RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM
PRICE AND DOUBLE AUCTIONS
IN-KOO CHO
Abstract. We analyze a broad class of auctions in which goods are traded at a single price, such as the uniform price auctions and the double auction (Satterthwaite and
Williams (1989)), while admitting the presence of both common and private value and
allowing players to trade more than one unit of the object. Instead of calculating a Nash
equilibrium, we first eliminate the weakly dominated strategies, and then, repeatedly
eliminate strictly dominated strategies among strictly increasing bidding strategies. We
show that the set of outcomes satisfying our criterion in large auctions converges to a
strategy profile, which is a natural generalization of the “truth telling” strategy in a
private value model. Because the “truth telling” strategy is an increasing function of the
private signal, the delivery price reveals the underlying common value, and the good is
delivered to someone with the highest private signal (Wilson (1977)). We offer a theoretical foundation to select a Nash equilibrium that aggregates the information, although we
do not calculate an equilibrium per se. The existing information aggregation results obtained for pure common value (e.g.,Wilson (1977) and Pesendorfer and Swinkels (1997))
and pure private value (e.g., Satterthwaite and Williams (1989), Jackson and Kremer
(2001) and Cripps and Swinkels (2003)) are robust against small perturbations of informational structure, and hold even without the full power of the equilibrium hypothesis.
Keywords. Large auctions, Private and Common Value, Information Aggregation, Rationalizability, Monotonic Bidding Strategies, Repeated elimination of weakly dominated
strategies
1. Introduction
This paper develops a general equilibrium selection criterion for large auctions by combining the weak dominance and the monotonicity of bidding strategies. We first eliminate the
weakly dominated strategies, and then, repeatedly eliminate strictly dominated strategies
from the set of all strictly increasing bidding strategies. Since the repeated elimination
of strictly dominated strategies is implied by the common knowledge of expected utility
maximization, our solution is directly implied by three pieces of common knowledge: every player is using a strictly increasing bidding function, no player uses weakly dominated
strategies, and every player maximizes von Neumann Morgenstern expected utility subject to his belief.1 This procedure is also implied by the almost common knowledge of
rationality (Dekel and Fudenberg (1990) and Borgers (1994)) in combination with the fact
that every bidder is using a strictly increasing bidding strategy.
Our approach complements, but also improves upon, the conventional equilibrium analysis in a number of important ways. First, the existence proof for outcomes surviving our
Date: July 6, 2004.
I am grateful for Eddie Dekel, Matt Jackson, Preston McAfee, Phil Reny, Jeroen Swinkels, and Steve
Williams for helpful conversations. Financial support from the National Science Foundation (SES-0004315)
is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author and do not necessarily reflect the views of the National Science Foundation.
1
I am grateful for Phil Reny for this interpretation.
1
2
IN-KOO CHO
procedure relies on the property of the nested sequence of compact sets rather than the
fixed point argument used to prove the existence of a Nash equilibrium. We can apply the
same logic to a larger class of models, including those for which we have yet to prove the
existence of a Nash equilibrium.2 Thus, our approach helps us look beyond the confines
of the equilibrium analysis.
Second, we impose a sufficiently mild restriction on the class of strictly increasing bidding strategies in order to make the symmetric Nash equilibrium strategy, if it exists,
remain a feasible strategy in symmetric auctions populated with ex ante identical bidders.
In particular, if the players are ex ante identical, then our criterion selects the symmetric
equilibrium as the most salient outcome of the model (Cho (2002)), in which the diverse
information about the value of the object is aggregated to reveal the underlying common
value (e.g., Wilson (1977) and Pesendorfer and Swinkels (1997)). The information aggregation result is a direct consequence of the rationality of the players over the set of strictly
increasing bidding strategies in combination with the elimination of weakly dominated
strategies.
Third, we obtain the information aggregation result under significantly milder requirements on the forecasting capability than we normally do in the equilibrium models. The
information aggregation properties of the competitive market are remarkable because a
decision maker needs to forecast only the market clearing price in order to achieve an
efficient allocation. In contrast, a decision maker in a Nash equilibrium of a competitive
bidding model has to perfectly foresee the equilibrium strategies of all other players. Thus,
if one obtains the information aggregation result through Nash equilibrium, it would be
natural to see whether one can obtain the same result while reducing the forecasting capability of a decision maker. We demonstrate that the information aggregation result obtains
without the full power of perfect foresight presumed under the equilibrium hypothesis.
Finally, one can interpret the monotonicity as a restriction on the beliefs and still
obtains the same result, if the space of private signals is finite. This interpretation is quite
useful in extending our result from the ex ante to the interim stage of decision making
process when each bidder knows his own type. At the interim stage, the same bidder
with a different signal is treated as a different decision maker. Thus, it is more difficult to
motivate the monotonicity of bidding strategies at the interim stage than at the ex ante
stage. Instead, suppose that each bidder believes that the other bidders are using strictly
increasing bidding strategies and this restriction on beliefs is common knowledge. We can
show that if the space of private signals is finite, then this belief is self-fulfilled when there
are many bidders: the best response against the belief that every player is using strictly
increasing bidding strategy is also strictly increasing. Then, we can invoke precisely the
same analysis to obtain the information aggregation result. Our analysis delineates the
sensible outcome in a large auction at the ex ante stage, but also at the interim stage.
We follow the lines of research pursued by Dekel and Wolinsky (2003), Battigalli and
Siniscalchi (2003) and Cho (2002) that investigate the set of rationalizable strategies in
the first price auction under different institutional and informational assumptions.3 Our
research is also closely related to Kremer (2002), and in particular, Jackson and Kremer
(2001) and Bali and Jackson (2001) that investigate the asymptotic properties of a class
of auctions by using the mechanism design approach. Our approach complements these
2
For recent progress, see Maskin and Riley (2000), Athey (2001), McAdams (2002), Jackson, Simon,
Swinkels, and Zame (2002), Reny and Zamir (2002), Fudenberg, Mobius, and Szeidl (2003), Jackson and
Swinkels (2003) and Perry and Reny (2003).
3
See Cho (1994) for an application to the bargaining problems.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS3
papers in the same way the strategic approach to the bargaining problem complements the
axiomatic approach. For example, Jackson and Kremer (2001) consider a class of auction
mechanisms that satisfy a series of “axioms”, including the symmetry of the allocation
rule. By suppressing the details of the trading rules, Jackson and Kremer (2001) obtain
very elegant results on the characterization of the outcomes for general trading rules. In
contrast, we investigate specific auctions by imposing restrictions on the strategy spaces.
While this approach reveals more about the decision making process, we end up with
potentially many equilibrium outcomes. Although we do not calculate an equilibrium per
se, our approach delineates sensible equilibria that are consistent with the fact that all
bidders are expected utility maximizers and the bidding strategies are strictly increasing.
In order to properly address the information aggregation properties of the competitive
equilibrium envisioned by Hayek (1948), we have to analyze a large multi-unit double auction populated with ex ante heterogeneous bidders under a general informational structure.
In this paper, we take first few steps toward the ultimate goal by analyzing models that
satisfy two restrictions.
First, although we examine models with some form of ex ante heterogeneity, this paper
will focus on the ex ante symmetric static auctions, in which every player has the same
utility function and the identical informational structure. The analysis of models with a
fully general form of ex ante heterogeneous players will be pursued in the ensuing projects.
Although we believe that our analytic method should apply to English auctions to identify
a sensible equilibrium, in which the bidders can update their information throughout the
auction by observing the other bidders’ move (Bikhchandani, Haile, and Riley (2002)),
the investigation of equilibrium selection in a dynamic auction remains beyond the scope
of this paper.
Second, our analysis covers the pure private value models but also those which are
“close” to the pure private or the pure common value models. We leave “intermediate” cases unexplored. Existing studies on information aggregation in double and multiunit auctions have been focused on the pure private value models (e.g., Satterthwaite
and Williams (1989), Swinkels (1999), Jackson and Kremer (2001), Cripps and Swinkels
(2003)). Our analysis demonstrates that the existing information aggregation result is robust against a small perturbation of the information structure and can hold even without
the full power of equilibrium hypothesis.4
As we use a significantly milder solution concept than Nash equilibrium, we have not
yet characterized the convergence rate of the set of rationalizable outcomes to the limit
outcome. Satterthwaite and Williams (1989) and Cripps and Swinkels (2003) demonstrated that the set of equilibrium outcomes converges to the limit outcome at the rate of
1/n, which shows the inefficiency arising from the asymmetric information vanishes very
quickly. As we move away from the equilibrium hypothesis, we drop the perfect foresight
capability from the individual players. Consequently, some of the techniques used in the
equilibrium models do not apply in our cases (e.g., Cripps and Swinkels (2003)). Still, in
the first price auctions, Cho (2002) proved that the set of rationalizable strategies (over
the restricted subset of strategy space) converges to the limit at the rate of 1/n. However, as we examine a much broader class of models populated with ex ante heterogeneous
players in which more than a single unit of object can be traded, the calculation of the
convergence rate becomes considerably more difficult.
4
We also analyze the models close to the pure common value models under the additional assumption
of the ex ante symmetric players.
4
IN-KOO CHO
In Section 2, we analyze the symmetric uniform price auctions as an extended example,
in which each buyer purchases a single unit. We formally describe the solution concept and
motivate the monotonicity restriction on the set of bidding strategies as a representation
of bounded rationality. We demonstrate that if we restrict the set of feasible strategies to
those which are strictly increasing with respect to the signal, then the set of rationalizable
strategies must converge to a bidding strategy, which is a natural generalization of the
truth telling strategy in the pure private value model. After completing the analysis for
the ex ante case, we re-interpret the restriction on the strategies as the restriction on the
beliefs to extend the result to the interim stage, where each decision maker knows his own
type but does not know the other players’ types.
In Section 3, we extend the insight obtained from analyzing the symmetric uniform
price auctions to k-double auctions (Chatterjee and Samuelson (1983) and Satterthwaite
and Williams (1989)). Although the players can influence the delivery price in k-double
auctions, the probability of doing so vanishes as the number of players increases. Thus,
the decision problem of the seller and the buyer “converges” to that in the uniform price
auction. Then, by invoking the result from the symmetric uniform price auction, we obtain
the information aggregation result.
To extend the information aggregation result to the multi-unit auctions, we have to
handle two important challenges. First, as the marginal utility from consuming an additional unit of an object changes, the bids for different units are generated from different
marginal utility functions, even though every bidder has the identical utility function.
Second, as illustrated by Ausubel and Cramton (1996) and Wilson (1979), the multiple
unit uniform price auctions are significantly different from the single unit counter parts,
as the multi-unit auctions admit inefficient Nash equilibria surviving repeated elimination
of weakly dominated strategies.
To address the first challenge, we examine, in Section 4, the uniform price auctions
in which each bidder can trade up to a single unit of the object, but bidders may have
different utility functions. Note that if the bidders can consume at most a single unit, the
utility function of a bidder is completely characterized by the marginal utility. Ex ante
heterogeneity of players influences the market outcome, only if the bidding strategies of
different players are different. Because we already obtain the key result while admitting
asymmetric bidding strategies, we can easily extend the information aggregation result to
the models with ex ante heterogeneous players.
In Section 5, we address the second challenge and examine the uniform price auctions
in which each buyer can purchase multiple units of the object. A bidder can influence the
delivery price for the objects he has already won if his bid is a pivotal bid. However, if
every bidder is using a strictly increasing bidding strategy, the probability that any bid
is a pivotal bid vanishes. In a large auction, each bidder who is bidding for multi-units
can be viewed as a “team” that has multiple agents with different (marginal) utility who
demand up to a single unit. Then, we can apply the results from Section 4 to obtain the
information aggregation result.
Throughout the paper, we focus on the auctions that are “close” to pure private value
models. Still, we can obtain the information aggregation result for the models close to
pure common values. In Section 6, we examine the asymptotic properties of auctions in
which the common value component in the utility function is sufficiently large under the
additional assumption that all bidders are ex ante identical. We develop an alternative
method of proof to obtain the information aggregation result. Section 7 concludes the
paper with a brief discussion about the future research.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS5
2. Uniform Price Auctions
2.1. Formal description. Consider an auction with n potential buyers and a seller who
has (≤ n) units of identical objects for sale. The number of objects may increase as the
number of bidders increases. We assume that
= α ∈ [0, 1]
lim
n→∞ n
is well-defined.
Each buyer demands exactly one unit. A buyer, often called a bidder, is identified by
i ∈ {1, . . . , n}. The value of the object is drawn from V = [v, v] according to distribution
function G(v). Conditioned on v, bidder type i receives a private signal si ∈ Si generated
according to a probability distribution function F (si |v).5
We impose fairly standard regularity conditions on the distribution of private signals.
Assumption 2.1. G and F are continuously differentiable over the support. Let f (si |v)
be the density function of F (si |v). For ∀i and ∀si in the support of F (·|v), F (si |v) is a
strictly decreasing continuous function of v, and F satisfies the monotonic likelihood ratio
property (MLRP). For a given v ∈ [v, v], si and sj are mutually independent for ∀i = j.
The utility function of player i is
u : Si × V → R.
By u(si , v), we mean the utility conditioned on si and v. If u(si , v) = si , then the auction
is perceived as the private value model by bidder i, in which si is interpreted as the private
value of the object. In case of the pure private value model, v is interpreted as the state of
the world. Our model covers the affiliated private value models. Since si is affiliated with
sj , our private value model is an affiliated private value model. We shall focus on the case
where the private value component exists, thus excluding the pure common value models.
Assumption 2.2. u(si , v) is continuously differentiable, satisfying
∂u(si ,v)
∂v
≥ 0.
∂u(si ,v)
∂si
> 0, and
For analytic convenience, we “parameterize” the amount of the common value component by λ ≥ 0 by assuming
u(si , v) = w(si ) + λv
for the rest of the paper where w (si ) > 0 is continuous. We are most interested in the
case where λ is close to 0 (close to pure private value models) or λ is sufficiently large
(close to pure common value models). The additive separable functional form is only for
analytical convenience, and can be relaxed.
Because we allow different bidders to use different bidding strategies, it is possible that
a bidder with the highest private signal may not place the highest bid. Thus, we need to
deal with the order statistics of the bids and those of the signals, separately. Because the
sample order statistics depend upon the size of the sample as well as the order among the
sample, it is convenient to spell out both the size of the sample and the order. Although it
5Our formulation of the private and common value components is different from Pesendorfer and
Swinkels (2000) and Goeree and Offerman (2002) in which each decision maker receives a multi-dimensional
signal: one for the private value and the other for the common value. We follow the framework of Wilson
(1977) in order to focus on the information aggregation properties, while avoiding potential conflicts between information aggregation and efficiency. Models with multi-dimensional signals will be investigated
after we resolve the multiplicity of the solution.
6
IN-KOO CHO
is a convention in statistics to count the order from below (e.g., David (1981)), it is more
convenient for our analysis to count the order statistics from the top among the bids and
the signals of players other than bidder i.
Let b1 , . . . , bn be the bids placed by the buyers, and s1 , . . . , sn be the private signals.
By b( k ) , we mean the k-th highest bid among n samples. Similarly, by s( k ) , we mean
n
n
the k-th highest signal among n samples. Because the statistical properties of the order
statistics depend on v, we sometimes write b( k ) (v) and s( k ) (v) in place of b( k ) and s( k ) .
n
n
n
n
Notice that under our convention, b( k ) need not be the same as b( 2k ) , although their
n
2n
asymptotic properties are identical. We shall use a number in [0, 1] instead of a fraction
to represent the population order statistics in the “limit” game which is populated with
infinitely many players. For example, s(α) (v) is the α-th population order statistic of the
private signals, conditioned on v:
F (s(α) (v)|v) = 1 − α.
Since every bidder’s distribution F (si |v) is strictly decreasing in v, s(α) (v) is strictly increasing in v.
Following Wilson (1977), we assume that the support of F (·|v) is increasing in v, and
every population order statistics is also increasing in v.
Assumption 2.3. For ∀α ∈ [0, 1], s(α) (v) > 0 is a continuous function of v ∈ [v, v].
We consider v and v as the extreme values. We assume that conditioned on the extreme
values of the object, the private signal reveals the underlying states.
Assumption 2.4. For v ∈ {v, v}, s(0) (v) = s(1) (v).
Under Assumptions 2.3 and 2.4, s(α) (v) is strictly increasing, but also onto. Thus, there
exists an inverse function
v(α) : Si → V
such that ∀si ,
s(α) (v(α) (si )) = si .
In a certain sense, we focus on the case in which much of uncertainty about the underlying value exists for the intermediate values. If the state is extremely good or bad, the
signal is so strong that everyone receives the best or the worst signal. This condition is
mild because we impose no restriction about how quickly the support of F (si |v) shrinks to
a degenerate point as v converges to either extreme point. Fix any F̃ (si |v) and G̃(v) that
satisfy Assumptions 2.1 and 2.3. We can find F (si |v) and G(v) satisfying Assumption 2.4
along with Assumptions 2.1 and 2.3 such that the joint distribution of (si , v) induced by
F (si |v) and G(v) is close to the one induced by F̃ (si |v) and G̃(v).
The main purpose of using Assumption 2.4 is to ensure that v(α) (si ) is well defined for
∀si ∈ Si . Alternatively, we could possibly replace Assumption 2.4 by a milder one. But,
we decide to use Assumption 2.4 for its simplicity.
As we allow different bidders to use different strategies, and also allow each bidder to
entertain virtually any conjecture about the other players bidding strategies, it is possible
that a bidder believes that he will win an object with probability 1 or 0. If he believes
that the other players use extremely conservative bidding strategies so that the delivery
price will be very low, then he believes he will win the object with probability 1. As a
result, he can use a wide range of bids, some of which are evidently unreasonable. To
avoid technical problems arising from the cases involving sure winning or losing bids, we
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS7
impose an assumption to ensure that every bidder faces a probability of winning an object
strictly between 0 and 1.
To this end, we assume that the ratio αn = /n between the total supply and the total
demand can vary.
Assumption 2.5. αn is a random variable that has a full support over [0, 1], and converges
in probability to a degenerate distribution concentrated at α ∈ (0, 1).
Because this assumption can be best motivated in the context of the expected utility
maximization, we shall revisit this assumption when we characterize the best response of
an individual player. (See Remark 2.13.)
If bidder i does not win the object, his payoff is 0. A strategy of player i is
bi : Si → R.
We call bi (si ) a bid conditioned on si , or simply a bid whenever the meaning is clear from
the context. Let ΣM
i be the set of all feasible bidding strategies. Instead of all bidding
strategies, we focus on a class of “well behaved” bidding strategies that meet the following
conditions:6
(2.1)
∀i, ∀bi ∈ ΣM
i , ∀si , si ∈ Si ,
bi (si ) − bi (si )
1
.
≥
si − si
M
For ∀α ∈ (0, 1), we choose M > 0 sufficiently large that
u(si , v(α) (si )) ∈ ΣM
i .
M
M =
M
Let ΣM
i be the set of bidding strategies that satisfy (2.1). Let Σ
i Σi , and Σ−i =
M
j=i Σj . From now on, by subscript −i, we mean a vector that does not have the i-th
component, or the Cartesian product except for the i-th element. For example, by b−i , we
mean the profile of bids (or bidding strategies) without the i-th bid (or bidding strategy).
Given b = (b1 , . . . , bn ), let Πi (b) be the ex ante expected payoff of player i.
M
Definition 2.6. bi is strictly dominated if ∃bi ∈ ΣM
i such that ∀b−i ∈ Σ−i , Πi (bi , b−i ) <
M
Πi (bi , b−i ). bi is weakly dominated, or simply, dominated if ∃bi ∈ ΣM
i such that ∀b−i ∈ Σ−i ,
Πi (bi , b−i ) ≤ Πi (bi , b−i ), and ∃b−i ∈ ΣM
−i , Πi (bi , b−i ) < Πi (bi , b−i ).
2.2. Elimination of weakly dominated strategies. We show that any bidding strategy below u(si , v(0) (si )) or above u(si , v(1) (si )) is weakly dominated. In fact, this is the
only step where we need the full power of elimination of weakly dominated strategies.
In the private value model, any bid above or below the true “private” value is a weakly
dominated strategy. Using the same logic, we can eliminate bids that are obviously too aggressive or too conservative. Recall that v(0) (si ) and v(1) (si ) are the smallest and the largest
common values conditioned on si , respectively. Thus, u(si , v(0) (si )) and u(si , v(1) (si )) are
the smallest and the largest utilities that can be generated conditioned on si , if player i
wins the object.
Lemma 2.7. If ∃si such that bi (si ) > u(si , v(1) (si )), then bi is weakly dominated. Similarly, if ∃si such that bi (si ) < u(si , v(0) (si )), then bi is weakly dominated.
6We admit mixed strategies over ΣM . However, in order to simplify notation, we always interpret b as
i
i
a pure strategy unless specified otherwise.
8
IN-KOO CHO
Proof. See Appendix A.
Lemma 2.7 implies that ∀i, ∀bi which is not weakly dominated, ∀si ,
(2.2)
u(si , v(0) (si )) ≤ bi (si ) ≤ u(si , v(1) (si )).
Notice that in order to obtain (2.2), we only need to invoke one round of eliminating
weakly dominated strategies. Under Assumption 2.4, F (si |v) and F (si |v) are degenerate.
Thus, (2.2) implies that
bi (s) = u(s, v)
and
bi (s) = u(s, v)
for ∀bi that survive one round of eliminating weakly dominated strategies. To simplify
notation, we shall regard for the rest of the paper
M
Σ̃M
i = bi ∈ Σi | ∀si , u(si , v(0) (si )) ≤ bi (si ) ≤ u(si , v(1) (si ))
as the set of feasible bidding strategies instead of ΣM
i . For example, by repeated elimination of strictly dominated strategies (RESDS), we mean RESDS applied over Σ̃M
i instead
M is a collection of uniformly bounded and strictly increasing func.
Note
that
Σ̃
of ΣM
i
i
tions. Thus, we can endow a weak topology to render Σ̃M
i a compact set, which ensures
has
a
convergent
subsequence.
that any sequence in Σ̃M
i
We have to examine the slope of increasing functions throughout the paper, which may
have jumps and kinks. To simplify the notation, by ϕ (x) for some increasing function ϕ,
we mean
ϕ(x + h+ ) − ϕ(x − h− )
.
lim inf
h+ + h−
h+ ,h− →0
In particular, if ϕ has a jump at x, then we write ϕ (x) = ∞.
2.3. Formal Results. Given b−i , let bri (b−i ) be an (ex ante) best response over Σ̃M
i :
Πi (bri (b−i ), b−i ) ≥ Πi (bi , b−i )
∀bi ∈ Σ̃M
i .
Because bi is strictly increasing for every i, Πi is continuous with respect to b = (b1 , b2 , . . .).
Since Σ̃M
i is compact, Πi has a maximum and therefore, the best response correspondence
is well defined.
We can extend the definition of the best response to cover the case when some other
players use randomized strategies, or even the case when others use correlated strategies.
M,0
= Σ̃M
, . . . , Σ̃iM,r−1 are defined. Let
Let Σ̃M,0
i . Suppose that Σ̃i
i
⎧
⎫
⎨
⎬
M,r−1 M,r−1
r
=
b
∈
Σ̃
∈
∆(
Σ̃
),
b
=
b
(ψ
)
∃ψ
Σ̃M,r
−i
i
−i
i
n,i
i
n,j
⎩ i
⎭
j=i
⊂ Σ̃iM,r−1
where ∆(Σ̃M,r−1 ) is the set of all probability distributions over Σ̃M,r−1 . Σ̃M,r
i
M,r
M,r ∞
for ∀i, ∀r ≥ 1. Since Σ̃i
is compact, {Σ̃i }r=0 is a nested sequence of compact sets,
which implies
∞
M
=
Σ̃M,r
Rn,i
n,i = ∅.
r=0
M
Definition 2.8. bi ∈ Σ̃M
i is ex ante rationalizable if bi ∈ Rn,i ≡
the set of ex ante rationalizable strategies.
∞
M,r
r=0 Σ̃n,i .
M
We call Rn,i
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS9
It is known that strategies survive RESDS if and only if they are ex ante rationalizable
(cf. Osborne and Rubinstein (1994)). Define
M
Rn,i
RiM =
n ≥1 n≥n
M . If b ≥ b
as the collection of limits of convergent sequences {bn,i } where bn,i ∈ Rn,i
i
( n ) ,
then bidder i wins the object to realize utility u(si , v) but pays
n
1
b( +1 ) = sup b |
1bi ≤b < 1 −
n
n
n
i=1
which is the highest losing bid, where 1X is the characteristic function of event X.
Let bn = (bn1 , . . . , bnn ) be a profile of bidding strategies in an auction populated with n
bidders, and bn( +1 ) be the associated delivery price. Consider the sequence of the profiles
n
n ∞
∗
of bidding strategies, {bn }∞
n=1 . Given {b1 }n=1 , let b1 be a limit of its convergent subsequence, which must exist because {bn1 } ⊂ Σ̃M
1 which is compact. By selecting convergent
subsequences recursively for each player i ≥ 1, we can define a profile of bidding strategies,
∗ ∞
{b∗i }∞
i=1 , for the limit game populated with infinitely many players. Given {bi }i=1 , let
n
1
∗
1b∗i ≤b < 1 −
b( +1 ) = sup b |
n
n
n
i=1
be the delivery price associated with the first n bidding strategies. By the strong law of
large numbers, we know that ∀v,
b∗( +1 ) → b(α) (v)
n
in probability where
P (bi ≤ b | v) di < 1 − α .
b(α) (v) = sup b |
i
For the rest of the paper, we treat b(α) (v) as a limit of b( +1 ) , and write
n
b( +1 ) → b(α) (v).
n
We refer to b(α) (v) as the delivery price in the limit game.
It is critical for us to understand the properties of b(α) (v) in order to examine the information aggregation properties of the large auctions. In Wilson (1977) and Pesendorfer and
Swinkels (1997) that analyze the pure common value models, the information aggregation
is a combination of two properties: first, the delivery price reveals the underlying value and
second, the market outcome converges to the underlying value, which essentially identifies
a solution in the limit game. As we analyze the models with private value component, it
would be convenient to examine the two properties separately.
Definition 2.9. A sequence of auctions reveals information if b(α) (v) is a strictly increasing function of v.
The revelation property stems from the monotonicity of the delivery price with respect
to the underlying state of the world rather than individual optimization problem. If
every bidder is using the same strictly increasing bidding function, then the information
revelation property follows directly from the monotonicity of the bidding function and the
monotonicity of s(α) (v) with respect to v. Note that if every bidder uses the same bidding
10
IN-KOO CHO
strategy b̃ ∈ Σ̃M
i , then the delivery price is determined by the bid placed by a bidder with
the pivotal signal:
b(α) (v) = b̃(s(α) (v)).
Therefore,
1
inf s (v) > 0
b(α) (v) = b̃ (s(α) (v))s(α) (v) =
M v (α)
since inf v s(α) (v) is bounded away from 0 by Assumption 2.3.
Our first task is to generalize this observation: establish the information revelation
property if bidder are using strictly increasing, but possibly asymmetric, bidding strategies.
Proposition 2.10. ∃ω > 0 such that ∀v,
b(α) (v) ≥ ω.
Proof. See Appendix B.
On the other hand, the second property that the delivery price converges to the true
value of an object in Wilson (1977) and Pesendorfer and Swinkels (1997) is intimately
related to the competitive pressure on the individual players. A natural counterpart of the
“true” value in the symmetric uniform price auction would be u(si , v(α) (si )), as α portion
of bidders wins an object in the limit game. If each bidder has a different marginal utility
function uq and q has non-degenerate distribution, one cannot treat uq (si , v(α) (si )) as the
“true” value of a bidder. In fact, bq (si ) = uq (si , v(α) (si )) is not a Nash equilibrium bidding
strategy in the limit game populated with infinitely many players if q has non-degenerate
distribution.
Although one can calculate the limit outcome in a fairly straightforward fashion, its interpretation is not as clear as we would expect from the ex ante symmetric models. Instead
of referring to a Nash equilibrium, we consider the second property as the convergence to
a particular rationalizable outcome of the limit game.
Definition 2.11. A sequence of auctions selects an outcome if there exists bi ∈ ΣM
i such
that
RiM = {bi }
Because any Nash equilibrium in the limit game which satisfies the monotonicity of the
bidding strategies is rationalizable, the selected rationalizable outcome is a Nash equilibrium outcome of the limit game.7
Theorem 2.12. If Assumptions 2.1, 2.2, 2.3, 2.4 and 2.5 hold in symmetric uniform
price auctions, then ∃λ , M > 0 such that ∀λ ∈ [0, λ ), ∀M > M , ∀i, RiM = {w(α) } where
w(α) (si ) = u(si , v(α) (si )).
Since w(α) is strictly increasing with respect to si , the delivery price converges to
u(α) (s(α) (v), v)
for ∀v. Since this function is strictly increasing in v, the selection result implies the
information revelation. In order to prove Theorem 2.12, we need to use Proposition 2.10.
In this sense, the information revelation implies the selection result. Within the confines of
the strictly increasing bidding functions and a single dimensional signal, the information
revelation turns out to be equivalent to the information aggregation result.
7For a finite n, our result implies the existence of Nash equilibrium, although we fall short of proving
the existence of a Nash equilibrium.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
11
For the rest of the section, we present the analysis of the uniform price auction, eventually leading to the proof of Theorem 2.12. Instead of jumping directly to the proof,
let us start with the optimization problem facing the individual players to illuminate the
implications of the restrictions on the bidding strategies.
2.4. Constrained vs. Unconstrained Optimization. Note that bri (b−i ) is a best response against b−i over the restricted set of strategies ΣM
i . For later reference, we call
max
{bi | Si →R}
Πi (bi , b−i )
the unconstrained optimization problem, and write a generic element of
arg
max
{bi | Si →R}
Πi (bi , b−i )
as b∗i (b−i ). Note that the set of feasible choices is compact in both cases. Because the other
player’s bidding strategies are strictly increasing, the objective function changes continr
∗
uously with respect to the choice of bidder i. Thus, for ∀b−i ∈ ΣM
−i , b (b−i ) and b (b−i )
∗
are well defined. We normally use the first order condition to characterize bi (b−i ). But,
because b∗i (b−i ) generally differs from bri (b−i ), the characterization of bri (b−i ) is difficult for
a finite n.
Let b∗i (si , b−i ) be the value of b∗i (b−i ) when player i’s private signal is si . We know that
the ex ante optimal solution induces an interim optimal solution in the unconstrained
optimization problem:
b∗i (si , b−i ) ∈ arg
max
{bi :Si →R}
Πi (bi , b−i | si )
∀si , ∀b−i , ∀i.
We can write the interim optimization problem of player i conditioned on si as
E u(si , v) | si , b( ) − b( ) dH(b( ) |si )
(2.3)
max
bi
b(
n−1
≤bi
n−1 )
n−1
n−1
where H(·|si ) is the conditional distribution of b( ) .
n−1
Conditioned on si , the probability of winning the object is
P b( ) ≤ bi (si ) | si .
n−1
If the probability of winning an object conditioned on si is 1, then we call the bid a sure
winning bid. If the probability is 0, then we call it a sure losing bid. If the probability
remains between 0 and 1, we call it an interior bid.
Let us first examine the interior bid bi in which the conditional expectation
E u(si , v) | si , bi = b( )
n−1
is well defined. Thus, if an unconstrained best response is an interior solution of the
optimization conditioned on si , then
(2.4)
b∗i (si , b−i ) = E u(si , v) | si , b∗i (si , b−i ) = b( )
n−1
along with the “second order” condition: ∃µ > 0 such that ∀µ ∈ (0, µ),
(2.5)
E u(si , v)|si , (b∗i (si , b−i ) + µ) = b( ) − (b∗i (si , b−i ) + µ) ≤ 0
n−1
(2.6)
E u(si , v)|si , (b∗i (si , b−i ) − µ) = b( ) − (b∗i (si , b−i ) − µ) ≥ 0.
n−1
If the equality holds in the second order condition, there could be multiple best responses.
12
IN-KOO CHO
Remark 2.13. As we dispense with the perfect foresight, we allow each player to entertain
M . As a result, b
can consist of very conservative bids so
any belief concentrated over Rn,−i
−i
that b( ) is so low so that P bi > b( ) | si = 1. In this case, the first order condition
n−1
n−1
does not characterize b∗i (si , b−i ), because {si , b∗i (si , b−i ) = b( ) } could be a null set. As a
n−1
result, a wide range of actions could be rationalizable if a bidder entertains an “extreme”
conjecture and many of them are “insensitive” to b−i . We use Assumption 2.5 to ensure
that the conditional probability on {si , bi = b( ) } is well defined and we can use the first
n−1
order condition to characterize the unconstrained best response for ∀n.
Notice that
u(si , v(0) (si )) ≤ b∗i (si , b−i ) ≤ u(si , v(1) (si )).
If b∗i (si , b−i ) is a winning bid with a very high probability, then it should be close to
u(si , v(1) (si )). Similarly, if b∗i (si , b−i ) is a losing bid with a very high probability, then it
should be close to u(si , v(0) (si )). Thus, in the limit, if b∗i (si , b−i ) is a winning bid with
probability 1, then
b∗i (si , b−i ) = u(si , v(1) (si )).
Similarly, if b∗i (si , b−i ) is a sure losing bid in the limit game, then
b∗i (si , b−i ) = u(si , v(0) (si )).
Otherwise, b∗i (si , b−i ) is characterized by the first order condition as described below.
By invoking the law of large numbers, we know that there exists s(α) (v) such that
s(
)
n−1
→ s(α) (v)
in probability. Thus, the interior bid in the limit game is characterized by
b∗i (si , b−i ) = E u(si , v) | si , b∗i (si , b−i ) = b(α)
∗
= w(si ) + λE v | si , b∗i (si , b−i ) = b(α) = w(si ) + λb−1
(α) (bi (si , b−i )).
The second order condition implies that b(α) (v) ≥ λ around the neighborhood of v where
b(α) (v) = b∗i (si , b−i ).
2.5. Monotonicity of b∗i (si , b−i ). If the bidders use different strategies, si and b(
)
n−1
may not be affiliated for a finite n (cf. Perry and Reny (2003)). Consequently, we cannot
claim that b∗i (si , b−i ) is a weakly increasing function of si for a finite n. However, we shall
show that in the limit, b∗i (si , b−i ) is strictly increasing in si and moreover, its slope is
bounded away from 0 uniformly.
First, observe that if b∗i (si , b−i ) is either a sure winning or a sure losing bid, then it
is strictly increasing in si . Second, if b∗i (si , b−i ) is an interior solution of the optimization problem, then the second order condition implies that b(α) (v) ≥ λ. If b(α) (v) = λ,
then there are multiple best responses and b∗i (si , b−i ) has a jump at si , implying that
∂b∗i (si , b−i )/∂si = ∞. A straightforward calculation shows that
w (si )
∂b∗i (si , b−i )
.
=
∂si
1 − λ/b(α) (v)
Thus, if b(α) (v) > λ, then the right hand side is bounded below by inf si w (si ) > 0.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
13
Choose M > 0 so that ∀M ≥ M ,
1
∂b∗ (si , b−i )
.
≥
∂si
M
For M > M , an unconstrained best response is a feasible bidding strategy. Thus,
b∗i (b−i ) = bri (b−i ).
(2.7)
For the rest of the paper, we choose M > M so that we can use the first order condition
to characterize bri (b−i ) in the limit game.
Suppose that b̂j (sj ) ≥ bj (sj ) for ∀j = i and ∀sj . Let b̂(α) and b(α) be the α-th highest
bids generated by (bi , b̂−i ) and (bi , b−i ). By invoking Proposition 2.10, we can easily show
(2.8)
E u(si , v)|si , bi = b(α) − b(α) ≥ E u(si , v)|si , bi = b̂(α) − b̂(α) .
and therefore,
b∗i (si , b̂−i ) ≤ b∗i (si , b−i )
(2.9)
∀si .
In the limit, b(α) (v) is strictly increasing in v, that offers the foundation for aggregating
the diverse information held by individual bidders into a single statistics, namely the
delivery price, that reveals v. The same information revelation property of b(α) (v) is
critical for establishing (2.9).
More precisely speaking, we can obtain (2.9) if we can find a strictly increasing bidding
function
b̃ : Sj → R
such that ∀v, ∀x,
(2.10)
P b(
)
n−1
≤ x | v = P b̃(s(
)
n−1
)≤x|v .
That is, if we can represent the delivery price as the bid of the pivotal bidder, then the
insight from the symmetric equilibrium applies to show that (2.9) must hold (Milgrom
and Weber (1982)).
While (2.10) may not hold for some v for a finite n, we can choose
∀j
(2.11)
b̃(sj ) = b(α) v(α) (sj )
in the limit. Since v(α) (sj ) is strictly increasing in sj and b(α) (v) is strictly increasing in
v, b̃(sj ) is strictly increasing. Then, we have
∀v.
(2.12)
P b(α) ≤ x | v = P b̃(s(α) ) ≤ x | v
By (2.12), we have
E u(si , v) | si , b̂(α) = x ≤ E u(si , v) | si , b(α) = x
which implies that (2.9) holds.
14
IN-KOO CHO
2.6. Monotonicity and Bounded Rationality. Under the standard informational assumptions of Wilson (1977) and Milgrom and Weber (1982), the monotonicity of the
bidding strategies is sensible, and the symmetric Nash equilibrium strategy is strictly
monotonic with respect to the private signal. Thus far, we motivated the monotonicity as
an intuitive behavioral assumption, as it nicely captures the idea that if a bidder receives
a stronger signal about the underlying value, he bids more aggressively. Still, it would
be instructive to see the full scope of the restriction over the individual decision making
process.
As Perry and Reny (2003) demonstrated, in an auction with a small number of players,
there may exist a Nash equilibrium with non-monotonic asymmetric bidding strategies.
However, as the number of bidders increases, (2.7) implies that the monotonicity restriction
hardly matters. Even if the constrained optimum br (b−i ) differs from the unconstrained
optimum b∗ (b−i ), the difference between br (b−i ) and b∗ (b−i ) vanishes in a large auction.
One can interpret the monotonicity of the bidding strategy as an expression of the
bounded rationality of a bidder who “underestimates” his strategic influence to the market
outcome. In a certain sense, our restriction shares the idea of of the cursed equilibrium of
Eyster and Rabin (2002): each decision maker underestimates the informational content
of the behavior of the other players with probability χ ∈ [0, 1], which Eyster and Rabin
(2002) uses as a measure of bounded rationality in a certain sense.
We can infer the same kind of behavior from (2.11). For a finite n, there is a random
variable ∆n such that
b̃(si ) = b(α) (v(α) (si )) + ∆n
and (2.11) implies that ∆n → 0 in distribution as n → ∞. The first term in the right
hand side is the “aggregate behavior” of the other players and ∆n measures the deviation
from the aggregate.
If ∆n is independent of v, or if every bidder uses the same bidding strategy, then (2.11)
holds for a finite n. If so, the unconstrained best response must be strictly increasing and
is a feasible strategy. Thus, the constrained best response must be an unconstrained best
response. In this case, monotonicity imposes no constraint on rationality.
But, in auctions with common value component combined with asymmetric bidding
strategies, ∆n is generally influenced by v, which generates the discrepancy between the
unconstrained and the constrained best responses. However, as ∆n → 0, the correlation
between ∆n and v vanishes, and so does the discrepancy between b∗ (b−i ) and br (b−i ).
Note that if a bidder “ignores” the informational content of ∆n about the underlying
v, then his best response must be strictly increasing. Using the terminology of Eyster
and Rabin (2002), our agents behave as if they have a positive χ ∈ [0, 1], ignoring the
correlation between ∆n and v. However, as the number of the agents increases, the
correlation between ∆n and v vanishes, and our agents behave more like those who have
a small χ. In the limit, our agents are observationally equivalent to fully rational agents
who have no restriction on the set of feasible bidding strategies. As we focus on the large
auctions, the monotonicity is intuitively and behaviorally appealing, but also presumes
only mild bounded rationality.
2.7. Ex ante vs. Interim Rationalizability. The set of ex ante rationalizable bidding
strategies converges to a single strategy if and only if the “upper bound” and the “lower
M converges to the same strategy. Because the maximum or the minimum of
bound” of Rn,i
two ex ante rationalizable bidding strategies may not be ex ante rationalizable, however,
M has the most and the least aggressive bidding strategy. To
it is not obvious whether Rn,i
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
15
M as n → ∞, we use interim rationalizability, which is
characterize the boundary of Rn,i
closed with respect to max and min operations.
M,0
= Σ̃M
, . . . , ΣiM,r−1
To define the interim rationalizability formally, let ΣM,0
i . Suppose that Σi
i
are defined. Let
⎧
⎫
⎨
⎬
M,r−1 M,r−1
r
=
b
∈
Σ
,
∃ψ
∈
∆(Σ
),
b
(s
)
=
b
(s
,
ψ
)
.
ΣM,r
∀s
i
i
−i
i
i
i
−i
i
n,i
i
n,j
⎩
⎭
j=i
M
Definition 2.14. bi ∈ Σ̃M
i is interim rationalizable if bi ∈ R̃n,i ≡
the set of interim rationalizable strategies.
∞
M,r
r=0 Σn,i .
M
We call R̃n,i
Clearly,
M
M
⊂ R̃n,i
.
Rn,i
M , then max(b , b ), min(b , b ) ∈ R̃M . Since the set of interim
Notice that if bi , bi ∈ R̃n,i
i i
i i
n,i
rationalizable bidding strategies is closed,
bn,i (si ) = sup bi (si )
bi ∈R̃M
n,i
and
bn,i (si ) = inf bi (si )
bi ∈R̃M
n,i
M . Define b (s ) and b (s ) as the pointwise limits of {b (s )}
are also elements of R̃n,i
i i
n,i i
i i
and {bn,i (si )} as n → ∞. Note that bi and bi are interim rationalizable strategies that
are rationalized by each other for ∀si . Thus, these two bidding strategies are ex ante
rationalizable.
Lemma 2.15. ∀i, ∃bi , bi ∈ RiM such that
bi (si ) ≤ bi (si ) ≤ bi (si )
∀bi ∈ RiM .
Recall that, for a given v, si and sj are mutually independent. Define b(
the -th order statistic of {bn,i (si )} conditioned on v. Similarly, define b(
)
n−1
)
n−1
(v) as
(v) as the
-th order statistic of {bn,i (si )}. Conditioned on v, {bn,i (si )} is a sequence of independent
random variables. By the law of large numbers, for each v, there exists b(α) (v) such that
b(
)
n−1
b(
)
n−1
(v) → b(α) (v)
in probability. Similarly,
(v) → b(α) (v)
in probability.
Because the monotonicity with respect to the other players’ bidding strategies holds
in the limit game populated with infinitely many players, the most aggressive bidding
strategy in RiM is rationalized by the least aggressive bidding strategy and vice versa: ∀i,
∀si , and ∀M ,
(2.13)
bi (si ) = E u(si , v) | si , bi (si ) = b(α) (v) = u si , (b(α) )−1 (bi (si ))
(2.14)
bi (si ) = E u(si , v) | si , bi (si ) = b(α) (v) = u si , (b(α) )−1 (bi (si ))
16
IN-KOO CHO
where bi and bi is the pointwise limits of bn,i and bn,i , if
b(α) (v(0) (si )) ≤ bi (si ) ≤ b(α) (v(1) (si ))
and
b(α) (v(0) (si )) ≤ bi (si ) ≤ b(α) (v(1) (si )).
The weak inequality may hold as an equality if bi (si ) or bi (si ) is a sure winning or losing
bid.
Note that bi (si ) is a best response against b−i for ∀si . Thus, bi is ex ante rationalizable.
Similarly, bi is ex ante rationalizable. One can infer from (2.13) and (2.14) that the set of
ex ante rationalizable strategies and the set of interim rationalizable strategies have the
same boundaries. Thus, Theorem 2.12 holds for the ex ante rationalizable strategies, but
also for the interim rationalizable strategies.
2.8. Monotonicity as a Restriction on Beliefs. Readers might wonder why I did
not opt for interim rationalizability to obtain a stronger result. Moreover, because the
bidders know their types when they participate the auctions, it would be more sensible to
investigate the interim stage than the ex ante stage. A natural question would be whether
we can extend the main result in terms of the decision problem at the interim stage.
At the interim stage, we treat the same bidder with different signals as different decision
makers. Thus, the monotonicity with respect to the signal is significantly less intuitive
than at the ex ante stage. At the interim stage, a more natural restriction would be to let
every bidder believes that his opponents are using strictly increasing bidding strategies and
this fact is common knowledge (Battigalli and Siniscalchi (2003)). We can then proceed
to show that if the type space is finite, then the main result obtained for the ex ante case
is carried over with little modification: for the same class of auctions, the set of interim
rationalizable bidding strategies (subject to the restricted beliefs) collapses to a single
profile of bidding strategies. In this sense, we can interpret the restriction on the set of
bidding strategies as a physical restriction but also as a restriction on the beliefs of the
players.
To formulate this observation, let us define
Si = {s = s1 , . . . , sR(h) = s}
where
sr+1 = sr + h
for ∀r = 1, . . . , R(h) − 1 and ∀i. We interpret h as the grid. As h → 0, Si becomes a dense
subset of [s, s]. Let F (si |v) be the distribution function over Si , that satisfies Assumption
2.1. All other details of the primitives remain the same as before.
Let ΣM
i be the set of bidding strategies whose slope is bounded below by 1/M . Because
the signal space is discrete,
h
∀bi ∈ ΣM
i .
M
If the signal space is continuous, we obtain (2.7) in the limit. If th signal space is finite,
we can show that (2.7) holds in the limit but also, for a finite, but sufficiently large, n.
(2.15)
bi (si + h) ≥ bi (si ) +
Lemma 2.16. ∀h > 0, ∃N and M such that ∀M ≥ M , ∀n ≥ N , ∀i,
b∗i (b−i ) = bri (b−i ) ∈ ΣM
i .
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
17
Proof. We know that in the limit,
against any profile b−i
M so that
∂b∗i (si , b−i )
≥ inf w (si ) > 0
si
∂si
of strictly increasing bidding functions by the other players. Choose
2
.
M
Note that b∗i (si , b−i ) converges to a strictly increasing function whose slope is bounded
from below by w (si ) as n → ∞. Fix M ≥ M . Since i Σ̃M
i is compact, we can choose a
sufficiently large N1 so that ∀n ≥ N , ∀b−i ,
inf w (si ) ≥
si
inf si w (si )
1
b∗i (s1 + h, b−i ) − b∗i (s1 , b−i )
≥
≥
.
h
2
M
Given N1 , . . . , Nr , choose Nr+1 such that ∀n ≥ Nr+1 ,
inf si w (si )
1
b∗i (sr+1 + h, b−i ) − b∗i (sr+1 , b−i )
≥
≥
.
h
2
M
Let N = max(N1 , . . . , NR(h)−1 ). Then, ∀M ≥ M , ∀n ≥ N , b∗i (b−i ) ∈ Σ̃M
i , and therefore,
Q.E.D.
b∗i (b−i ) = bri (b−i ).
In this sense, the belief that every other player is using a strictly increasing bidding
M ⊂ ΣM . Let b and b be the
strategy is self-fulfilled for a sufficiently large n, because Rn,i
i
i
i
most and the least aggressive rationalizable bidding strategies (subject to the constraint on
the beliefs that every other player is using a strictly increasing bidding strategy). Clearly,
bi , bi ∈ RiM and also (2.9) holds. Hence, we obtain (2.13) and (2.14), which imply Theorem
2.12 in combination with Proposition 2.10.
While the interim rationalizability is more intuitive and in some case, more sensible,
it is rather cumbersome to analyze models with discrete type spaces. We can better
motivate our solution concept in the context of elimination of dominated strategies, which
is equivalent to ex ante rationalizability. For the rest of this paper, in order to simplify
notation, we shall interpret the monotonicity as a restriction on the set of feasible bidding
strategies defined over a continuous space of private signals.
2.9. Benchmark. Before proving the convergence result, it is instructive to calculate a
M , which is in fact a (symmetric) Nash equilibrium of the model.8
particular solution in Rn,i
Define
wn (si ) = E u(si , v) | si = s( )
n−1
and
(2.16)
w(α) (si ) = u(si , v(α) (si ))
It is straightforward to verify that wn and w(α) are Lipschitz continuous and strictly
increasing functions of si . Hence, wn , w(α) ∈ ΣM
i for a sufficiently large M . It is straightforward to verify that
∀i, ∀si .
wn (si ) = b∗i (si , wn,−i )
M
Since wn ∈ ΣM
i , the unconstrained optimal response must be optimal over Σi :
wn (si ) = bri (si , wn,−i ).
8A Nash equilibrium in the asymmetric model can be constructed by following a similar logic.
18
IN-KOO CHO
Clearly,
(2.17)
bn,i (si ) ≤ wn (si ) ≤ bn,i (si )
∀si , ∀n.
and
bi (si ) ≤ w(α) (si ) ≤ bi (si )
∀si .
M
Proposition 2.17. (wn , . . . , wn ) is a Nash equilibrium for ∀n ≥ 1 and therefore, wn ∈ Rn,i
for ∀n, ∀i and for any sufficiently large M .
2.10. Cobweb Property. By using the information revelation property of the delivery
price, we can show that unless the common value component is too large, the set of
rationalizable outcomes collapses to a single strategy profile, which is in fact the symmetric
Nash equilibrium of the limit game.
Let · be the sup norm over the space of function. By Proposition 2.10, we can choose
λ > 0 and > 0 such that ∀λ ∈ [0, λ ),
0≥
λ
> −1 + .
λ − b(α) (v)
Choose M such that ∀M > M 1
si
M
so that an unconstrained optimal response is a feasible bidding strategy in the limit game.
We know
−1
bi (si ) = b∗i (si , b−i ) = w(si ) + λb(α)
(bi (si ))
inf w (si ) >
and
−1
bi (si ) = b∗i (si , b−i ) = w(si ) + λb(α) (bi (si ))
Although bi (si ) is a best response to b−i in principle, b−i influences bi (si ) only through
b(α) (v).
Suppose that b(α) (v) changes to b(α) (v) + dh. If bi (si ) is a corner solution, then bi (si )
does not respond to such a small change. If bi (si ) is an interior solution, then
0≥
λ
dbi (si )
> −1 + dh
λ − b(α) (v)
for λ ∈ [0, λ ). A simple calculation shows that
db(α) ≤ sup dbi (si ).
i,si
We can repeat the same exercise for b(α) and bi (si ) to have
0≥
λ
dbi (si )
> −1 + dh
λ − b(α) (v)
and
db(α) ≤ sup dbi (si ).
i,si
Hence, the difference between the two best responses cannot be larger b(α) − b(α) .
Formally,
bi − bi ≤ b(α) − b(α) (1 − ) < b(α) − b(α) ≤ sup bi − bi i
∀i
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
19
Thus,
sup bi − bi ≤ b(α) − b(α) (1 − ) < b(α) − b(α) ≤ sup bi − bi i
i
which is a contradiction. This contradiction proves Theorem 2.12.
It is important to note that we do not take the full advantage of the ex ante symmetry
of the bidders other than the fact that the private signal is drawn from the distribution
functions which share the same support. As a result, Proposition 2.10 and Theorem 2.12
can be extended to a larger class of auctions in which some players have different utility
functions from others such as double auctions and auctions with heterogeneous bidders,
or in which different units generate different utilities such as the multi-unit auctions. To
see how the insight from the uniform price auctions is carried over to the other trading
models, let us examine three important examples: double auctions, uniform price auctions
with bidders with different utility functions and multi-unit auctions.
3. Double Auctions
Imagine a market populated with n buyers, each of whom demands at most 1 unit of
goods, and n sellers, each of whom has one unit for sale. Each buyer and seller submits a
number, which is interpreted as a bid to buy or an ask price for sale. Then, the auctioneer
arranges 2n prices from the highest to the lowest:
b(
1
)
2n
≥ b(
2
)
2n
≥ · · · ≥ b( 2n ) .
2n
The delivery price is determined as the convex combination of the n-th and n+1-st highest
prices,
n
p = kb( 2n
) + (1 − k)b( n+1 )
2n
where each price could be either a seller’s ask price or a buyer’s bid, and k ∈ [0, 1].9 At
n
the end of the auction, a player who places a price which is as large as b( 2n
) gets out of
the auction with a single unit of the good. If the player is a buyer, this means that he
successfully purchases the good, paying p. If the player is a seller, then she fails to sell
the good, and keeps the object at the end of the game. Because the delivery price is a
convex combination of two bids, we call this auction a k-double auction (Chatterjee and
Samuelson (1983)). Let us keep other primitives of the model such as the utility function
and the informational structure of each player the same as in the symmetric uniform price
auction investigated in the previous section.
For k ∈ (0, 1), both parties can influence the delivery price, and consequently, some
potential gains from trading may not be realized. However, a careful examination of the
payoff function reveals that the probability that a player can influence the market price
vanishes as the number of players increases.
Let Πbi (bi , b−i |si ) be the interim expected payoff of the buyer when he places bid bi ∈ R
conditioned on si against b−i . Since the buyer can influence the delivery price only if
n
b( 2n−1
) ≤ bi < b( n−1 ) ,
2n−1
9Satterthwaite and Williams (1989) as well as most textbooks on order statistics rank the random
variable from the smallest to the largest. However, to remain consistent with the auction literature, we
rank the random variable from the largest to the smallest.
20
IN-KOO CHO
it would be convenient to decompose the interim payoff of the buyer into two components,
Πui (bi , b−i )|si ) and Πfi (bi , b−i |si ) where
n
(3.18)Πui (bi , b−i |si ) = E u(si , v) − kb( n−1 ) − (1 − k)b( 2n−1
) | si , bi ≥ b( n−1 ) ,
2n−1
2n−1
f
n
n
(3.19)Πi (bi , b−i |si ) = E u(si , v) − kbi − (1 − k)b( 2n−1
) | si , b( 2n−1
) ≤ bi ≤ b( n−1 )
2n−1
and
Πbi (bi , b−i |si )
⎞
⎛
n
Πui (bi , b−i |si )P bi ≥ b( n−1 ) | si , bi ≥ b( 2n−1
)
2n−1
⎠ P bi ≥ b( n ) si
= ⎝
2n−1
n
n
+Πfi (bi , b−i |si )P b( 2n−1
) ≤ bi < b( n−1 ) | si , bi ≥ b( 2n−1
)
2n−1
Since every player uses a strictly increasing bidding strategy,
n
n
≤
b
<
b
|s
,
b
≥
b
lim P b( 2n−1
n−1
i
)
( 2n−1 ) = 0
(
) i i
n→∞
2n−1
which implies
Πbi (bi , b−i |si ) − Πui (bi , b−i |si ) → 0
uniformly as n → ∞. Both objective functions are continuous over a compact set, since
every player uses a strictly increasing strategy. Thus, the limit of the sequence of optimal
solutions for Πbi (bi , b−i |si ) must be an optimal solution for Πui (bi , b−i |si ) in the limit, which
is characterized by the first order condition. In particular, bi (si ) and bi (si ) that are the
M as n → ∞ must satisfy
limit of the most and the least aggressive rationalizable bids in Rn,i
(3.20)
bi (si ) = u si , (b(0.5) )−1 (bi (si ))
(3.21)
bi (si ) = u si , (b(0.5) )−1 (bi (si ))
where b(0.5) (v) and b(0.5) (v) as the median bids if every seller and buyer uses the most or
the least aggressive rationalizable bidding strategies.
Note that despite the difference in their bidding strategies for a finite n, the buyers and
the sellers place their bids according to
b∗i (si , b−i ) = E u(si , v) | si , b∗ (si , b−i ) = b(0.5)
in the limit as n → ∞. Thus, the upper and the lower boundaries of the set of rationalizable
bidding strategies for every player can be characterized by (3.20) and (3.21). Following
exactly the same logic as we used in analyzing the uniform price auctions, we can show
that
bi (si ) = bi (si ) = u(si , v(0.5) (si )).
Theorem 3.1. If Assumptions 2.1, 2.2, 2.3, 2.4 and 2.5 hold in a k-double auction, then
∃λ , M > 0 such that ∀λ ∈ [0, λ ), ∀M > M , ∀i,
RiM = {w(0.5) }
where w(0.5) (si ) = u(si , v(0.5) (si )).
Theorem 3.1 does not tell us the rate of convergence. Although the buyer’s bidding
strategy converges at the rate of 1/n to what he would have used in the uniform price
auction, Theorem 2.12 does not tell us the convergence speed of the rationalizable bidding
strategies to u(si , v(0.5) (si )).
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
21
4. Ex Ante Heterogeneous Players
Because we have already obtained the information aggregation result while admitting
asymmetric bidding strategies across different players, we can easily extend Proposition
2.10 from the ex ante symmetric uniform auctions to the models where bidders may have
different ex ante characteristics. The examination of a fully general model will be pursued
in the ensuing projects. In this section, we shall explore an important special case where
we expand the auctions by “replicating” the bidders of the same ex ante characteristics.
In this case, we shall only consider the heterogeneous utility functions.
Let us normalize the index of the characteristics to a subset of [0, 1]. Let Q ⊂ [0, 1]
be the collection of indexes. The difference between the two indexes can be used as a
measure for the ex ante difference between the two players. For each q, there are n ex ante
identical bidders who share the same utility function uq (si , v). We call the group of the
ex ante identical bidders a cohort. Let bi,q (si ) be the bid placed by i-th agent in cohort q.
To maintain consistency with the models in the previous sections, we assume that
(4.22)
uq (si , v) = wq (si ) + λv
where λ ≥ 0 measures the size of the common value, and wq (si ) is a strictly increasing
continuous function of si and strictly decreasing continuous function of q. All other primitives remain the same as the symmetric uniform price auctions. In particular, every bidder
in the game receives his private signal si drawn from the same probability distribution
F (si | v). This assumption is mainly to simplify the model, and can be substantially
relaxed.
The rest of the analysis is identical to the uniform price auctions populated with ex ante
identical bidders. Because we assume that the distribution of the private signal of every
bidder in the game is identical, the ex ante heterogeneity of players makes a difference
only if the bidding strategies are different. As we already address issues arising from the
heterogeneous bidding function, we can invoke exactly the same analysis to obtain the
information aggregation result.
Proposition 4.1. ∃ω > 0 such that ∀v,
b(α) (v) ≥ ω.
Once we establish Proposition 4.1, the next step is to derive the cobweb properties in
the limit model by choosing a sufficiently small λ > 0. Because every bidder in the game
has the same marginal utility from the common value component, we can follow exactly
the same steps to find λ > 0 such that ∀λ ∈ [0, λ ), we can narrow down the set of
rationalizable bidding strategies to a single strategy profile.
Theorem 4.2. ∃λ , M > 0 such that ∀λ ∈ [0, λ ), ∀M > M , ∀i, RiM is singleton.
5. Multi-unit Auctions
We can easily describe the multi-unit version of the symmetric uniform price auction
examined in Section 2, following the framework of Jackson and Kremer (2001) by reinterpreting cohort q ∈ Qn as the number of units each bidder can purchase. Let Qn be
a finite set.
Imagine a uniform price auction populated with n bidders, each of whom can purchase
up to Kn units. The marginal utility of consuming q-th unit is uq (si , v) = wq (si ) + λv
22
IN-KOO CHO
where wq (si ) is strictly increasing in si and strictly continuously decreasing in q. A bidding
strategy of bidder i is bi = (bi,q ) where
bi,q : Si → R
is player i’s bid for q-th unit. Without loss of generality, assume that bi,q (si ) is a decreasing
function of q for ∀i and ∀si . Let bi (si ) = (bi,q (si ))q∈Q be the bid profile of player i
conditioned on si . We call bi the bidding function of player i, or sometimes, the demand
curve of player i.
After collecting n demand curves, the auctioneer constructs the aggregate demand curve
and chooses as the delivery price the highest price among those which induce non-negative
excess demand, which is the highest losing bid. We normalize the total “quantity” of
demand into 1 so that
Qn ⊂ [0, 1].
If bidder i wins ki units, his expected payoff is
⎡
⎤
E⎣
uq (si , v) − b( +1 ) | si ⎦
q∈Qn ,q≤ki
nKn
where is the total number of units for sale. In the limit game, we have to replace the
summation by the integration. Following Jackson and Kremer (2001), we assume that
if b̃i,q (si ) ≥ bi,q (si ) for some q, then b̃i,q (si ) ≥ bi,q (si ) for ∀q so that we can “rank” the
bidding strategies of a player unambiguously. Define ΣM
i as the set of bidding functions
satisfying
bi,q (si ) − bi,q (si )
1
∀si = si , ∀q ∈ [0, 1].
≥
si − si
M
The main difference of the decision problem from the single unit demand model analyzed
in the previous section is that bidder i can influence the delivery price if ∃q such that bi,q (si )
is a pivotal bid:
b( −q+1 ) ≤ bi,q (si ) ≤ b( −q ) .
(n−1)Kn
(n−1)Kn
Thus, bidder i has incentive to lower the losing bid as an attempt to lower the delivery
price. However, under Assumption 2.5, every bid is a winning bid with a positive probability. The positive probability of winning renders the excessively aggressive or conservative
bids to a weakly dominated strategies.
M , ∀i, ∀s , ∀q ∈ Q ,
Proposition 5.1. ∀bi ∈ Rn,i
i
n
uq (si , v(1) (si )) ≥ bi,q (si ) ≥ uq (si , v(0) (si )).
Proof. See Appendix C.
As we have observed in the double auctions, the probability that a bid becomes a pivotal
bid vanishes to 0 as the number of bidders increases, as long as every player is using a
strictly increasing bidding function. Thus, in the limit game, the decision problem of a
bidder in the multi-unit auctions is essentially to maximize the profit from the individual
bid, ignoring that each bid can influence the delivery price. Hence, in the limit game, any
rationalizable bi,q is characterized by the same way as the rationalizable bid by bidder i in
cohort q in the unit demand uniform price auction populated with ex ante heterogeneous
bidders analyzed in the last section. By invoking exactly the same analysis, we obtain the
information aggregation result as well as the convergence result.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
23
b(α) (v)
b(α) (v)
u(si , v)
b(α) (v(α) (si ))
u(si , v)
u(si , v(α) (si )) = b(α) (v(α) (si ))
b(α) (v(α) (si )) = u(si , v(α) (si ))
v(α) (si )
v(α) (si )
v
Figure 1: b(si ) = b(si ) for some si < si
Proposition 5.2. ∃ω > 0 such that ∀v,
b(α) (v) ≥ ω.
Theorem 5.3. ∃λ , M > 0 such that ∀λ ∈ [0, λ ), ∀M > M , ∀i, RiM is singleton.
6. Models with Large Common Value Component
To see how we can obtain the information aggregation result for models with a large
common value component, let us re-visit the uniform price auction populated with ex ante
symmetric players. If λ > 0 is large, then we can no longer establish the cobweb properties
between b(α) and bi . However, if the agents are ex ante identical, then bi and bi must be
identical for ∀i. Hence, ∀i, ∀v,
b(α) (v) = bi (s(α) (v))
b(α) (v) = bi (s(α) (v)).
Since s(α) (v) is 1-1, ∀si ,
s(α) (v(α) (si )) = si .
Thus, ∀i, ∀si ,
(6.23)
b(α) (v(α) (si )) = bi (si )
(6.24)
b(α) (v(α) (si )) = bi (si ).
Choose M sufficiently large so that b∗i (b−i ) = br (b−i ). If all bidders are ex ante identical,
then we have to solve 4 simultaneous equations: (2.13), (2.14), (6.23) and (6.24). Note
that w(α) defined as (2.16) is a solution of the equations. Our task is to show that in the
limit game, w(α) is the only solution.
Theorem 6.1. In the symmetric uniform price auctions, ∃λ , M > 0 such that ∀λ ≥ λ ,
∀M > M , ∀i, RiM = {w(α) } where w(α) (si ) = u(si , v(α) (si )).
Proof. Choose M such that ∀M ≥ M , b∗i (b−i ) = br (b−i ) in the limit. Choose λ such
that ∀λ > λ ,
(6.25)
w(si ) + λv(α) (si ) = w(si ) + λv(α) (si )
∀si = si .
(s ) are continuous over a compact set, such λ > 0 exists.
Since w (si ) and v(α)
i
24
IN-KOO CHO
We prove the theorem by way of contradiction. Suppose that ∃i, ∃si ∈ Si such that
bi (si ) > bi (si )
or equivalently,
(6.26)
b(α) (v(α) (si )) > b(α) (v(α) (si ))
as depicted in Figure 1. Recall that
bi (si ) ≥ w(α) (si ) ≥ bi (si ).
Under the hypothesis of the proof, at least one inequality must hold with a strict inequality.
By the definition,
−1
bi (si ) = u(si , b(α)
(bi (si )))
(6.27)
and
−1
bi (si ) = u(si , b(α) (bi (si ))).
Since w(α) (si ) = u(si , v(α) (si )), (6.26) implies that
−1
−1
(bi (si )) ≥ v(α) (si ) ≥ b(α) (bi (si ))
b(α)
with at least one strict inequality in the above inequalities. Let us consider the case where
−1
v(α) (si ) > b(α) (bi (si )).
The remaining case follows from a symmetric argument.
By Assumption 2.3, ∀v, ∃si and si such that v(α) (si ) = v and
−1
b(α) (bi (si )) = v(α) (si ).
(6.28)
Since v(α) (si ) > v(α) (si ), si > si . By the choice of si ,
u(si , v(α) (si )) = bi (si ).
From (6.28),
bi (si ) = b(α) (v(α) (si )).
By (6.23),
bi (si ) = b(α) (v(α) (si ))
which implies
bi (si ) = u(si , b−1
(b (s ))) = u(si , v(α) (si ))
(α) i i
because of (6.27). Thus,
(6.29)
which contradicts (6.25).
u(si , v(α) (si )) = u(si , v(α) (si ))
Q.E.D.
Because the asymmetry between sellers and buyers in k-double auctions vanishes in the
limit, Theorem 6.1 is easily extended to symmetric k-double auctions with a large common
value component, where every buyer shares the same characteristics, and so does every
seller.
Theorem 6.2. In the symmetric k-double auctions, ∃λ , M > 0 such that ∀λ ≥ λ ,
∀M > M , ∀i, RiM = {w(0.5) } where w(0.5) (si ) = u(si , v(0.5) (si )).
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
25
7. Concluding Remarks
In a large private value double auction, the area below the “demand” curve and above
the “supply” curve constructed from the bids represents the social surplus realized from
the trading. Because some common value auctions are observationally equivalent to private value models (Laffont and Vuong (1996)), the area may over-estimate the social
contribution of a particular trading mechanism. In order to properly evaluate the social
contribution of an auction mechanism, we have to find a way to decompose the total revenue from an auction into the private and common value components, because the social
contribution of an auction mechanism is generated from the private value component.
Recent experimental studies show the impact of the private and common value components in auctions in which the experimenter can observe and control the “relative contribution” of each component to utility (Goeree and Offerman (2002)). Roughly speaking, our
exercise is to reverse engineer an estimate of the amount which each component contributes
to the observed outcome.
To describe our main idea, let us focus on the k-double auction populated with ex ante
identical bidders which has both private and common value components. Assume that
every player has a linear utility function u(si , v) = si + λv where 0 ≤ λ ≤ ∞ so that both
pure private and pure common value models are covered. Imagine that the government
has just introduced a new market based upon the k-double auction, and many sellers and
buyers have participated in the market to generate substantial trading volume. Our task
is to evaluate the social contribution of the new market.
For a moment, let us assume that we know the true value of λ in order to illustrate
how to isolate the net social gains of trading from the total revenue generated by the
auction. From the analysis of the k-double auction, we know that any rationalizable
bidding strategy must converge to b(si ) = si + λv(0.5) (si ). If the true common value is v ∗ ,
then the delivery price is
b(s(0.5) (v ∗ )) = s(0.5) (v ∗ ) + λv ∗ .
Let q ∗ be the normalized quantity of trade: the proportion of buyers who succeed in
purchasing a good. Since s(0.5) (v) is strictly increasing in v, we can infer the true common
value v ∗ by observing the market clearing price in the limit. Because the bidding function
b(si ) is strictly increasing in si , the trading volume in the limit auction is bounded away
from 0. At the same time, the delivery price reveals the true common value of the object.
If the true common value v ∗ were known, then each bidder’s bidding strategy in the
limit auction should have been
b∗ (si ) = si + λv ∗
instead of b(si ). This induces a new aggregate demand D∗ (p) and supply S ∗ (p) curves. If
every buyer uses b∗ (si ), then the aggregate demand curve is flatter than D(p) induced by
b(si ), but passes through (q ∗ , b(s(0.5) (v ∗ ))). Similarly, the aggregate supply curve passes
through the same point, but is flatter than S(p) induced by b(si ). We believe that the
area below D∗ (p) and above S ∗ (p) properly measures the social gains from trading.
26
IN-KOO CHO
Appendix A. Proof of Lemma 2.7
We only prove the first part, because the second part follows from the same logic. Recall that bi is strictly
increasing and u(si , v(1) (si )) is a continuous function. Notice that if bi (si ) > u(si , v(1) (si )) for some si , then
there is > 0 such that ∀si ∈ [si , si + ], bi (si ) > u(si , v(1) (si )). Suppose that bi (si ) is a winning bid and
the delivery price is above u(si , v(1) (si )), and therefore generates a negative payoff. Then, player i is better
off by placing the bid u(si , v(1) (si )), because he will either lose and get payoff 0, or still win with the same
(negative) payoff. Thus, for ∀b−i , Πi (bi , b−i ) ≤ Πi (w(1) (si ), b−i ) where w(1) (si ) = u(si , v(1) (si )), while a
strict inequality holds if player i wins the object with a positive probability conditioned on si ∈ [si , si + ]
by placing bi (si ).
Appendix B. Proof of Proposition 2.10
B.1. Preliminaries. Given a profile of strictly increasing bidding strategies,
b1 , . . . , b n ,
we shall construct a new profile of bidding strategies that can be “ranked.” Recall that every bidding
strategy is defined over the same domain. Define
p( 1 ) , . . . , p( nn )
n
as follows:
p( 1 ) (s) = max {b1 (s), . . . , bn (s)}
n
which is the collection of the most aggressive bids. Given p( 1 ) , . . . , p( k ) are defined, define
n
n
%
&
p( k+1 ) (s) = max {b1 (s), . . . , bn (s)} \ p( 1 ) (s), . . . , p( k ) (s) .
n
n
n
Clearly,
p( k ) ∈ ΣM
i
n
for ∀i, ∀n. By the construction,
p( k+1 ) (s) ≤ p( k ) (s).
n
n
We can write
p( k+1 ) ≤ p( k ) .
n
n
Following the convention of notation of this paper, we write p(r) (s) in the limit game if
k
→ r.
n
Note that the construction process only changes the “ownership” of the bid from bidder i to a hypothetical
bidder who is ranked according to how aggressive the bidding strategies are. Let us endow the uniform
distribution over the set of players. Thus, ∀b ∈ R,
%
&
P ({(s, i) | bi (s) ≤ b}) = P (s, k) | p( k ) (s) ≤ b .
n
Thus, in the limit game,
P ({(s, i) | bi (s) ≤ b}) = P
(s, r) | p(r) (s) ≤ b
.
Since {p(r) } preserves the order statistics of {bi }, it suffices to prove the statement with respect to {p(r) }.
B.2. Weakly Increasing b(α) (v). Since p(r) is a strictly increasing function, F (p−1
(r) (b) | v) is a continuous
function of b. Thus, ∀v, there exists b(α) (v) such that
1 − α = F (b−1
i (b(α) (v)) | v)di.
−1
If ṽ > v, then F (b−1
i (b) | ṽ) ≤ F (bi (b) | v). Thus, if
1 − α ≥ F (b−1
i (b) | v)di,
then
1−α≥
F (b−1
i (b) | ṽ)di.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
27
Thus,
b(α) (ṽ) ≥ b(α) (v).
While it is straightforward to prove that b(α) (v) is weakly increasing, we need some preliminary steps to
prove that it is strictly increasing.
B.3. Existence of an Interior Solution. By the definition,
1 1
P p(r) (si ) ≤ b(α) (v) | v dr =
F p−1
1−α =
(r) (b(α) (v)) | v dr.
0
0
Since b(α) (v) is weakly increasing and therefore, almost everywhere differentiable, we take the total derivative to obtain
⎤
⎡
1 f p−1 (b(α) (v)) | ṽ
1 ∂F p−1 (b(α) (v)) | ṽ (r)
(r)
dr + ⎣
dr ⎦ b(α) (v) = 0
∂ṽ
p
(s)
ṽ=v
0
0
(r)
where p(r) (s) = b(α) (v). Note that such s exists for some r, if and only if
0 < F p−1
(r) (b(α) (v)) | v < 1
which is equivalent to
p(r) (s(0) (v)) < b(α) (v) < p(r) (s(1) (v)).
It is possible that for r = 0, the right inequality might be violated. Similarly, for r = 1, the left inequality
may not hold. Still, we can show that there must exist at least one r that satisfies both inequalities.
Lemma B.1. ∀v, ∃r(v) ∈ (0, 1) such that
p(r) (s(0) (v)) < b(α) (v) < p(r) (s(1) (v)).
Proof. We know that b(α) (v) is weakly increasing in v, with possible jump discontinuities where b(α) (v)
may have multiple values. The lack of continuity of b(α) (v) or p(r) (s) makes the proof considerably more
complicated.
Case 1. {(s, r) | p(r) (s) ≥ b(α) (v)} and {(s, r) | s ≥ s(α) (v)} differ only by a null set:
P (s, r) | p(r) (s) ≥ b(α) (v), s < s(α) (v) ∪ (s, r) | p(r) (s) < b(α) (v), s ≥ s(α) (v) = 0.
Since the joint probability distribution over [s(0) (v), s(1) (v)] × [0, 1] has no atom, we have the case in which
the boundary of
{(s, r) | p(r) (s) ≥ b(α) (v)}
is equal to
{(s, r) | s = s(α) (v)}
which is the boundary of
{(s, r) | s ≥ s(α) (v)}.
Hence, ∀r, ∀ > 0,
p(r) (s(α) − ) < b(α) (v) < p(r) (s(α) + )
since p(r) (s) is a strictly increasing function of s. Thus, ∀r,
p(r) (s(0) (v)) < b(α) (v) < p(r) (s(1) (v))
since
s(0) ≤ s(α) ≤ s(1) .
Case 2. Let us consider the remaining cases where the boundaries of
{(s, r) | p(r) (s) ≥ b(α) (v)}
and
{(s, r) | s ≥ s(α) (v)}
must differ. Define
r α (v) = inf r | p(r) (s(α) (v)) < b(α) (v) .
We claim that
r α (v) > 0
∀v.
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IN-KOO CHO
Suppose that r α (v) = 0 for some v. Then, ∀r > 0,
p(r) (s(α) (v)) < b(α) (v).
Unless Case 1 holds, the inequality implies that the small neighborhood around the borderline {(s, r) | s =
s(α) (v)} is inside of the area where
{(s, r) | p(r) (s(α) (v)) < b(α) (v)}.
Since the joint probability distribution over [s(0) (v), s(1) (v)] × [0, 1] has a full support over its domain,
1 − α = P {(s, r) | p(r) (s(α) (v)) ≤ b(α) (v)}
≥ P {(s, r) | p(r) (s(α) (v)) < b(α) (v)}
> P {(s, r) | s < s(α) (v)} = 1 − α
which is impossible. Thus, r α (v) > 0.
Similarly, define
rα (v) = sup r | p(r) (s(α) (v)) > b(α) .
Following a similar logic as above, we can show that
r α (v) < 1
∀v.
Next, we claim that
∀v
r α (v) ≤ r α (v)
from which the conclusion of the lemma follows. Suppose that ∃v such that
r α (v) > r α (v).
Consider
rα (v) > r =
r α (v) + r α (v)
> r α (v).
2
The first and the last inequalities imply that
p(r ) (s(α) (v)) > b(α) (v) > p(r ) (s(α) (v))
which is impossible. Clearly, ∀r ∈ [r (α) (v), r (α) (v)],
p(r) (s(0) (v)) < b(α) (v) < p(r) (s(1) (v))
as desired.
B.4. Uniform Lower Bound of the Slope. Define
r(v) = inf r | p(r) (s(0) (v)) ≤ b(α) (v)
and
r(v) = sup r | p(r) (s(1) (v)) ≥ b(α) (v) .
Since
p(r) (s(0) ) > p(r) (s(α) ) > p(r) (s(1) ),
r(v) ≥ r α (v) ≥ r α (v) ≥ r(v).
(B.30)
Clearly, ∀r ∈ [r(v), r(v)],
p(r) (s(0) (v)) < b(α) (v) < p(r) (s(1) (v)).
Lemma B.2. ∃∆ > 0 such that ∀v, r(v) − r(v) ≥ ∆ or b(α) (v) ≥ ∆.
Proof. Suppose the contrary. Since [v, v] is compact, we can find v ∗ such that
r(v ∗ ) = r(v ∗ )
and
(B.31)
By (B.30),
Since p(r∗ )
b(α) (v ∗ ) = 0.
r(v ∗ ) = r(v ∗ ) = rα (v ∗ ) = rα (v ∗ ) = r ∗ .
is strictly increasing, the definitions of r α (v ∗ ) and r α (v ∗ ) imply that
p(r∗ ) (s(1) (v ∗ )) < b(α) (v ∗ ) < p(r∗ ) (s(0) (v ∗ )).
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
29
Moreover, ∀r > r ∗ = r(v ∗ ),
p(r) (s(0) (v ∗ )) ≤ b(α) (v ∗ )
and ∀r < r ∗ ,
Thus,
p(r) (s(1) (v ∗ )) ≥ b(α) (v ∗ ).
P ({(s, r) | r ≥ r ∗ }) = P
∗
Hence, r = α. Since
ω̂ > 0 such that ∀v,
p(r∗ )
≥ 1/M and
s(α) (v)
(s, r) | p(r) (s) ≤ b(α) (v ∗ )
= 1 − α.
is continuous over a compact interval [v, v], there exists
p(r∗) (s(0) (v))s(0) (v) >
ω̂
M
and
ω̂
.
M
∗
Thus, if b(α) (v ) = 0, then there exists a small h > 0 such that
p(r∗) (s(1) (v))s(1) (v) >
r(v ∗ + h) > r(v ∗ )
and
r(v ∗ + h) > r(v ∗ ).
Recall that r(v) ≤ r(v) for ∀v. Since the joint probability distribution over (s, r) has a full support over
its domain,
1−α = P (s, r) | p(r) (s) ≤ b(α) (v ∗ + h) ≤ P ({(s, r) | r ≥ r(v ∗ + h)}) < P ({(s, r) | r ≥ r(v ∗ ) = r ∗ }) = 1−α
Q.E.D.
which is impossible.
B.5. Main Proof. Suppose that
r(v) − r(v) ≥ ∆.
Recall that
∂F p−1
(r) (b(α) (v)) | ṽ −
dr
∂ṽ
ṽ=v
0
.
b(α) (v) =
1 f p−1 (b(α) (v)) | v
(r)
dr
p(r)
0
1
Let us check the denominator first. Since p(r) ∈ ΣM
i ,
p(r) ≥
1
.
M
Since f (·) is continuous over a compact set, it is uniformly bounded from above, say f . Thus, the denominator is not larger than M f .
Next, let us check the nominator. Note that if r(v) > r > r(v), then ∃ω > 0 such that ∀v,
∂F p−1
(r) (b(α) (v)) | ṽ > ω
−
∂ṽ
ṽ=v
by Assumption 2.1. Hence,
1 ∂F p−1 (b(α) (v)) | ṽ (r)
−
dr ≥ ∆ω .
∂ṽ
ṽ=v
0
Combining the two observations, we conclude that if r(v) − r(v) ≥ ∆, then
b(α) (v) ≥
By Lemma B.2,
∆ω .
Mf
(
'
∆ω .
b(α) (v) ≥ min ∆,
Mf
The conclusion of the proposition follows from the observation that the right hand side is independent of
λ.
30
IN-KOO CHO
Appendix C. Proof of Proposition 5.1
Suppose that there exists q̃(si ) > 0 such that
bi,q̃ (si ) > uq̃ (si , v(1) (si )).
Define
q(si ) = inf{q | bi,q (si ) > uq (si , v(1) (si ))}
and
q(si ) = sup{q | bi,q (si ) > uq (si , v(1) (si ))}.
Since uq (si , v(1) (si ) is continuous in si ,
q(si ) < q̃ ≤ q(si ).
Construct an alternative bidding strategy
uq (si , v(1) (si )) if q(si ) ≤ q ≤ q(si )
b̃i,q (si ) =
otherwise.
bi,q (si )
Since uq (·) is continuous in q and strictly decreasing in q,
bi,q(si ) (si ) = uq(si ) (si , v(1) (si )).
Since both bi (·) and uq (·) are feasible strategies, b̃i ∈ ΣM
i .
Since b̃i,q (si ) ≤ bi,q (si ), player i cannot win more units by b̃i than bi . If bidder i wins the same unit
conditioned on si , then the expected payoff is the same under the two bidding strategies. If b̃i (si ) brings
fewer units than bi (si ), then the delivery price, say p, should satisfy that ∃q ∈ (q(si ), q(si )) such that
bi,q (si ) > p > uq (si , v(1) (si )).
The continuity of uq (·) with respect to q implies that around a small open neighborhood of q, the right
inequality must continue to hold. For each lost q-th unit, bidder i saves the loss of
uq (si , v) − p ≤ uq (si , v(1) (si )) − p < 0.
Thus, the expected payoff from b̃i (si ) is higher, which implies that b̃i weakly dominates bi . We can apply
a similar argument to prove that any bid below uq (si , v(0) (si )) for some q and si is weakly dominated.
RATIONALIZABILITY AND MONOTONICITY IN LARGE UNIFORM PRICE AND DOUBLE AUCTIONS
31
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Department of Economics, University of Illinois, 1206 S. 6th Street, Champaign, IL 61820
USA
E-mail address: inkoocho@uiuc.edu
URL: http://www.cba.uiuc.edu/inkoocho