the proof of Lemma 1 that it is optimal for
o
n
player 2 to demand u2 (t) = 1 and reject at any time t < min T~; t when player 1 accepts
his demand of 1 at rate
2.
and probability p until time T~; and that in addition if t < T~
then player 2 is indi¤erent between accepting and rejecting at time t when player 1 accepts
with this rate and probability until time T~. Therefore, it is optimal for player 2 to demand
u2 (t) = 1 and reject at time t when player 1 accepts with this rate and probability only until
n
o
time min T~; t .
h
If t 2 T~; t and ht is consistent with , then under 2 player 2 is certain that player
1 is playing
at ht . Since t
T , this implies that it is optimal for player 2 to reject. If
a history ht is not reached under strategy pro…le (
with
2;
2)
(as is the case if ht is inconsistent
or if t > t ), then any continuation strategy of player 2’s is optimal.
see that accepting
Finally, to
(t ) (i.e., accepting at the more favorable of dates (t ; 1) and (t ; 1))
is optimal, note that the fact that t 2 argmint
all t 2 [t ; T ]. Hence, t 2 argmaxt2[t
;T ]
e
rt
1
24
T
e
rt
(t) implies that
(t)
(t) . Because ~ coincides with
(t ) for
after
time t , it follows that, conditional on having reached time t , player 2 receives at most
supt2(t
;T ]
e
rt
1
(t) if he rejects, and receives e
weakly more. Therefore,
4.3
2
2
2
(
rt
(t ) if he accepts, which is
1
2 ).
Maxmin Strategies
This subsection shows that
ticular u1 ( ) = u1 ( ;
2)
itself is a maxmin strategy given posture , and that in par-
= mint
T( )
e
rt
(t), where I have made the dependence of T on
explicit.
Lemma 3 For any posture , u1 ( ) = u1 ( ;
Proof. By Lemma 2, (
2;
2)
2 argmin u1 ( ;
12
she conforms to
2,
= mint
T( )
e
rt
(t).
is a solution to (2), so
2
Under strategy
2)
1) .
(10)
1
player 2 always demands 1 and only accepts player 1’s demand if
through time t . Hence, sup 1 u1 ( 1 ;
2)
=e
rt
(t ) = u1 ( ;
2 ),
and
therefore
2 argmax u1 ( 1 ;
2) .
(11)
1
Now (10) and (11) imply the following chain of inequalities:
sup inf u1 ( 1 ;
1
12
1)
inf u1 ( ;
12
1
1)
1
= u1 ( ;
2)
(by (10))
= max u1 ( 1 ;
1
2)
max min u1 ( 1 ;
1
12
(by (11))
1) .
1
This is possible only if both inequalities hold with equality (and the supremum and in…mum
in the …rst line are attained at
mint
T( )
e
rt
and
2,
respectively).
Therefore, u1 ( ) = u1 ( ;
2)
=
(t).
As an aside, a similar argument establishes the equivalence between the maxmin approach
of De…nitions 2 and 3 and the Bayesian approach of De…nition 4.
25
= u1 , and
( ) = u1 ( ) for all , upess
Corollary 1 upess
1
1
is a maxmin posture if and only
if it is a pessimistic posture.
Proof. (10) and (11) imply that inf
12
1
of inequalities that proves that sup 1 inf
sup 1 u1 ( 1 ;
12
1
u1 ( 1 ;
1)
1)
2 ),
= u1 ( ;
= u1 ( ;
by the same chain
Hence, upess
( ) =
1
2 ).
= u1 ( ) for all , and the remainder of the result follows from De…nition 4.
u1 ( ;
2)
4.4
Proof of Theorem 1
I now sketch the remainder of the proof of Theorem 1. The details of the proof are deferred
to the appendix.
The …rst part of the proof is constructing a sequence of postures f
log ") and f
1= (1
…ne
n
n
(t)g converges to
min fert = (1
(t)
ng
such that limn!1 u1 (
log ") ; 1g for all t 2 R+ . De-
by
Let Tn1 be the time where
n
ert
;1
1 log "
n
n+1
n (t) = min
for all t 2 R+ .
(t) reaches 1. It can be shown that Tn1 > T~ (
n)
for all n 2 N,
where T~ is de…ned as in Lemma 1 and I have emphasized the dependence of T~ on . This
implies that
n
n
(t) =
n
n+1
(t) is non-decreasing and
T(
n)
= T~ (
n ).
ert
1 log "
n
T~ (
for all t
T~ (
n)
n)
=
t T(
=
Therefore, limn!1 u1 (
n)
= 1= (1
n
< 1.
Since
n)
that
(t)
log ").
guarantees more than 1= (1
such that
T 1 , where T 1 is the time at which
(t), this implies that any posture
must satisfy T~ ( )
n)
1
n
n + 1 1 log "
T~( n )
n
1
.
n + 1 1 log "
the crucial observation is that any posture
T ( ) satis…es T~ ( )
rt
n)
The second part is showing that no posture
rt
T~ (
min
t
e
n
< 1, it follows from the de…nition of T (
min e
=
T( )
and that
Thus, by Lemma 3,
u1 (
mint
n ),
ert = (1
log ") for all t
(t) reaches 1. Since u1 ( ) =
that guarantees at least 1= (1
T 1 ; in particular, player 2 may reject
26
(t)
log "). Here,
log ")
(t) until time T 1 . But receiving
n)
=
the entire surplus at T 1 is worth only 1= (1
more than 1= (1
log "), so it follows that no posture guarantees
log "). The appendix shows that in addition no sequence of postures f
converging pointwise to any posture other than
payo¤s fu1 (
5
0
n )g
converging to 1= (1
0
ng
can correspond to a sequence of maxmin
log ").
Extensions
This section presents …ve extensions of Theorem 1. Section 5.1 characterizes the maxmin
payo¤ when player 1 can only announce constant postures; Section 5.2 extends Theorem 1 to
general convex bargaining sets; Section 5.3 considers heterogeneous discounting; Section 5.4
extends Theorem 1 to higher-order knowledge of rationality; and Section 5.5 allows player
1’s credibility to vary with her announcement.
5.1
Constant Postures
Theorem 1 shows that the unique maxmin posture is non-constant.
In this subsection, I
determine how much lower a player’s maxmin payo¤ is when she is required to announce
a constant posture. The purpose of this study is, …rst, to establish that announcing nonconstant postures allows a player to guarantee herself a signi…cantly higher payo¤; second,
to determine the share of the surplus that a player can guarantee herself in settings where
announcing a non-constant posture might not be credible; and, third, to facilitate comparison
with the existing reputational bargaining literature, in which typically players can only
announce constant postures.
A posture
is constant if
notation by writing
constant posture
(t) =
(0) for all t.
for the constant demand
If
is constant, I slightly abuse
(t) in addition to the posture itself. The
that maximizes u1 ( ) is the maxmin constant posture, denoted
,28 and
the corresponding payo¤ is the maxmin constant payo¤, denoted u1 . These can be derived
using Lemmas 1 through 3, leading to the following:
28
Such a posture exists for all " < 1, so there is no need for a limit de…nition like De…nition 3. When
" = 1, such a de…nition would imply that the maxmin constant posture equals 1.
27
Proposition 2 For all " < 1, the unique maxmin constant posture is
and the maxmin constant payo¤ is u1 (") = exp ( (1
Proposition 2 solves for
"
" ))
"
=
2 log "
p
(log ")2 4 log "
,
2
".
and u1 ("), but it does not yield a clear relationship between
the maxmin constant payo¤, u1 ("), and the (overall) maxmin payo¤, u1 ("). Therefore, I
graph the ratio of u1 (") to u1 (") in Figure 2. In addition, the following analytical result
regarding the ratio of u1 (") to u1 (") is straightforward:
Corollary 2 u1 (") =u1 (")
is
decreasing
in
",
lim"!1 u1 (") =u1 (")
=
1,
and
lim"!0 u1 (") =u1 (") = e.
3.0
2.5
2.0
1.5
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 2: The Ratio u1 (") =u1 (") for " 2 [0; 1].
The most interesting part of Corollary 2 is that a player’s maxmin payo¤ is approximately
e
2: 72 times greater when she can announce non-constant postures than when she can only
announce constant postures, when her commitment probability is small. Thus, there is a
large advantage to announcing non-constant postures. However, a player can still guarantee
herself a substantial share of the surplus when she can only announce constant postures, and
her maxmin payo¤ goes to 0 with " at the same rate in either case.
5.2
General Convex Bargaining Sets
This subsection shows that the maxmin payo¤ derived in Theorem 1 is a lower bound on the
maxmin payo¤ with general convex bargaining sets, normalized so that each player’s lowest
28
and highest feasible payo¤s are 0 and 1. More generally, taking a concave transformation
of the Pareto frontier of the bargaining set weakly increases the maxmin payo¤.
Formally, a decreasing function
: [0; 1] ! [0; 1] is the Pareto frontier (of the bargaining
rt
set) if the game ends with payo¤s (e
rt
u1 (t) ; e
1’s demand u1 (t), and ends with payo¤s e
rt
1
(u1 (t))) when player 2 accepts player
(u2 (t)) ; e
rt
u2 (t) when player 1 accepts
player 2’s demand u2 (t). Note that the de…nition of player 1’s maxmin payo¤ is valid for
any bargaining set. The result is the following:
Proposition 3 Suppose that
(0) = 1 and
satisfying
(1) = 0, and that
(0) = 0 and
frontier is , and let u1
Then u1
: [0; 1] ! [0; 1] is a decreasing and concave function satisfying
(1) = 1.
: [0; 1] ! [0; 1] is an increasing and concave function
Let u1 be player 1’s maxmin payo¤ when the Pareto
be player 1’s maxmin payo¤ when the Pareto frontier is
.
u1 .
Proposition 3 shows that taking any concave transformation
of a Pareto frontier
weakly increases player 1’s maxmin payo¤. The intuition is that any …xed demand of player
1’s leaves more for player 2 when the Pareto frontier is
than when it is
, which
implies that player 2 must believe that player 1 is conceding more rapidly in order for him
to reject player 1’s demand when the Pareto frontier is
.
This in turn lets player 1
build reputation more quickly and thus guarantee herself a higher payo¤.
5.3
Heterogeneous Discounting
I have assumed that the players have the same discount rate. This simpli…ed notation and
led to simple formulas for u1 (") and
"
in Theorem 1. However, it is straightforward to gen-
eralize the model to the case where player i has discount rate ri and ri 6= rj ; one must only
keep track of whose discount rate “r”stands for in the above analysis. Introducing heterogeneous discounting yields interesting comparative statics with respect to the players’relative
patience, r1 =r2 (as will become clear, u1 depends on r1 and r2 only through r1 =r2 ). First,
the standard result in the reputational bargaining literature (Abreu and Gul, 2000; Compte
and Jehiel, 2002; Kambe, 1999) that player 1’s sequential equilibrium payo¤ converges to 1
as r1 =r2 converges to 0, and converges to 0 as r1 =r2 converges to 1, also applies to player
29
1’s maxmin payo¤. Thus, this important limit uniqueness result continues to hold under
knowledge of rationality, and in particular does not rely on equilibrium. This is analogous
to the …nding of Watson (1993) and Battigalli and Watson (1997) that the limit uniqueness
result of Fudenberg and Levine (1989) holds under knowledge of rationality. However, I also
derive player 1’s maxmin payo¤ for …xed r1 =r2 (rather than only in the limit). This leads
to a second comparative static result, which indicates that a change in relative patience has
a larger e¤ect on the maxmin payo¤ than a much larger change in commitment probability.
An analogous result holds in equilibrium in existing reputational bargaining models.
I …rst present the analog of Theorem 1 for heterogeneous discount rates, and then state
the two comparative statics results as corollaries, omitting their proofs.
Proposition 4 If player i’s discount rate is ri , then player 1’s maxmin payo¤, u1 ("), is the
unique number u1 that solves
u1 =
1
1
r1
r2
log "
r1
r2
.
(12)
1 log u1
Corollary 3 shows that the standard limit comparative statics on r1 =r2 in reputational
bargaining models require only …rst-order knowledge of rationality.
Corollary 3 limr1 =r2 !0 u1 (") = 1. If " < 1, then in addition limr1 =r2 !1 u1 (") = 0.
Corollary 4 shows that the commitment probability " must decrease exponentially to
(approximately) o¤set a geometric increase in relative patience (r1 =r2 ) 1 .
stated for the case r1 =r2
The result is
1, where even an exponential decrease in " does not fully o¤set
a geometric increase in (r1 =r2 ) 1 . If r1 =r2 > 1, then an exponential decrease in " can more
than o¤set a geometric increase in (r1 =r2 ) 1 .
Corollary 4 Suppose that r1 =r2
1 and that r1 =r2 and " both decrease while (r1 =r2 ) log "
remains constant. Then u1 (") increases.
5.4
Rationalizability
Theorem 1 derives the highest payo¤ that player 1 can guarantee herself under …rst-order
knowledge of rationality, the weakest epistemic assumption consistent with the possibility
30
of reputation-building. I now show that player 1 cannot guarantee herself more than this
under the much stronger assumption of normal-form rationalizability (or under any …niteorder knowledge of rationality), which reenforces Theorem 1 substantially.
I consider the following de…nition of (normal-form) rationalizability:
De…nition 7 A set of bargaining phase strategy pro…les
under rational behavior given posture
such that
that
2
( )
1
2
1
( 1 ); and for all
2
if for all
2
2
1
2
1
=
1
2
there exists some belief
there exists some belief
", with strict inequality only if
2
1,
1
and
2
2
2
2
2
(
1
2
is closed
1
2
(
2)
[ f g) such
( 2 ).
The set of rationalizable strategies given posture is
[
RAT
( )
f : is closed under rational behavior given posture g .
Player 1’s rationalizable maxmin payo¤ given posture
uRAT
( )
1
sup
1
inf
22
RAT (
2
)
is
u1 ( 1 ;
2) .
Player 1’s rationalizable maxmin payo¤ is
uRAT
1
A posture
f
ng
RAT
such that
sup uRAT
( ).
1
is a rationalizable maxmin posture if there exists a sequence of postures
n
(t) !
RAT
(
(t) for all t 2 R+ and uRAT
1
n)
.
! uRAT
1
The result is the following:
Proposition 5 Player 1’s rationalizable maxmin payo¤ equals her maxmin payo¤, and the
unique rationalizable maxmin posture is the unique maxmin posture. That is, uRAT
= u1 ,
1
and the unique rationalizable maxmin posture is
Any rationalizable strategy given posture
RAT
=
.
is also rational given posture . Therefore,
Lemma 1 applies under rationalizability. The only additional fact used in the proof of Theorem 1 is that u1 ( ) = mint
T( )
e
rt
(t) for any posture
(Lemma 3). Supposing that the
analogous equation holds under rationalizability (i.e., that uRAT
( ) = mint
1
T( )
e
rt
(t)),
the proof of Theorem 1 goes through as written. Hence, to prove Proposition 5 it su¢ ces to
prove the following lemma, the proof of which shows that the -o¤setting belief and strategy
are not only rational but rationalizable:
31
Lemma 4 For any posture , uRAT
( ) = mint
1
T( )
e
rt
(t).
Proposition 5 shows that the characterization of the maxmin payo¤ and posture under
…rst-order knowledge of normal-form rationality extends to normal-form rationalizability.
A natural question is whether it also extends to various forms of knowledge of sequential
rationality or extensive-form rationalizability (e.g., Pearce, 1984; Battigalli and Siniscalchi,
2003).
I partially address this question in the supplementary appendix by showing that
Theorem 1 extends at least to a simple notion of iterated conditional dominance. I summarize the argument here. The -o¤setting belief assigns positive probability to player 1’s
accepting a demand of 1 at certain times, and this behavior is ruled out by iterated conditional dominance.
Hence, the proof of Theorem 1 (or Proposition 5) does not extend to
iterated conditional dominance. However, the -o¤setting belief can be approximated by
a belief that replaces player 1’s accepting a demand of 1 with player 1’s quickly increasing
her own demand to almost 1 and then quickly reducing it to almost 0.
From player 2’s
perspective, this is almost as good as her accepting a demand of 1, as he can accept when
her demand is close to 0. In addition, this behavior of player 1’s is consistent with iterated
conditional dominance, since it can be shown that almost any behavior at histories that are
inconsistent with
is consistent with iterated conditional dominance, and therefore player
1 may expect that player 2 will accept her demand when it is close to 1.
Therefore, the
approximating belief assigns positive probability only to strategies that are consistent with
iterated conditional dominance, and one can prove that Theorem 1 applies under iterated
conditional dominance by replacing the -o¤setting belief with the approximating belief.
5.5
Varying Credibility and “Exogenous” Postures
I brie‡y discuss how the results would change if the credibility of player 1’s announcement
depended on the posture she announced, or if player 1 was restricted to announcing one of an
exogenously given vector of postures and the credibility of her announcement was determined
by Bayesian updating on a vector of prior commitment probabilities (as in Abreu and Gul
(2000)).
The extension to varying credibility is straightforward. Let u1 ( ; ") be player 1’s maxmin
32
payo¤ given posture
and credibility ", let T ( ; ") be the time T de…ned in Lemma 1 with
the dependence on
and " made explicit, and let " ( ) be the credibility of announcement
(i.e., replace " by " ( ) in De…nition 1, leaving all other de…nitions unchanged).
The
following result is an immediate consequence of Lemma 3.
Corollary 5 When player 1’s credibility varies with her announcement, her maxmin payo¤
is
u1 = sup u1 ( ; " ( )) = sup
min
t T ( ;"( ))
e
rt
(t) .
Since u1 ( ; " ( )) is non-decreasing in " ( ), Corollary 5 captures a tradeo¤ between
announcing a more credible posture and announcing a posture that guarantees a higher
payo¤ for a given level of credibility.
The model with an exogenous vector of postures can be thought of as a special case of
varying credibility where " ( ) is determined endogenously. Formally, let there be a vector of
postures ( k )K
k=1 (assumed …nite for simplicity) with corresponding commitment probabilities
("k )K
k=1 . Assume that player 2’s assessment of the probability that player 1 is committed to
posture
k
after she announces such a posture is determined by Bayesian updating given his
belief about the probability that she announces each posture, denoted (~ 2 ( k ))K
k=1 . Thus,
the resulting commitment probability is
"k
PK
~"k =
"k + 1
k0 =1
De…ne player 1’s maxmin payo¤ to be
~2:
PK inf
k=1 ~ 2 (
k )=1
.
"k 0 ~ 2 ( k )
max u1 ( k ; ~"k ) .29
k
The analog of Corollary 5 for this model is the following.
Corollary 6 In the model with exogenous postures, player 1’s maxmin payo¤ is
u1 =
29
P inf
~2: K
k=1 ~ 2 (
This de…nition is “minmax” over ~ 2 and
max
k )=1
k
k
min
t T(
"k )
k ;~
e
rt
k
(t) .
rather than “maxmin.” With a maxmin de…nition (i.e.,
reversing the order of the inf and max in the de…nition), player 1’s maxmin payo¤ would simply equal
max
k
u1 (
k ; "k ),
because the worst belief ~ 2 would always be that ~ 2 (
player 1 announced.
33
k)
= 1 for whichever posture
k
I remark that the considerations that determine the “worst case”belief about player 1’s
announcement, ~ 2 , are similar to those that determine the equilibrium distribution of player
1’s choice of posture in Abreu and Gul. In particular, it must be the case that u1 ( k ; ~"k ) =
u1 (
"k 0 )
k0 ; ~
u1 (
k0 ; ")
whenever
k
and
k0
both receive positive weight under ~ 2 and u1 ( k ; ") and
are strictly increasing in ", as otherwise transferring weight to whichever posture
yields a higher maxmin payo¤ would reduce player 1’s (overall) maxmin payo¤. Similarly,
in equilibrium player 1’s mixing probabilities are determined by the condition that any two
postures she chooses with positive probability must yield the same payo¤ given player 2’s
resulting posterior.
6
Two-Sided Commitment
This section introduces the possibility that both players may announce— and become committed to— postures prior to the start of bargaining. I show that each player i’s maxmin
payo¤ is close to that derived in Section 4 when her opponent’s commitment probability,
"j , is small in absolute terms (even if "j is large relative to "i ). In addition, each player’s
maxmin posture is exactly as in Section 4. This shows that the analysis of Section 4 provides a two-sided theory of reputational bargaining. The results of this section contrast with
the existing reputational bargaining literature, which emphasizes that relative commitment
probabilities are crucial for determining equilibrium behavior and payo¤s.
Formally, modify the model of Section 3 by assuming that in the announcement stage
players simultaneously announce postures ( 1 ;
probabilities "1 and "2 , respectively.30
2 ),
to which they become committed with
The bargaining phase is unaltered.
Thus, at the
beginning of the bargaining phase, player i believes that player j is committed to posture
with probability "j and is rational with probability 1
j
"j (though this fact is not common
knowledge). The following de…nitions are analogs of De…nitions 1 through 3 that allow for
the fact that both players may become committed to the postures they announce:
De…nition 8 A belief
postures
30
i;
j
if
i
i
j
of player i’s is consistent with knowledge of rationality given
"j ;
i
j
> "j only if there exists
j
such that
The events that player 1 and player 2 become committed need not be independent.
34
j
( i)
"i and
j
2
and
j
j
( j ); and, for all
2
j
( j ). Let
j
i; j
6=
j,
2 supp ( i ) only if there exists
j
j
such that
j
( i)
"i
be the set of player i’s beliefs that are consistent with knowledge
i
of rationality given postures
i;
ui
j
. Player i’s maxmin payo¤ given postures
i;
j
sup
inf ; ui ( i ;
i2
i
i
j
i;
j
is
i) .
i
Player i’s maxmin payo¤ is
sup inf ui
ui
i
A posture
i
such that
n
i;
.
j
j
is a maxmin posture (of player i’s) if there exists a sequence of postures f
(t) !
i
(t) for all t 2 R+ and inf
j
ui
n;
j
ng
! ui .
Note that if "j = 0 then all of these de…nitions (for player i) reduce to the corresponding
Thus, writing ui ("i ; "j ) for player i’s
de…nitions in the one-sided commitment model.
maxmin payo¤ in the two-sided commitment model when the commitment probabilities
are "i and "j , it follows that ui ("i ; 0) = ui ("i ), player i’s maxmin payo¤ in the one-sided
commitment model.
I now show that ui ("i ; "j ) is approximately equal to ui ("i ) whenever "j is small, and
that the maxmin posture is exactly as in the one-sided commitment model. This is simply
because player i cannot guarantee herself anything in the event that player j is committed
(e.g., if player j’s announced posture always demands the entire surplus), which implies that
player i guarantees herself as much as possible by conditioning on the event that player j is
not committed. In this event, which occurs with probability 1
"j , player i can guarantee
herself ui ("i ), and the only way she can guarantee herself this much is by announcing
Theorem 2 Player i’s maxmin payo¤ is ui ("i ; "j ) = (1
maxmin posture is
i;("i ;"j )
Proof. Let
0
j
0 for all
Therefore, inf
i.
Next, let
=
i; j
i
"j ) ui ("i ), and player i’s unique
"i .
0
j
be the posture of player j’s given by
j
"i .
ui
i;
j
= 0 for all
("i ; "j ) be the set of beliefs
(t) = 1 for all t. Note that ui
i
=
that are consistent with knowledge of
I claim that if
35
0
j
i.
rationality for commitment probabilities ("i ; "j ), and let
the one-sided commitment model.
i;
i
2
i
i
("i ) be the analogous set in
i; j
i
("i ; "j ), then there exists
0
i
2
i
i
("i ) such that
i
puts probability 1
strategy . To see this, note that
puts probability 1
i; j
i
("i ; "j ),
j
"j on
0
i
i
"j on strategy
j
equals
j ).
and puts probability "j on
0
i
"j , so there exists a strategy
j
such that
i
and puts probability "j on . Furthermore, by de…nition of
2 supp ( 0i ) only if there exists
(whether or not
0
i
such that
j
By de…nition of
i
i
j
( i)
"i and
("i ), this implies that
0
i
j
2
2
i
i
j
( j)
("i ).
Combining the above observations,
i;
inf ui
j
= inf sup
j
j
= inf sup
j
i
= sup
i
inf
;
i
ui ( i ;
("i ;"j )
inf
(1
02
i
i
i ("i )
inf
02
i
j
i
i2
i
i
i ("i )
(1
i)
"j ) ui ( i ;
"j ) ui ( i ;
0
i)
0
i)
+ "j u i
i;
j
.
+ "j (0)
"j ) u i ( i ) .
= (1
Therefore, the de…nitions of ui ("i ; "j ) and ui ("i ) imply that ui ("i ; "j ) = sup i (1
(1
"j ) u i ( i ) =
"j ) ui ("i ). Similarly, the de…nition of a maxmin posture in the one-sided commitment
model implies that
i;("i ;"j )
is a maxmin posture in the two-sided commitment model if and
only if it is a maxmin posture in the one-sided commitment model with " = "i .
Theorem 2 implies that the qualitative insights of Theorem 1 also apply with two-sided
commitment. For example, …xing any "2 bounded away from 0, u1 ("1 ; "2 ) goes to 0 at a
logarithmic rate in "1 . Thus, Theorem 2 says much more than that u1 ("1 ; "2 ) is continuous
in "2 at "2 = 0. Table 2 displays the maxmin payo¤ for both the one-sided commitment
model and the two-sided commitment model in the case where "1 = "2 = ".
"
ui (")
(1
") ui (")
:25
:42
:31
:1
:30
:27
10
3
:13
:13
10
6
:07
:07
10
9
:05
:05
Table 2: The Maxmin Payo¤ for Di¤erent Commitment
Probabilities " with One- and Two-Sided Commitment
36
Finally, note that the assumption that both players best-respond to beliefs that are consistent with knowledge of rationality does not determine how bargaining proceeds. However,
it is interesting to compare a player’s opponent’s worst-case conjecture (i.e., o¤setting belief) about her strategy with her equilibrium strategy in the existing reputational bargaining
literature, where players are restricted to announcing constant postures. Both in equilibrium and in her opponent’s worst-case conjecture, a player mixes between mimicking her
announced posture and (in e¤ect) conceding. In the worst-case conjecture, a player concedes
at the rate that makes her opponent indi¤erent between accepting and rejecting her demand
when she follows her maxmin posture and he demands the entire surplus. In equilibrium,
a player concedes at the rate that makes her opponent indi¤erent between accepting and
rejecting her demand when both players make their equilibrium demands.
There is no
general way to order the concession rates in equilibrium and in the worst-case conjecture,
because a player’s demand is often higher in equilibrium (implying a lower concession rate),
while her opponent’s demand is always lower in equilibrium (implying a higher concession
rate).
Thus, a player’s concession rate in equilibrium and in her opponent’s worst-case
conjecture are determined by similar indi¤erence conditions, but one cannot predict whether
agreement will be reached sooner in equilibrium or under knowledge of rationality (for either
the players’true strategies or for their opponents’worst-case conjectures).
7
Conclusion
This paper analyzes a model of reputational bargaining in which players initially announce
postures to which they may become committed and then bargain over a unit of surplus. It
characterizes the highest payo¤ that a player can guarantee herself under …rst-order knowledge of rationality, along with the bargaining posture that she must announce in order to
guarantee herself this much. A key step in the characterization is showing that this maxmin
payo¤ is the payo¤ a player receives when her opponent holds the “o¤setting belief” that
she mixes between following her announced posture and accepting her opponent’s demand
at a speci…c rate. This intermediate result lets one evaluate a posture in terms of its performance against an opponent who holds the corresponding o¤setting belief, rather than having
37
to check its performance against every rational opposing strategy.
I …nd that a player can guarantee herself a relatively high share of the surplus even if her
probability of becoming committed is very small, and that the unique bargaining posture
that guarantees this much is simply demanding this share of the surplus in addition to
compensation for any delay in reaching agreement. These insights apply for one- or twosided commitment, for general bargaining sets, for heterogeneous discount factors, for any
level of knowledge of rationality or iterated conditional dominance, and for any bargaining
procedure with frequent o¤ers. In addition, if a player could only announce postures that
always demand the same share of the surplus (as in most of the existing literature), her
maxmin payo¤ would be approximately e times lower.
These results are intended to complement the existing equilibrium analysis of reputational
bargaining models. Consider the fundamental question, “What posture should a bargainer
stake out?” In equilibrium analysis, the answer to this question depends on her opponent’s
beliefs about her continuation play following every possible announcement. Yet it may be
impossible for either the bargainer or an outside observer to learn these beliefs, especially
when bargaining is one-shot.
Hence, an appealing alternative approach is to look for a
posture that guarantees a high payo¤ against any belief of one’s opponent, and for the
highest payo¤ that each player can guarantee herself. This paper shows that this approach
yields sharp and economically plausible results, while addressing important concerns about
robustness.
The results of this paper are particularly applicable in models where the division of the
surplus is of primary importance (rather than the details of how bargaining proceeds, which
depend on additional behavioral assumptions).
A leading example is the class of models
where two parties make costly ex ante investments and then bargain over the resulting surplus. It is a direct consequence of Theorem 2 that, if the players’commitment probabilities
are small, both players bene…t from comparable increases in their commitment probabilities.31 Note that this is distinct from the idea that both players bene…t from reducing delay;
31
To see this somewhat more formally, recall that Theorem 2 states that player i’s maxmin pay-
o¤ is ui ("i ; "j ) =
lim"i !0 "i (1
1 "j
1 log "i .
Therefore,
log "i ) = 0, it follows that
@ui ("i ;"j )
@"i
1 "j
,
"i (1 log "i )2
@ui ("i ;"j )=@"i
lim"i !0 @u ("i ;"j )=@"j =
i
=
38
while
1.
@ui ("i ;"j )
@"j
=
1
1 log "i .
Since
In this sense, an increase in "i
rather, both players bene…t because higher commitment probabilities reduce the scope for
pessimism about how bargaining will proceed (which would not be possible in an equilibrium
analysis). This implies that, for example, comparably increasing both players’commitment
powers increases investments whenever investments are complementary. It seems likely that
additional insights could be derived by further studying non-equilibrium models of bargaining
both in this class of models and in other applied theory models involving bargaining.
Finally, I discuss two additional issues for future research. First, an earlier version of this
paper extends the model to multilateral bargaining, where n
agree on the division of the surplus.
3 players must unanimously
In such a model, player j may reject any proposal
if he expects player k to do so as well, and vice versa. Hence, a player with commitment
power cannot guarantee herself a positive payo¤ under knowledge of rationality. It therefore
remains to be seen whether reputational models can make sharp and robust predictions in
multilateral bargaining.
Second, it would be interesting to analyze commitment and reputation-building under
knowledge of rationality in dynamic games other than bargaining, noting that the de…nition
of a player’s maxmin payo¤ and posture extends to general games. One intriguing observation is that the rate at which the reputation-builder’s maxmin payo¤ converges to her lowest
feasible payo¤ as her commitment probability converges to 0 is slower in my model than in
existing repeated game models. In particular, the reputation-builder’s (player 1’s) maxmin
payo¤ converges to her minimum payo¤ of 0 at a logarithmic rate in " in my model, while
in repeated game models this convergence is at a polynomial rate in ".32
To understand
this di¤erence, recall from the proof of Proposition 2 that if player 1 announces a constant
posture , then her maxmin payo¤ given posture
equals "
=(1
)
, which is polynomial in
". However, the maxmin constant posture is increasing in " (and goes to 0 as " ! 0), and
Corollary 2 shows that player 1 can guarantee herself a payo¤ that goes to 0 at a logarithmic
increases player i’s maxmin payo¤ by much more than an increase in "j decreases it.
32
Fudenberg and Levine (1989) show that if player 1 is committed to her Stackelberg action with probability
" and player 2 is myopic, then player 1’s payo¤ in any Nash equilibrium is at least "r1 u
~1 + (1
some constant
"r1 ) u1 , for
> 0, where u
~1 is her Stackelberg payo¤ and u1 is her lowest feasible payo¤. This bound
is the basis for most of the subsequent literature; for example, convergence to u1 is also polynomial in " in
Schmidt (1993), Cripps, Schmidt, and Thomas (1996), and Evans and Thomas (1997).
39
rate in " by appropriately recalibrating her announced posture as " ! 0.
Thus, roughly
speaking, the reason why reputation bounds in the repeated games literature converge to 0
more quickly than in my model is that there is generally no way to continuously moderate
one’s posture as one’s commitment probability decreases in repeated games.
Appendix: Omitted Proofs
Proof of Proposition 1. The proof uses results from Section 4, and therefore should not
be read before reading Section 4.
Fix a posture
and payo¤ u1 2 [u1 ( ) ; 1). If u1 6=
(0), then let ^ 2 be identical to the
-o¤setting strategy de…ned in De…nition 6, with the modi…cation that player 1’s demand
is accepted at any history ht at which player 1 has demanded u1 at all previous dates.33 If
(0), then let ^ 2 be identical to the -o¤setting strategy de…ned in De…nition 6, with
u1 =
the modi…cation that player 1’s demand is accepted at date ( log (u1 ) =r; 1) if player 1
has demanded 1 at all previous dates. In either case, let
that
2;
2
( )
".
If u1 6=
similarly, if u1 =
2
be as in De…nition 6, and note
(0), no strategy under which u1 (0) = u1 is in the support of
(0), no strategy under which u1 (0) = 1 is in the support of
u1 < 1). Therefore, the same argument as in the proof of Lemma 2 shows that ^ 2 2
Hence, the belief ^ 1 given by ^ 1 (^ 2 ) = 1 is an element of
1.
2
(since
2
(
2 ).
Furthermore, under strategy
^ 2 , player 2 always demands 1 and only accepts player 1’s demand if player 1 has either
conformed to
u1 6=
through time t (de…ned in Section 4.2) or has always demanded u1 (in the
(0) case) or 1 (in the u1 =
(0) case).
Note that exp ( r ( log (u1 ) =r)) = u1 .
Hence, in either case, u1 ( 1 ; ^ 1 ) 2 f0; u1 ( ) ; u1 g for every strategy
strategy of player 1’s that always demands u1 (if u1 6=
1.
Let ^ 1 be the
(0)) or 1 (if u1 =
(0)) and never
accepts player 2’s demand. Then u1 (^ 1 ; ^ 1 ) = u1 = max 1 u1 ( 1 ; ^ 1 ), completing the proof.
Proof of Lemma 1. I prove the result for pure strategies
2,
which immediately implies
the result for mixed strategies.
Fix
33
2
such that
2
( )
" and pure strategy
2
2
2
( 2 ). The plan of the proof is to
That is, modify the second part of De…nition 6 to include this contingency.
40
show that if agreement is not reached by T~ under strategy pro…le ( ;
2
2
2
then player 2 must
at any time t > T~. This su¢ ces to prove the lemma,
be certain that player 1 is playing
because
2 ),
( 2 ) implies that player 2 accepts
(t) no later than time t = T if at any
time t > T~ agreement has not been reached and he is certain that player 1 is playing .
(
I begin by introducing some notation. Let
assigns to player 1 not playing
2; 2)
(t) be the probability that player 2
at date (t; 1) when his initial belief is
until date (t; 1) is given by player 1’s following strategy
strategy
2;
this is determined by Bayes’rule, because
(
agreement is reached at time , let
2; 2)
(
(t) =
2; 2)
1 if agreement is never reached under ( ;
2)
t^( ;
2)
(
sup t :
2; 2)
2
" > 0. By convention, if
( )
( ) for all t > . Let t ( ;
2 ));
2)
2)
with belief
2.
2)
be the
(with the convention that
and let
(t) > 0 ,
the latest time at which player 2 is not certain that player 1 is playing
pro…le ( ;
and play up
and player 2’s following (pure)
time at which agreement is reached under strategy pro…le ( ;
t( ;
2
under strategy
Let
T^
t^( ;
sup
( 2 ; 2 ): 2 ( ) "; 2 2 2 ( 2 );t( ; 2 ) t^( ;
2) .
(13)
2)
That is, T^ is the latest possible time t at which player 2 is not certain that player 1 is
following
and agreement is not reached by t.
The remainder of the proof consists of
showing that T^ = T~.
There are three steps involved in showing that T^ = T~, that is, that the value of the
program (13) is T~. Step 1 shows that in solving (13) one can restrict attention to a simple
class of belief-strategy pairs ( 2 ;
t( ;
t^( ;
2)
2)
2 ).
Step 2 reduces the constraints that
2
2
2
( 2 ) and
to an in…nite system of inequalities involving player 1’s concession rate
and probability. Step 3 solves the reduced program.
Step 1:
( 2;
2)
In the de…nition of T^ it is without loss of generality to restrict attention to
such that
2
always demands 1,
any history ht+ at which u1 (t) 6=
(t), and
2
puts probability 1 on player 1 conceding at
2
puts probability 0 on player 1 conceding at
any history ht . That is, that the right-hand side of (13) continues to equal T^ when this
additional constraint is imposed on ( 2 ;
2 ).
41
Proof. Suppose that ( 02 ;
0
2)
constraints of (13)). Let
2
0
2
satis…es
( )
0
2
",
2
2
( 02 ), and t ( ;
be the belief under which player 1 demands
accepts player 2’s demand at every history of the form ( ( ) ; 1)
probability at which player 1 deviates from
under strategy pro…le ( 02 ;
there exists a strategy
2
t
t^( ;
0
2)
0
2)
(the
(t) for all t 2 R+ ;
at the same rate and
at time t (i.e., at date (t; 1), (t; 0), or (t; 1))
0
2 );
and rejects player 2’s demand at every other history. Clearly,
2
( 2 ) that always demands 1 and that in addition rejects player
2
1’s demand whenever rejection is optimal under belief
2.
Note that player 1’s rate and
probability of deviating from
at history ( ( ) ; 1)
under strategy pro…le ( 02 ;
and that player 2’s continuation payo¤ after such a deviation
0
2 ),
t
under belief
is weakly higher in the former case. Recall that strategy
2
is the same as at time t
never accepts player 2’s demand,
so agreement is reached only if player 2 accepts player 1’s demand or if player 1 has deviated
from .
is optimal for all t < t ( ;
( ( ) ; 1)
t
0
2 ),
it follows that rejecting player 1’s demand
is optimal under belief
2,
for all t < t ( ;
whenever it is optimal, this implies that t ( ;
( 02 ; 02 ) (t) for all t 2 R , so t^( ; ) = t^( ;
+
2
2
2
". Therefore, ( 2 ;
( )
2)
0
2)
(t) under strategy pro…le ( 02 ;
Therefore, since rejecting player 1’s demand
0
2 ).
Since
0
2 ).
prescribes rejection
2
t( ;
0
2 ).
Hence, t ( ;
2
t^( ;
2)
2; 2)
(
Furthermore,
2)
satis…es the constraints of (13),
(t) at history
(t) =
Finally,
2 ).
always demands u2 (t) = 1,
puts probability 1 on player 1 conceding at any history ht+ at which u1 (t) 6=
puts probability 0 on player 1 conceding at any history ht , and t^( ;
t^( ;
2)
(t),
0
2 ).
2
So the
right-hand side of (13) continues to equal T^ when the additional constraint is imposed.
Step 2 of the proof builds on Step 1 to further simplify the constraint set of (13). For
any belief
2
satisfying the conditions of Step 1, let
and probability of player 1 at history ( ( ) ; 1)
t
2
(t) and p 2 (t) be the concession rate
when her strategy is given by
be the (countable) set of times s such that p 2 (s) > 0; and let t^( 2 )
t^( ;
0
2 ),
2;
let S
2
0
2
is
where
the strategy that always demands 1 and always rejects player 1’s demand. Fixing a strategy
2
2
2
( 2 ) that always demands 1 (which exists, by Step 1), note that ( ;
induce the same path of play until time t ( ;
but not under
0
2 ),
and therefore t ( ;
there exists a strategy
2
2
2
2)
2)
t^( ;
2)
and ( ;
(at which point player 2 accepts under
2)
if and only if t ( ;
2)
( 2 ) that always demands 1 and satis…es t ( ;
0
2)
2
t^( 2 ). Hence,
2)
t^( ;
2)
if and only if it is optimal for player 2 to reject player 1’s o¤er until time t^( 2 ) when he
42
always demands 1, player 1 plays , and his initial belief is
2.
I now use this observation
to simplify the constraints of (13).
Step 2a:
2
2
2
For any belief
2
satisfying the conditions of Step 1, there exists a strategy
( 2 ) that always demands 1 and satis…es t ( ;
1
Z
(t)
t^(
2)
exp
r(
t)
Z
2
t
t
X
+
s2S
2\
+ exp
(t;t^(
exp
2)
r (s
0
(s) ds @
Z
t)
2
t
)
r t^( 2 )
s
t
Z
t^(
2)
2
t
for all t < t^( 2 ) .
t^( ;
2)
s2S
Y
0
(q) dq @
!
B
(s) ds @
s2S
if and only if
1
(1
2 \(t;
0
2)
)
q2S
Y
(1
2 \(t;s)
Y
2\
p 2 (s))A
(t;t^(
(1
)
2)
2
( )d
1
p 2 (q))A p 2 (s)
1
C
p 2 (s))A v t^( 2 )
(14)
Proof. By the above discussion, it su¢ ces to show that (14) holds if and only if it is
optimal for player 2 to reject player 1’s o¤er until time t^( 2 ) when he always demands 1,
player 1 plays , and his initial belief is
2.
The left-hand side of (14) is player 2’s payo¤
from accepting player 1’s demand at date (t; 1) when p 2 (t) = 0.
The right-hand side
of (14) is player 2’s continuation payo¤ from rejecting player 1’s demand until time t^( 2 )
when p 2 (t) = 0.
Thus, (14) must hold if it is optimal for player 2 to reject until time
t^( 2 ), and (14) implies that it is optimal for player 2 to reject at times before t^( 2 ) where
p 2 (t) = 0. It remains to show that (14) implies that it is optimal for player 2 to reject at
times before t^( 2 ) where p 2 (t) > 0. Suppose that p 2 (t) > 0. At date (t; 1), the fact
that S
lim
"t
2
(1
is countable and (14) holds at all times before t that are not in S
2
implies that
( )) is weakly less than player 2’s continuation payo¤ from rejecting player 1’s
demand until time t^( 2 ). Furthermore, the fact that player 1 concedes with probability 0 at
date (t; 1) (as
2
satis…es the conditions of Step 1) implies that lim
"t
(1
( )) is indeed
player 2’s payo¤ from accepting at date (t; 1). Thus, rejecting is optimal at date (t; 1).
At date (t; 1), player 2’s payo¤ from accepting is (1
p 2 (t) =2) (1
(t)) + (p 2 (t) =2) (1),
while his continuation payo¤ from rejecting until time t^( 2 ) is 1 p 2 (t) times the right-hand
side of (14) plus p 2 (t) (1). Hence, (14) implies that rejecting is optimal at date (t; 1) as
43
well. So (14) implies that it is optimal for player 2 to reject at times before t^( 2 ) where
p 2 (t) > 0 (when he always demands 1, player 1 plays , and his initial belief is
Step 2b:
2
2
2
For any belief
2
satisfying the conditions of Step 1, there exists a strategy
( 2 ) that always demands 1 and satis…es t ( ;
v (t)
Z t^( 2 )
exp
r(
Z
t)
+
s2S
2
t
t
X
2\
exp
(t;t^(
2)
r (s
0
Z
t)
s
2
t
t
Z
t^(
2)
2
t
for all t < t^( 2 ) .
t^( ;
2)
(s) ds @
)
r t^( 2 )
+ exp
s2S
Y
0
(q) dq @
!
B
(s) ds @
s2S
if and only if
1
(1
2 \(t;
0
2)
)
q2S
Y
2\
Y
p 2 (s))A
(1
2 \(t;s)
(t;t^(
(1
2)
)
immediately implies (14) because v (t)
Now v (t) > 1
v ( ; 1) = 1
( )d
1
p 2 (q))A p 2 (s)
1
C
p 2 (s))A v t^( 2 )
Note that (15)
(t) for all t. For the converse, suppose that
1
(t) (as (14) and (15) are identical at t if v (t) = 1
(t) implies that v (t) = e
( ).
2
(15)
Proof. By Step 2a, it su¢ ces to show that (14) is equivalent to (15).
(14) holds, and that v (t) > 1
2 ).
r(
t)
1
( ) for some
(t)).
> t such that
Therefore, v ( ; 1) is weakly less than the limit as s "
of the
right-hand side of (15) evaluated at time s (with the convention that the right-hand side of
(15) equals v (s) if s
t^( 2 )). But the right-hand side of (15) at time t is at least e
r(
t)
times as large as this limit, which implies that the right-hand side of (15) at time t is at least
e
r(
t)
v ( ; 1) = v (t). Hence, (15) holds.
( 2 ; 02 ) . By Step 1 and Step 2b (and the use of Bayes’rule to compute
Let 2 (t)
(t)), (13) may be rewritten as
8
9
Rt 2
Q
2
<
=
exp
(s)
ds
(1
p
(s))
"
2
s2S \[0;t)
0
2
^
T = sup sup t :
(t) =
>0 .
Rt 2
Q
:
;
2 (s))
exp
(s)
ds
(1
p
2 : 2 ( ) ";
s2S 2 \[0;t)
0
2
(15) holds
(16)
The last step of the proof is solving this simpli…ed program. Steps 3a and 3b show that there
exists some belief
2
that both attains the (outer) supremum in (16) (with the convention
44
that the supremum is attained at
over all beliefs
2
2
2
if t^( 2 ) = T^ = 1) and also maximizes limt"T^
that attain the supremum (note that this limit exists for all
2,
2
(t)
because
(t) is non-increasing). I then show that (15) must hold with equality (at all t < T^) under
any such belief
2,
which implies that (16) may be solved under the additional constraint
that (15) holds with equality. This …nal program is in fact solved as a step in the proof of
Step 3a, and has value T~.
Step 3a: There exists a belief
2
that attains the supremum in (16).
Proof. The plan of the proof is as follows. I …rst construct a belief
and (15) holds at all times t < T^ under
2.
2
such that
2
( )
"
I then show that if there exists a time t < T^ at
which (15) holds with strict inequality under
2,
0
2
then there exists an alternative belief
that attains the supremum in (16). Finally, I show that if (15) holds with equality at all
times t < T^ under
2,
n
2
Fix a sequence
n
2
Note that
then
2
itself attains the supremum in (16).
such that t^(
n
2)
" T^,
n
2
( )
" for all n, and (15) holds for all n.
(t) is non-increasing in t, for all n. Since the space of monotone functions from
R+ to [0; 1] is sequentially compact (by Helly’s selection theorem; see, e.g., Billingsley (1995)
m
2
Theorem 25.9), there exists a subsequence
increasing) function
Let
2
2
at rate
2
lim
(
2
1)
2
.34 Furthermore,
be a belief such that
(t) =
2 0 (t)
1
2 (t)
if
2
2
( )
2
(0)
that converges pointwise to some (non1
", because
m
2
1
" for all m.
" and player 1 demands (t) for all t, concedes
is di¤erentiable at t and
2
(t) > 0 and concedes at rate
(t) = 0 otherwise, and concedes with discrete probability p 2 (t) =
"t
(0)
lim
(1
2( )
"t
2 (t)) lim "t
2 (t)
2(
)
if
(t) > 0 and concedes with probability p 2 (t) = 0 otherwise. Note that such a belief
exists, because it can be easily veri…ed that for any belief corresponding to concession rate
34
Showing that the space of monotone functions from R+ ! [0; 1] is sequentially compact requires a slightly
di¤erent version of Helly’s selection theorem than that in Billingsley (1995), so here is a direct proof: If ffn g
is a sequence of monotone functions R+ ! [0; 1], then there exists a subsequence ffm g
on Q+ to a monotone function f : Q+ ! [0; 1].
ffn g that converges
Let f~ : R+ ! [0; 1] be given by f~ (x) = liml!1 f (xl ),
1
where fxl gl=1 " x and xl 2 Q+ for all l. Then f~ is monotone, which implies that there is a countable set S
such that f~ is continuous on R+ nS. Since S is countable, there exists a sub-subsequence ffk g
ffm g such
that ffk g converges on S. Finally, let f^ (x) = f~ (x) if x 2 R+ nS and f^ (x) = limk!1 fk (x) if x 2 S. Then
ffk g ! f^.
45
and probability
2
and p 2 ,
exp
2
(t) =
Rt
2
0
exp
(s) ds
Rt
2
0
Q
s2S
(s) ds
2 \[0;t)
Q
p 2 (s))
(1
(1
2
(0))
for all t,
2 \[0;t) (1
s2S
(17)
p 2 (s))
and therefore player 1 never concedes with probability at least 1
2
" under any
(0)
belief with this concession rate and probability.
Observe that (15) holds at all times t < T^ under
m
2
exp
(t) !
Z t
2
2.
To see this, note that the fact that
(t) for all t implies that
m
2
(s) ds
0
s2S
Y
1
p
m
2
2 \[0;t)
Z
(s) ! exp
t
2
(s) ds
0
s2S
Y
(1
2 \[0;t)
for all t. Since for all t < T^, there exists M > 0 such that (15) holds at time t under
all m > M , this implies that (15) holds at all times t < T^ under
p 2 (s))
m
2
for
2.
I now show that if there exists a time t < T^ at which (15) holds with strict inequality
under belief
2,
then there exists an alternative belief
0
2
that attains the supremum in (16).
I claim that there then exists a time t1 < t^( 2 ) at which
Rt +
2
(15) holds with strict inequality and in addition either t11
(s) ds > 0 for all > 0 or
P
2
> 0. To see this, note that there must exist a time
s2S 2 \[t1 ;t1 + ) p (s) > 0 for all
R
0
t+
2
t0 2 t; t^( 2 ) such that either t0
(s) ds > 0 for all
> 0 or p 2 (t0 ) > 0 (because
Suppose such a time t exists.
otherwise (15) could not hold with strict inequality at t). Let t1 be the in…mum of such
Rt +
P
2
times t0 , and note that either t11
(s) ds > 0 for all > 0 or s2S 2 \[t1 ;t1 + ) p 2 (s) > 0
for all
> 0. Then the fact that (15) holds with strict inequality at time t implies that
Rt
(15) holds with strict inequality at time t1 , because otherwise the fact that t 1 2 (s) ds = 0
and p 2 (t00 ) = 0 for all t00 2 [t; t1 ) would imply that (15) could not hold with strict inequality
at time t. This proves the claim.
Thus, let t0 < t^( 2 ) be such that (15) holds with strict inequality at time t0 and in
Rt +
P
2
addition t00
(s) ds > 0 for all
> 0 (the case where s2S 2 \[t0 ;t0 + ) p 2 (s) > 0 is
similar, and thus omitted). Since v is continuous but for downward jumps, there exist
>0
and
2
(t)
2
(t)
> 0 such that (15) holds with strict inequality at t for all t 2 [t0 ; t0 +
is replaced by (1
)
for all t 2
= [t0 ; t0 +
) and
2
(t) for all t 2 [t0 ; t0 +
20
(t)
(1
)
46
2
).
De…ne
20
(t) by
(t) for all t 2 [t0 ; t0 +
).
) when
20
(t)
Next, I claim
that at time t0 player 2’s continuation payo¤ from rejecting
when player 1’s concessions are given by (
(
20
1
(t) ; p
2
(t) ; p 2 (t)) than when they are given by
(t)), where p 2 00 (t) is de…ned by p 2 00 (t) p 2 (t) for all t 6= t0 , and p 2 00 (t0 )
R t+
2
(s) ds (1 p 2 (t0 )) > 0. This follows because the total probability
t
2 00
exp
with which player 1 concedes in the interval [t0 ; t0 +
20
and under (
t0 under (
20
(t) ; p
(t) ; p
2 00
2 00
(t)).
concessions are given by (
p
2 00
20
(t0 ).
) is the same under (
2
(t) ; p 2 (t))
(t)), and some probability mass of concession is moved earlier to
Therefore, there exists
t0 player 2’s continuation payo¤ from rejecting
where p
until t^( 2 ) is strictly lower
(t) is de…ned by p
2
2 (0; p
2 00
(t0 )) such that at time
until t^( 2 ) is the same when player 1’s
20
(t) ; p 2 (t)) and when they are given by (
20
p 2 (t) for all t 6= t0 , and p
(t)
20
(t0 )
p
(t) ; p
2 00
20
(t0 )
(t)),
<
The fact that (15) holds at all t < T^ when player 1’s concessions are given by
(t) ; p 2 (t)) now implies that (15) holds at all t < T^ when player 1’s concessions are
R T^ 2 0
Q
given by ( 2 0 (t) ; p 2 0 (t)). Furthermore, exp
(t)
dt
p 2 0 (s)) >
s2S 2 \[0;T^) (1
0
R T^ 2
Q
0
p 2 (s))
exp
(t) dt
". Therefore, sup t : 2 (t) > 0
T^,
s2S 2 \[0;T^) (1
0
0
so by the de…nition of T^ it must be that sup t : 2 (t) > 0 = T^. Thus, 02 attains the
(
2
supremum in (16).
Finally, suppose that (15) holds with equality at all t < T^ under belief
above). Then (15) holds with equality at all t < t^( 2 ) under belief
(by de…nition of T^).
2,
2
(de…ned
because t^( 2 )
Let t < t^( 2 ) be a time at which v is di¤erentiable.
derivative of the right-hand side of (15) at t must exist and equal v 0 (t).
T^
Then the
This implies
that p 2 (t) = 0, and, by Leibniz’s rule, the derivative of the right-hand side of (15) equals
2
(t) + (r +
2
(t)) v (t). Hence,
2
(t) =
rv (t) v 0 (t) 35
.
1 v (t)
Since v is di¤erentiable almost everywhere, this implies that
Z
Z
2
(s) ds =
(s) ds for all < t^( 2 ) ,
0
35
(18)
0
Note that v (t) = 1 is impossible. For v ( ) 2 [0; 1] implies that v 0 (t) = 0 if v 0 (t) exists and v (t) = 1.
But then the equation v 0 (t) =
2
(t) + (r +
2
(t)) v (t) would reduce to 0 = r, violating the assumption
that r > 0.
47
where
is de…ned by (4). Similarly, if (15) holds with equality then the di¤erence between
the limit as s " t of the right-hand side of (15) evaluated at s and the limit as s # t of the
v (t), for all t < t^( 2 ).
right-hand side of (15) evaluated at s must equal v (t; 1)
inspection, this di¤erence equals p 2 (t)
p 2 (t) =
By
p 2 (t) v (t). Hence,
v (t; 1) v (t)
1 v (t)
for all t < t^( 2 ) at which v is discontinuous, and p 2 (t) = 0 otherwise.36 Therefore,
s2S
for all
Y
(1
2 \[0;
Y
p 2 (s)) =
(1
(19)
p (s))
s2S\[0; )
)
< t^( 2 ), where S is the set of discontinuity points of v, and p is de…ned by (5).
Combining (18) and (19), I conclude that if (15) holds with equality under belief
9
8
Z t
=
<
Y
^
(s) ds
(1 p (s)) > " ,
t ( 2 ) = sup t : exp
;
:
0
2,
then
s2S\[0;t)
which equals T~. In addition,
2
0 for all t > T~ (by (17), recalling that 1
(t)
so T^ = T~ and the supremum in (16) is attained by
Step 3b:
limt"T^
2
(t) over all beliefs
satis…es limt"T^
2
that attain the supremum in (16).
2
If
= 0, then any belief
2
2
(t) = . Thus, suppose that
(t) over all beliefs
n
2g
> 0. Let f
n
2
(t) "
(t) is non-increasing, so limt"T^
exists
2
(t)
2
m
2
(t) exists. Because
T^ such that
2
(t0 ) <
2
2
for all m, so this implies that
m
2
The above sequential
m
2 g
(t0 ) >
m
(t) >
(t) for all t.
48
and a belief
Furthermore,
satis…es the constraints of (16),
. And
for all m > M .
The fact that v cannot jump up rules out v (t) = 1.
n
2g
f
2
(t) < . Then there
. Since limm!1 limt"T^
exists M > 0 such that, for all m > M , limt"T^
36
(t) !
. Now suppose, toward a contradiction, that limt"T^
> 0 and t0
that attain the
be a sequence of beliefs
.
compactness argument implies that there exists a subsequence f
satisfying the constraints of (16) such that
2
that attains the supremum in (16) also
that all attain the supremum in (16) such that limt"T^
limt"T^
"),
2.
2 [0; 1] be the supremum of limt"T^
supremum in (16).
2
(0)
There exists a belief that both attains the supremum in (16) and maximizes
Proof. Let
2
2
m
2
m
2
(t) = , there
(t) is non-increasing
Now
m
2
(t0 ) !
2
(t0 )
implies that
2
(t0 )
the fact that limt"T^
, a contradiction. Therefore, limt"T^
2
(t) > 0 implies that
2
(t) = . Furthermore,
attains the supremum in (16).
2
I now complete the proof of Lemma 1. If (15) holds with strict inequality at some time
t < T^ under a belief
2
such that t^( 2 ) = T^, then the procedure for modifying
0
2
limt"T^
(t) > limt"T^
2
such that t^( 02 ) = T^ and
0
2
in the …fth paragraph of the proof of Step 3a yields a belief
This implies that the only beliefs
(t).
supremum in (16) and maximize limt"T^
2
described
2
that both attain the
2
(t) (over all beliefs that attain the supremum in
(16)) satisfy the additional constraint that (15) holds with equality.
Since such a belief
exists by Step 3b, the value of (16) equals the value of (16) under this additional constraint,
which I have shown to equal T~ in the last paragraph of the proof of Step 3a.
Let
Proof of Theorem 1.
converges pointwise to
show that Tn1 > T~ (
Since
1
n
n
n+1
(t) =
ert
1 log "
n
n+1
n)
exp
for all t
Tn1 and
n
Note that f
ng
log "), it remains only to
= 1= (1
1
r
log
n+1
n
log ") .
(1
is non-decreasing, it follows that v (t) =
Tn1 . Therefore,
Z
Tn1
Z
!
rv (t) v 0 (t)
dt
1 v (t)
Tn1
n+1
n
r
0
n+1
n
n+1
n
= exp
= exp
= exp
n)
for all n 2 N. To see this, note that Tn1 =
0
= exp
be de…ned as in Section 4.4.
. To show that limn!1 u1 (
ert
1 log "
for all t
and
n
1
(1
n
(1
Y
1
s2S\[0;Tn1 ]
log ") e
(1
log ") 1
(1
log ") 1
e
rt
v (s; 1)
1 v (s)
!
dt
rTn1
n
n+1
1
log "
1
log ") "
< ".
Hence, by the de…nition of T~ (
is strictly decreasing in
1
n ), Tn
for all
T~ (
n ). Furthermore, the fact that exp
2 [0; Tn1 ] implies that Tn1 > T~ (
To complete the proof of Theorem 1, I must show that if f
tures converging pointwise to some posture
satisfying u1 (
49
n)
n ).
ng
R
0
rv(t) v 0 (t)
dt
1 v(t)
is any sequence of pos-
! u1
1= (1
log "), then
=
.37 There are two steps. First, letting fvn g be the continuation value functions corre-
sponding to the f
n g,
and letting v be the continuation value function corresponding to
ert = (1
(that is, v (t) = max f1
Second, I show that this implies that
Step 1a: For all
supt
T~( )
jv (t)
T1
1= (1
rt
min e
Furthermore, if u1 ( )
1= (1
I will show that, for all
Fix
T( )
e
rt
T~ ( ) that
T~ ( )
(1
log ")) (1) =
1
log "
1
.
ert (1= (1
(t) would be strictly less than 1= (1
> 0, there exists
2 (0; 1= (1
T ( ) and supt
log ")).
Then it must be that T~ ( )
, it must also be that (t)
log ")
) for all t
T~( )
jv (t)
log ")
)
log ")
log ")) such that if
.
(t)
v (t)j > , then T~ ( ) > T 1
log ")). This completes the proof of Step 1a.
(1
> 0 and
2 (0; 1= (1
for all t
T ( ) and supt
v (t)
ert (1= (1
1
rT~( )
e
rT 1 + log (1
T ( ), for otherwise mint
(1=r) log (1
, then
log ")
for some posture and some 2 (0; 1= (1
log ")
(t)
t T( )
< exp
log ")
1= (1
log ")), for otherwise it would follow from T ( )
(1
u1 ( ) =
ert (1= (1
> 0 such that if u1 ( )
log ") (which equals limn!1 Tn1 ).
(1=r) log (1
for all t
vn (t)j ! 0.
(where v is the continuation value function corresponding to ).
v (t)j
(1=r) log (1
jv (t)
.
=
> 0, there exists
Proof. Suppose that u1 ( )
Let T 1
rt
log ") ; 0g), I show that supt2R+ e
log ")
Z
exp
T~( )
T~( )
0
log ")), and suppose that
jv (t)
v (t)j > .
ert (1= (1
(t)
log ")
)
A straightforward implication is that
T ( ). Also, if T~ ( ) is …nite then
!
Y
rv (t) v 0 (t)
1 v (s; 1)
dt
".
1 v (t)
1 v (s)
~
s2S\[0;T ( )]
) for all t
Thus, if T~ ( ) is …nite then it must be that
inf
v:R+ !R+ :
1
v(t) 1 ert ( 1 log
);
"
supt T~ ( ) jv (t) v(t)j>
37
exp
Z
0
T~( )
!
rv (t) v 0 (t)
dt
1 v (t)
Technically, I must also show that u1 (
observation that T (
)
1= (1
) = 1 (which follows because
Y
s2S\[0;T~( )]
log "). In fact, u1 (
(t) = 1 for all t
50
1
T~ (
v (s; 1)
1 v (s)
",
(20)
) = 0, by Lemma 3 and the
)).
where T~ ( ) is viewed as a parameter and the in…mum is taken over all functions v : R+ ! R+
satisfying the constraints. I will show that if
only if T~ ( ) > T 1
(1=r) log (1
(1
> 0 is su¢ ciently small then (20) can hold
log ")).
I …rst show that any attainable value of the program on the left-hand side of (20) can
be arbitrarily closely approximated by the value attained by a continuous function v (t)
satisfying the constraints of (20); hence, in calculating the in…mum over such values, attention
may be restricted to continuous functions. To see this, …x
[
S
[s
; s] .
s2S\[0;T~( )]
De…ne the function v (t) by v (t)
v (t)
1
t
(s
)
2 (0; 1) and let
v (t) for all t 2
= S , and
v (s
)+
t
(s
)
v (s) for all t 2 S .
Observe that v is continuous, and that v satis…es the constraints of (20) if
small.38 Furthermore, for all s 2 S,
Z s
v 0 (t)
exp
dt
1 v (t)
s
Also, since v (t)
goes to 0 as
1
(1= (1
log ")
! 0,
!0
Therefore,
Z
lim exp
!0
0
Z
= lim exp
!0
Z
0
Proof:
subset of
T~( )
S
rv (t)
dt
1 v (t)
!
rv (t) v 0 (t)
dt exp
1 v (t)
!
rv (t) v 0 (t)
dt
1 v (t)
The constraint v (t)
R2+ .
1
!
rv (t) v 0 (t)
dt
1 v (t)
[0;T~( )]nS
= exp
38
T~( )
v (s
)
1 v (s
)
=
.
1 v (s)
1 v (s)
h
i
) < 1 for all t 2 0; T~ ( ) , and the measure of S
=
Z
lim exp
1
Y
1
s2S\[0;T~( )]
ert
1
1 log "
= 1.
Z
S
rv (t)
dt
1 v (t)
v (s; 1)
1 v (s)
s2S\[0;T~( )]
1
v (s
)
1 v (s)
speci…es that the graph of v (t) lies in a convex
Since the graph of v (t) lies in the convex hull of the graph of v (t), the graph of v (t) lies
> 0). In addition, the fact that supt
T~ ( )
Y
.
in any convex subset of R2+ that contains the graph of v (t). Hence, v (t)
supt
is su¢ ciently
jv (t)
v (t)j >
T~ ( )
jv (t)
v (t)j >
for su¢ ciently small .
51
1
ert
1
1 log "
(for all
and v (t) ! v (t) for all t 2 R+ implies that
Thus, the value of the program in (20) attained by any function v is arbitrarily closely approximately by the value attained by the continuous function v as
small
! 0, and for su¢ ciently
> 0 this function also satis…es the constraints of (20).
I now derive a lower bound on the left-hand side (20) under the additional constraint
that v is continuous. Using the fact that v (s; 1) = v (s) for all s when v is continuous and
integrating the v 0 (t) = (1
v (t)) term, this constrained program may be rewritten as
1
!0
Z T~( )
rv (t)
1 v (0) A
inf
dt @
exp
.
v:R+ !R+ continuous:
1 v (t)
~
0
1
v
T
(
)
1
v(t) 1 ert ( 1 log
);
"
supt
Since v (t)
T~ ( ) jv
(t) v(t)j>
0 for all t, the value of this program is bounded from below by the value of the
program:
inf
v:R+ !R+ continuous:
1
v(t) 1 ert ( 1 log
);
"
supt T~ ( ) jv (t) v(t)j>
Z
exp
T~( )
0
!
rv (t)
dt (1
1 v (t)
v (0)) .
(21)
Note that the value of the program (21) is continuous and non-increasing in the parameter
T~ ( ).
Let T~
0 if the value of the program (21) is less than " when T~ ( ) = 0, let T~
1
if the value of the program is greater than " for all T~ ( ) 2 R+ , and otherwise let T~ be
the value of the parameter T~ ( ) such that (21) equals " (which exists by the Intermediate
Value Theorem). I will show that T~ > T 1
2 (0; 1= (1
(1=r) log (1
(1
log ")) for su¢ ciently small
log ")).
I …rst show that this inequality holds for
= 0, that is, that T~0 > T 1 .
To see this,
h
i
note that (21) decreases whenever the value of v (t) is increased on a subset of 0; T~ ( ) of
positive measure. Hence, the unique solution to the program (21) without the constraint
supt
T~( )
jv (t)
v (t)j >
is v (t) = 1
ert = (1
T~ ( ) (when
log ") = v (t) for all t
=
0). Using this observation, it is straightforward to check that the value of the parameter T~ ( )
such that the value of the program (21) without the constraint supt
equals " is T 1 .
Therefore, the constraint supt
T~0 > T 1 .
T~( )
jv (t)
v (t)j >
T~( )
jv (t)
v (t)j >
binds in (21), and
Next, T~ is continuous in , by the Maximum Theorem (which implies that the value
of (21) is continuous in , for …xed T~ ( )) and the Implicit Function Theorem. And T 1
52
as well. Hence, the fact that T~0 > T 1 implies
log ")) is continuous in
(1=r) log (1
(1
that T~ > T 1
(1=r) log (1
ert (1= (1
By (20), if (t)
log ")) for some
(1
2 (0; 1= (1
) for all t
log ")
log ")).
T ( ) and supt
then T~ is a lower bound on T~ ( ). Thus, the fact that T~ > T 1
for some
2 (0; 1= (1
rt
jv (t)
jv (t)
v (t)j > ,
(1=r) log (1
(1
log "))
log ")) completes the proof.
Step 1b: For all
supt2R+ e
T~( )
> 0, there exists
> 0 such that if u1 ( )
1= (1
, then
log ")
.
v (t)j
Proof. Step 1a implies that, for any > 0 and K > 1, there exists (K) 2 (0; 1= (1
such that if u1 ( )
1= (1
(K), then supt
log ")
0
now argue that, for K su¢ ciently large, there exists
1= (1
0
log ")
rt
, then in addition supt>T~( ) e
jv (t)
K ! 1, T~ ( ) ! T 1 uniformly over all postures
=K. Choose K > 1 such that e
rT~( )
posture , and suppose that a posture
e
rt0
jv (t0 )
rt0
e
v (t0 ) > + e
max e
rt
rt1
, or equivalently
(t1 )
1
u1 ( ) =
such that supt
T~( )
Therefore, taking
then supt
supt2R+ e
T~( )
rt
e
jv (t)
Step 2: If
0, then
=
rt
n
0
(t1 )
min e
< e
T~( )
rt
e
v (t)j
v (t)j
(t) !
rt
~
)
I
v (t)j
for any such
=K but
v (t)j
~
T~ ( )
v
jv (t)
erT (
jv (t)
rt0
e
erT (
)
ert0 ,
.
v (t0 )
rT~( )
rT 1
ert1 . Hence,
1
rt
1
(t)
e
~
er T (
)
=2 = 1= (1
rt1
i
T~ ( ) ; T ( ) such that
ert1
1
=e
h
rT~( )
log ")
=2.
min f (K ) ; =2g, it follows that if u1 ( )
jv (t)
e
T~ ( )
< =2 and v
(t)
t T( )
e
=K.
. To see this, note that as
By the de…nition of T ( ), this implies that there exists t1 2
e
v (t)j
v (t0 ). Therefore,
1
t T~( )
jv (t)
v (t)j
is such that supt
rt0
rt
2 (0; (K)) such that if u1 ( )
for some t0 > T~ ( ). Now v (t0 )
v (t0 )j >
so it follows that e
rT 1
e
T~( )
log "))
=K and supt>T~( ) e
rt
jv (t)
1= (1
log ")
0
, and hence
v (t)j
.
(t) for all t 2 R+ for some posture , and supt2R+ e
.
53
rt
jv (t)
vn (t)j !
Proof. First, note that if
such that
vn (t)
n
(t) <
1
for all n > N .
(t)
(t) +
(t) for some t 2 R+ , then there exist N > 0 and
(t) <
Since vn (t)
It is more di¢ cult to rule out the possibility that
Since
and
degenerate closed interval I0
the case that
supt2R+ e
rt
n
(t) for some t 2 R+ . Sup-
(t) >
are right-continuous, there exist
R+ such that
(t) >
> 0 and a non-
for all t 2 I0 .
(t) +
vn (t)j ! 0 would fail, so this is not possible.39
jv (t)
0 such that
n1
(t1 ) + =2. Since
(t1 ) <
there exists a non-degenerate closed interval I1
all t 2 I1 .
(t), this implies that
If it were
(t) + =2 for all t 2 I0 and n su¢ ciently large, then the condition
(t)
t1 2 I0 and n1
n
for all n > N , a contradiction.
= v (t) +
pose that this is so.
1
>0
n1
and
I0 such that
Next, it cannot be the case that
n
Hence, there exists
are right-continuous,
n1
(t) + =2 for
(t) <
(t) + =2 for all t 2 I1 and
(t)
n > n1 (by the same argument as above), so there exists t2 2 I1 and n2 > n1 such that
n2
(t2 ) <
I1 such that
I2
As above, this implies that there exists a non-degenerate closed
(t2 ) + =2.
n2
(t)+ =2 for all t 2 I2 . Proceeding in this manner yields an in…-
(t) <
nite sequence of non-degenerate closed intervals fIm g and integers fnm g such that Im+1
nm+1 > nm , and
(t) + =2 for all t 2 Im and m 2 N. Let I
(t) <
nm
empty set (possibly a single point), and …x t 2 I. Then
nm
Im ,
\m2N Im , a non-
(t) + =2 for all m 2 N,
(t) <
and since nm+1 > nm for all m 2 N this contradicts the assumption that
n
(t) !
(t).
Proof of Proposition 2.
Lemmas 1 through 3 apply to any posture, whether or not it
is constant. In addition, if
is constant then T ( ) = T~ ( ). Thus, Lemma 3 implies that
u1 ( ) = mint
39
T( )
e
rt
=e
rT~( )
Proof: Suppose that supt2R+ e
rt
. Furthermore,
jv (t)
vn (t)j ! 0. Fix N > 0, suppose that
for all t 2 I0 and n > N , and denote the length of I0 by 2
player 2 cannot receive a payo¤ above e
when facing posture
all n > N .
vn (t0 +
n
for any n > N .
)+ e
r
1 (1
(t0 )
=2; 1
(t0 )
max 1
(t0 )
(1
(t0 )
(t0 )
1
=2; e
e
r
which contradicts the hypothesis that supt2R+ e
r
=2
rt
(t)
(t) + =2
and the midpoint of I0 by t0 .
Then
max 1
(t0 )
=2; e
r
vn (t0 +
)
=2, there exists N 0 > 0 such that vn (t0 +
(t0 ) < 1
=2) = 1
n
=2) from accepting at any time t 2 [t0 ; t0 +
Hence, vn (t0 )
In addition, noting that
N 0 . Therefore, vn (t0 )
max 1
rt0
) = and p (t) = 0 for all
(t) = r (1
(t0 +
)+ e
(1
(t0 +
= v (t0 )
jv (t)
54
r
1 (1
)) + 1
1
e
vn (t)j ! 0.
r
(t0 )
e
r
(1
]
for
) <
=2) for all n >
(t0 )
=2)
=2 for all n > max fN; N 0 g,
t, so, by the de…nition of T~ ( ),
exp
r
1
T~ ( )
= ",
or
1
r
T~ ( ) =
if
< 1, and T~ ( ) = 1 if
log "
1
= 1. Therefore,
u1 =
=
max e
rT~( )
2[0;1]
max exp
log "
1
2[0;1)
.
(22)
Note that (22) is concave in . Hence, the …rst-order condition
1=
"
(1
")
2
(23)
log ",
which has a solution if " < 1, is both necessary and su¢ cient.
equation yields
"
=
2
log "
q
(log ")2
4 log "
2
Finally, substituting (23) into (22) yields u1 = exp ( (1
" ))
Solving this quadratic
.
".
Proof of Proposition 3. The proof of Theorem 1 goes through for any decreasing Pareto
frontier , with the modi…cations that v (t; 1) now equals max
than max
t
e
r(
t)
log ").
e
( ) rather
Therefore, it su¢ ces to show that, for any u1 2 [0; 1], the posture
Pareto frontier is
(t), for all t, when the
than when it is . Note that the set of times at which
by (4) is the same for either Pareto frontier, because
and
t)
(t) = min fert u1 ; 1g for a value of u1 that may di¤er
(t) = min fert u1 ; 1g corresponds to a weakly higher concession rate
Hence, since
r(
(t) (with the analogous modi…cation for v (t)), and that the
1
maxmin posture is now given by
from 1= (1
t
(t) is given
(u) = 1 if and only if
(u) = 1.
are concave and thus di¤erentiable almost everywhere, it su¢ ces to
show that
0
r ( (ert u1 ))
1
( (ert u1 )) 0 (ert u1 ) rert u1
( (ert u1 ))
55
r (ert u1 )
1
0
(ert u1 ) rert u1
(ert u1 )
for all t 2 R+ and u1 2 [0; 1]; or, dividing both sides by r and writing u for ert u1 ,
0
( (u))
1
0
( (u))
( (u))
(u) u
0
(u)
1
(u) u
(u)
for all u 2 [0; 1]. This inequality may be rearranged as
( (u))
(u)
((1
(u))
0
( (u))
(1
The maintained assumptions on
imply that
(x)
all x 2 [0; 1], so
0 and (1
(u))
0
( (u))
0 and u
(u)
(u)
x and (1
0
0
( (u))))
( (u))
(u) u
0
x)
(1
(24)
0.
1
(x) for
( (u)))
0. Since
(x)
0, it follows that (24) holds.
Lemmas 1 through 3 continue to hold, replacing r with r1 or
Proof of Proposition 4.
r2 as appropriate. In particular,
(t) =
r2 v(t) v 0 (t)
,
1 v(t)
and the same argument as in the proof
of Theorem 1 implies that the unique maxmin posture
(t) = min fer1 t u1 ; 1g,
satis…es
where u1 is the (unique) number such that the time at which (t) reaches 1 equals T~ ( ).
r2 (1 er1 t u1 )+r1 er1 t u1
r t
Thus, given posture , it follows that (t) =
= r2 e u 1 + r1 r2 . Now
er1 t u
1
Z
exp
T~(
)
r2
e
0
!
r1 t
u1
+ r1
r2 dt
1
u1
= exp
r2
r1
1
1
e
r1 T~(
)
r2 ) T~ ( ) .
+ (r1
Setting this equal to " and rearranging implies that T~ ( ) is given by
e
r1 T~(
r1
u log " +
r2 1
)
~
Using the condition that er1 T (
)
r1
r2
1 u1 r1 T~ ( ) = 1.
u1 = 1, this can be rearranged to yield (12).
~
there is a unique pair u1 ; T~ ( ) that satis…es both (25) and er1 T (
curve in u1 ; T~ ( )
~
er1 T (
)
(25)
)
Finally,
u1 = 1, because the
space de…ned by (25) is upward-sloping, while the curve de…ned by
u1 = 1 is downward-sloping.
Proof of Lemma 4.
u1 ( ) = mint
Let T_
T( )
e
rt
The fact that
RAT
2
( )
1
immediately implies that uRAT
( )
1
(t). Therefore, it su¢ ces to show that uRAT
( )
1
min argmaxt e
rt
1
mint
T( )
e
rt
(t).
(t) . Note that T_ is well-de…ned and …nite because (t)
is lower semi-continuous and limt!1 e
rt
(t) = 0. Let _ 2 2
1
2
be the strategy that
always demands 0, rejects up to time T_ , accepts at date T_ ; 1 if and only if lim
T_ , and accepts at all dates T_ ; 1 and later (for all histories). Let _ 2 2
56
1
"T_
( )
be identical
to the -o¤setting belief
2
with the modi…cation that _ 2 always accepts demands of 0.
Since p (0) = 0 for any posture , it follows that u1 ( _ 2 ; _ 2 ) = 1, and therefore _ 2 2
In addition, it is clear that u2 (
2
2
2
(
2)
2;
2)
(by Lemma 2) and u2 (
it is clear that _ 2 2
2
( ), and
2
u2 ( _ 2 ;
2;
1
(
2)
for all
2)
2
2
2,
1
( _ 2 ).
so the observations that
imply that
2
( _ 2 ). Finally,
2)
= u2 ( _ 2 ;
2)
by Lemma 3. Summarizing, I have established
2
2
that the arrows in the following diagram may be read as “is a best-response to”:
!
Therefore, the set f ; _ 2 g
f
implies that f
RAT
2
mint
T( )
e
rt
2;
_ 2g
2;
2
"
# .
_2
_2
_ 2 g is closed under rational behavior given posture , which
( ).
( )
Hence, uRAT
1
sup 1 u1 ( 1 ;
2)
= u1 ( ;
2)
=
(t).
References
[1] Abreu, D., and Gul, F. (2000), “Bargaining and Reputation,” Econometrica, 2000, 85117.
[2] Abreu, D., and Pearce, D. (2007), “Bargaining, Reputation, and Equilibrium Selection
in Repeated Games with Contracts,”Econometrica, 75, 653-710.
[3] Battigalli, P., and Watson, J. (1997), “On ‘Reputation’Re…nements with Heterogeneous
Beliefs,”Econometrica, 65, 369-374.
[4] Battigalli, P., and Siniscalchi, M. (2003), “Rationalization and Incomplete Information,”
Advances in Theoretical Economics, 3, Iss. 1, Article 3.
[5] Bergin, J., and MacLeod, W.B. (1993), “Continuous Time Repeated Games,”International Economic Review, 34, 21-37.
[6] Billingsley, P. (1995), Probability and Measure. New York, NY: John Wiley & Sons, Inc..
57
[7] Chatterjee, K., and Samuelson, L. (1987), “Bargaining with Two-Sided Incomplete
Information: An In…nite-Horizon Model with Alternating O¤ers,”Review of Economic
Studies, 59, 175-192.
[8] Chatterjee, K., and Samuelson, L. (1988), “Bargaining Under Two-Sided Incomplete
Information: the Unrestricted O¤ers Case,”Operations Research, 36, 605-618.
[9] Cho, I.-K. (1994), “Stationarity, Rationalizability, and Bargaining,”Review of Economic
Studies, 61, 357-374.
[10] Compte, O., and Jehiel, P. (2002), “On the Role of Outside Options in Bargaining with
Obstinate Parties,”Econometrica, 70, 1477-1517.
[11] Crawford, V. (1982), “A Theory of Disagreement in Bargaining,” Econometrica, 50,
607-637.
[12] Cripps, M.W., Schmidt, K., and Thomas, J.P. (1996), “Reputation in Perturbed Repeated Games,”Journal of Economic Theory, 69, 387-410.
[13] Ellingsen, T., and Miettinen, T. (2008), “Commitment and Con‡ict in Bilateral Bargaining,”American Economic Review, 98, 1629–1635.
[14] Evans, R., and Thomas, J.P. (1997), “Reputation and Experimentation in Repeated
Games with Two Long-Run Players,”Econometrica, 65, 1153-1173.
[15] Feinberg, Y., and Skrzypacz, A. (2005), “Uncertainty about Uncertainty and Delay in
Bargaining,”Econometrica, 73, 69-91.
[16] Fershtman, C., and Seidmann, D.J. (1993), “Deadline E¤ects and Ine¢ cient Delay in
Bargaining with Endogenous Commitment,”Journal of Economic Theory, 60, 306-321.
[17] Fudenberg, D., and Levine, D.K. (1989), “Reputation and Equilibrium Selection in
Games with a Patient Player,”Econometrica, 57, 759-778.
[18] Green, J.R., and Scotchmer, S. (1995), “On the Division of Pro…t in Sequential Innovation,”RAND Journal of Economics, 26, 20-33.
58
[19] Grossman, S.J. and Hart, O.D. (1986), “The Costs and Bene…ts of Ownership: A Theory
of Vertical and Lateral Integration,”Journal of Political Economy, 94, 691-719.
[20] Hosios, A. (1990), “On the E¢ ciency of Matching and Related Models of Search and
Unemployment,”Review of Economic Studies, 57, 279-298.
[21] Kambe, S. (1999), “Bargaining with Imperfect Commitment,” Games and Economic
Behavior, 28, 217-237.
[22] Kreps, D.M. and Wilson, R. (1982), “Reputation and Imperfect Information,”Journal
of Economic Theory, 27, 253-279.
[23] Knoll, M.S. (1996), “A Primer on Prejudgment Interest,” Texas Law Review, 75, 293374.
[24] Milgrom, P., and Roberts, J. (1982), “Predation, Reputation, and Entry Deterrence,”
Journal of Economic Theory, 27, 280-312.
[25] Muthoo, A. (1996), “A Bargaining Model Based on the Commitment Tactic,” Journal
of Economic Theory, 69, 134-152.
[26] Myerson, R. (1991), Game Theory: Analysis of Con‡ict. Cambridge, MA: Harvard
University Press.
[27] Pearce, D. (1984), “Rationalizable Strategic Behavior and the Problem of Perfection,”
Econometrica, 52, 1029-1050.
[28] Royden, H.L. (1988), Real Analysis, Third Edition. Upper Saddle River, NJ: Prentice
Hall.
[29] Rubinstein, A. (1982), “Perfect Equilibrium in a Bargaining Model,”Econometrica, 50,
97-109.
[30] Schelling, T.C. (1956), “An Essay on Bargaining,” American Economic Review, 46,
281-306.
59
[31] Schmidt, K. (1993), “Reputation and Equilibrium Characterization in Repeated Games
of Con‡icting Interests,”Econometrica, 61, 325-351.
[32] Simon, L.K., and Stinchcombe, M.B. (1989), “Extensive Form Games in Continuous
time: Pure Strategies,”Econometrica, 57, 1171-1214.
[33] Watson, J. (1993), “A ‘Reputation’ Re…nement without Equilibrium,” Econometrica,
61, 199-205.
[34] Watson, J. (1998), “Alternating-O¤er Bargaining with Two-Sided Incomplete Information,”Review of Economic Studies, 65, 573-594.
[35] Wolitzky, A. (2011), “Indeterminacy of Reputation E¤ects in Repeated Games with
Contracts,”Forthcoming, Games and Economic Behavior.
[36] Yildiz, M. (2003), “Bargaining without a Common Prior: An Immediate Agreement
Theorem,”Econometrica, 71, 793-811.
[37] Yildiz, M. (2004), “Waiting to Persuade,” Quarterly Journal of Economics, 119, 223248.
60
Supplementary Appendix (NOT FOR PUBLICATION)
This appendix shows that the characterization of the maxmin payo¤ and posture (Theorem
1) continues to apply when the solution concept is strengthened from …rst-order knowledge
of rationality to iterated conditional dominance, or when the continuous-time bargaining
protocol of the text is replaced by any discrete-time bargaining protocol with su¢ ciently
frequent o¤ers. However, the characterization does not apply with both iterated conditional
dominance and discrete-time bargaining, as the fact that (complete-information) discretetime bargaining is solvable by iterated conditional dominance implies that the predictions of
the model with iterated conditional dominance and discrete-time bargaining depend on the
order and relative frequency of o¤ers.40
Iterated Conditional Dominance
This section shows that Theorem 1 continues to hold under a natural notion of iterated conditional dominance. Because the model has incomplete information and is not a multistage
game with observed actions (as players do not observe each other’s choice of demand paths
on the integers), no o¤-the-shelf version of iterated conditional dominance is applicable, and
even the simplest version that is applicable requires some new notation.
For integer t, let
and history ht .
btc
ui 2 supp
btc
btc
u1 ; u2
i
i
(ht ) be the element of
I …rst introduce the idea that a triple
btc
btc
ht ; u1 ; u2
i
at date (t; 0)
is “ i -coherent” if
hbtc and at ht the path of realized demands between btc and t coincides with
.
btc
btc
De…nition 9 A triple ht ; u1 ; u2
40
(U t ) prescribed by strategy
is
i -coherent
btc
if ui 2 supp
i
btc
btc
hbtc and u1 ( ) ; u2 ( ) =
Informally, there is a race between the number of rounds of iterated conditional dominance and the
frequency of o¤ers.
I conjecture that, for any number of rounds of iterated conditional dominance, the
maxmin payo¤ and posture in discrete-time bargaining converge (in the sense of Proposition 7) to the
maxmin payo¤ and posture in continuous-time bargaining as o¤ers become frequent. This is consistent with
Rubinstein bargaining, where the round at which any demand other than 0 or 1 is deleted goes to in…nity as
the time between o¤ers vanishes (so that iterated conditional dominance has no “bite”in the continuous-time
limit). I thank Je¤ Ely for helpful comments on this point.
61
2 [btc ; t], where ht = (u1 ( ) ; u2 ( ))
(u1 ( ) ; u2 ( )) for all
btc
btc
2)
btc
btc
btc
such that ht ; u1 ; u2
if there exist demand paths u1 ; u2
For any strategy pro…le ( 1 ;
btc
t.
is
payo¤ under strategy pro…le ( i ;
j)
i -coherent
i -coherent.
btc
and any triple ht ; u1 ; u2
btc
such that u1 ( ) ; u2 ( ) =
2 [btc ; t], where ht = (u1 ( ) ; u2 ( ))
(u1 ( ) ; u2 ( )) for all
A history ht is
t,
each player i’s expected
btc
btc
conditional on reaching the triple ht ; u1 ; u2
is well-
de…ned, and is denoted
ui
I also write ui ( i ;
t
j jh )
i;
btc
btc
t
j jh ; u1 ; u2
.
for player i’s expected payo¤ conditional on reaching ht at date
(t; 0) for integer t.
I now de…ne iterated conditional dominance. Informally, the idea is that a strategy is
conditionally dominated if it is either strictly dominated or is “conditionally weakly dominated.” The di¤erence between the de…nition of iterated conditional dominance for the
two players re‡ects the fact that player 1 is committed to strategy
with probability ",
and therefore that player 2 is restricted to assigning probability at least " to strategy
histories that are consistent with . Note that the support of the -o¤setting belief
at
2
in-
cludes strategies that are iteratively conditionally dominated, as it is easy to verify that any
strategy of player 1’s that ever accepts a demand of 1 is iteratively conditionally dominated.
De…nition 10 For any posture
1
2,
a strategy
1
2
1
and set of bargaining phase strategy pro…les
=
1
2
is conditionally dominated with respect to ( ; ) if either of the
following conditions hold
1. There exists a strategy
0
1
2
such that
1
u1 ( 01 ;
for all beliefs
1
2
(
2. There exists a strategy
u1
for all
some
1 -coherent
1 -coherent
1)
> u1 ( 1 ;
1)
2 ).
0
1
2
0
1;
1
btc
btc
t
1 jh ; u1 ; u2
btc
btc
ht ; u1 ; u2
btc
such that
btc
ht ; u1 ; u2
u1
and all beliefs
and some belief
62
btc
btc
t
1 jh ; u1 ; u2
1;
1
1
2
(
2
(
2 ),
2 ).
with strict inequality for
A strategy
2
2
2
is conditionally dominated with respect to ( ; ) if either of the
following conditions hold
0
2
1. There exists a strategy
2
2
such that
u2 ( 02 ;
for all beliefs
2
2
2
(
1
> u2 ( 2 ;
[ f g) such that
2
2)
" with strict inequality only if
( )
1.
0
2
2. There exists a strategy
u2
for all
(
2)
with
2
such that
btc
btc
t
2 jh ; u1 ; u2
0
2;
btc
btc
ht ; u1 ; u2
2 -coherent
1 ),
2
u2
2
2
(
2 -coherent
conditional dominance given posture
if every
=
1
2
i
i
not conditionally dominated) with respect to ( ; ).
undominated strategies given posture
( )
[
f :
and all beliefs
btc
btc
ht ; u1 ; u2
2
2
that is inconsistent
1 ).
A set of bargaining phase strategy pro…les
ICD
btc
btc
t
2 jh ; u1 ; u2
that are inconsistent with
with strict inequality for some
and some belief
2;
2
1
2
is closed under
is conditionally undominated (i.e.,
The set of iteratively conditionally
is
is closed under conditional dominance given posture g .
Player 1’s maxmin payo¤ under iterated conditional dominance given posture
uICD
( )
1
sup
1
inf
ICD (
2
22
)
u1 ( 1 ;
is
2) .
Player 1’s maxmin payo¤ under iterated conditional dominance is
uICD
1
A posture
ICD
( ).
sup uICD
1
is a maxmin posture under iterated conditional dominance if there exists a
sequence of postures f
ng
such that
n
(t) !
ICD
63
(t) for all t 2 R+ and uICD
(
1
n)
! uICD
.
1
This version of iterated conditional dominance is stronger than rationalizability, in that
ICD
( )
RAT
( ) for any posture
.
This can be seen by noting that every set
that is closed under conditional dominance is also closed under rationalizability, because
rationalizability is equivalent to imposing only the …rst of the two conditions in the de…nition
of conditional dominance (for both player 1 and player 2). An immediate consequence of this
observation is that the maxmin payo¤ under iterated conditional dominance is weakly greater
than the maxmin payo¤ (under …rst-order knowledge of rationality), that is, uICD
1
u1 . In
fact, the two payo¤s are equal, as are the corresponding maxmin postures.
Proposition 6 Player 1’s maxmin payo¤ under iterated conditional dominance equals her
maxmin payo¤, and the unique maxmin posture under iterated conditional dominance is the
= u1 , and the unique maxmin posture under iterated
unique maxmin posture. That is, uICD
1
conditional dominance is
ICD
.
=
The rest of this section is devoted to proving Proposition 6. The proof builds on that
of Proposition 5. This is because it can be shown that the set of iteratively conditionally
undominated strategies and the set of rationalizable strategies are identical up to strategies
that are “exceptional”in the following sense.
De…nition 11 A strategy
i
2
i
is exceptional given posture
if either of the following
conditions hold:
i 2 f1; 2g and
i
ever accepts a demand of 1, rejects a demand of 0, makes a demand
of 0 or a path of demands converging to 0 (i.e., lim
"t
ui ( ) = 0), or makes a demand
of 1 at every successor of some history ht .
i = 1 and
i
ever accepts a demand u2 (t)
consistent with
with t
history ht consistent with
1
e
r(t
t , rejects a demand u2 (t)
with t
t , or demands
t)
1
(t ) > 0 at any history ht
e
r(t
t)
(t ) < 1 at any
(t ) at any history ht consistent
with .
The relationship between iterated conditional dominance and rationalizability is formalized in the following lemma, which is a key step in the proof of Proposition 6.
64
Lemma 5 For any posture , every strategy that is rationalizable and non-exceptional given
posture
is also iteratively conditionally undominated given posture .
The proof of the lemma uses the concept of a unique optimal action: an action (accepting, rejecting, or choosing a demand path for the next integer) is the unique optibtc
btc
mal action at a triple ht ; u1 ; u2
ui
btc
btc
t
i jh ; u1 ; u2
i;
under a belief
i
if every strategy
i
that maximizes
btc
btc
prescribes that action at history ht (where the arguments u1 ; u2
are omitted in the case of choosing a demand path for the next integer).
Proof of Lemma 5.
Fix a posture ; for the duration of the proof, I omit the modi…er
“given posture .”To prove the lemma, I show that for every non-exceptional strategy
every
i -coherent
and belief
i
btc
and
btc
history ht that is inconsistent with , there exist demand paths u1 ; u2
with support on strategies that are rationalizable and non-exceptional such that
ht ; u1 ; u2
btc
is
under belief
2.
btc
i
i -coherent
and
i
If i = 1, this conclusion also holds at
consistent with .
btc
btc
prescribes the unique optimal action at ht ; u1 ; u2
i -coherent
histories that are
This implies that the second of the two conditions in the de…nition
of conditional dominance can never hold if
i
is non-exceptional, for i = 1; 2. Therefore,
every non-exceptional strategy that is conditionally dominated is also strictly dominated, and
hence every non-exceptional strategy that is rationalizable is also iteratively conditionally
undominated.
I start by establishing a statement with the important implication that, starting from
a history that is inconsistent with , any continuation strategy is part of a rationalizable
strategy.
Step 1: Any strategy
1
2
1
history ht that is consistent with
that demands
(t) and rejects player 2’s demand at every
is rationalizable. Any strategy
2
2
2
that demands 1
and accepts at (and not before) the more favorable of dates (t ; 1) and (t ; 1) if player 1
follows
until time t is rationalizable.
Proof. By the proof of Lemma 4, strategy _ 2 is rationalizable for player 1, and strategy
2
is rationalizable for player 2. Now if strategies
1
2
1
(
2)
and
2
2
2
( _ 2 ), so
1
and
Step 2: For i = 1; 2, if a strategy
and inconsistent with
i
2
1
and
2
are as in the statement, then
are rationalizable as well.
is non-exceptional and a history ht is
, then there exist demand paths
65
btc
btc
u1 ; u2
i -coherent
and a belief
i
with
btc
i -coherent
and
i
prescribes the unique optimal action at
Proof. Fix a non-exceptional strategy
i
btc
btc
ht ; u1 ; u2
and a history ht that is
btc
ht ; u1 ; u2
support on strategies that are rationalizable and non-exceptional such that
is
under belief
i -coherent
i.
and inconsistent
with . Step 1 implies that any continuation strategy of player j’s is part of a rationalizable
strategy. Hence, the restriction that
i
has support on strategies that are rationalizable and
non-exceptional implies only that continuation strategies are non-exceptional.
Suppose that
i -coherent
i
accepts at ht .
Then the fact that
i
is non-exceptional and ht is
btc
btc
imply that ui (t) > 0 and uj (t) < 1. Let u1 ; u2
continue to demand ui (t) and uj (t) until dte, and let
i
specify that the players
assign probability 1 to a rationalizable
strategy under which at every successor history of ht player j demands
1+uj (t)
2
(after time dte)
and rejects any strictly positive demand; such a strategy exists by the previous paragraph,
and is clearly non-exceptional (in particular, player j always chooses demand paths that
always make demands u2 ( ) 2 (0; 1)). Then it is clear that accepting at ht is the optimal
btc
btc
action at ht ; u1 ; u2
Suppose that
i
under belief
rejects at ht .
41
i.
Then the fact that
coherent imply that ui (t) > 0 and uj (t) > 0. Let
i
i
is non-exceptional and ht is
i-
assign probability 1 to a rationalizable
and non-exceptional strategy under which player j reduces his demand to uj (t) =2 by some
time
such that
e
r(
t)
1
uj (t)
2
uj (t) ,
>1
subsequently demands uj (t) =2 forever, and rejects player i’s demand at every successor
btc
btc
history of ht unless player i demands 0 (or let u1 ; u2
specify that player j’s demand
btc
btc
follows such a path, in case there is no integer between t and ). Choose any u1 ; u2
btc
btc
ht ; u1 ; u2
such that
is
i -coherent
and player j’s demands follow such a path.
Now
rejecting until time
and then accepting (while never demanding 0) is strictly better for
player i under belief
i
btc
btc
ht ; u1 ; u2
than is accepting at ht , so rejecting is the unique optimal action at
under belief
i.
Finally, suppose that t is an integer and that
fact that
41
i
i
chooses demand path uti at ht .
is non-exceptional implies that uti ( ) > 0 for all
The
2 [t; t + 1) and that
Note that the possibility that player i could reject at ht but accept “immediately” after ht is ruled out
by the assumption that the probability that a player accepts by date (t; 1) is right-continuous in t.
66
lim
"t+1
uti ( ) > 0.
Let
i
assign probability 1 to a rationalizable and non-exceptional
r
e
strategy under which player j demands 1
lim
"t+1
uti ( )
2
at ht and at every successor his-
tory of ht ; accepts at date (t + 1; 1) if ui ( ) = uti ( ) for all
2 [t; t + 1); and otherwise
rejects any strictly positive demand at every successor history of ht . Now choosing demand
path uti at ht and rejecting player j’s demand until time t + 1 yields payo¤ e
under belief
under belief
i,
i,
r
while every other continuation strategy yields payo¤ at most
so choosing demand path
uti
lim
e
r
"t+1
lim
uti ( )
"t+1
uti ( )
2
t
is the unique optimal action at h under belief
i.
Step 3:
If strategy
1
2
1
is non-exceptional and a history ht is
btc
btc
consistent with , then there exist demand paths u1 ; u2
and a belief
btc
prescribes the unique optimal action at
1
btc
btc
ht ; u1 ; u2
1
btc
ht ; u1 ; u2
strategies that are rationalizable and non-exceptional such that
and
1 -coherent
under belief
with support on
is
1 -coherent
1.
Proof. If t > t , then if player 2 plays a rationalizable and non-exceptional strategy
accepts at time t under strategy pro…le ( ;
2)
and
2
that
(which exists), then player 2’s continuation
play starting from ht is restricted only by the requirement that it is non-exceptional. Hence,
the proof in this case is just like the proof of Step 2. I therefore assume that t
Suppose that
1
accepts at ht . Then the fact that
and consistent with , and t
De…ne the strategy • 2 2
2
1
is non-exceptional, ht is
t implies that u1 (t) > 0 and u2 (t) < 1
e
t.
1 -coherent
r(t
t)
(t ).
as follows:
If h is consistent with , then demand 1 until time dt + 1e, subsequently demand 21 ,
reject all positive demands until the more favorable of dates (t ; 1) and (t ; 1), and
subsequently accept all demands of less than 1.
If h is inconsistent with , then demand
Note that • 2 2
Let
1
2
1+u2 (t)
2
( ), so • 2 is rationalizable. In addition, • 2 is clearly non-exceptional.
btc
btc
assign probability 1 to • 2 , and let u1 ; u2
( ) for all
btc
btc
btc
rejects at ht ; u1 ; u2
btc
btc
specify that player 1 demands u1 ( ) =
2 [t; dte) and that player 2 continues to demand u2 (t) until dte.
btc
accepting at ht ; u1 ; u2
ht ; u1 ; u2
and reject all positive demands.
yields payo¤ 1
yields strictly less.
under belief
u2 (t) under belief
1,
Then
while any strategy that
So accepting is the unique optimal action at
1.
67
Suppose that
rejects at ht . Then the fact that
1
and consistent with , and t
•2,
1,
btc
btc
and u1 ; u2
rather than
1+u2 (t)
2
t implies that u1 (t) > 0 and u2 (t) > 1
btc
btc
yields payo¤ e
btc
that rejects at ht ; u1 ; u2
at
r(t
t)
under belief
(t ). Let
e
r (t
t)
(t )
2
(t ) under belief
1,
while any strategy
1.
1
chooses demand path ut1 at ht . Since
are continuous on [t; t + 1), there are three cases.
1. ut1 ( ) =
( ) for all
2. ut1 (t) =
(t) but ut1 ( ) 6=
3. ut1 (t) 6=
(t).
Start with case 1.
and t
t)
yields strictly less. So rejecting is the unique optimal action
Finally, suppose that t is an integer and that
ut1 and
r(t
e
1 -coherent
at histories h that are inconsistent with .42 Then rejecting and following
btc
btc
btc
ht ; u1 ; u2
is non-exceptional, ht is
be as above, with the modi…cation that • 2 demands 1
at ht ; u1 ; u2
strategy
1
2 [t; t + 1).
( ) for some
Here, the fact that
t implies that in fact t + 1
consistent with . Now for all
non-exceptional strategy
2
2 [t; min ft + 1; t g).
1
is non-exceptional, ht is consistent with ,
t , as
1
never demands
(t ) at a history ht
> 0, Step 1 implies that there exists a rationalizable and
that demands 1 at all times
at (but not before) time t under strategy pro…le ( ;
2 ),
such that e
r(
(t+1))
, accepts
rejects all strictly positive demands
at dates (t + 1; 1) and earlier, rejects all strictly positive demands at dates after (t + 1; 0)
such that u1 ( ) 6=
( ) for some
after (t + 1; 0) such that u1 ( ) =
e
r(t
(t+1))
< t + 1, and accepts all demands less than 1 at dates
( ) for all
< t + 1 but u1 (t + 1) 6=
(t + 1). Since
(t ) < 1 (which follows from the de…nition of t ), choosing demand path ut1 and
then deviating from
to a demand close to 1 at time t + 1 yields a strictly higher payo¤
under any belief that assigns probability 1 to such a strategy than does choosing any other
demand path, for
su¢ ciently small.
For case 2, the fact that
that t < t . Let
42
0
1
is non-exceptional, ht is consistent with , and t
be the in…mum over times
t implies
2 [t; min ft + 1; t g) such that ut1 ( ) 6=
( ).
This modi…cation serves only to ensure that • 2 does not demand 1 forever after some history, and thus
that • 2 is non-exceptional.
68
Now for all
strategy
2
> 0, Step 1 implies that there exists a rationalizable and non-exceptional
that demands 1 at all times
such that e
not before) time t under strategy pro…le ( ;
histories that are inconsistent with either
maxft+1;t g)
, accepts at (but
rejects all strictly positive demands at all
n
j
k
o
t+1 0
t
or u1 , and, for all k 2 0; 1; : : : ;
1 ,
2 ),
) if u1 ( ) = ut1 ( ) for all 2 [t; 0 + k ],
t+1
0
c if player 1’s demands are consistent
and accepts at date (t + 1; 1) with probability b
accepts at time
0
k
r(
+ k with probability
(1
with ut1 on [t; t + 1). Since ut1 is continuous, player 1 receives a strictly higher payo¤ from
choosing ut1 and then rejecting until time t + 1 than from mimicking , under the belief that
player 2 plays such a strategy for su¢ ciently small
(as player 1 strictly prefers to have her
demand accepted at any time prior to t than at t , by de…nition of t ). In addition, player
1 receives a strictly higher payo¤ from choosing ut1 than from choosing any demand path
that coincides with ut1 until some time
2 [t; t + 1) and then diverges from ut1 . Therefore,
choosing demand path ut1 is player 1’s unique optimal action at ht under the belief that
player 2 plays such a strategy for su¢ ciently small .
For case 3, Step 1 implies that for all
exceptional strategy
2
> 0 there exists a rationalizable and non-
that accepts at (but not before) time t under strategy pro…le ( ; t ),
rejects all strictly positive demands at all histories that are inconsistent with , and, for
n
j k
o
1
all k 2 0; 1; : : : ;
2 , with probability k (1
) demands 1 until time t + k and
1
reduces its demand to by time t + (k + 1) , and with probability b c demands 1 until
time t + 1
and reduces its demand to
by time t + 1. Step 1 also implies that there
exists a rationalizable and non-exceptional strategy that accepts at (but not before) time t
under strategy pro…le ( ; t ), demands 1
e
r
lim
"t+1
2
uti ( )
at ht and at every successor
history of ht ; accepts at date (t + 1; 1) if ui ( ) = uti ( ) for all
2 [t; t + 1); and otherwise
rejects any strictly positive demand at every successor history of ht (as in the last part of
the proof of Step 2). Let
1
assign probability 1
…rst kind and assign probability
claim that, for
1.
to player 2’s playing a strategy of the
to player 2’s playing a strategy of the second kind.
I
su¢ ciently small, ut1 is player 1’s unique optimal action at ht under belief
To see this, …rst note that e
r(t
t)
(t ) < 1 implies that choosing ut1 is strictly better
than choosing any demand path that coincides with
until time t , for
su¢ ciently small.
Finally, any strategy that chooses a demand path other than ut1 that diverges from
69
before
time t does no better than choosing demand path ut1 and rejecting until time t + 1 in the
event that player 2 plays a strategy of the …rst kind, and does strictly worse in the event
that player 2 plays a strategy of the second kind. Therefore, choosing demand path ut1 is
player 1’s unique optimal action at ht under belief
I now complete the proof of the lemma.
and Step 2, strategy
strategy
0
2
2
1,
for
> 0 su¢ ciently small.
By the de…nition of conditional dominance
can be conditionally dominated (with respect to some ( ; )) by
0
2
only if either
strictly dominates
2
(with respect to ( ; ); i.e., if the …rst
condition in the de…nition of conditionally dominance with respect to ( ; ) holds) or
agrees with
2
at all
2 -coherent
histories that are inconsistent with . But, again by the
de…nition of conditional dominance, if
all
2 -coherent
0
2
0
2
conditionally dominates
0
2
histories that are inconsistent with , then
2
and agrees with
2
must strictly dominate
at
2.
The same argument applies for player 1, noting that Steps 2 and 3 imply that a strategy
1
0
1
can be conditionally dominated by a strategy
agrees with
1
at all
1 -coherent
0
1
only if
strictly dominates
1
or if
histories ht , whether or not ht is consistent with
0
1
(which
is needed for the argument given the di¤erence in the de…nitions of conditional dominance
for players 1 and 2).
Therefore, for i = 1; 2, if
i
is non-exceptional then it cannot be
conditionally dominated unless it is also strictly dominated. Finally, if
and non-exceptional, then it is not strictly dominated with respect to
not conditionally dominated with respect to
dominated with respect to the smaller set
;
ICD
;
RAT
i
is rationalizable
;
RAT
, hence
, and hence also not conditionally
. This proves that every rationalizable
and non-exceptional strategy is iteratively conditionally undominated.
I now prove Proposition 6.
Proof of Proposition 6.
uICD
( )=
1
belief
2
As was the case for Proposition 5, it su¢ ces to show that
(t ) for every posture . The proof proceeds by approximating the -o¤setting
with beliefs f
2
( )g
>0
that have support on rationalizable and non-exceptional
strategies only (unlike the o¤setting belief
itself, which assigns positive probability to
2
the exceptional strategy ~ ), and by approximating the -o¤setting strategy
exceptional strategies f
2
( )g
then implies that the strategy
>0
2
such that
2
( ) 2
2
(
2
( )) for all
> 0.
2
with nonLemma 5
( ) is iteratively conditionally undominated for all
70
> 0.
Finally, as
! 0,
sup u1 ( 1 ;
2
1
which implies that uICD
( )
1
ICD
RAT
( )
( )) ! u1 ( ) ,
Since uICD
( )
1
u1 ( ).
( ), this shows that uICD
( ) = u1 ( ) =
1
u1 ( ) is immediate because
(t ), completing the proof
of the proposition.
I now present an argument leading to the construction of the beliefs f
strategies f
2
( )g
weight under belief
>0 .
2
2
( )g
>0
and
I start by de…ning the strategies of player 1’s that receive positive
2 0; 12 . Let
( ). Fix t 2 (0; t ) and
0
8
n
o
< min ; r(t t) ; (t ) if (t ) > 0
2
o
n 3
,
r(t
t)
:
if (t ) = 0
min ; 3
and let ~ (t; ) be the strategy that demands u1 ( ) =
u1 ( ) =
t + 0 =r
0 =r
( ) for all
t + 0 =r
0 =r
(t) + 1
(1
2 [0; t); demands
0
)
2 [t; t + 0 =r]; demands
for all
u1 ( ) =
t + 2 0 =r
0 =r
0
(1
)+ 1
2 [t + 0 =r; t + 2 0 =r]; demands u1 ( ) =
for all
0
if
t + 2 0 =r
0 =r
0
> t + 2 0 =r; and accepts a demand of
player 2’s if and only if it equals 0. Intuitively, ~ (t; ) mimics
until time t and then quickly
rises to almost one before quickly falling to almost zero, where “quickly”and “almost zero”
are both measured by
(the point of having
0
rather than
in the formulas will become
clear shortly).
I claim that ~ (t; ) is iteratively conditionally undominated. To see this, observe that
~ (t; ) is a best response to any strategy
2
2
with the following properties:
demands 1 and accepts at (and not before) date (t ; 1) if player 1 follows
until
time t .
If h is inconsistent with
if and only if
but consistent with ~ (t; ), then
t + 0 =r.
71
2
demands 1 and accepts
If h is inconsistent with both
and ~ (t; ), then
2
demands 1 and rejects player 1’s
demand.
This follows because playing ~ (t; ) against such a strategy
0
rt
e
0
(1
2
yields payo¤
),
while the only other positive payo¤ that can be obtained against strategy
e
and e
rt
0
(1
0
)
e
rt
rt
0
because
(t )
min
n
e
rt
is
,
1 r(t t)
; 3
2
o
(as can be easily checked). Now,
by Step 1 of the proof of Lemma 5, there exists a rationalizable strategy
so ~ (t; ) is rationalizable. In addition, ~ (t; ) is non-exceptional, because
t 2 [0; t ) (recalling the de…nition of
2
) and ~ (t; ) always demands
0
6=
2
of this form,
(t) > 0 for all
(t ) at time t ,
so Lemma 5 implies that ~ (t; ) is iteratively conditionally undominated.
I now introduce versions of some of the key objects of Section 4.2, indexed by . Let
rv (t)
2
e 0 (1
(t; ) =
if v is di¤erentiable at t and v (t) < e
2
p (t; ) =
if v (t) < v (t; 1)
e
2
0
0
(1
0
^ (t; ) #
Let
), and let
(t; ) = 0 otherwise; and let
v (t; 1) v (t)
0)
e 2 0 (1
v (t)
), and let p (t; ) = 0 otherwise. De…ne T~ ( ), T ( ), t ( ),
^ (t; ), and p^ (t; ) as in Section 4.2, with
de…nitions. Note that, as
0
(1
v 0 (t)
0)
v (t)
! 0,
(t; ) and p (t; ) replacing
(t; ) #
(t) and p (t) in the
(t) and p (t; ) # p (t) for all t 2 R+ . Hence,
(t), p^ (t; ) # p (t), T~ ( ) " T~, T ( ) " T , and t ( ) " t .
( ) be the belief that player 1 rejects all non-zero demands of player 2’s and that
her path of demands begins by following
(t) and then switches to following ~ (t; ) at time
t with hazard rate ^ (t; ) and discrete probability p^ (t; ), for all t < t ( ).43 Let
the belief that coincides with
43
1,
( ) be
( ) until date (t; 1) and subsequently coincides with the
There is a technical problem here because it is not clear that
distribution over
;t
that is, as an element of
(
1 ).
However, it should be clear that
be approximated by a …nite-dimensional distribution over
72
( ) can be written as a …nite-dimensional
1
( ) can in turn
in a way that su¢ ces for the proof.
belief that player 1 follows . Let
1
" on strategy
;t
( ). Let
2
2
( ) put probability " on strategy
( ) be some best response to
2
and put probability
( ) with the following
properties:
t and e
rt
2
( ) demands 1 at all times t such that t
.
2
( ) rejects player 1’s demand at those histories that are consistent with
2
( ) where
accepting and rejecting are both optimal actions.
2
( ) rejects all positive demands at histories that are inconsistent with .
2
( ) is non-exceptional.
It is clear that such a strategy exists. Furthermore, any such strategy is rationalizable,
by Step 1 of the proof of Lemma 5, and hence any such strategy is iteratively conditionally
undominated, by Lemma 5.
I claim that as
( ;
2
! 0, the time at which agreement is reached under strategy pro…le
( )) converges to t , uniformly over possible choices of
2
( ) satisfying the above
properties. To see this, observe that, as in Section 4.2, if v (t) < e
2
0
0
(1
) then
(t; )
and p (t; ) are the rate and probability of player 1’s switching to ~ (t; ) that make player 2
indi¤erent between accepting and rejecting . And, under belief
2
( ), player 2 believes that
player 1 switches to ~ (t; ) with rate and probability (t; ) and p (t; ) if v (t) < e
and t < t ( ). Furthermore, since
0
(1
0
)
(t) is positive and continuous on [0; t ), it follows that
n
lim inf t : v (t)
e
!0
Hence, for small
2
2
0
(1
0
o
) =t .
player 2 is indi¤erent between accepting and rejecting
to time min ft ; t ( )g, and therefore
2
until close
( ) speci…es that he rejects until close to time
min ft ; t ( )g. Since t ( ) ! t , this shows that the time at which agreement is reached
under strategy pro…le ( ;
2
( )) converges to t .
The proof is nearly complete. Strategy
for all
> 0.
When facing strategy
2
2
( ) is iteratively conditionally undominated
( ), the highest payo¤ that player 1 can receive
when player 2 accepts at a history that is consistent with
Furthermore, for any
converges to
(t ) as
! 0.
> 0, the most player 2 accepts at a history that is inconsistent with
73
is
0
; and the highest payo¤ player 1 can receive by accepting a demand of player 2’s is
(since
2
( ) demands 1 at all times t such that e
uICD
( )
1
lim sup u1 ( 1 ;
!0
2
( )) = max
rt
). It follows that
(t ) ; lim 0 ; lim
!0
1
!0
= max
(t ) ; 0; 0 =
(t ) ,
completing the proof.
Discrete-Time Bargaining with Frequent O¤ers
This section shows that Theorem 1 continues to hold when the continuous-time bargaining
protocol of the text is replaced by any discrete-time bargaining protocol with su¢ ciently
frequent o¤ers.
More precisely, for any sequence of discrete-time bargaining games that
converges to continuous time (in that each player may make an o¤er close to any given time),
the corresponding sequence of maxmin payo¤s and postures converges to the continuoustime maxmin payo¤ and posture given by Theorem 1.
Abreu and Gul (2000) provide a
similar independence-of-procedures result for sequential equilibrium outcomes of reputational
bargaining. Because my result concerns maxmin payo¤s and postures rather than equilibria,
my proof is very di¤erent from Abreu and Gul’s.
Formally, replace the (continuous time) bargaining phase of Section 3 with the following
procedure: There is a (commonly known) function g : R+ ! f0; 1; 2g that speci…es who
makes an o¤er at each time. If g (t) = 0, no player takes an action at time t. If g (t) = i 2
f1; 2g, then player i makes a demand ui (t) 2 [0; 1] at time t, and player j immediately accepts
or rejects.
If player j accepts, the game ends with payo¤s (e
rt
ui (t) ; e
rt
(1
ui (t))); if
player j rejects, the game continues. Let Iig = ft : g (t) = ig, and assume that Iig \ [0; t] is
…nite for all t and that Iig is in…nite. The announcement phase is correspondingly modi…ed
so that player 1 announces a posture
posture
: Iig ! [0; 1], and if player 1 becomes committed to
(which continues to occur with probability "), she demands
rejects all of player 2’s demands.
(t) at time t and
I refer to the function g as a discrete-time bargaining
game.
I now de…ne convergence to continuous time. This de…nition is very similar to that of
Abreu and Gul (2000), as is the above model of discrete-time bargaining and the corresponding notation.
74
De…nition 12 A sequence of discrete-time bargaining games fgn g converges to continuous
time if for all
Iign \ [t; t +
> 0, there exists N such that for all n
N , t 2 R+ , and i 2 f1; 2g,
] 6= ;.
The maxmin payo¤ and posture in a discrete-time bargaining game are de…ned exactly
as in Section 3. Let u1;g be player 1’s maxmin payo¤ in discrete-time bargaining game g,
and let u1;g ( ) be player 1’s maxmin payo¤ given posture
in g.
The independence-of-
procedures result states that, for any sequence of discrete-time bargaining games converging
to continuous time, the corresponding sequence of maxmin payo¤s fu1;gn g converges to u1 ,
and any corresponding sequence of postures f
, where u1 and
gn
g such that u1;gn (
gn
) ! u1 “converges”to
are the maxmin payo¤ and posture identi…ed in Theorem 1. The nature
of the convergence of the sequence f
gn
g to
is slightly delicate. For example, there may
be (in…nitely many) times t 2 R+ such that limn!1
gn
(t) exists and is greater than
(t),
because these demands may be “non-serious”(in that they are followed immediately by lower
demands).44
Thus, rather than stating the convergence in terms of f
gn
g and
, I state
it in terms of the corresponding continuation values of player 2, which are the economically
more important variables. Formally, given a posture
gn
in discrete-time bargaining game
g n , let
v gn (t)
maxgn e
r(
t)
(1
gn
( )) .
t: 2I1
Let v (t) = max f1
ert = (1
log ") ; 0g, the continuation value corresponding to
in the
continuous-time model of Section 3. The independence-of-procedures result is as follows:
Proposition 7 Let fgn g be a sequence of discrete-time bargaining games converging to con-
tinuous time. Then u1;gn ! u1 , and if f
gn and u1;gn (
gn
gn
g is a sequence of postures with
gn
a posture in
) ! u1 , then v gn (t) ! v (t) for all t 2 R+ .
The key fact behind the proof of Proposition 7 is that for any sequence of discretetime postures f
limn!1 u1 (
44
gn
gn
g converging to some continuous-time posture
) (where u1 (
gn
, limn!1 u1;gn (
) is the maxmin payo¤ given a natural embedding of
The reason this complication does not arise in Theorem 1 is that the assumption that
at non-integer times rules out “non-serious” demands.
75
gn
) =
gn
in
is continuous
continuous time, de…ned formally in the proof). This fact is proved by constructing a belief
gn
that is similar to the
-o¤setting belief in each discrete-time game gn and then showing
that these beliefs converge to the -o¤setting belief in the limiting continuous-time game.
Observe that a posture
Proof of Proposition 7.
in discrete-time bargaining game
g induces a “continuous-time posture” ^ (i.e., a map from R+ ! [0; 1]) according to
^ (t) =
(min f
t:
2 Iig g).
That is, ^ ’s time-t demand is simply ’s next demand
in g. I henceforth refer to a posture
in g as also being a continuous-time posture, with
the understanding that I mean the posture ^ de…ned above.
However,
may not be a posture in the continuous-time bargaining game of Section
3, because it may be discontinuous at a non-integer time.
To avoid this problem, I
now introduce a modi…ed version of the continuous-time bargaining game of Section 3.
Formally, let the continuous-time bargaining game g cts be de…ned as in Section 3, with
the following modi…cations:
Most importantly, omit the requirement that player i’s de-
mand path uti : [t; t + 1) ! [0; 1] (which is still chosen at integer times t) is continuous. Second, specify that the payo¤s if player i accepts player j’s o¤er at date (t; 1) are
(e
rt
(1
lim inf
"t
uj ( )) ; e
rt
lim inf
"t
uj ( )) (because lim
Third, add a fourth date, (t; 2) to each instant of time t.
"t
uj ( ) may now fail to exist).
At date (t; 2), each player i
announces accept or reject, and, if player i accepts player j’s o¤er at date (t; 2), the game
ends with payo¤s (e
rt
(1
lim inf
#t
uj ( )) ; e
rt
lim inf
#t
uj ( )).
Adding the date (t; 2)
ensures that each player has a well-de…ned best-response to her belief, even though uj (t)
may now fail to be right-continuous. One can check that the analysis of Sections 3 and 4,
including Lemmas 1 through 3 and Theorem 1, continue to apply to the game g cts , with the
exception that in g cts the maxmin posture
is not in fact unique; however, every maxmin
posture corresponds to the continuation value function v (by the same argument as in Step
1 of the proof of Theorem 1).45
Because of this, for the remainder of the proof I slightly
abuse notation by writing u1 ( ) for player 1’s maxmin payo¤ given posture
45
in the game
The reason I did not use the game g cts in Sections 3 and 4 is that it is di¢ cult to interpret the assumption
that player i can accept the demand lim inf
made at time t for all
#t
uj ( ) at time t, since the demand uj ( ) has not yet been
> t. Thus, I view the game g cts as a technical construct for analyzing the limit of
discrete-time games, and not as an appealing model of continuous-time bargaining in its own right.
76
g cts , rather than in the model of Section 3. Importantly, u1 ( ) equals player 1’s maxmin
in both g cts and in the model of Section 3 when
payo¤ given
of Section 3, but u1 ( ) is well-de…ned for all
is a posture in the model
: R+ ! [0; 1].
Similarly, I write u1 (v)
for player 1’s maxmin payo¤ given continuation value function v : R+ ! [0; 1].
well-de…ned because u1 ( ) = mint
T
e
rt
(t) by Lemma 3, T depends on
v (by Lemma 1), and it can be easily veri…ed that mint
(and thus depends on
T
e
rt
(t) = mint
This is
only through
T
e
rt
(1
v (t))
only through v). A similar argument, which I omit, implies that
one may write u1;gn (v gn ) for player 1’s maxmin payo¤ given continuation value function v gn
in discrete-time bargaining game gn .
I now establish two lemmas, from which Proposition 7 follows. Their proofs require some
additional notation. Let
g
i
be the set of player i’s strategies in g cts with the property that
player i’s demand only changes at times t 2 Iig , player i only accepts player j’s o¤er at times
2 Iig [ Ijg . One
t 2 Ijg , and player i’s action at time t only depends on past play at times
can equivalently view
g
i
as player i’s strategy set in g itself. Thus, any belief
2
in g may
g
i ).
also be viewed as a belief in g cts (with supp ( 2 )
Lemma 6 For any sequence of discrete-time bargaining games converging to continuous
time, fgn g, there exists a sequence of postures f
gn 0
g with
gn 0
a posture in gn and limn!1 u1;gn (
gn 0
)
u1 .
gn 0
Proof. Let
gn 0
be given by
with the convention that max f
…rst claim that limn!1 u1 (
gn 0
exp
0
46
T~(
)
u1 .46
) are non-decreasing, sup
by Lemma 1, T~ (
Z
2
t:
n
n+1
I1gn g
(max f
t:
0 if the set f
2 I1gn g) for all t 2 R+ ,
2 I1gn g is empty. I
t:
To show this, I …rst establish that T~ (
gn 0
)
2 I1gn g for all n, where T 1 is de…ned as in the proof of Theorem 1. Since
min f > T 1 :
(and thus
gn 0
(t) =
gn 0 )
r
gn 0
t
e
r(
t)
(1
gn 0
gn 0
( )) = 1
(t). Therefore,
) satis…es
n+1
n
!
(max f
t : 2 I1gn g)
dt
(max f
t : 2 I1gn g)
Theorem 1 implies that limn!1 u1 (
gn 0
)
t2I1gn \
Y
[0;T~(
)
gn 0 )
(max f < t :
(t)
u1 , so this inequality must hold with equality.
the inequality is needed for the proof.
77
2 I1gn g)
(26)
But only
".
Now
Z
exp
T~(
gn 0 )
n+1
n
r
0
Z
exp
T~(
gn 0 )
r (1
(t))
(t)
0
Z
exp
!
(max f
t : 2 I1gn g)
dt
(max f
t : 2 I1gn g)
maxf <T~(
gn 0 ):
2I1gn g
!
dt
max
r (1
Observe that if T~ (
gn 0
1
) > min f > T :
< T~ (
0
I1gn g
2
t2I1gn \[0;T~(
(0)
n
(t)) +
(t)
0
Y
(t)
gn 0 )
!
2 I1gn
:
gn 0
that T (
gn 0
In addition,
Since T~ (
) = T~ (
gn 0
gn 0
then max
limn!1 T~ (
gn 0
)
lim u1 (
gn 0
) = lim
n!1
gn 0
T 1 . In addition, limn!1 supt2R+ j
min e
n!1 t T~(
rt gn 0
(t)
lim
(27)
n
< T~ (
gn
)
u1 (
gn
gn 0
Therefore, if
gn ), then
1
2
gn ;g cts
1
1
2
gn ;g
n
1
gn
1
(i.e., if
1
1
gn
1
) =
sup
12
gn
1
12
rt
12
gn
and
gn
2
gn
1
g cts
1 ;
T~(
gn 0 )
gn
2
e
rt gn 0
(t).
(
gn
1
12
= u1
gn
;
= u1 (
gn
),
gn
;
gn
(t) = u1 .
in discrete-time bargaining
;gn
2
2
rt
t T1
( 2 ), then
gn
1 )
2
2
2
( 2 ) as
with a strategy outside
. Now
1
;gn
u1 ( 1 ;
inf
g
n ;g cts
1)
u1 ( 1 ;
1)
1
gn
is as in De…nition 6, and the second line follows because
third line follows because u1
2 I1gn g
min f > T 1 :
)
min e
(t)
gn 0 )
and
infgn
sup
where
> T 1,
is consistent with knowledge of rationality in
gn ;g cts
gn ;g
n
; that is,
u1;gn (
o
(t)j = 0, so it follows that
well (i.e., there is no bene…t to responding to a strategy in
gn
2 ).
I1gn
2
):
) = mint
(t)
) for any posture
game g n . To see this, note that if supp ( 2 )
of
gn 0
gn 0
(t) < 1 for all t, which implies
min e
n!1 t T~(
gn 0 )
Next, I claim that u1;gn (
o
2 I1gn g for all n, and fgn g converges to continuous time,
min f > T 1 :
)
gn 0
(t) is non-decreasing and
Therefore, by Lemma 3, u1 (
).
)
gn 0 )
= sup
12
g cts
1
inf
12
gn ;g cts
gn ;g
u1 ( 1 ;
gn ;g cts
n
1
1
1)
; the
by Lemma 3,
1
and the fourth line follows by Lemma 3.
Combining the above claims, it follows that limn!1 u1;gn (
78
2 I1gn g)
dt .
and therefore (27) is less than ", which contradicts (26). Hence, T~ (
for all n.
(max f < t :
(t)
gn 0
)
limn!1 u1 (
gn 0
)
u1 .
Lemma 7 For any sequence of discrete-time bargaining games converging to continuous
time, fgn g and any sequence of functions fv gn g such that v gn is a continuation value function
in gn and limn!1 v gn (t) exists for all t 2 R+ , it follows that limn!1 u1;gn (v gn ) exists and
equals limn!1 u1 (v gn ).
Proof. Fix a sequence of continuation value functions fv gn g (with v gn a continuation
value function in discrete-time game gn ) converging pointwise to some function v : R+ !
I have already shown that u1;gn (
[0; 1].
gn
)
u1 (
gn
gn
) for any posture
in game gn ,
u1 (v gn ). This immediately implies that limn!1 u1;gn (v gn )
or equivalently u1;gn (v gn )
lim supn!1 u1 (v gn ) for every convergent subsequence of fu1;gn (v gn )g. Hence, I must show
that limn!1 u1;gn (v gn )
lim inf n!1 u1 (v gn ) for every convergent subsequence of fu1;gn (v gn )g.
I establish this inequality by assuming that there exists
> 0 such that limn!1 u1;gn (v gn ) >
for some convergent subsequence of fu1;gn (v gn )g and then deriving
lim inf n!1 u1 (v gn ) +
a contradiction. The approach is to …rst de…ne analogs of the continuous-time -o¤setting
belief and the time T~ (de…ned in Section 4.2) for game gn , denoted
and then show that T~n ! T~.
n
2
2
I must introduce some additional notation before de…ning the belief
min f > t :
gn
2
n
2.
and T~n 2 R+ ,
Let tnext
gn (i) =
2 Iign g be the time of player i’s next demand at t. Given continuation value
function v gn and any corresponding posture
n
gn (ht ) =
demands f
gn
gn
n
gn accepts player 2’s demand at
Second, f
(ht ) for all t 2 I1gn .
time t 2 I2gn with probability
p^n (t)
n
n
gn be de…ned as follows: First, f
gn
, let f
pn (t)
;1 ,
n (t)
min
where
pn (t)
2I1gn ;
if
pn (t)
<t:
2 I1gn ;
next
gn
er(t
max
<t:
next (2)=t
gn
) gn
v
1
( ) v gn (t)
v gn (t)
(2) = t is non-empty and v gn ( ) < 1 for all time
0 otherwise; and
n
(t)
max
(
)
(1 pn ( )) "
;0 .
pn ( ))
2I2gn (1
<t: 2I2gn
<t:
79
in this set, and
Let T~n be the supremum over times t at which
n
n
tnext
^n tnext
tnext
gn (2) p
gn (2) = p
gn (2) ,
and let
Tn
sup argmax e
rt gn
v
(t) .
t T~n :
t2I1gn
gn
By an argument similar to the proof of Lemma 2, if
exists a belief
the demand
2
gn
2
gn
1 )
(
and strategy
2
gn
2
2
such that
(t) is accepted under strategy pro…le (
gn
2
;
T n , then there
for some t
(t) <
gn
(
2 ).
)
",
2
;gn
2
2
In particular, ug1n (
( 2 ), and
gn
;
2)
<
Thus, by the hypothesis that limn!1 u1;gn (v gn ) > lim inf n!1 u1 (v gn ) + , there must
.
gn
exist N > 0 such that
T n and all n > N , and hence v gn (t)
for all t
(t)
for
1
T n and all n > N .
all t
Let
n
2
gn
assign probability " to
n
2
with the property that
n
gn , and …x
" to f
and probability 1
n
2
2
;gn
2
(
n
2)
always demands 1 and rejects player 1’s demand at any history
gn
at which player 1 has deviated from
n
2
(which is possible because
assigns probability 0
to such histories, except for terminal histories), as well as at any history at which player 2
n
2.
is indi¤erent between accepting and rejecting player 1’s demand under belief
gn
is a best-response to
n
2
in gn .
This implies that u1;gn (
Thus, to show that limn!1 u1;gn (v gn )
it su¢ ces to show that limn!1 ug1n (
gn
lim inf n!1 u1 (v gn ) +
gn
gn
;
n
2)
for all n.
(the desired contradiction),
lim inf n!1 u1 (v gn ) + .
n
2)
;
ug1n (
)
Note that
Observe that pn (t) satis…es
exp ( r (t
for all
t such that
player 1’s demand
(under belief
mint
T~ (v)
T (v gn )
e
rt
)) (pn (t) (1) + (1
2 I1gn and
at any time
next
gn
pn (t)) v gn (t))
v gn ( )
(2) = t. Hence, it is optimal for player 2 to reject
n
at which
next
gn
(2) p^n
next
gn
(2) = pn
next
gn
(2)
n
2 ).
Therefore, ug1n (
(1
v gn (t)), and limn!1 T~ (v gn ) = T~ (v). Hence, showing that limn!1 T~n =
gn
;
T~ would imply that limn!1 ug1n (
n
2)
gn
mint
;
n
2)
Tn
e
rt
(1
v gn (t)).
Now u1 (v gn ) =
lim inf n!1 u1 (v gn ), yielding the desired
contradiction. The remainder of the proof shows that limn!1 T~n = T~.
To see that limn!1 T~n = T~, …rst …x t0
such that, for all t
t
t0 and all n
T~ and note that for all
N 0 , if gn (t) = 2 then min
(if this set is non-empty). Next, since both e
80
rt
v (t) and e
> 0 there exists N 0 > 0
t:
rt gn
v
2 I1gn ;
next
gn
(2) = t
(t) are non-increasing
(as is easily checked) and v gn (t) ! v (t) for all t 2 R+ , it follows that for all
> 0 such that t
1
t0 and
t) gn
v
( )
> 0 there exists
v (t; 1) <
0
. Since
T~, combining these observations and letting S be the (countable)
for all t
v (t)
; t] implies that er(
2 [t
0
0
set of discontinuity points of v (t), for all
some s 2 S \ [0; t0 ], and n
lim
n!1
> 0 there exists N 00 such that if t = snext
gn (2) for
N 00 , then pn (t)
Y
v(t; 1) v(t)
1 v(t)
pn snext
gn (2)
1
=
s2S\[0;t0 ]
< 0 .47 Hence,
Y
(1
(28)
p (s))
s2S\[0;t0 ]
T~, where p is as in Section 4.2.
for all t0
Finally, I establish that whenever v is continuous on an interval [t0 ; t1 ] with t1 T~,
Z t1
Z t1
Y
rv (t) v 0 (t)
n
(1 p (t)) = exp
dt = exp
(t) dt , (29)
lim
n!1
1 v (t)
t0
t0
gn
t2I2 \[t0 ;t1 ]
where
is as in Section 4.2. I will prove this fact by showing that the limit as n ! 1 of a
R t1 rv(t) v0 (t)
Q
…rst-order approximation of the logarithm of t2I gn \[t0 ;t1 ] (1 pn (t)) equals
.
1 v(t)
t0
2
= ft 2 [t0 ; t1 ] : pn (t) > 0g, with tk;gn < tk+1;gn for all k 2
Let t1;gn ; t2;gn ; : : : ; tK(n);gn
f1; : : : ; K (n)
1g and all n 2 N, and let t0;gn = max
:
2 I1gn ;
next
gn
(2) = t1;gn . Note
that K (n) is …nite because I2gn \[t0 ; t1 ] is …nite, and that in addition tnext
k;gn (1) < tk+1;gn for all
tnext
k;gn ;gn (1) to avoid redundant notation). Furthermore, since e
k (where tnext
k;gn (1)
r
v gn ( )
is non-increasing,
tnext
k;gn (1) 2
2I1gn ;
for all k 2 f0; 1; : : : ; K (n)
K(n)
Y
(1
p (tk;gn )) =
k=1
1g. Therefore,
Y
k=1
K(n)
=
Y
k=1
0
= @
47
K(n) 1
Y 1
k=1
1
1
2I1gn ;
<tk;gn :
next (2)=t
k;gn
gn
er(tk;gn
next
e r(tk;gn (1)
next
er(tk+1;gn tk;gn (1)) v gn
next
e r(tk;gn (1) tk;gn ) v gn
S is countable because e
rt
1
min
1
min
<tk;gn :
2I1gn ; next
gn (2)=tk;gn
) v gn ( )
<tk+1;gn :
next (2)=t
k+1;gn
gn
K(n)
n
er(tk+1;gn
argmax
er(tk;gn ) v gn ( )
1 v gn (tk;gn )
) v gn ( )
tk ) gn
v
tnext
k;gn (1)
1
next
next
tk;gn (1)
1 er(t1;gn t0;gn (1)) v gn tnext
0;gn (1)
A
(30).
next
r tK(n);g (1) tK(n);gn
gn tnext
tnext
(1)
n
(1)
1
e
v
k;gn
K(n);gn
v (t) is non-increasing, and monotone functions have at most countably many
discontinuities. Unlike in Section 4, S need not be a subset of N here.
81
Next, taking a …rst-order Taylor approximation of log (1
rx gn
log (1
e v
(t)) = log (1
v
gn
erx v gn (t)) at x = 0 yields
rxv gn (t)
+ O x2 .
g
n
1 v (t)
(t))
Therefore, a …rst-order approximation of the logarithm of (30) equals
0
1
K(n) 1
g
next
n
X
rv tk;gn (1)
@
A
(tk+1;gn tk;gn )
gn tnext (1)
1
v
k;g
n
k=1
+ log 1
er(t1;gn
tnext
0;gn (1)) gn
v
tnext
0;gn (1)
next
e r(tK(n);gn (1)
log 1
tK(n);gn ) gn
v
tnext
K(n);gn (1)
.
I now show that
K(n) 1
lim
n!1
X
tk;gn )
(tk+1;gn
k=1
rv gn tnext
k;gn (1)
1
v gn
tnext
k;gn
(1)
Z
=
t1
t0
rv (t)
dt
1 v (t)
(31)
and
lim log 1 er(t1;gn
n!1
Z t1
v 0 (t)
=
dt,
1 v (t)
t0
tnext
0;gn (1)) gn
v
tnext
0;gn (1)
log 1
next
e r(tK(n);gn (1)
tK(n);gn ) gn
v
tnext
K(n);gn (1)
(32)
which completes the proof of (29). Equation (32) is immediate, because, since v is continuous
on [t0 ; t1 ], both the left- and right-hand sides equal
log (1
v (t0 ))
v (t1 )) .
log (1
To establish (31), let
f n (t)
exp
r
f (t)
exp
r
1+
t
rv gn (t)
1 v gn (t)
t
rv (t)
.
1 v (t)
and let
1+
For all n > N , it can be veri…ed that both f n (t) and f (t) are non-increasing on the interval
[t0 ; t1 ], using the facts that e
for all n > N and t
t1
t 2 R+ , there exists N 000
rt gn
v
(t) and e
T~.
Fix
rt
v (t) are non-increasing and that v gn (t)
> 0 and m 2 N.
Because v gn (t) ! v (t) for all
N such that, for all n > N 000 , jf n (t)
f (t)j <
set
t0 ;
(m
1) t0 + t1 (m
;
m
82
1
2) t0 + 2t1
; : : : ; t1 .
m
for all t in the
Since both f n and f are non-increasing on [t0 ; t1 ], this implies that
jf n (t)
f (t)j < +
max
f
k2f1;:::;K(n) 1g
(m
k) t0 + kt1
m
(m
f
k
1) t0 + (k + 1) t1
m
for all t 2 [t0 ; t1 ]. Since f is continuous on [t0 ; t1 ], taking m ! 1 implies that jf n (t)
rv gn (t)
2 for all t 2 [t0 ; t1 ]. Therefore
rv(t)
1 v(t)
1 v gn (t)
1+
2 exp r
f (t)j <
t1 for all t 2 [t0 ; t1 ].
Hence,
K(n) 1
lim
n!1
X
(tk+1;gn
tk;gn )
k=1
K(n) 1
rv gn tnext
k;gn (1)
=
v gn tnext
k;gn (1)
1
X
lim
n!1
(tk+1;gn
tk;gn )
(tk+1;gn
tk;gn )
k=1
K(n) 1
=
lim
n!1
Z
=
X
k=1
t1
PK(n)
1
k=1
(tk+1;gn
v tnext
k;gn (1)
1
rv (tk;gn )
1 v (tk;gn )
rv (t)
dt,
1 v (t)
t0
where the …rst equality follows because
rv tnext
k;gn (1)
tk;gn )
t1
t0 for all n 2 N, the
second follows because tnext
k;gn (1) 2 [tk;gn ; tk+1;gn ] and v is continuous on [t0 ; t1 ], and the third
follows by de…nition of the (Riemann) integral.
Combining (28) and (29), it follows that
lim
n!1
for all t
Y
(1
Z
n
p (s)) = exp
Y
t
(s) ds
0
s2I2gn \[0;t]
(1
p (s))
s2S\[0;t]
T~. This implies that limn!1 T~n = T~, completing the proof of the lemma.
I now complete the proof of Proposition 7.
Let fgn g be a sequence of discrete-time bargaining games converging to continuous time.
Recall that u1;gn = sup
gn
gn
u1;gn (
gn
). Thus, there exists a sequence of postures f
a posture in gn , such that limn!1 ju1;gn
u1;gn (
gn
gn
g, with
)j = 0. Let fv gn g be the correspond-
ing sequence of continuation value functions. Because e
rt gn
v
(t) is non-increasing and the
space of monotone functions from R+ ! [0; 1] is sequentially compact (by Helly’s selection
theorem or footnote 34), this sequence has a convergent subsequence fv gk g converging to
some v on R+ .
I claim that v = v . Toward a contradiction, suppose not. Since v is the unique maxmin
continuation value function in g cts , there exists
> 0 such that u1 > limk!1 u1 (v gk ) + .
By Lemma 7, limk!1 u1;gk (v gk ) = limk!1 u1 (v gk ). Finally, by Lemma 6, there exists an
83
alternative sequence of postures f
gk 0
g such that limk!1 u1;gk (
gk 0
observations implies that there exists K > 0 such that, for all k
u1;gk (
gk 0
) > u1
u1 . Combining these
)
K,
=3 > u1 (v gk ) + 2 =3 > u1;gk (v gk ) + =3,
which contradicts the fact that limk!1 ju1;gk
u1;gk (
gk
)j = 0.
Therefore, v = v .
In
addition, since this argument applies to any convergent subsequence of fv gn g, and every
subsequence of fv gn g has a convergent sub-subsequence, this implies that v gn ! v pointwise.
A similar contradiction argument shows that limk!1 u1;gk (v gk ) = u1 , for any convergent
subsequence fv gk g
fv gn g. Since limk!1 ju1;gk
u1;gk (
gk
)j = 0, it follows that u1;gk ! u1 .
And, since this argument applies to any convergent subsequence of fv gn g, this implies that
u1;gn ! u1 .
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