Sequentially Optimal Mechanisms
Vasiliki Skretay
UCLA
October 2005
Abstract
This paper establishes that posting a price in each period is a revenue maximizing
allocation mechanism in a …nite period model without commitment. A risk neutral
seller has one object to sell and faces a risk neutral buyer whose valuation is private
information and drawn from an arbitrary bounded subset of the real line. The seller has
all the bargaining power; she designs a mechanism to sell the object at t, but if trade does
not occur at t; she can propose another mechanism at t+1: We show that posting a price
in each period is an optimal mechanism. A methodological contribution of the paper is
to develop a procedure to characterize optimal dynamic incentive schemes under noncommitment that is valid irrespective of the structure of the agent’s type. Keywords:
mechanism design, non-commitment, bargaining under incomplete information, optimal
auctions, durable good monopoly. JEL Classi…cation Codes: C72, D44, D82.
This work is based on Chapter 1 of my doctoral thesis submitted to the Faculty of Arts and Sciences
at the University of Pittsburgh. I am indebted to Masaki Aoyagi and Phil Reny for superb supervision,
guidance and support. I am grateful to an editor of this journal and to two anonymous referees, whose
comments signi…cantly improved the paper. Special thanks to Kim-Sau Chung and Radu Saghin. I would
like to thank Andrew Atkeson, David Levine, Ellen McGrattan and especially John Riley for very helpful
comments. I had very helpful discussions with Marco Bassetto, Andreas Blume, Roberto Burguet, Hal Cole,
Dean Corbae, Jacques Cremer, Jim Dana, Peter Eso, Philippe Jehiel, Patrick Kehoe, Erzo Luttmer, Mark
Satterthwaite, Kathy Spier, Dimitri Vayanos, Asher Wolinsky and Charles Zheng. I would also like to thank
the Department of Management and Strategy, KGSM, Northwestern University for its warm hospitality, the
Andrew Mellon Pre-Doctoral Fellowship, the Faculty of Arts and Sciences at the University of Pittsburgh
and the TMR Network Contract ERBFMRXCT980203 for …nancial support. A signi…cant amount of this
work was performed while I was at the University of Minnesota and the Federal Reserve Bank of Minneapolis.
All remaining errors are my own.
y
Department of Economics, University of California, Los Angeles; Box 951477, Los Angeles, CA 900951477, USA. skreta@econ.ucla.edu
1
1. Introduction
This paper establishes that posting a price in each period is a revenue maximizing allocation
mechanism in a …nite horizon model, where the seller behaves sequentially rationally, and
faces one buyer whose valuation is private information. It also develops a methodology to
derive optimal mechanisms without commitment in asymmetric information environments
that allows for the agent’s type to be a continuum or …nite or any bounded measurable subset
of the real line. Our characterization extends the literature on optimal negotiation/selling
mechanisms by requiring the seller to behave sequentially rationally, and it provides a
foundation for take-it-or-leave-it o¤ers in the bargaining and in the durable goods monopoly
literatures.
Riley and Zeckhauser (1983) provide a characterization of an optimal selling procedure
under commitment when the seller faces one buyer whose valuation is private information,
and show that at the optimum the seller posts a price. Commitment requires that the seller
will never propose another mechanism in the future, even in the event that the mechanism
she initially chose failed to realize any of the existing gains of trade.1
Without commitment the seller can choose optimally another mechanism at t+1 if trade
did not occur up to t: Sequential rationality implies that if it is common knowledge that
gains of trade exist, parties cannot credibly commit to stop negotiating at a point where
no agreement is reached. This is an old idea in economics and it dates at least back to
Coase (1972), who conjectured that a monopolist of a durable good, who lacks commitment
looses all her monopoly pro…ts. In the durable good monopoly literature,2 the seller behaves
sequentially rationally, but her strategy is restricted to be a sequence of posted prices.3 The
same is true in the bargaining under incomplete information literature,4 where the proposer
makes deterministic o¤ers at each round.5 Often the right to make o¤ers is assigned to one
of the negotiating parties. Suppose that the uninformed one makes the o¤ers. Would it
be bene…cial for her instead of making a take-it-or-leave-it o¤er at each round, to employ
more sophisticated bargaining procedures? Would that possibility allow her to learn the
other party’s private information faster? What is the optimal negotiating process from the
uninformed party’s point of view? The aim of this paper is to answer these questions.
We show that an optimal allocation mechanism without commitment is to post a price
in each period. Our result provides a foundation for posted prices in the durable good
1
Real world examples about the inability of the sellers to commit can be found in McAfee and Vincent
(1997).
2
See Stokey (1979), Bulow (1982) and Gul, Sonnenschein and Wilson (1986).
3
McAfee and Vincent (1997) examine a closely related problem, where there are …nitely many buyers.
They characterize sequentially optimal auctions under the assumption that the seller’s strategy is a sequence
of reservation prices, and the buyers follow a stationary strategy.
4
See, for instance, Sobel and Takahashi (1983) and Fudenberg, Levine and Tirole (1985).
5
Bargainers do commit to the protocol within each negotiation round. “Non-commitment” means that
bargainers do not stop negotiating if they know that gains of trade exist.
2
monopoly literature,6 and for take-it-or-leave-it o¤ers in the bargaining literature, since it
establishes that posted prices perform better in terms of expected revenue than all possible
alternative institutions.
Another contribution of this work is methodological. Without commitment one cannot
use the revelation principle in order to …nd an optimal mechanism. Moreover the extended
revelation principle for environments with limited commitment by Bester and Strausz (2001)
does not apply, because it requires that the type space be …nite. Here we develop a solution
method that is based on looking for equilibrium outcomes and is valid irrespective of the
structure of the type space; it can be …nite or a continuum, or any measurable bounded
subset of the real line.7 This is the …rst paper that provides a complete characterization
of optimal mechanisms without commitment, in an asymmetric information environment
where the agent’s type can be a continuum.
2. The Environment
In this section we describe the formal setup considered in this paper. We describe players
and their payo¤s; the timing of the game; we then de…ne mechanisms, assessments, social
choice functions, outcomes of the game, what we mean by implementation, and …nally our
equilibrium concept.
A risk neutral seller owns a unit of an indivisible object. Her valuation for the object is
normalized to zero. She faces one risk neutral buyer whose valuation v is private information
and is distributed on V according to F . The convex hull of V is the interval [0; 1]:8 Time
is discrete and the game lasts T periods, t = 1; :::; T < 1 : Both the seller and the buyer
discount the future with the same discount factor, . The seller’s goal is to maximize
expected discounted revenue. The buyer aims to maximize surplus. All elements of the
game except the realization of the buyer’s valuation are common knowledge. We now
describe the timing of the game.
Timing
At the beginning of period t = 1 nature determines the valuation of the buyer, which
remains constant over time. Subsequently the seller proposes a mechanism: The mech6
As shown there, the optimal sequences of prices depends on the length of the horizon and on the discount
factor, and it is decreasing overtime.
7
If one wants to depart from the …nite type case, the nature of the problem forces such generality. Even
if one starts with a continuum of types, because of the complexity of the strategy spaces the support of the
principal’s posterior beliefs at t = 2 can be arbitrarily complicated, (whereas in the case where one starts
with a …nite type space, the type space at the beginning of a subsequent period is again …nite). And since
the game that starts at t = 2 is isomorphic to the whole game, with the di¤erence that it lasts one period
less, we write our model directly for arbitrary type spaces.
8
All the analysis goes through, if the convex hull of V is [a; b]; where -1 < a < b < 1:
3
anism is played and if the buyer obtains the object the game ends, else we move on
to period t = 2:
At t = 2 the seller proposes a mechanism. The mechanism is played and if the buyer
obtains the object the game ends, else we move on to period t = 3:
....
At t = T the seller proposes a mechanism. The mechanism is played and the game
ends at the end of period T; irrespective of whether trade takes place or not.
We continue by de…ning what we mean by the word “mechanism.”
Mechanisms
A mechanism is a game form. A game form Mt = (St ; gt ) consists of a set of actions St
and a mapping gt : St ! [0; 1] R that maps an action st 2 St into outcomes. An outcome
of a mechanism is a probability that the buyer obtains the object at period t, rt (st ) 2 [0; 1]
and an expected payment zt (st ) 2 R: A pair (rt (st ); zt (st )) is called a contract. The set of
all possible mechanisms is denoted by M.
The buyer can always choose not to participate in a mechanism, in which case he gets a
payo¤ of 0: We include the choice of non-participation in the de…nition of each mechanism,
by assuming that there is an option s 2 S such that (r(s); z(s)) = (0; 0): Now we can move
on and talk about strategies and assessments.
Assessments
An assessment consists of a strategy pro…le and a belief system. A strategy pro…le
= ( i )i=S;B ; speci…es a strategy for each player. In order to talk about strategies we
need some additional notation. Let IS and IB denote the information sets of the seller
and the buyer respectively. A strategy for the seller, S ; is a sequence of maps from IS
to M. A behavioral strategy of the buyer, B ; consists of a mapping from V
IB to
a probability distribution over actions, (St ): A belief system, ; maps IS to the set of
probability distributions over V: Let F (vjitS ) denote the seller’s beliefs about the buyer’s
valuation at information set itS ; t = 1; :::; T: Sometimes when we are referring to a particular
information set we simplify the notation by setting F (vjitS ) = Ft (v); t = 1; :::; T .9 For t = 1
we have F (vji1S ) = F (v), that is, the seller has correct prior beliefs. A strategy pro…le
and a belief system is an assessment.
9
But the reader should keep in mind that there are many di¤erent histories that lead to no trade up to
period t, and for each such history there is in general a di¤erent posterior.
4
Allocation Rules and Payment Rules
Given an assessment, (not necessarily an equilibrium), the outcome from the ex-ante
point of view is an allocation rule p( ; ) and a payment rule x( ; ): The rule p( ; )(v) is
the expected, discounted probability that a v type buyer will obtain the object given the
assessment ( ; );
T
X
t
p( ; )(v) =
1ftrade at tg j( ; ); v ;
t=1
and x( ; )(v) is the expected, discounted payment that a v type buyer will incur given
( ; ),
x( ; )(v) =
T
X
t=1
t
1ftrade at tg fexpected payment at tg j( ; ); v :
It is possible that di¤erent strategy pro…les lead to the same allocation and payment rules.
It will be useful to de…ne the outcomes of assessments at a continuation game that starts
at t when the seller’s information set is iSt . The situation is isomorphic to the game as a
whole, with the only di¤erence that the seller’s beliefs are now given by Ft (:jiSt ): Then we
can talk about the allocation rule pt;iSt ( ; ) and the payment rule xt;iSt ( ; ) that arise by
the assessment ( ; ) at the continuation game that starts at t; when the seller’s information
set is iSt : They are the analogs of p( ; ); x( ; ) for the particular continuation game. As
before, we will often suppress the notation ( ; ):
In order to talk about implementation we de…ne social choice functions.
Social Choice Functions
In the environment under consideration, a social choice function speci…es for each valuation of the buyer v and each period t; a probability of trade rt (v) 2 [0; 1] and an expected
payment zt (v) 2 R: Given a social choice function frt (v); zt (v)gTt=1 we can de…ne the corresponding allocation rule p and payment rule x by
p(v) = r1 (v) + (1
r1 (v)) [r2 (v) + (1
r2 (v)) [:::::]] and
x(v) = z1 (v) + (1
r1 (v)) [z2 (v) + (1
r2 (v)) [:::::]] :
Allocation rules map valuations to expected discounted probabilities of trade and payment
rules map valuations to expected discounted payments. The expected payo¤ of the seller
and buyer respectively, from the interim point of view, are given by
Z
Expected Payo¤ of the seller =
x(v)dF (v) and
V
Expected Payo¤ of type v buyer = p(v)v
5
x(v);
where the integral in the expression of the seller’s expected payo¤ comes from the fact
that the seller does not know the valuation of the buyer. There are many di¤erent social
choice functions frt (v); zt (v)gTt=1 that lead to the same p(v) and x(v); and hence to the
same payo¤s to the buyer and the seller. All such social choice functions are equivalent for
our purposes and hence when we talk about a social choice function we will simply mean
their “reduced versions” given by p and x: Now that we have speci…ed what we mean by a
social choice function we can talk about implementation.
An assessment ( ; ) implements the (reduced) social choice function p and x if for all
v 2 V p( ; )(v) = p(v) and x( ; )(v) = x(v).
The set of implementable social choice functions depends on the solution concept. In
this paper we require assessments to be Perfect Bayesian Equilibria.
Solution Concept
A Perfect Bayesian Equilibrium, (PBE), is a strategy pro…le, ; and a belief
system, ; that satisfy:
1. For all v 2 [0; 1] the buyer’s strategy is a best response at each itb and t;
given the seller’s strategy.
2. Given Ft (:jitS ) and the buyer’s strategy, the seller chooses at each itS and
t an optimal sequence of mechanisms for the remainder of the game.
3. Ft (:jitS ) is derived from Ft 1 given itS using Bayes’rule whenever possible.
A P BE requires strategies to dictate optimal behavior at each information set. The
nature the buyer’s and the seller’s information sets depends on what they observe during
play. The buyer observes the mechanism that the seller proposes at each stage as well as
whether trade takes place or not. We assume that the seller observes the action that the
buyer chooses, s, as well as whether trade takes place or not. What the seller observes at
each stage is called the degree of transparency of mechanisms. It determines, is some sense,
her commitment power.10
Our objective is to …nd an assessment that is a P BE; and guarantees highest expected
revenue for the seller among all P BE 0 s. In order to do so, we search for a social choice
function that maximizes expected revenue among all social choice functions that are implemented by assessments that are P BE 0 s:
The following section describes the technical di¢ culties of the problem at hand, as well
as our solution method.
If the seller does not observe anything, then all BN E 0 s are P BE 0 s and the “commitment solution” is
sequentially rational. In “Extensions on Sequentially Optimal Mechanisms” we establish that our result is
robust to the case where the seller observes only whether trade took place or not.
10
6
3. Technical Difficulties & The Procedure
In this section we explain why one cannot employ the revelation principle in order to
characterize a revenue maximizing P BE; and how we sidestep this di¢ culty.
When the mechanism designer behaves sequentially rationally, information revelation
is very di¢ cult because the mechanism designer, the seller in this case, cannot commit
not to use it and exploit the buyer in the future. The buyer of course anticipates this,
so in all but the last stage of the game, he will typically behave in a way that does not
reveal his information.11 The earlier literature on mechanism design without commitment,
(Freixas, Guesnerie and Tirole (1985), FGT (1985), Hart and Tirole (1988), La¤ont and
Tirole (1988), and (1993)), observed that, with the exception of the …nal period of the game,
one cannot use the standard revelation principle in order to …nd the optimal mechanism
in each stage. Without the help of the revelation principle one may need to consider
mechanisms with arbitrary message spaces, which is indeed what we do here, but then it
becomes quite challenging even to write down the seller’s optimization problem. For this
reason FGT (1985) characterize the optimal incentive schemes among the class of linear
incentive schemes. LT (1988) consider arbitrary schemes but examine only special classes
of equilibria, without characterizing the optimum.
A remarkable result is derived in a recent paper by Bester and Strausz (2001), BS, who
show that when the principal faces one agent whose type space is …nite, she can, without
loss of generality, restrict attention to mechanisms where the message space has the same
cardinality as the type space. This result does not apply to our problem because the type
space can be a continuum. Indeed, it seems di¢ cult to translate the idea that it is enough to
have one message per type in the case of a continuum of types. Moreover as BS illustrate,
in order to …nd an optimal mechanism one has to check which incentive compatibility
constraints are binding. In an environment with limited commitment, constraints may be
binding “upwards” and “downwards.” Even if one could obtain an analog of the BS result
for the continuum type case, which is indeed challenging, it does not seem straightforward to
generalize the procedure of checking which incentive compatibility constraints are binding.
When there is no a priori restriction to a canonical class of mechanisms, looking for equilibria is indeed very ambitious given the size of strategy spaces, (the set of possible mechanisms is enormous), and the complexity of the game. Mechanisms employed at t = 2; :::; T
11
To see why, suppose that at period one the seller employs a direct revelation mechanism, the buyer has
claimed to have valuation v; and according to this mechanism no trade takes place. If the seller behaves
sequentially rationally, she will try to sell the object at t = 2 using a di¤erent mechanism. And in the case
that the buyer has revealed his true valuation at t = 1, the seller has complete information at t = 2: She
can therefore use this information to extract all the surplus from the buyer. In this situation the buyer will
have an incentive to manipulate the seller’s beliefs. One would expect, and it is indeed true, that he will
not always reveal his valuation truthfully at the beginning of the relationship. This is established formally
in Proposition 1 in La¤ont and Tirole (1988), where they show that there is no P BE where each type of
the buyer chooses a di¤erent action at t = 1: Hence truth-telling with probability one cannot occur at any
P BE at t = 1:
7
depend on the seller’s posterior. Along the equilibrium path the posterior is determined
by Bayes’ rule from the buyer’s strategy, the mechanism proposed by the seller, and the
action chosen by the buyer. There can be in…nitely many choices at t that end up in no
trade since lotteries are allowed. Each of these choices leads to a di¤erent posterior and an
optimal t + 1 mechanism. At an equilibrium the mechanism at t has to be optimally chosen
taking into account, not only revenue at t; but also what beliefs the seller will have after
each history where there is no trade at up to t, which in turn will determine the optimal
mechanism for t+1 and so on: And in order for someone to …nd a revenue maximizing P BE;
one has to …nd all P BEs of the game, then calculate the corresponding revenue, and …nally
compare the seller’s revenue at each of these P BEs: A paper that looks for equilibria is
La¤ont and Tirole (1988). That paper looks for Perfect Bayesian Equilibria of a two-period
regulation game without commitment, where the …rm’s type is drawn from a continuum.
LT (1988) provide some properties of equilibria but they do not obtain a characterization of
an optimal equilibrium.12 Indeed looking for equilibria is very di¢ cult, even in our model
which is in some dimensions simpler than LT (1988).
We are able to sidestep the di¢ culties that arise in the analysis of LT (1988) by looking at
equilibrium outcomes instead of equilibria. Investigating properties of equilibrium outcomes
is much simpler. In one sentence, what we do is to characterize the properties of outcomes,
p( ; )0 s and x( ; )0 s that are implemented by assessments that are P BE 0 s; and then
choose the ones that the seller prefers: This idea was inspired from Riley and Zeckhauser
(1983). Obviously the set of P BE implementable allocation and payment rules is a subset
of the BN E implementable ones. In order to characterize this subset we examine what
restrictions the requirement that ( ; ) be a P BE implies on p and x: In this way we
obtain allocation rules and payment rules that satisfy necessary conditions of being P BE
implementable. We then choose the p and x that the seller likes best among those that
satisfy the necessary conditions of being P BE implementable and verify that there exists
indeed a P BE that implements them. This is in line with the approach of the revelation
principle which prescribes a way to obtain all the set of BN E implementable allocation and
payment rules: “just look at the social choice functions that can be implemented by truthtelling equilibria of direct revelation mechanisms.” While our focus is still on equilibrium
outcomes, rather than equilibria, we do not work with a “canonical family”of mechanisms.
12
First they show that there is no separating equilibrium, that is, there exists no equilibrium where each
type of the agent chooses a di¤erent action at t = 1: Then they consider the situation where the uncertainty
about the agents’type is very small, that is, the agent’s type belongs in a very small interval. In this case the
distortion of a full pooling continuation equilibrium compared to the commitment optimum, goes to zero,
as the length of the interval of possible types goes to zero. They also show that there exist other equilibria
that have the property that the distortion compared to the commitment case goes to zero as the length of
the interval of possible types goes to zero. These equilibria exhibit, what they call in…nite reswitching or
pooling over a large scale. Hence for small uncertainty, that is when the agent’s type is almost known, these
three kinds of continuation equilibria are candidates for an optimum. For the case that uncertainly is not
trivial the authors do not say which equilibria may be optimal.
8
This alternative route in the case that we are interested in BN E implementable social
choice functions is as simple as working with direct revelation mechanisms, but has the
additional advantage that it allows us to obtain properties of P BE-implementable social
choice functions. Focusing on outcomes renders mechanism design “without commitment”
a tractable problem, even if one works with mechanisms with arbitrary message spaces.
In the following section we formulate the seller’s problem.
4. Formulation of the Problem
Here we employ the idea sketched in the previous section in order to write down the seller’s
maximization problem. The seller seeks to solve
Z
max
x(v)dF (v)
p;x
V
subject to p; x being P BE implementable.
We begin by examining the implementability constraints. First of all, p has to satisfy
resource constraints. Since there is only one object to be allocated, it must be the case that
p(v) 2 [0; 1] for all v: Second, at a P BE the buyer’s and the seller’s strategy must be best
responses at each information set. For the buyer this implies, at the very least, that there
is no type v that can bene…t by behaving as type v 0 does. For the seller this implies that
at each information set her strategy speci…es an optimal sequence of mechanisms employed
in the remainder of the game. Finally, from the fact that the buyer can always choose not
to participate in a mechanism, we get that the buyer’s expected payo¤ for the continuation
of the game must be non-negative. It follows that if p and x are implemented by a P BE
they must, at the very least, satisfy the constraints of the following Program, which we call
Program A:
Z
max
p;x
x(v)dF (v)
V
subject to:
IC “incentive constraints,” the buyer’s strategy is such that
p(v)v x(v) p(v 0 )v x(v 0 ); for all v; v 0 2 V
P C “voluntary participation constraints,”
p(v)v x(v) 0 for all v 2 V
RES “resource constraints” for all v 2 V 0 p(v) 1
SRC(t; itS ) “sequential rationality constraints,”for all t; t = 2; :::; T; and for
each information set of no trade at t, itS ; the seller chooses a mechanism that
maximizes revenue:
Z
max
xt;itS (v)dFt;itS (v)
pt;it ;xt;it
S
S
Yt;it
S
9
subject to:
IC(t; itS ) pt;itS (v)v xt;itS (v) pt;itS (v 0 )v xt;itS (v 0 ); for all v; v 0 2 Yt;itS
P C(t; itS ) pt;itS (v)v xt;itS (v) 0 for all v 2 Yt;itS
RES(t; itS ) “resource constraints” 0 pt;itS (v) 1 for all v 2 Yt;itS ;
where Yt;itS is the support of the posterior Ft;itS :
Belief s posterior beliefs Ft;itS are derived using the buyer’s strategy and
Bayes’rule whenever possible.
Summarizing, if p; x are implemented by an assessment ( ; ) that is a P BE; then
conditions IC; P C, RES and SRC will be satis…ed. These are necessary conditions for p
and x to be P BE implementable. In what follows, we will obtain a solution of Program A, p
and x; and establish that there exists indeed an assessment that is a P BE and it implements
p and x: We present the solution when T = 2: First we solve the problem without imposing
the sequential rationality constraints and show that an optimum is implemented by posting
the same price in each period. Then, we impose the sequential rationality constraints, and
establish again that at an optimum the seller posts a price in each period. With sequential
rationality constraints, the optimal sequence of prices depends on the discount factor and
on the length of the game, and as it is shown in the durable good monopoly literature,
it is decreasing overtime. By induction one can show that the same result holds for any
T < 1.13 Omitted proofs are in the Appendix, unless otherwise noted.
5. Optimal Mechanisms without Sequential Rationality Constraints
In this section we solve the problem without sequential rationality constraints. This problem
provides a benchmark to compare the e¤ect of sequential rationality on the solution. It is
also the problem that the seller solves at the beginning of the …nal period of the game when
we impose the sequential rationality constraints.
Without sequential rationality constraints the seller’s problem reduces to
Z
x(v)dF (v)
max
p;x
V
subject to:
IC
p(v)v x(v) p(v 0 )v x(v 0 ) for all v; v 0 2 V
PC
p(v)v x(v) 0 for all v 2 V
RES
0 p(v) 1 for all v 2 V:
Even though this problem is isomorphic to the problem in the classical works of Myerson
(1981) or Riley and Zeckhauser (1983), their solution approach does not go through because
13
The complete analysis of the case where T > 2 can be found in “Extensions on Sequentially Optimal
Mechanisms,” which can be downloaded from the author’s website.
10
it requires that the type space be an interval. The key step there is to rewrite revenue as a
function of the allocation rule, which is the gist of the famous revenue equivalence theorem.
That step relies on the type space being an interval, which is not necessarily true in our
model, where we work with probability measures that have support arbitrary measurable
subsets of the real line. The reason we allow for such generality is the following. When
T > 1 and we impose sequential rationality constraints, SRC; then SRC at T imply that
for all histories iTS , pT;iT and xT;iT solve a problem that is isomorphic to the current one:
S
S
And because we allow for the use of general mechanisms by the seller and mixed strategies
by the buyer, posteriors may be quite complicated indeed.
How does one obtain a solution when the type space can be a “messy” measurable
set? The idea is to solve an arti…cial problem where the type space is an interval, but its
solution restricted to V solves the actual problem of interest, Program A. In particular, in
the Proposition that follows we establish that we can obtain a solution of Program A by
solving an arti…cial problem, Program B, where we require the constraints to hold on the
whole convex hull of V; which is the interval [0; 1]:
Program B:
max
p;x
Z
1
x(v)dF (v)
0
subject to:
IC
p(v)v x(v) p(v 0 )v x(v 0 ) for all v; v 0 2 [0; 1]
PC
p(v)v x(v) 0 for all v 2 [0; 1]
RES
0 p(v) 1 for all v 2 [0; 1]:
Proposition 1 14 Let pA ; and pB denote solutions of Program A and Program B respectively, and let R(pA ) and R(pB ) denote the seller’s expected revenue at each of these
solutions: Then
R(pA ) = R(pB ):
Before proceeding let us sketch brie‡y the proof of Proposition 1.15 Programs A and
B have the same objective function since dF is zero on [0; 1]nV and they di¤er only in the
constraint set. Program B has a lot more constraints than Program A therefore R(pA )
R(pB ): We establish that the reverse inequality is true by showing that a solution pA and
xA of Program A appropriately extended on [0; 1] satis…es all the constraints of Program
B. In particular we extend a solution of A, call it pA ; as follows. For a type v in [0; 1]nV
we set the value of pA equal to the value of pA at the largest type in V less or equal to v:
It is then easy to see that because pA is feasible on V and no real new options have been
14
15
The conjecture that such a result may be available arose from discussions with Kim-Sau Chung.
More details can be found in the Appendix. See also Skreta (2005a) for full details.
11
added, the resulting allocation rule is feasible for Program B. It follows that the values of
these programs are the same.
Now that we have replaced the type space with its convex hull, we can obtain properties
of feasible allocation and payment rules using standard techniques. Let U ; ( B (v); v) =
p(v)v x(v) denote the buyer’s expected discounted payo¤ when his valuation is v given
the assessment ( ; ): From Lemma 2 in Myerson (1981) we have:
Lemma 1 If p; x are implemented by a BN E; they satisfy IC; P C and RES constraints
and the following conditions must hold: for all v 2 [0; 1] (a) p(v) is increasing in v (b)
Rv
U ; ( B (v); v) = 0 p(s)ds + U ; ( B (0); 0) (c) U ; ( B (0); 0) 0 and (d) p(v) 2 [0; 1] for
all v 2 [0; 1]:
Using these properties and standard arguments, we can write the seller’s expected revenue as a function only of the allocation rule and the payo¤ that accrues to the lowest type
of the buyer16
Z 1
Z 1
Z 1
x(v)dF (v) =
p(v)vdF (v)
p(v)[1 F (v)]dv U ; ( B (0); 0):
0
0
0
From the analysis in Myerson (1981), we know that if p solves
max
p2=
Z
Z
1
p(v)vdF (v)
0
1
p(v)[1
F (v)]dv;
(1)
0
where
= = fp : [0; 1] ! [0; 1]: p is increasingg ;
Z v
and x(v) = p(v)v
p(s)ds;
0
then (p; x) are optimal.
Our objective is to choose a function p that is increasing and such that (1) is maximized.
Because (1) is linear in p it follows that the maximizer is an extreme point of the feasible
set. Extreme points of the set of increasing functions from [0; 1] to [0; 1] are step functions
that jump from zero to one. The optimal allocation rule has the bang-bang property: the
seller trades with zero probability with types below a cuto¤ of v ; p(v) = 0 for v 2 [0; v );
whereas trades with probability 1, that is p(v) = 1 with types v 2 [v ; 1]. The Proposition
that follows states this result and provides a formal de…nition of the cut-o¤ v :
16
U
;
If F has a strictly positive density, then we obtain the familiar expression
(
B (0); 0):
12
R1
0
h
p(v) v
[1 F (v)
f (v)
i
f (v)dv
Proposition 2 17 Without sequential rationality constraints the seller maximizes revenue
by posting at each t the same price v ; where
Z v~
Z v~
v
inf v 2 [0; 1] s.t.
sdF (s)
[1 F (s)]ds 0; for all v~ 2 [v; 1] :
(2)
v
v
The revenue maximizing allocation and payment rules are given by
p(v) = 1 if v v
= 0 if v < v
and
x(v) = v if v v
:
= 0 if v < v
The proof of this result is relatively standard and can be found in the “Technical Appendix for Sequentially Optimal Mechanisms.”The revenue maximizing allocation rule without
sequential rationality constraints can be implemented by the following strategy pro…le. The
seller makes a take-it-or-leave-it o¤er in each period, t = 1; :::; T of v : The buyer’s strategy
is as follows: for v
v the buyer accepts the seller’s o¤er at t = 1 and for v < v the
buyer rejects the seller’s o¤er at t = 1; :::; T: It is very easy to see that players’strategies
are mutual best responses hence the given assessment is a BN E:
This characterization is a generalization of the analysis in Myerson (1981) and Riley and
Zeckhauser (1983), “commitment solution,” for distributions that have arbitrary support
and not necessarily satisfy the monotone hazard rate property. When T > 1 and we impose
sequential rationality constraints, SRC; it will be crucial for our analysis to be able to
say how the price that the seller will post at the …nal period of the game depends of the
posterior (see the proof of Proposition 8). The de…nition of an optimal cuto¤ in (2) allows
us to obtain such comparative statics results, something that is not that obvious with the
“ironing technique” in Myerson (1981). We now continue with the characterization of a
revenue maximizing rule with sequential rationality constraints.
6. Optimal Mechanisms with Sequential Rationality
6.1
Formulation of the Problem
In this section we solve for optimal mechanisms taking into account sequential rationality
constraints. We analyze the case when T = 2: As in the case without sequential rationality
constraints; we …rst show that it is without loss of generality to solve an arti…cial program
where we replace the type space with its convex hull. Program B for T = 2 is given by
Z 1
max
x(v)dF (v)
p;x
subject to:
IC
p(v)v
17
x(v)
p(v 0 )v
0
x(v 0 ); for all v; v 0 2 [0; 1]
I thank Phil Reny for suggesting parts of the proof of Proposition 2.
13
PC
p(v)v x(v) 0 for all v 2 [0; 1]
RES
0 p(v) 1 for all v 2 [0; 1]:
SRC(t = 2; i2S ) “sequential rationality constraints.”For each information set
of no trade at t = 2, i2S ; the seller chooses a mechanism that maximizes revenue:
Z
x2;i2S (v)dF2;i2 (v)
max
p2;i2 ;x2;i2
S
S
S
Y2;i2
S
subject to
IC(t = 2; i2S ) p2;i2 (v)v x2;i2 (v) p2;i2 (v 0 )v x2;i2 (v 0 );
S
S
S
S
for all v; v 0 2 Y2;i2
S
P C(t = 2; i2S ) p2;i2 (v)v x2;i2 (v) 0 for all v 2 Y2;i2
S
S
S
RES(t = 2; i2S ) 0 p2;i2 (v) 1 for all v 2 Y2;i2
S
S
Belief s posterior beliefs F2;i2 are derived using the buyer’s strategy
S
and Bayes’rule whenever possible.
Program B is exactly the same as Program A for T = 2 but with V and YT;iT replaced
S
by their convex hulls [0; 1] and YT;iT , respectively.
S
Proposition 3 The value of Program A and Program B is the same.
Proposition 3 is analogous to Proposition 1, but now we have to take into account
the sequential rationality constraints. We sketch why Proposition 3 is true. By applying
Proposition 1 to the seller’s problem at t = 2; we know that at each information set a
solution of SRC(t = 2; i2S ) extended on Y2;i2 = [v; v] solves Program B at information set i2S :
S
Therefore, essentially nothing changes on the set of constraints described in SRC(t = 2; i2S ):
Then, using exactly the same arguments as in the proof of Proposition 1, one can show that
a solution p and x of Program A appropriately extended on [0; 1] satis…es all the constraints
of Program B. It follows that the values of these programs must be the same.
Using standard arguments, Program B can be rewritten as
Z 1
Z 1
max
p(v)vdF (v)
p(v)[1 F (v)]dv
(3)
p;x
0
0
subject to:
=[0;1] = fp : [0; 1] ! [0; 1]: p is increasingg ;
Z v
x(v) = p(v)v
p(s)ds and
0
p(v) 2 [0; 1] for all v 2 [0; 1]:
SRC(t = 2; i2S ) “sequential rationality constraints.” For each history i2S
14
max
p2;i2 ;x2;i2
S
S
Z
Y2;i2
p2;i2 (v)vdF2;i2 (v)
S
S
S
Z
Y2;i2
p2;i2 (v)[1
S
F2;i2 (v)]dv
S
S
subject to
n
o
p
:
Y
!
[0;
1]:
p
is
increasing
;
2
2
2
2;iS
2;iS
2;iS
2;i2
S
Z v
x2;i2 (v) = p2;i2 (v)v
p2;i2 (s)ds and
=Y
S
=
S
v
S
p2;i2 (v) 2 [0; 1] for all v 2 Y2;i2 :
S
S
Belief s posterior beliefs F2;i2 are derived using the buyer’s strategy
S
and Bayes’rule whenever possible.
We now describe properties that (p; x) satisfy if they are implemented by strategy pro…les
that are P BE 0 s.
6.2
Necessary Conditions at a P BE
Fix an assessment ( ; ) that is a P BE and call p; x the allocation and the payment rule
implemented by it. We start by drawing p from the smallest type, type 0. Let s denote
an action that leads to contract (r; z): This is the contract with the smallest r; among
all contracts lead by actions weakly preferred by type 0 at t = 1 at the P BE under
consideration: Let [0; v]; v
1; denote the convex hull of types that choose action s at
t = 1; and let F2 denote the seller’s posterior at t = 2 after she observes action s at t = 1
and no trade takes place. Since T = 2 is the …nal period of the game, we know from
Proposition 2 that the seller will post a price, which we denote by z2 (F2 ):
The Proposition that follows describes properties of P BE implementable (p; x):
Proposition 4 If (p; x) are implemented by an assessment that is a P BE they satisfy (i)
Lemma 1 and (ii)
p(v) = r for v 2 [0; z2 (F2 ))
p(z2 (F2 )) 2 [r; r + (1 r) ]
(4)
p(v) = r + (1 r) for v 2 (z2 (F2 ); v)
p(v) 2 [r + (1 r) ; 1]
where v 2 [0; 1]; r 2 [0; 1]; and z2 (F2 ) solves (2) given F2 ; where F2 is a probability measure
that can arise at a P BE and has support a set with convex hull the interval [0; v].
De…nition 1 We call P the set of allocation rules that satisfy the properties described in
Proposition 4:18
18
The de…nition allows for v = z2 (F2 ) in which case p(v) 2 [r; 1]:
15
For v 2 [0; v) it is surprising to see that the shape of p is the same as if all types in [0; v]
choose action s with probability one. In that case the price at t = 2 after the buyer chooses
s, call it z2 (v); would have been optimal given
(
F (v)
F
(v) ; for v 2 [0; v] :
(5)
F2 (v) =
0; otherwise
In general, z2 (F2 ) is optimally chosen given some posterior F2 whose support has convex
hull [0; v]; but it is not a mere truncation of the original distribution. Now, for types in
[v; 1]; the shape of p depends on the seller’s and the buyer’s strategy. But for certain, if p
is implemented by an assessment that is a P BE; then it is also BN E implementable and
it satis…es all the properties stated in Lemma 1. Hence it has to be an increasing function.
With the help of Proposition 4, the following section establishes that a solution of (3)
is implemented by a P BE of the game where the seller posts a price in each period.
6.3
Revenue Maximizing P BE
The solution will proceed as follows. First, we restrict attention to allocation rules implemented by strategy pro…les, where the seller at t = 1 employs simple mechanisms that
separate types into two groups: “high” and “low” ones. We call this set of allocation rules
P : We show that a revenue maximizing allocation rule among P is implemented by a
P BE of the game where the seller posts a price in each period. Second, we consider the
general case, where the mechanism consists of a game form and the buyer may employ
mixed strategies, where potentially non-convex sets of types choose the same action with
positive probability at t = 1. We show that under certain conditions, the seller can without
loss restrict attention to allocation rules in P : We conclude that if these conditions are
satis…ed, at an optimum the seller posts a price in each period. In the …nal, and most
delicate, step we show that the previously imposed conditions are satis…ed by all P BEs
and our quasi solution is a real solution. We start our exploration with the case where at
t = 1 the seller employs mechanisms that contain two options.
Step 1: Revenue Maximizing P BE among 2-Option Mechanisms
Here we look for the revenue maximizing allocation rule among the ones implemented by
strategy pro…les where the seller at t = 1 employs a mechanism that contains two contracts:
one targeted to the “low” types, (r; z); and one targeted to the “high” types, (1; z1 ). We
show that at the optimum this kind of mechanism reduces to a posted price: the options
available are (0; 0) and (1; z1 ):
We look at assessments of the following form. The seller at t = 1 proposes a mechanism
that apart from the outside option (0; 0); contains two contracts. Contract (r; z) is targeted
to “low”types and option (1; z1 ) is targeted to “high”valuation types, where r 2 [0; 1] and
z; z1 2 R. The buyer behaves as follows. At t = 1 types v 2 [0; v) choose (r; z) and types in
16
(v; 1] choose (1; z1 ). Type v is indi¤erent between choosing: (r; z) and choosing (1; z1 ) at
t = 1; and may be randomizing between the two. Now at t = 2 after the history where the
buyer chose (r; z) at t = 1 and no trade took place, beliefs are given by (5), and the seller
chooses a price z2 that satis…es z2 z2 (v); (note that this allows for suboptimal behavior
at t = 2 on the part of the seller): The buyer at t = 2 accepts z2 for v 2 [z2 ; v) and rejects
z2 for v 2 [0; z2 ): This assessment implements an allocation rule of the form
p(v) = r for v 2 [0; z2 )
p(v) = r + (1 r) for v 2 [z2 ; v)
p(v) 2 [r + (1 r) ; 1]
p(v) = 1 for v 2 (v; 1]:
(6)
De…nition 2 We call P the set of allocation rules have the shape given by (6), for some
r 2 [0; 1]; v 2 [0; 1]; z2 z2 (v); where z2 (v) solves (2) given (5).
Now we move on to establish that a revenue-maximizing element of P can be implemented by a P BE of the game where the seller posts a price in each period.
Proposition 5 Let p denote a solution of maxp2P R(p): Then p can be implemented by
a P BE of the game where the seller posts a price in each period.
This step establishes that if the seller restricts attention to period one mechanisms
that apart from the outside option (0; 0); contain two options: one targeted to the “low”
types, (r; z); and one targeted to the “high”types, (1; z1 ), then at an optimum this kind of
mechanism reduces to a posted price: the options available are (0; 0) and (1; z1 ): Now we
proceed to solve the general problem.
Step 2: If for all p 2 P it holds z2 (F2 )
z2 (v); then posted prices are optimal
Here we assume that for all p 2 P it holds z2 (F2 ) z2 (v). Given this assumption, we
show that a solution of
Z 1
Z 1
max
p(v)vdF (v)
p(v)[1 F (v)]dv
(7)
p;x
0
0
subject to:
p 2 P and x(v) = p(v)v
Z
v
p(s)ds.
0
is an allocation rule that is implemented by a P BE of the game where the seller posts a
price in each period. In particular, we show that each element of P, for which it holds
that z2 (F2 ) z2 (v); is dominated in terms of expected revenue by an element of P . Then
our result follows from Proposition 5. The formulation of (7) is analogous to program (1),
where we do not impose sequential rationality constraints. There the only requirement of
p is to be increasing and to satisfy p(v) 2 [0; 1].
We will employ the following auxiliary result.
17
Lemma 2 If z2 (v) solves (2) given (5); then z2 is increasing in v:
We are now ready to state the main result of this step.
Proposition 6 For each p 2 P with the property that z2 (F2 )
allocation rule p~ 2 P that generates higher revenue then p:
z2 (v); there exists an
Proof. We establish that each allocation rule p 2 P where z2 (F2 ) z2 (v), is dominated in
terms of expected revenue by an allocation rule in p~ 2 P : Take an element of p 2 P; for
which we have that z2 (F2 ) z2 (v): First suppose that 0 < z2 (F2 ) < v: Expected revenue
from p is then given by
Z
0
+
+
Z
z2 (F2 )
rvdF (v)
Z
z2 (F2 )
r[1
F (v)]dv
Z v
[r + (1
0
v
[r + (1
r) ]vdF (v)
z2 (F2 )
Z 1
p(v)vdF (v)
v
Z
r) ][1
F (v)]dv
z2 (F2 )
1
p(v)[1
F (v)]dv:
v
We construct an allocation rule p~; that is an element of P and it generates higher revenue
than p. For the range [0; v) we set p~(v) = p(v): For types in v 2 [v; 1]; we choose p~ optimally
imposing the constraint that the resulting allocation rule be increasing on [0; 1] and ignoring
all sequential rationality constraints: For types [v; 1] it is, in some sense, as if we are solving
a “commitment problem.” Let
Z v~
Z v~
v
inf v 2 [v; 1] s.t.
sdF (s)
[1 F (s)]ds 0; for all v~ 2 [v; 1] ;
v
v
then for the same reasons as in the case without sequential rationality constraints, discussed
in Proposition 2, we would like to set p~ equal to its lowest possible value for the types
v 2 [v; v ); which is now r + (1 r) ; and set it equal to its largest possible value, which is
1, for the region where the virtual valuation is on average positive, that is on [v ; 1]: The
resulting allocation rule is
p~(v) = r for v 2 [0; z2 (F2 ))
p~(v) = r + (1
r) for v 2 [z2 (F2 ); v )
p~(v) = 1 for v 2 [v ; 1]:
Now because v
v; from Lemma (
2 we have that z2 (v) z2 (v ); where z2 (v ) is optimal
F (v)
F
(v ) ; for v 2 [0; v ] . Moreover, since we assumed
at T = 2 given a posterior F2 (v) =
0 otherwise
that z2 (F2 ) z2 (v), we also have z2 (F2 ) z2 (v ). Hence the resulting allocation rule is
18
an element of P . Moreover, by construction it generates higher revenue for the seller than
p.
So far we have assumed that 0 < z2 (F2 ) < v: If 0 = z2 (F2 ) < v all previous arguments go
though since for for v 2 (z2 (F2 ); v) we have that p(v) = r + (1 r) : Now if z2 (F2 ) = v it is
possible that p(v) < r+(1 r) : Again, all our arguments go through as before, since it turns
out p(v + ") r + (1 r) ; for " > 0; arbitrarily small. To see that p(v + ") r + (1 r) ;
we argue by contradiction. First, observe that type v makes zero surplus at t = 2 since
z2 (F2 ) = v; hence p(v)v x(v) = rv z = [r + (1 r) )]v [z + (1 r) z2 (F2 )]: Now
let us consider type v + "; and suppose that p(v + ") < r + (1 r) ; then since at a
P BE the buyer’s strategy is a best response, it follows that p(v + ")(v + ") x(v + ")
[r + (1 r) ](v + ") [z + (1 r) z2 (F2 )]; but since p(v + ") < r + (1 r) we have that
p(v + ")v x(v + ") > [r + (1 r) ]v [z + (1 r) z2 (F2 )]: This implies that there exists an
interval of types (~
v ; v] such that p(v + ")v x(v + ") > [r + (1 r) ]v [z + (1 r) z2 (F2 )]
rv z for v 2 (~
v ; v]; contradicting the fact that [0; v] is the convex hull of types that choose
s with strictly positive probability at t = 1. Hence p(v + ") r + (1 r) and all arguments
are as in the case of z2 (F2 ) < v:
Let us recap the arguments used to establish Proposition 6. We started with an allocation rule p 2 P where z2 (F2 )
z2 (v): Talking as given the shape of p for [0; v); we
optimally choose p~(v) for v 2 [v; 1] ignoring all sequential rationality constraints and imposing only the requirement that the resulting allocation rule be monotonic on [0; 1]. Using
the assumption that z2 (F2 ) z2 (v) and Lemma 2, we establish that p~ is an element of P :
From Proposition 5 and Proposition 6 we can then conclude that:
Proposition 7 If for all p 2 P we have that z2 (F2 ) z2 (v); then under non-commitment
the seller maximizes expected revenue by posting a price in each period.
Step 3: For all p 2 P that are P BE implementable it holds z2 (F2 )
z2 (v)
The set P contains allocation rules that satisfy minimal requirements of being implemented by a P BE, since in establishing Proposition 4, we have required the seller to behave
optimally at t = 2 only after the history where the buyer choose s at t = 1: Therefore,
strictly speaking, P is a superset of P BE implementable allocation rules. Here we study
further restrictions of P BE-implementability, and establish that for all p 2 P that are
P BE implementable we have z2 (F2 ) z2 (v): Then our “quasi”solution of Step 2 is a real
solution. The following Proposition essentially establishes that the seller cannot bene…t
from “sophisticated” strategies of the buyer. The proof is lengthy so we give an overview
here, and place the details in the Appendix.
Proposition 8 For all p 2 P that are P BE implementable we have z2 (F2 )
z2 (v):
Overview of Proof: Fix an assessment that is a P BE: From Proposition 4 we know
that it implements an allocation rule in P. Recall that s denotes an action that leads to the
19
contract (r; z); the contract with the smallest r; among all contracts lead by actions weakly
preferred by type 0 at t = 1 at the P BE under consideration: Recall also that we use [0; v];
v
1; to denote the convex hull of types that choose action s; and F2 denote the seller’s
posterior at t = 2 after she observes action s at t = 1 and no trade takes place. Since T = 2
is the …nal period of the game, an optimal mechanism derived in Proposition 2 is to post a
price, which we denote by z2 (F2 ):
Clearly if all types in [0; v] choose s with probability one, then the posterior is (5) and it
is immediate that z2 (F2 ) z2 (v): The only possibility then that z2 (F2 ) > z2 (v); is when a
strictly positive measure of types in [0; v] choose an action, other then action s; with strictly
positive probability. Types in [0; v] can be choosing with strictly positive probability an
action s^ that leads to the same contract as action s; namely (r; z); and/or can be choosing
an action s~ that leads to a contract (~
r; z~); which is di¤erent from (r; z):
For v 2 [0; v] let m(v) 2 [0; 1] denote the probability that type v is choosing at t = 1
actions that lead to contract (r; z): This implies that with probability (1 m(v)) type v
is choosing actions that lead to a contract di¤erent from (r; z) . We assume that m is a
measurable function of v; with m(v) 2 [0; 1] for all v 2 [0; v]: Now, for simplicity suppose
that other than s; there is only one more action that leads to (r; z); call it s^:19 A fraction
(v) of m(v) type v chooses action s and a fraction 1
(v) of m(v) chooses action s^:
We assume that is a measurable function of v; with (v) 2 [0; 1] for all v 2 [0; v]: Thus
the seller’s posteriors at the beginning of the
period of the game after observing s;
( R…nal
s
(t)m(t)dF (t)
R0v
; for v 2 [0; v]
(t)m(t)dF (t)
and respectively s^, are given by F2 (s) =
; and F^2 (s) =
0
0; otherwise
( R s (1 (t))m(t)dF (t)
R0v
; for v 2 [0; v]
(t))m(t)dF (t)
. Since both actions are chosen with strictly positive
0 (1
0; otherwise
Rv
Rv
probability, we have that 0 (t)m(t)dF (t) > 0 and 0 (1
(t))m(t)dF (t) > 0: When all
types that choose either s or s^; all coordinate on one
of
the
two,
( R s m(t)dF (t) say s, the posterior after
R0v
for v 2 [0; v]
the seller observes s at t = 1 is given by F2m (s) =
:
0 m(t)dF (t)
0; otherwise
Let z2 (F2 ); respectively z2 (F^2 ) and z2 (F2m ); solve (2) given posterior F2 ; respectively F^2
and F2m : In terms of t = 1 allocation, actions s and s^ are equivalent since they lead to the
same contract (r; z), but they may lead to di¤erent t = 2 options since it is possible that
z2 (F2 ) 6= z2 (F^2 ):
In the …rst step of our proof we show that this is impossible; in order for both actions
s and s^ to be chosen with strictly positive probability at t = 1, they must lead to the same
t = 2 options, namely z2 (F2 ) = z2 (F^2 ): We also show that z2 (F2m ) = z2 (F^2 ) = z2 (F2 ): In
the second and …nal step of our proof, we show that z2 (F2 )
z2 (v): The details can be
found in the Appendix.
19
The case where more than two actions lead to (r; z) can be handled analogously.
20
Proposition 8 is a key step in our characterization: complicated strategies or mixed
strategies of the buyer do not pay, in the sense that the set of possible posterior beliefs
induced by such strategies does not allow the seller to support higher prices at t = 2;
compared to the case where convex set of types choose the same action with probability
one: We are now ready to state our result.
Theorem 1 Under non-commitment the seller maximizes expected revenue by posting a
price in each period.
Proof. This follows from Proposition 7 and Proposition 8.
Intuitively, the reason why simply posting a price is optimal, is that proposing a mechanism at t = 1 with more options has higher cost than bene…t. The potential bene…t from
o¤ering a mechanism with many options, is that it may allow for more possibilities for
types to self-select at t = 1; which in turn, in case that no trade takes place at period t = 1;
provides the seller at t = 2 with more precise information about the buyer’s valuation: The
cost is that by providing more possibilities to self-select also increases the possibilities to
masquerade as another type. It turns out, that the cost of having higher types behaving as if
they are lower ones, is greater than the gain obtained from having more precise information
at t = 2. We proceed to compare what is the revenue loss for the seller, given her inability
to commit.
7. Commitment and Non-Commitment: Revenue Comparisons
In this section we compare expected revenue for the seller when she employs a revenue
maximizing mechanism with and without commitment. Without commitment the set of
feasible allocation and payment rules must satisfy additional constraints, since we require
the seller and the buyer to behave sequentially rationally. From this observation it follows
that in general RC RN C ( ); where RC and RN C denote the highest revenue that the seller
can achieve with and without commitment respectively. The di¤erence in RC and RN C ( )
depends on the discount factor. When the buyer and the seller are very patient, (in this
model the buyer and the seller have the same discount factor), the seller will …nd it bene…cial
to postpone all trade to the last period of the game where she has commitment power. It
follows that when = 1 expected revenue with and without commitment coincide. The
same is true for = 0, which is essentially as if the game lasts one period. The seller posts
at t = 1 the revenue maximizing price as in the environment with commitment; therefore
RN C (0) = RC . For intermediate values of the discount factor it holds that RN C < RC :
To get some idea about the magnitude of the di¤erence in expected revenue we present an
example.
Example 1 Suppose that T = 2 and that v is uniformly distributed on the interval [0,1].
The optimal mechanism with commitment is to post a price v = 0:5 in each period, and the
21
corresponding expected revenue is RC = 0:25: Now let us determine RN C ( ). Our theorem
tells us that the seller will maximize revenue by posting a price in each period. Let v denote
the valuation of the buyer who is indi¤ erent between accepting the price at t = 1; denoted
by z1 ; and accepting the price at t = 2; denoted by z2 (v); that is v z1 = v
z2 (v)
z1 z2 (v)
or v = 1
: For the assumed prior we have that, if the buyer rejects the price o¤ er at
(
v
v ; for v 2 [0; v] ; and the price posted at t = 2 is given by z (v) = v :
t = 1; then F2 (v) =
2
2
0; otherwise
Substituting this expression of z2 (v) into v we get that z1 = v (1 0:5 ) : Given the above
relationship between z1 ; v and z2 (v) the seller will pick
v 2 arg max(1
v)v (1
0: 5 ) +
v
v v
:
2 2
The following table gives the solution for di¤ erent values of the discount factor.
Discount Factor
0
0:3
0:5
0:7
0:9
1
Price at t=1, z1 Price at t=2, z2 (v)
0:5
0:5
0:46
0:27
0:45
0:3
0:44
0:34
0:46
0:42
0:5
0:5
v
0:5
0:54
0:6
0:68
0:84
1
RN C
0:25
0:23
0:225
0:222
0:232
0:25
8. Generality and Robustness of the Result
Our objective has been to characterize a revenue maximizing P BE; in a game where the
seller can choose a di¤erent mechanism in each period. The set of P BE-implementable
social choice functions depends (i) on the generality of the buyer’s strategy (ii) on the
generality of mechanisms that the seller employs (iii) on the degree of transparency of
mechanisms, (or degree of commitment), and …nally (iv) on the length of the time horizon.
Regarding the generality of the strategy that the buyer maybe employing, we have not
imposed any restrictions. The buyer may be employing mixed strategies and non-convex
sets of types may be choosing the same actions.
Regarding the generality of de…nition of mechanisms we have assumed that a mechanism
consists of a deterministic game form. The same is true in Bester and Strausz (2001)20 and
in the earlier literature from Hart and Tirole (1988) to Rey and Salanie (1996).
With respect to the degree of transparency, in this paper the principal observes the
actions chosen by the agent, namely s. This is also assumed by Bester and Strausz (2001)
20
In BS (2001) the agent reports m and the outcome is x(m) in a set of decisions X: In our terminology
m is s; x(m) is g(s) and the set X is [0; 1] R:
22
and by the earlier literature. In this case, the assumption that game forms are deterministic
is without any loss.21 A recent paper Bester and Strausz (2004) studies mechanism design
with limited commitment assuming that the mechanism designer does not observe s and
allowing for the game form to be random. When the mechanism designer does not observe
s; whether the game form is random or not, is crucial for the set of outcomes. In some sense,
the ability to employ random devices, together with the ability to commit not to observe
the action that the agent chooses, (or in the terminology of BS (2004), the message that the
agent reports), increases the commitment power of the mechanism designer.22 BS (2004)
show that in this case, the principal can do better compared to the case where she observes
s: In the “Extensions on Sequentially Optimal Mechanisms” we establish that our result is
robust to the case that the seller does not observe s; as in BS (2004), but in addition she
does not observe g(s); 23 and observes only whether trade took place or not. In this case,
the mechanism designer has more commitment than BS (2004), because she does not even
observe g(s). The optimality of posted prices is therefore robust to alternative assumptions
regarding the degree of commitment of the mechanism designer.24
Finally, in the “Extensions on Sequentially Optimal Mechanisms”one can also …nd the
complete analysis for T > 2: The main structure of the proof for the T = 2 case extends to
longer …nite horizons, but some of the details of the arguments become more involved.
9. Concluding Remarks
This paper establishes that a revenue maximizing allocation mechanism in a T -period model
under non-commitment is to post a price in each period. It also develops a procedure to
…nd optimal mechanisms under non-commitment in asymmetric information environments.
This method does not rely on the revelation principle.
Previous work has assumed that the seller’s strategy is to post a price and the problem
21
Suppose that given an action s, g(s) is random, and with probability the outcome is (r; z) and with
probability (1 ) the outcome is (^
r; z^): Given that the seller observes s her beliefs about the buyer’s valuation
will be the same irrespective of whether the outcome of the game form is (r; z) or (^
r; z^) - in other words the
second period price will be simply a function of s; call it z2 (s): Types below z2 (s) reject this price at t = 2;
hence for those types the expected outcome from choosing s is p = r + (1
)^
r and x = z + (1
)^
z . Now
types above z2 (s) accept the price at t = 2 and for those types the expected outcome from choosing s is given
by p = (r+(1 r) )+(1
)(^
r +(1 r^) ) and x = (z+(1 r) z2 (s))+(1
)(^
z +(1 r^) z2 (s)); which can be
also rewritten as p = r +(1
)^
r+ [1
r (1
)^
r)] and x = z +(1
)^
z + [1
r (1
)^
r)] z2 (s).
These outcomes can arise by a deterministic mapping that maps s to r~ = r + (1
)^
r and z~ = z + (1
)^
z:
22
Recall that if the mechanism designer can commit not to observe anything, the commitment solution is
sequentially rational.
23
Recall that g(s) is a probability of trade r and an expected payment z:
24
The result is also robust to the case where the mechanism is de…ned as a game form endowed with a
communication device, mediator, so long as (1) either the seller observes the exchange of messages between
the mediator and the buyer or (2) the mediator cannot force actions to the buyer. Details available from
the author upon request.
23
of the seller is to …nd what price to post. We provide a reason for the seller’s choice to post
a price, even though she can use in…nitely many other possible institutions: posted price
selling is an optimal strategy in the sense that it maximizes the seller’s revenue. We hope
that the methodology developed in this paper will prove useful in deriving optimal dynamic
incentive schemes under non-commitment in other asymmetric information environments.
In the future we plan to study the problem in an in…nite-horizon framework, which
may be a more appropriate model to study mechanism design without commitment. This
problem is involved with issues which require careful analysis beyond the scope of this paper.
10. Appendix
Proof of Proposition 1
Let pA ,xA denote a solution of Program A and pB ,xB denote a solution of Program B.
Let R(pA ); (respectively R(pB )); denote the expected maximized revenue at a solution of
Program A, (respectively Program B). A priori it is not obvious, how R(pA ) and R(pB )
compare to each other, since a solution of Program A is an allocation and a payment rule
de…ned on V; whereas a solution of Program B is an allocation and a payment rule de…ned
on [0; 1]: We …rst reinterpret Program A in an appropriate way to make its value comparable
to the value of Program B, and then show that R(pA ) = R(pB ):
Step 1: Make R(pA ) and R(pB ) comparable.
In order to make R(pA ) and R(pB ) comparable, we slightly modify Program A, by
de…ning pA and xA on an “extended type space,”namely the convex hull of V; which is [0; 1].
So long as the value of pA and xA for a type in [0; 1]nV is an element of fpA (v); xA (v)gv2V ;
it is …ne for our purposes. The “extended”allocation and payment rule still have to satisfy
IC, P C and RES only on V: Essentially the only thing that has changed is the region of
integration in the expression of objective function, but since dF is zero on [0; 1]nV , we have
that
Z
Z
1
xA (v)dF (v) =
xA (v)dF (v):
V
0
Clearly this program is equivalent to Program A. Hence, from now on, when we refer to
Program A we mean its reinterpretation.
Step 2: We show that R(pA ) = R(pB ).
A solution of Program A has to satisfy IC, P C and RES on V: A solution of Program
B has to satisfy all these constraints on the extended type space, that is on [0; 1]. Since
Program B has more constraints, it follows that
R(pA )
R(pB ):
R(pA )
R(pB ):
We will be done if we establish that
24
(8)
We argue by contradiction. Suppose not, then
R(pA ) > R(pB ):
(9)
Now consider the allocation and payment rule de…ned as follows. For v 2 V we set
pA (v) = pA (v); and xA (v) = xA (v)
and for v 2 [0; 1]nV we do the following.25 Suppose that (vL ; vH ) is an interval in [0; 1]nV
and let v^ 2 (vL ; vH ) denote the type that is indi¤erent between choosing the actions speci…ed
for vL and the actions speci…ed for vH ; that is
pA (vH )^
v
xA (vH ) = pA (vL )^
v
xA (vL ):
(10)
Such a type exists, since for vL we have that
pA (vL )vL
xA (vL )
pA (vH )vL
xA (vH )
pA (vH )vH
xA (vH )
pA (vL )vH
xA (vL );
and for vH we have that
then by continuity, there exists v^ 2 (vL ; vH ) such that (10) holds.
On (vL ; vH ), pA and xA are de…ned as
pA (v) = pA (vL ) and xA (v) = xA (vL ) if v 2 (vL ; v^)
(11)
pA (v) = pA (vH ) and xA (v) = xA (vH ) if v 2 [^
v ; vH ):
Proceed similarly for all subintervals of types that are not contained in V: Now since dF (v) =
0 for v 2 [0; 1]nV we have that
R(pA ) = R(pA ):
(12)
From (9) and (12) we obtain that
R(pA ) > R(pB ):
We establish that pA and xA satisfy all the constraints of Program B. First, from (11) one
can see that from the buyer’s perspectives no ‘real’new options have been added since the
menu fpA (v); xA (v)gv2[0;1] is exactly the same as fpA (v); xA (v)gv2V : Then using the fact
that pA and xA satisfy IC on V; equality (10), and the observation that no real options have
been added, it follows that pA ; xA satisfy IC on V . Now to check IC on [0; 1]; consider types
in (vL ; vH ): From the fact that pA ; xA satisfy IC on V; we have that pA (vH ) pA (vL ): Then
25
It is without loss to take V to be closed. Otherwise, we can de…ne pA and xA on the closure of V by
setting pA and xA equal to their limit values. It is then trivial to establish that the resulting pA and xA
are feasible on the closure of V:
25
we know from (10) that all types below v^ prefer (pA (vL ); xA (vL )) and all types above v^ prefer
(pA (vH ); xA (vH )): By the de…nition of pA ; xA it then follows that deviations within (vL ; vH )
are not pro…table. Now from the fact that pA and xA satisfy IC on V we can conclude
that type vL prefers outcome (pA (vL ); xA (vL )) to all other options in fpA (v); xA (v)gv2V
and type vH prefers outcome (pA (vH ); xA (vH )) to all other options in fpA (v); xA (v)gv2V :
Then using (10) it is very easy to see that no type in v 2 (vL ; vH ) can pro…tably deviate by
behaving like a type in [0; 1]n(vL ; vH ). Similarly, we can check that IC is satis…ed for all
v 2 [0; 1]nV: It is also immediate that pA and xA satisfy voluntary participation constraints
and resource constraints.
Hence pA ; xA are feasible for Program B and moreover generate strictly higher revenue
then pB , contradicting the optimality of pB and xB . It follows that the maximized values
of Programs A and B are the same.
Proof of Proposition 4
Consider a P BE assessment ( ; ) and let p; x denote the allocation and payment rules
implemented by it. Point (i) in obvious. All P BE 0 s are BN E 0 s hence p and x satisfy
Lemma 1.
To see (ii) let s denote an action that leads to contract (r; z): This is the contract with
the smallest r; among all contracts lead by actions weakly preferred by type 0 at t = 1 at
the P BE under consideration: Also let Y denote the set of types of the buyer that choose s
at t = 1 with strictly positive probability, and let [0; v]; with 0 v; denote its convex hull.
From Proposition 2, we have that after the history that the buyer chose action s; and no
trade took place at t = 1; the seller will maximize revenue at t = 2 by posting a price. Let
us call this price as z2 (F2 ) and de…ne
vL = inf fv 2 Y s.t. v accepts z2 (F2 ) at t = 2g :
First we show that vL = z2 (F2 ); then we establish that for v 2 (vL ; v) we have that
p(v) = r + (1 r) . Finally, we show that for v 2 [0; vL ) we have that p(v) = r:
Step 1: We show that vL = z2 (F2 ): At a P BE the buyer’s strategy must be a best
response to the seller’s strategy. This implies that
[r + (1
r) ]vL
[z + (1
r) z2 (F2 )]
rvL
z:
We now show that this inequality must hold with equality. We argue by contradiction.
Suppose that [r + (1 r) ]vL [z + (1 r) z2 (F2 )] > rvL z; then the seller can increase
z2 (F2 ) by z such that
[r + (1
r) ]vL
[z + (1
r) z2 (F2 )]
z = rvL
z
and raise higher revenue at the continuation game that starts at t = 2: All types v vL still
prefer to choose (1; z2 (F2 )) at t = 2 then to choose (0; 0): Hence the seller has a pro…table
26
deviation at t = 2; contradicting the fact that we are looking at a P BE: At a P BE we
therefore have that
[r + (1
r) ]vL
[z + (1
r) z2 (F2 )] = rvL
z;
(13)
from which it is immediate that vL = z2 (F2 ):
Step 2: We claim that for v 2 (z2 (F2 ); v); where z2 (F2 ) < v; we have that p(v) =
r + (1 r) : We will establish this result by observing that if a v 2 (z2 (F2 ); v) is choosing a
sequence of actions with positive probability, then it must be that the expected discounted
probability from these alternative actions, p^; is such that p^ = r + (1 r) : Otherwise
depending on whether p^ < r + (1 r) or p^ r + (1 r) ; either a type smaller than v,
say z2 (F2 ); or greater than v; say v; has a pro…table deviation. Take, for instance, the case
that p^ > r + (1 r) : Then, for v it must hold p^v x
^ [r + (1 r) ]v [z + (1 r) z2 (F2 )];
which implies together with (13), that all types in (v; v) strictly prefer an action di¤erent
from s; contradicting the fact that [0; v] is the convex hull of the set of types that choose s:
Step 3: For v 2 [0; z2 (F2 )); z2 (F2 ) 6= 0 we have that p(v) = r: Again if a v 2 (0; z2 (F2 )) is
choosing a sequence of actions with positive probability, then it must be that the expected
discounted probability from these alternative actions p^; is such that p^ = r: Otherwise
depending on whether p^ < r or p^ > r either a type smaller than v, say 0; or greater than
v; say z2 (F2 ); has a pro…table deviation. Suppose that p^ < r; then for type v it must be
p^v x
^ rv z, but then 0 has a pro…table deviation since p^0 x
^ > r0 z; contradicting
the fact that action s is weakly preferred by type 0: The fact that p(0) = r follows from the
fact that s induces the contract with the smallest r among all actions weakly preferred by
type 0, so that p(0) cannot be less than r.
Summing up, if 0 < z2 (F2 ) < v; from Step 2 we have that p(v) = r + (1 r) for
v 2 (z2 (F2 ); v); and from Step 3 we have that p(v) = r; for v 2 [0; z2 (F2 )). Now if z2 (F2 ) = v;
then p(v) 2 [r; 1]; (since p(z2 (F2 )) 2 [r; r + (1 r) ] and p(v) 2 [r + (1 r) ; 1]). It follows
that if an allocation rule is implemented by an assessment that is a P BE it is given by
(4).
Proof of Proposition 5
Consider an assessment that implements an element of P : The seller’s expected revenue
is given by
Z z2
Z z2
R(p) =
rsdF (s)
r[1 F (s)]ds
0
0
Z v
Z v
+
[r + (1 r) ]sdF (s)
[r + (1 r) ][1 F (s)]ds
z2
1
+
Z
v
sdF (s)
Z
z2
1
[1
F (s)]ds:
v
In order to …nd an optimal allocation rule out of P we have to choose (i) z2 ; (ii) r and (iii)
v optimally.
27
Step 1: At an optimum z2 = z2 (v):
From the de…nition of an allocation rule in P we have that z2 must be less or equal to
z2 (v): We now establish that at an optimum of P it will be z2 = z2 (v): Since z2 (v) solves
(2) given beliefs (5) we have that for all z2 < z2 (v)
Z z2 (v)
Z z2 (v)
F (s)
dF (s)
[1
]ds < 0,
s
F (v)
F (v)
z2
z2
which since F (v) > 0 implies that
Z z2 (v)
sdF (s)
z2
from which it is immediate that
Z z2 (v)
[r + (1 r) ]sdF (s)
z2
Z
z2 (v)
[F (v)
F (s)]ds < 0,
z2
Z
z2 (v)
[r + (1
r) ][1
F (s)]ds < 0.
z2
It follows that at an optimum z2 = z2 (v):
Step 2: At an optimum r = 0 or r = 1:
Di¤erentiating with respect to r we get that
Z z2 (v)
Z z2 (v)
@R(p)
=
sdF (s)
[1 F (s)]ds
@r
0
0
Z v
Z v
+
(1
)sdF (s)
(1
)[1
z2 (v)
F (s)]ds;
z2 (v)
which does not depend on r. If @R(p)
0; then at an optimum it must be r = 1 for
@r
@R(p)
all v 2 [0; 1]: If @r < 0 then at an optimum it must be r = 0; and the participation
constraints imply that z = 0:
Step 3: Optimal v determines optimal sequence of prices.
From Step 1 and 2 we have that an optimal allocation rule of P can be implemented by
an assessment where the seller proposes a mechanism with two contracts (0; 0) and (1; z1 );
where z1 denotes the price that the seller must post at t = 1 to keep type v; indi¤erent
between z1 at t = 1 and z2 (v) at t = 2; that is v z1 = (v z2 (v)) or v = z1 1 z2 (v) :
Types [0; v] of the buyer choose (0; 0) at t = 1; whereas types (v; 1] choose (1; z1 ): Each
v 2 [0; 1] determines z1 at t = 1 via v = z1 1 z2 (v) and the optimal price at t = 2; z2 (v);
2)
via maxz2 2[0;v] [1 FF(z(v)
]z2 : In order to …nd an optimal allocation rule out of P ; the seller
chooses v optimally. It is very easy to see that this is a P BE of the game where the seller
posts a price in each period. This completes the proof.
Proof of Lemma 2
We argue by contradiction. Suppose that v^ > v but z2 (^
v ) < z2 (v); then
[F (^
v)
F (v)]z2 (v) > [F (^
v)
28
F (v)]z2 (^
v ):
(14)
Since z2 (v) is the optimal price given beliefs (5) it solves z2 (v) maxz2 2[0;v] [1
F (z2 (v))
F (v) ]z2 (v)
follows that [1
[F (v)
[1
F (z2 (v))]z2 (v)
1
F (^
v)
Multiplying (14) and (15) by
>
F (z2 (^
v ))
v)
F (v) ]z2 (^
1
[(F (^
v)
F (^
v)
1
[(F (^
v)
F (^
v)
F (z2 )
F (v) ]z2 :
It
or
[F (v)
F (z2 (^
v ))]z2 (^
v ):
(15)
and adding them up we get
F (v)) z2 (v) + (F (v)
F (z2 (v))) z2 (v)]
F (v)) z2 (^
v ) + (F (v)
F (z2 (^
v ))) z2 (^
v )] :
(16)
Now (16) can be rewritten as,
1
[F (^
v)
F (^
v)
F (z2 (v))z2 (v)] >
1
[F (^
v)
F (^
v)
F (z2 (^
v ))z2 (^
v )] ;
contradicting the de…nition of z2 (^
v ):
Proof of Proposition 8
Step 1: We establish z2 (F2m ) = z2 (F^2 ) = z2 (F2 )
Step 1.1: First we show that z2 (F2 ) = z2 (F^2 ):
Suppose not, and without any loss, assume that z2 (F2 ) < z2 (F^2 ): Then for all v 2 V we
have that
[r + (1
r) ]v
[z + (1
r) z2 (F2 )] > [r + (1
r) ]v
[z + (1
r) z2 (F^2 )];
hence for all v z2 (F2 ) the buyer strictly prefers s to s^: But then when the seller sees s^;
she can infer that the valuation of the buyer is below z2 (F2 ); which in turn implies that
a price of z2 (F^2 ) > z2 (F2 ) cannot be optimal. Contradiction. This completes the proof of
Step 1.1.
Step 1.2: This step establishes that z2 (F2m ) = z2 (F^2 ) = z2 (F2 ):
Assume wlog that z2 (F2m ) < z2 (F2 ): By the de…nition of z2 (F2m ); (given by (2), with F
replaced by F2m (v)); it follows that
Z z2 (F2 )
Z z2 (F2 )
Z s
m
sdF2 (s)
[1
dF2m (t)]ds 0:
(17)
z2 (F2m )
z2 (F2m )
0
R z (F2 )
R z (F2 )
But since z2 (F2m ) is not a maximizer for beliefs F2 we have that z22(F m
sdF2 (s) z22(F m
[1
)
2
2 )
nR
o
R
R
R
Rs
z2 (F2 )
z2 (F2 )
v
s
1
s
(s)m(s)dF
(s)
[
(t)m(t)dF
(t)
(t)m(t)dF
(t)]ds
<
dF
(t)]ds
<
0;
or
m
m
2
A
0
0
z2 (F )
z2 (F ) 0
0; where
Z
1
A
z2 (F2 )
z2 (F2m )
2
Rv
0
1
;
(t)m(t)dF2 (t)
s (s)m(s)dF (s)
2
which in turn implies that
Z
z2 (F2 )
z2 (F2m )
Z
[
v
(t)m(t)dF (t)
0
Z
0
29
s
(t)m(t)dF (t)]ds < 0:
(18)
Similarly, since z2 (F2m ) is not a maximizer for beliefs F^2 we have that
R z2 (F^2 )
Rs
^
0 dF2 (t)]ds < 0; which implies
z2 (F m ) [1
2
Z
Z
z2 (F^2 )
s(1
(s))m(s)dF (s)
z2 (F2m )
z2 (F^2 )
z2 (F2m )
Z v
[ (1
(t))m(t)dF (t)
0
Z
R z2 (F^2 )
^
z2 (F2m ) sdF2 (s)
s
(1
(t))m(t)dF (t)]ds < 0
0
(19)
adding (18) and (19) and recalling that z2 (F2 ) = z2 (F^2 ) we obtain that
Z
z2 (F^2 )
sm(s)dF (s)
z2 (F2m )
which since
Rv
0
Z
Z
z2 (F^2 )
z2 (F2m )
Z
[
v
m(t)dF (t)
0
Z
s
m(t)dF (t)]ds < 0;
0
m(t)dF (t) > 0 also implies that
z2 (F^2 )
z2 (F2m )
m(s)dF (s)
sR v
Z
0
m(t)dF (t)
z2 (F^2 )
z2 (F2m )
sdF2m (s)
Z
Rs
z2 (F^2 )
[1
z2 (F2m )
Z
0
z2 (F^2 )
[1
z2 (F2m )
R0v
m(t)dF (t)
]ds < 0; or
m(t)dF (t)
Z s
dF2m (t)]ds < 0;
(20)
0
But (20) contradicts (17).
Now we argue that z2 (F2m ) > z2 (F2 ) is not possible either. To see this simply, reverse
the roles of z2 (F2m ) and z2 (F2 ); z2 (F^2 ) and replace in the above inequalities “ ” with “<”
and “<”with “ ”: It follows then that z2 (F2m ) = z2 (F^2 ) = z2 (F2 ): This completes the proof
of Step 1.2.
So far we have established that z2 (F2m ) = z2 (F^2 ) = z2 (F2 ): Now, we move on to compare
z2 (F2m ) and z2 (v).
Step 2: We establish that z2 (F2 )
z2 (v)
First we …nd that only types above z2 (F2 ) can be choosing with strictly positive probability actions that lead to contracts other than (r; z): Then m(v) = 1 for all v 2 [0; z2 (F2 )):
Step 2.1: We show that m(v) = 1 for all v 2 [0; z2 (F2 )):
We argue by contradiction. Suppose that there exists v~ 2 [0; z2 (F2 )) that is choosing
with strictly positive probability an action s~ that leads to a contract (~
r; z~) for which either
r~ 6= r and/or z~ 6= z: Let z~2 denote the price that the seller will post at t = 2 after the history
that the buyer choose s~ at t = 1: Now, from Proposition 4 we know that for v 2 [0; z2 (F2 )) we
have that p(v) = r; therefore it must be the case that either (a) r = r~ or (b) r = r~ + (1 r~) :
We show that case (a) is impossible, since r~ = r implies z~ = z; which implies in turn,
that (~
r; z~) is the same contract as (r; z): To see this we argue by contradiction. Suppose
not, and wlog let z~ > z: Depending on whether z + (1 r) z2 (F2 ) < z~ + (1 r~) z~2 or
30
z + (1 r) z2 (F2 ) z~ + (1 r~) z~2 ; there are two cases to consider: If z + (1 r) z2 (F2 ) <
z~ +(1 r~) z~2 then, since we also have that z < z~; all types of the buyer prefer (r; z) to (~
r; z~):
The reason is that the probability of obtaining the object is always the same under these
two options, r = r~ and of course r + (1 r) = r~ + (1 r~) ; but payments are always lower
at contract (r; z): But then the types that are choosing with strictly positive probability
action s~, which leads to (~
r; z~); are not best-responding, contradicting the supposition that
we are looking at a P BE: Now in the case that z + (1 r) z2 (F2 ) z~ + (1 r~) z~2 ; then
all types above z2 (F2 ) prefer (~
r; z~) over (r; z): This implies that the seller posts at t = 2 a
price z~2 strictly lower than z2 (F2 ) given a posterior with support [z2 (F2 ); v]: This cannot
be optimal, which contradicts the fact that we are looking at P BE: Hence r~ = r implies
that z~ = z.
We now show that (b) is impossible. First if r = r~ + (1 r~) then it must be the
case that z = z~ + (1 r~) z^2 : If z > z~ + (1 r~) z~2 than no type below z2 (F2 ) would
choose s; contradicting the fact that [0; v] is the convex hull of types that choose s: If
z < z~ + (1 r~) z~2 ; than no type would choose the sequence of actions that leading to
r~ + (1 r~) ; and z~ + (1 r~) z~2 : Hence it must be z = z~ + (1 r~) z~2 : Now since v~ is choosing
(~
r; z~) at t = 1 and accepts (1; z~2 ) at t = 2; it must be the case that z~2 v~: For type z~2 it
holds that r~z~2 z~ = [~
r + (1 r~) ]~
z2 [~
z + (1 r~) z~2 ] = r~
z2 z: But since 0 z~2 and r~ r
we have that r~0 z~
r0 z contradicting the de…nition of (r; z); which is the contract
with the smallest r that is weakly preferred by type 0 at t = 1. This completes the proof
of Step 2.1.
From Step 2.1 it follows that m(v) = 1 for v 2 [0; z2 (F2 )): Then, it is immediate that
F2m (v) =
8
>
>
>
<
>
>
>
:
F (z2 (F2 ))+
F (v)
Rv
F (z2 (F2 ))+
F (z2 (F2 ))+
z2 (F2 )
Rv
z (F2
R v2
m(s)dF (s)
; v 2 [0; z2 (F2 ))
;
) m(s)dF (s)
z2 (F2 )
m(s)dF (s)
(21)
; v 2 [z2 (F2 ); v]
Rv
where F (z2 (F2 )) + z2 (F2 ) m(s)dF (s) > 0 since action s is chosen by strictly positive probability.
Now that we have more information about F2m , and we know that z2 (F2 ) = z2 (F2m ); we
investigate whether it is possible to have z2 (F2 ) > z2 (v):
Step 2.2: We show that z2 (F2 )
z2 (v):
Case 1: z2 (F2 ) = 0
It is then immediate that z2 (F2 ) z2 (v):
Case 2: z2 (F2 ) > 0
Since z2 (v) is the optimal price at T = 2; it solves (2) for beliefs given by (5). From
the previous step we have that z2 (F2 ) = z2 (F2m ), where z2 (F2m ) solves (2), for (21). The
31
relevant expressions are26
1
= F (v)
F (v)
F
v; v~
F (v)
Z
Z
v~
sdF (s)
v
v~
(F (v)
F (t))dt ;
(22)
v
and
(v; v~ jF2m )
"
= F (z2 (F2 )) +
Z
#
v
m(t)dF (t)
z2 (F2 )
Z
v~
v
Since m(v) = 1 for v; v~ 2 [0; z2 (F2 )); we obtain that
"Z
Z
Z v~
v~
m
(v; v~ jF2 ) =
tdF (t)
F (z2 (F2 )) +
v
v
sdF2m (s)
Z
v~
F2m (s)]ds :
[1
v
v
m(s)dF (s)
!
#
F (t) dt :
z2 (F2 )
F
F (v)
more easily comparable with (v; v~ jF2m ), let us do some rewriting.
R v~
By adding and subtracting v F (z2 (F2 ))dt to
v; v~ FF(v) we get that
To make
v; v~
F
v; v~
F (v)
=
Z
Z
v~
tdF (t)
v
v~
(F (z2 (F2 )) + F (v)
F (z2 (F2 ))
F (t)) dt:
v
Hence for v; v~ 2 [0; z2 (F2 )) (v; v~ jF2m ) and
v; v~
F
F (v)
di¤er by a constant
Rv
z2 (F2 ) m(s)dF (s)
in the case of (v; v~ jF2m ); versus F (v) F (z2 (F2 )) in the case of
v; v~ FF(v) . Now because m(v) 2 [0; 1] and F
0; we have that F (z2 (F2 )) + F (v) F (z2 (F2 )) F (z2 (F2 )) +
Rv
z2 (F2 ) m(t)dF (t) which implies that
v; v~
F
F (v)
(v; v~ jFTm ); for all v; v~ 2 [0; z2 (F2 )):
(23)
Suppose that z2 (v) < z2 (F2 ); then by the de…nition of z2 (v); it follows that
z2 (v); v~
F
F (v)
0; for all v~ 2 [z2 (v); z2 (F2 )];
which together with (23) implies that
(z2 (v); v~ jF2m )
0; for all v~ 2 [z2 (v); z2 (F2 )];
(24)
and by the de…nition of z2 (F2 ) we have that
(z2 (F2 ); v~ jF2m )
0; for all v~ 2 [z2 (F2 ); v];
Rv
26
Multiplying by a constant, F (v), in the case of (5) and by F (z2 ) +
does not change anything, because we care about the sign of the integral.
32
z2
(25)
m(t)dF (t) in the case of (21);
but (24) and (25) imply
(z2 (v); v~ jF2m )
0; for all v~ 2 [z2 (v); v];
contradicting the de…nition of z2 (F2 ): Hence we have shown that z2 (v) z2 (F2 ):
The reason why this is true follows from the fact that only types greater than z2 (F2 )
maybe be choosing with positive probability an action that leads to a contract other then
(r; z). To put it very roughly, (21) puts less weight on the higher types of [0; v] compared
to (5): This completes the proof of Step 2.2.
References
[1] Bester, H., and R. Strausz (2001): “Contracting with Imperfect Commitment
and the Revelation Principle: The Single Agent Case.” Econometrica, 69, 1077-1098.
[2] Bester H. and R. Strausz (2004): “Contracting with Imperfect Commitment and
Noisy Communication,” mimeo.
[3] Bulow, J. (1982): “Durable Goods Monopolists,” Journal of Political Economy, 90,
314-322.
[4] Coase, R. (1972): “Durability and Monopoly,” Journal of Law and Economics, 15,
143-149.
[5] Freixas, X., R. Guesnerie, and J. Tirole (1985): “Planning under Incomplete
Information and the Ratchet E¤ect,” Review of Economic Studies, 52, 173-192.
[6] Fudenberg, D., D. Levine, and J. Tirole (1985): “In…nite-horizon Models of
Bargaining with One-sided Incomplete Information,” In ‘Game Theoretic Models of
Bargaining’, edited by Alvin Roth. Cambridge University Press.
[7] Gul, F., Sonnenschein, H., and R. Wilson (1986): “Foundations of Dynamic
Monopoly and the Coase Conjecture,” Journal of Economic Theory 39, 155-190.
[8] Hart, O., and J. Tirole (1988): “Contract Renegotiation and Coasian Dynamics,”
Review of Economic Studies, 55, 509-540.
[9] Laffont,J.J., and J. Tirole (1988): “The Dynamics of Incentive Contracts,”
Econometrica, 56, 1153-1175.
[10] Laffont,J.J., and J. Tirole (1993): A Theory of Incentives in Procurement and
Regulation, MIT Press.
[11] McAfee, R. P. and D. Vincent (1997): “Sequentially Optimal Auctions,” Games
and Economic Behavior, 18, 246-276.
33
[12] Myerson, R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6, 58-73.
[13] Riley, J. G. and W. F. Samuelson (1981): “Optimal Auctions,” American Economic Review, 71, 381-392.
[14] Riley, J. and R. Zeckhauser (1983): “Optimal Selling Strategies: When to Haggle,
When to hold Firm,” Quarterly Journal of Economics, 98, 267-287.
[15] Rey, P. and B. Salanie (1996): “On the Value of Commitment in Contracting with
Asymmetric Information,” Econometrica, 64, 1395-1414.
[16] Salanie, B. (1997): The Economics of Contracts, MIT Press.
[17] Skreta, V. (2005a): “Mechanism Design for Arbitrary Type Spaces,”mimeo UCLA.
[18] Skreta, V. (2005b): “Technical Appendix for Sequentially Optimal Mechanisms,”
mimeo UCLA.
[19] Skreta, V. (2005c): “Extensions on Sequentially Optimal Mechanisms,” mimeo
UCLA.
[20] Sobel, J. and I. Takahashi (1983): “A Multi-stage Model of Bargaining,” Review
of Economic Studies, 50, 411-426.
[21] Stokey, N. (1979): “Rational Expectations and Durable Goods Pricing,”Bell Journal
of Economics, 12, 112-128.
34