Review of Economic Studies (2006) 73, 1085–1111
c 2006 The Review of Economic Studies Limited
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0034-6527/06/00421085$02.00
Sequentially Optimal Mechanisms1
VASILIKI SKRETA
University of California
First version received November 2002; final version accepted December 2005 (Eds.)
This paper establishes that posting a price in each period is a revenue-maximizing allocation mechanism in a finite 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 non-commitment that is valid irrespective of the
structure of the agent’s type.
1. INTRODUCTION
This paper establishes that posting a price in each period is a revenue-maximizing allocation
mechanism in a finite 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 finite 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
offers 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.2
Without commitment the seller can optimally choose 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 loses all her monopoly profits. In the
durable good monopoly literature,3 the seller behaves sequentially rationally, but her strategy is
restricted to be a sequence of posted prices.4 The same is true in the bargaining under incomplete
1. This work is based on Chapter 1 of my doctoral thesis submitted to the Faculty of Arts and Sciences at the
University of Pittsburgh.
2. Real-world examples about the inability of the sellers to commit can be found in McAfee and Vincent (1997).
3. See Stokey (1979), Bulow (1982), and Gul, Sonnenschein and Wilson (1986).
4. McAfee and Vincent (1997) examine a closely related problem, where there are finitely 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.
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information literature,5 where the proposer makes deterministic offers at each round.6 Often, the
right to make offers is assigned to one of the negotiating parties. Suppose that the uninformed
one makes the offers. Would it be beneficial for her instead of making a take-it-or-leave-it offer
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 monopoly
literature7 and for take-it-or-leave-it offers 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 find 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 finite. 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 finite or a continuum or any measurable bounded subset of the real line.8 This is the first
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 set-up considered in this paper. We describe players and
their pay-offs and the timing of the game; we then define mechanisms, assessments, social choice
functions, outcomes of the game, what we mean by implementation, and finally our equilibrium
concept.
A risk-neutral seller owns a unit of an indivisible object.9 Her valuation for the object is
normalized to 0. 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].10 Time is discrete and
the game lasts T periods, t = 1, . . . , T < ∞. Both the seller and the buyer discount the future with
the same discount factor, δ. 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.
2.1. 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 mechanism
is played, and if the buyer obtains the object the game ends, else we move on to period
t = 2.
5. See, for instance, Sobel and Takahashi (1983) and Fudenberg, Levine and Tirole (1985).
6. 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.
7. 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.
8. If one wants to depart from the finite 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 finite type space, the type
space at the beginning of a subsequent period is again finite). And since the game that starts at t = 2 is isomorphic to the
whole game, with the difference that it lasts one period less, we write our model directly for arbitrary type spaces.
9. Female pronouns are used for the seller and male for the buyer.
10. All the analysis goes through, if the convex hull of V is [a, b], where −∞ < a < b < ∞.
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SEQUENTIALLY OPTIMAL MECHANISMS
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• 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.
We continue by defining what we mean by the word “mechanism”.
2.2. 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 ∈ St into outcomes. An outcome of a
mechanism is a probability that the buyer obtains the object at period t,rt (st ) ∈ [0, 1] and an
expected payment z t (st ) ∈ R. A pair (rt (st ), z t (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
pay-off of 0. We include the choice of non-participation in the definition of each mechanism, by
assuming that there is an option s ∈ st such that (r (s), z(s)) = (0, 0). Now we can move on and
talk about strategies and assessments.
2.3. Assessments
An assessment consists of a strategy profile and a belief system. A strategy profile σ = σi where
i is either S or B, specifies 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. An element of IS , (IB ) is denoted by i S , (i B ). A strategy for the seller, σS , is a
sequence of maps from IS to M. A behavioural strategy of the buyer, σB , consists of a mapping
from V × IB to a probability distribution over actions, #(St ). A belief system, µ, maps IS to the
set of probability distributions over V . Let F(v | i St ) denote the seller’s beliefs about the buyer’s
valuation at information set i St at period t, t = 1, . . . , T . Sometimes, when we are referring to a
particular information set, we simplify the notation by setting F(v | i St ) = Ft (v), t = 1, . . . , T .11
For t = 1, we have F(v | i S1 ) = F(v), that is, the seller has correct prior beliefs. A strategy profile
σ and a belief system µ is an assessment.
2.4. 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(σ, µ) and a payment rule x(σ, µ). The rule p(σ, µ)(v) is the expected,
discounted probability that a v-type buyer will obtain the object given the assessment (σ, µ),
! T
#
"
p(σ, µ)(v) = E
δ t 1{trade at t} | (σ, µ), v ,
t=0
and x(σ, µ)(v) is the expected, discounted payment that a v-type buyer will incur given (σ, µ),
#
! T
"
t
δ 1{no trade up to t} {expected payment at t} | (σ, µ), v .
x(σ, µ)(v) = E
t=0
It is possible that different strategy profiles lead to the same allocation and payment rules.
11. But the reader should keep in mind that there are many different histories that lead to no trade up to period t,
and for each such history there is, in general, a different posterior.
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It will be useful to define the outcomes of assessments at a continuation game that starts
at t when the seller’s information set is i St . The situation is isomorphic to the game as a whole,
with the only difference that the seller’s beliefs are now given by Ft (· | i St ). Then we can talk
about the allocation rule pt,i t (σ, µ) and the payment rule xt,i t (σ, µ) that arise by the assessment
S
S
(σ, µ) at the continuation game that starts at t, when the seller’s information set is i St . They are
the analogues of p(σ, µ) and x(σ, µ) for the particular continuation game. As before, we will
often suppress the notation (σ, µ).
In order to talk about implementation we define social choice functions.
2.5. Social choice functions
In the environment under consideration, a social choice function specifies, for each valuation of
the buyer v and each period t, a probability of trade rt (v) ∈ [0, 1] and an expected payment z t (v) ∈
T
R. Given a social choice function {rt (v), z t (v)}t=1
we can define the corresponding allocation
rule p and payment rule x by
p(v) = r1 (v) + (1 − r1 (v))δ[r2 (v) + (1 − r2 (v))δ[· · · ]] and
x(v) = z 1 (v) + (1 − r1 (v))δ[z 2 (v) + (1 − r2 (v))δ[· · · ]].
Allocation rules map valuations to expected discounted probabilities of trade, and payment rules
map valuations to expected discounted payments. The expected pay-off of the seller and buyer,
respectively, from the interim point of view, are given by
$
Expected pay-off of the seller = x(v)dF(v) and
V
Expected pay-off of the type-v buyer = p(v)v − x(v),
where the integral in the expression of the seller’s expected pay-off comes from the fact that the
seller does not know the valuation of the buyer. There are many different social choice functions
T that lead to the same p(v) and x(v) and hence to the same pay-offs to the buyer
{rt (v), z t (v)}t=1
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 specified what we mean by a social choice function we can talk about
implementation.
An assessment (σ, µ) implements the (reduced) social choice function p and x if for all
v ∈ V, p(σ, µ)(v) = p(v) and x(σ, µ)(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.
2.6. Solution concept
A Perfect Bayesian Equilibrium (PBE), is a strategy profile, σ , and a belief system, µ, that satisfy
1. For all v ∈ [0, 1] the buyer’s strategy is a best response at each i Bt and t, given the seller’s
strategy.
2. Given Ft (· | i St ) and the buyer’s strategy, the seller chooses at each i St and t an optimal
sequence of mechanisms for the remainder of the game.
3. Ft (· | i St ) is derived from Ft−1 given i St using Bayes’ rule whenever possible.
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A PBE requires strategies to dictate optimal behaviour at each information set. The nature of
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. We assume that the seller observes the action that the buyer chooses, s, as well as whether
trade takes place. What the seller observes at each stage is called the degree of transparency of
mechanisms. It determines, in some sense, her commitment power.12
Our objective is to find an assessment that is a PBE and guarantees highest expected revenue
for the seller among all PBEs. 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 PBEs.
The following section describes the technical difficulties of the problem at hand, as well as
our solution method.
3. TECHNICAL DIFFICULTIES AND THE PROCEDURE
In this section we explain why one cannot employ the revelation principle in order to characterize
a revenue-maximizing PBE and how we sidestep this difficulty.
When the mechanism designer behaves sequentially rationally, information revelation is
very difficult 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.13 The earlier
literature on mechanism design without commitment (Freixas, Guesnerie and Tirole, 1985; Hart
and Tirole, 1988; Laffont and Tirole, 1988, 1993) observed that, with the exception of the final
period of the game, one cannot use the standard revelation principle in order to find 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 Freixas
et al. (1985) characterize the optimal incentive schemes among the class of linear incentive
schemes. Laffont and Tirole (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 finite, 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 difficult 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 find an optimal mechanism one has to check the incentive compatibility constraints that are binding. In an
environment with limited commitment, constraints may be binding “upwards” and “downwards”.
Even if one could obtain an analogue of the BS result for the continuum type case, which is
indeed challenging, it does not seem straightforward to generalize the procedure of checking
12. If the seller does not observe anything, then all BNEs are PBEs and the “commitment solution” is sequentially
rational. In Skreta (2005c) we establish that our result is robust to the case where the seller observes only whether trade
took place or not.
13. To see why, suppose that at period 1 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 different 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 Laffont and Tirole (1988), where they show that there is no PBE where each type of the
buyer chooses a different action at t = 1. Hence truth telling with probability 1 cannot occur at any PBE at t = 1.
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those incentive compatibility constraints that 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 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 infinitely many choices
at t that end up in no trade, since lotteries are allowed. Each of these choices may lead to a different 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 up to t, which in turn will determine the optimal
mechanism for t + 1 and so on. And in order for someone to find a revenue-maximizing PBE, one
has to find all PBEs of the game, then calculate the corresponding revenue, and finally compare
the seller’s revenue at each of these PBEs. A paper that looks for equilibria is Laffont and Tirole
(1988). That paper looks for PBEs of a two-period regulation game without commitment, where
the firm’s type is drawn from a continuum. Laffont and Tirole (1988) provide some properties of
equilibria, but they do not obtain a characterization of an optimal equilibrium.14 Indeed, looking for equilibria is very difficult even in our model, which is in some dimensions simpler than
Laffont and Tirole (1988).
We are able to sidestep the difficulties that arise in the analysis of Laffont and Tirole (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(σ, µ)s and x(σ, µ)s that are implemented by assessments that are PBEs, and
then choose the ones that the seller prefers. This idea was inspired from Riley and Zeckhauser
(1983). Obviously the set of PBE-implementable allocations and payment rules is a subset of
the BNE-implementable ones. In order to characterize this subset, we examine what restrictions the requirement that (σ, µ) be a PBE implies on p and x. In this way we obtain allocation rules and payment rules that satisfy necessary conditions of being PBE implementable. We
then choose the p and x that the seller likes best among those that satisfy the necessary conditions of being PBE implementable and verify that there exists indeed a PBE that implements
them. This is in line with the approach of the revelation principle, which prescribes a way to
obtain all the set of BNE-implementable allocation and payment rules: “just look at the social
choice functions that can be implemented by truth-telling 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. This alternative route in the case that we are
interested in BNE-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
PBE-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.
14. First they show that there is no separating equilibrium, that is, there exists no equilibrium where each type of
the agent chooses a different action at t = 1. Then they consider the situation where the uncertainty about the agent’s
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 0, as the length of the interval of possible types
goes to 0. They also show that there exist other equilibria that have the property that the distortion compared to the
commitment case goes to 0 as the length of the interval of possible types goes to 0. These equilibria exhibit, what they
call infinite 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.
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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
$
max x(v) dF(v)
p,x
V
subject to p, x being PBE 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) ∈
[0, 1] for all v. Second, at a PBE, 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
benefit by behaving as type v ' does. For the seller, this implies that at each information set her
strategy specifies 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 pay-off for the continuation of the game must be non-negative. It follows that if p and x are implemented by a PBE they must, at the very least, satisfy the constraints
of the following program:
$
max x(v)dF(v)
p,x
V
subject to
IC “incentive constraints”, the buyer’s strategy is such that
p(v)v − x(v) ≥ p(v ' )v − x(v ' ); for all v, v ' ∈ V
PC “voluntary participation constraints”,
p(v)v − x(v) ≥ 0 for all v ∈ V
RES “resource constraints” for all v ∈ V 0 ≤ p(v) ≤ 1.
SRC (t, i St ) “sequential rationality constraints”, for all t, t = 2, . . . , T , and for each information set of no trade at t, i St , the seller chooses a mechanism that maximizes revenue:
$
xt,iSt (v)dFt,i t (v)
max
pt,i t ,xt,i t
S
S
S
Yt,i t
S
subject to
IC (t, i St ) pt,iSt (v)v − xt,i t (v) ≥ pt,i t (v ' )v − xt,i t (v ' ), for all v, v ' ∈ Yt,i t
S
S
S
S
PC (t, i St ) pt,i t (v)v − xt,i t (v) ≥ 0 for all v ∈ Yt,i t
S
S
S
RES (t, i St ) “resource constraints” 0 ≤ pt,i t (v) ≤ 1 for all v ∈ Yt,i t ,
S
S
where Yt,i t is the support of the posterior Ft,i t .
S
S
Beliefs, posterior beliefs Ft,i t are derived using the buyer’s strategy and Bayes’ rule whenS
ever possible.
Summarizing, if p, x are implemented by an assessment (σ, µ) that is a PBE, then conditions
IC, PC, RES, and SRC will be satisfied. These are necessary conditions for p and x to be PBE
implementable. In what follows, we will obtain a solution, p and x, and establish that there
exists indeed an assessment that is a PBE 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
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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 over time. By induction one can show that
the same result holds for any T < ∞.15 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 effect of sequential rationality on the solution. It is also the
problem that the seller solves at the beginning of the final period of the game when we impose
the sequential rationality constraints.
Without sequential rationality constraints the seller’s problem reduces to
$
max x(v)dF(v)
p,x
V
subject to
IC
PC
RES
p(v)v − x(v) ≥ p(v ' )v − x(v ' ) for all v, v ' ∈ V
p(v)v − x(v) ≥ 0 for all v ∈ V
0 ≤ p(v) ≤ 1 for all v ∈ V .
We call it Program A. 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 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 i ST ,
pT,i T , and x T,i T solve a problem that is isomorphic to the current one. And because we allow for
S
S
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 artificial 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 artificial 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:
$1
max x(v)dF(v)
p,x
0
15. The complete analysis of the case where T > 2 can be found in Skreta (2005c), which can be downloaded from
the author’s website.
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subject to
p(v)v − x(v) ≥ p(v ' )v − x(v ' ) for all v, v ' ∈ [0, 1]
p(v)v − x(v) ≥ 0 for all v ∈ [0, 1]
0 ≤ p(v) ≤ 1 for all v ∈ [0, 1].
IC
PC
RES
Proposition 1. 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. Then16
R( pA ) = R( pB ).
Before proceeding let us sketch briefly the proof of Proposition 1.17 Programs A and B have
the same objective function since d F is 0 on [0, 1]\V , and they differ 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 x A of Program A appropriately extended on [0, 1] satisfies all the constraints of 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 Uσ,µ (σB (v), v) = p(v)v −
x(v) denote the buyer’s expected discounted pay-off when his valuation is v given the assessment
(σ, µ). From Lemma 2 in Myerson (1981) we have
Lemma 1. If p,x are implemented by a BNE, they satisfy IC, PC, and RES constraints
and the following% conditions must hold: for all v ∈ [0, 1] (a) p(v) is increasing in v, (b)
v
Uσ,µ (σB (v), v) = 0 p(s)ds + Uσ,µ (σB (0), 0), (c) Uσ,µ (σB (0), 0) ≥ 0, and (d) p(v) ∈ [0, 1] for
all v ∈ [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 pay-off that accrues to the lowest type of the buyer18
$1
0
x(v)dF(v) =
$1
0
p(v)vdF(v) −
$1
0
p(v)[1 − F(v)]dv − Uσ,µ (σB (0), 0).
From the analysis in Myerson (1981), we know that if p solves
max
p∈*
$1
0
p(v)vdF(v) −
$1
0
p(v)[1 − F(v)]dv,
(1)
where
* = {p : [0, 1] → [0, 1]: p is increasing},
$v
and x(v) = p(v)v − p(s)ds,
0
then ( p, x) are optimal.
16. The conjecture that such a result may be available arose from discussions with Kim-Sau Chung.
17. More details can be found in the Appendix. See also Skreta (2005a) for full details. &
'
%
18. If F has a strictly positive density, then we obtain the familiar expression 01 p(v) v − [1−F(v)]
f (v)dv −
f (v)
Uσ,µ (σB (0), 0).
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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 0 to 1. The optimal allocation rule has the bang-bang property: the seller trades with
zero probability with types below a cut-off of v ∗ , p(v) = 0 for v ∈ [0, v ∗ ), whereas trades with
probability 1, that is, p(v) = 1 with types v ∈ [v ∗ , 1]. The proposition that follows states this
result and provides a formal definition of the cut-off v ∗ .
Proposition 2. Without sequential rationality constraints the seller maximizes revenue by
posting at each t the same price v ∗ , where19


$ṽ
$ṽ


v ∗ ≡ inf v ∈ [0, 1] s.t. sdF(s) − [1 − F(s)] ds ≥ 0, for all ṽ ∈ [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 Skreta (2005b). The
revenue-maximizing allocation rule without sequential rationality constraints can be implemented
by the following strategy profile. The seller makes a take-it-or-leave-it offer in each period,
t = 1, . . . , T of v ∗ . The buyer’s strategy is as follows: for v ≥ v ∗ the buyer accepts the seller’s
offer at t = 1, and for v < v ∗ the buyer rejects the seller’s offer at t = 1, . . . , T . It is very easy to
see that players’ strategies are mutual best responses hence the given assessment is a BNE.
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
that do 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 final period of the game depends on the posterior (the proof
of Proposition 8 in the Appendix). The definition of an optimal cut-off 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 analyse the case when T = 2. As in the case without sequential rationality constraints,
we first show that it is without loss of generality to solve an artificial program where we replace
the type space with its convex hull. Program B for T = 2 is given by
max
p,x
$1
x(v)dF(v)
0
19. I thank Phil Reny for suggesting parts of the proof of Proposition 2.
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subject to
IC
PC
RES
p(v)v − x(v) ≥ p(v ' )v − x(v ' ); for all v, v ' ∈ [0, 1]
p(v)v − x(v) ≥ 0 for all v ∈ [0, 1]
0 ≤ p(v) ≤ 1 for all v ∈ [0, 1].
SRC (t = 2, i S2 ) “sequential rationality constraints”. For each information set of no trade at
t = 2, i S2 , the seller chooses a mechanism that maximizes revenue:
$
max
x2,i 2 (v)dF2,i 2 (v)
p2,i 2 ,x2,i 2
S
S
S
S
Ȳ2,i 2
S
subject to
IC (t = 2, i S2 ) p2,i 2 (v)v − x2,i 2 (v) ≥ p2,i 2 (v ' )v − x2,i 2 (v ' ),
S
S
S
S
for all v, v ' ∈ Ȳ2,i 2
S
PC (t = 2, i S2 ) p2,i 2 (v)v − x2,i 2 (v) ≥ 0 for all v ∈ Ȳ2,i 2
S
S
S
RES (t = 2, i S2 )0 ≤ p2,i 2 (v) ≤ 1 for all v ∈ Ȳ2,i 2 .
S
S
Beliefs posterior beliefs F2,i 2 are derived using the buyer’s strategy and Bayes’ rule whenS
ever possible.
Program B is exactly the same as Program A for T = 2 but with V and YT,i T replaced by
S
their convex hulls [0, 1] and ȲT,i T , respectively.20
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, i S2 )
extended on Ȳ2,i 2 = [v, v̄] solves Program B at information set i S2 . Therefore, essentially nothS
ing changes on the set of constraints described in SRC (t = 2, i S2 ). 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] satisfies 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
max
p,x
$1
0
p(v)vdF(v) −
$1
0
p(v)[1 − F(v)]dv
(3)
subject to
*[0,1] = {p : [0, 1] → [0, 1]: p is increasing},
$v
x(v) = p(v)v − p(s)ds and
0
p(v) ∈ [0, 1] for all v ∈ [0, 1].
20. There is a slight abuse of terminology because now Program A and Program B include also the sequential
rationality constraints.
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SRC (t = 2, i S2 ) “sequential rationality constraints”. For each history i S2
$
$
max
p2,i 2 (v)vdF2,i 2 (v) −
p2,i 2 (v)[1 − F2,i 2 (v)]dv
p2,i 2 ,x2,i 2
S
S
S
S
Ȳ2,i 2
S
S
Ȳ2,i 2
S
S
subject to
*Ȳ
2,i S2
= {p2,i 2 : Ȳ2,i 2 → [0, 1]: p2,i 2 is increasing},
S
S
x2,i 2 (v) = p2,i 2 (v)v −
S
S
S
$v
p2,i 2 (s)ds and
S
v
p2,i 2 (v) ∈ [0, 1] for all v ∈ Ȳ2,i 2 .
S
S
Beliefs posterior beliefs F2,i 2 are derived using the buyer’s strategy and Bayes’ rule whenever
S
possible.
We now describe properties that ( p, x) satisfy if they are implemented by strategy profiles
that are PBEs.
6.2. Necessary conditions at a PBE
Fix an assessment (σ, µ) that is a PBE 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 PBE under consideration. Let [0, v̄], v̄ ≤ 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 final
period of the game, we know from Proposition 2 that the seller will post a price, which we denote
by z 2 (F2 ).
The proposition that follows describes properties of PBE-implementable ( p, x).
Proposition 4. If ( p, x) are implemented by an assessment that is a PBE they satisfy (i)
Lemma 1 and (ii)
p(v) = r for v ∈ [0, z 2 (F2 ))
p(z 2 (F2 )) ∈ [r,r + (1 − r )δ]
p(v) = r + (1 − r )δ for v ∈ (z 2 (F2 ), v̄)
(4)
p(v̄) ∈ [r + (1 − r )δ, 1]
where v̄ ∈ [0, 1], r ∈ [0, 1], and z 2 (F2 ) solves (2) given F2 , where F2 is a probability measure
that can arise at a PBE and has support a set with convex hull the interval [0, v̄].
Definition 1.
Proposition 4.21
We call P the set of allocation rules that satisfy the properties described in
21. The definition allows for v̄ = z 2 (F2 ) in which case p(v̄) ∈ [r, 1].
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For v ∈ [0, v̄) it is surprising to see that the shape of p is the same as if all types in [0, v̄]
choose action s with probability 1. In that case the price at t = 2 after the buyer chooses s, call it
z 2 (v̄), would have been optimal given

 F(v) , for v ∈ [0, v̄]
F(v̄)
F2 (v) =
.
(5)
0,
otherwise
In general, z 2 (F2 ) is optimally chosen given some posterior F2 whose support has convex
hull [0, v̄], but it is not necessarily a mere truncation of the original distribution. Now, for
types in [v̄, 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 PBE, then it is also BNE implementable, and it satisfies 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 PBE of the game where the seller posts a price in each period.
6.3. Revenue-maximizing PBE
The solution will proceed as follows. First, we restrict attention to allocation rules implemented
by strategy profiles, where the seller at t = 1 employs simple mechanisms that separate types
into two groups: “high” and “low”. We call this set of allocation rules P ∗ . We show that a
revenue-maximizing allocation rule among P ∗ is implemented by a PBE 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 satisfied, at an optimum the seller posts a
price in each period. In the final, and most delicate, step we show that the previously imposed conditions are satisfied by all PBEs 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 PBE among 2-option mechanisms
Here we look for the revenue-maximizing allocation rule among the ones implemented by
strategy profiles 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, z 1 ). We show that at
the optimum this kind of mechanism reduces to a posted price: the options available are (0, 0)
and (1, z 1 ).
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, z 1 ) is targeted to “high” valuation types, where r ∈ [0, 1] and z, z 1 ∈ R.
The buyer behaves as follows. At t = 1 types v ∈ [0, v̄) choose (r, z) and types in (v̄, 1] choose
(1, z 1 ). Type v̄ is indifferent between choosing: (r, z) and choosing (1, z 1 ) 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 z 2 that satisfies
z 2 ≤ z 2 (v̄) (note that this allows for suboptimal behaviour at t = 2 on the part of the seller). The
buyer at t = 2 accepts z 2 for v ∈ [z 2 , v̄) and rejects z 2 for v ∈ [0, z 2 ). This assessment implements
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REVIEW OF ECONOMIC STUDIES
an allocation rule of the form
p(v) = r for v ∈ [0, z 2 )
p(v) = r + (1 − r )δ for v ∈ [z 2 , v̄)
p(v̄) ∈ [r + (1 − r )δ, 1]
(6)
p(v) = 1 for v ∈ (v̄, 1].
Definition 2. We call P ∗ the set of allocation rules having the shape given by (6), for some
r ∈ [0, 1], v̄ ∈ [0, 1]; z 2 ≤ z 2 (v̄), where z 2 (v̄) solves (2) given (5).
Now we move on to establish that a revenue-maximizing element of P ∗ can be implemented
by a PBE of the game where the seller posts a price in each period.
Proposition 5. Let p ∗ denote a solution of max p∈P ∗ R( p). Then p ∗ can be implemented
by a PBE 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, z 1 ), then at an optimum this kind of mechanism reduces to a
posted price: the options available are (0, 0) and (1, z 1 ). Now we proceed to solve the general
problem.
Step 2. If for all p ∈ P it holds z 2 (F2 ) ≤ z 2 (v̄), then posted prices are optimal
Here we assume that for all p ∈ P it holds z 2 (F2 ) ≤ z 2 (v̄). Given this assumption, we show
that a solution of
$1
$1
max p(v)vdF(v) − p(v)[1 − F(v)]dv
(7)
p,x
0
0
subject to
p ∈ P and x(v) = p(v)v −
$v
p(s)ds
0
is an allocation rule that is implemented by a PBE of the game where the seller posts a price in
each period. In particular, we show that each element of P, for which it holds that z 2 (F2 ) ≤ z 2 (v̄),
is dominated in terms of expected revenue by an element of P ∗ . Then our result follows from
Proposition 5.
We will employ the following auxiliary result.
Lemma 2. If z 2 (v̄) solves (2) given (5), then z 2 is increasing in v̄.
We are now ready to state the main result of this step.
Proposition 6. For each p ∈ P with the property that z 2 (F2 ) ≤ z 2 (v̄), there exists an
allocation rule p̃ ∈ P ∗ that generates higher revenue than p.
Proof. We establish that each allocation rule p ∈ P where z 2 (F2 ) ≤ z 2 (v̄) is dominated in
terms of expected revenue by an allocation rule in p̃ ∈ P ∗ . Take an element of p ∈ P, for which
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we have that z 2 (F2 ) ≤ z 2 (v̄). First suppose that 0 < z 2 (F2 ) < v̄. Expected revenue from p is then
given by
z$
2 (F2 )
r vdF(v) −
0
$v̄
+
z 2 (F2 )
+
$1
z$
2 (F2 )
0
r [1 − F(v)]dv
[r + (1 − r )δ]vdF(v) −
$v̄
z 2 (F2 )
p(v)vdF(v) −
v̄
$1
[r + (1 − r )δ][1 − F(v)]dv
p(v)[1 − F(v)]dv.
v̄
We construct an allocation rule p̃, that is an element of P ∗ and it generates higher revenue than p.
For the range [0, v̄) we set p̃(v) = p(v). For types in v ∈ [v̄, 1], we choose p̃ optimally imposing
the constraint that the resulting allocation rule be increasing on [0, 1] and ignoring all sequential
rationality constraints. For types [v̄, 1] it is, in some sense, as if we are solving a “commitment
problem”. Let


$ṽ
$ṽ


v ∗∗ ≡ inf v ∈ [v̄, 1] s.t. sdF(s) − [1 − F(s)]ds ≥ 0, for all ṽ ∈ [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 p̃ equal to its lowest possible value for the types v ∈ [v̄, v ∗∗ ),
which is now r + (1 − r )δ, 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
p̃(v) = r for v ∈ [0, z 2 (F2 ))
p̃(v) = r + (1 − r )δ for v ∈ [z 2 (F2 ), v ∗∗ )
p̃(v) = 1 for v ∈ [v ∗∗ , 1].
Now because v ∗∗ ≥ v̄, from Lemma. 2 we have that z 2 (v̄) ≤ z 2 (v ∗∗ ), where z 2 (v ∗∗ ) is optimal
F(v)
, for v ∈ [0, v ∗∗ ]
. Moreover, since we assumed that
at T = 2 given a posterior F2 (v) = F(v ∗∗ )
0,
otherwise
∗∗
z 2 (F2 ) ≤ z 2 (v̄), we also have z 2 (F2 ) ≤ z 2 (v ). Hence, the resulting allocation rule is an element
of P ∗ . Moreover, by construction, it generates higher revenue for the seller than p.
So far we have assumed that 0 < z 2 (F2 ) < v̄. If 0 = z 2 (F2 ) < v̄ all previous arguments
go through since for v ∈ (z 2 (F2 ), v̄) we have that p(v) = r + (1 − r )δ. Now if z 2 (F2 ) = v̄ it is
possible that p(v̄) < r + (1 − r )δ. Again, all our arguments go through as before, since it turns
out p(v̄ + ε) ≥ r + (1 − r )δ, for ε > 0, arbitrarily small. To see that p(v̄ + ε) ≥ r + (1 − r )δ, we
argue by contradiction. First, observe that type v̄ makes zero surplus at t = 2 since z 2 (F2 ) = v̄,
hence p(v̄)v̄ − x(v̄) = r v̄ − z = [r + (1 − r )δ]v̄ − [z + (1 − r )δz 2 (F2 )]. Now let us consider type
v̄ + ε, and suppose that p(v̄ + ε) < r + (1 − r )δ, then since at a PBE the buyer’s strategy is a best
response, it follows that p(v̄ + ε)(v̄ + ε) − x(v̄ + ε) ≥ [r + (1 −r )δ](v̄ + ε) − [z + (1 −r )δz 2 (F2 )],
but since p(v̄ + ε) < r + (1 −r )δ we have that p(v̄ + ε)v̄ − x(v̄ + ε) > [r + (1 −r )δ]v̄ − [z + (1 −
r )δz 2 (F2 )]. This implies that there exists an interval of types (ṽ, v̄] such that p(v̄ + ε)v − x(v̄ +
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REVIEW OF ECONOMIC STUDIES
ε) > [r + (1 − r )δ]v − [z + (1 − r )δz 2 (F2 )] ≥ r v − z for v ∈ (ṽ, v̄], contradicting the fact that
[0, v̄] is the convex hull of types that choose s with strictly positive probability at t = 1. Hence
p(v̄ + ε) ≥ r + (1 − r )δ, and all arguments are as in the case of z 2 (F2 ) < v̄. Let us recap the arguments used to establish Proposition 6. We start with an allocation rule
p ∈ P where z 2 (F2 ) ≤ z 2 (v̄). Taking as given the shape of p for [0, v̄), we optimally choose p̃(v)
for v ∈ [v̄, 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 z 2 (F2 ) ≤ z 2 (v̄)
and Lemma 2, we establish that p̃ is an element of P ∗ . From Propositions 5 and 6 we can then
conclude that
Proposition 7. If for all p ∈ P we have that z 2 (F2 ) ≤ z 2 (v̄), then under non-commitment
the seller maximizes expected revenue by posting a price in each period.
Step 3. For all p ∈ P that are PBE implementable it holds that z 2 (F2 ) ≤ z 2 (v̄)
The set P contains allocation rules that satisfy minimal requirements of being implemented
by a PBE, 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 PBE-implementable allocation rules. Here we study further restrictions of
PBE implementability and establish that for all p ∈ P that are PBE implementable we have
z 2 (F2 ) ≤ z 2 (v̄). Then our “quasi” solution of Step 2 is a real solution. The following proposition
essentially establishes that the seller cannot benefit 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 ∈ P that are PBE implementable we have z 2 (F2 ) ≤ z 2 (v̄).
Overview of Proof: Fix an assessment that is a PBE. From Proposition 4 we know that
it implements an allocation rule in P. Recall that s denotes an action that leads to the 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 PBE under consideration. Recall also that we use [0, v̄], v̄ ≤ 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 final period of the game,
an optimal mechanism derived in Proposition 2 is to post a price, which we denote by z 2 (F2 ).
Clearly if all types in [0, v̄] choose s with probability one, then the posterior is (5) and it is
immediate that z 2 (F2 ) ≤ z 2 (v̄). The only possibility then that z 2 (F2 ) > z 2 (v̄), is when a strictly
positive measure of types in [0, v̄] choose an action, other then action s, with strictly positive
probability. Types in [0, v̄] can be choosing with strictly positive probability an action ŝ that
leads to the same contract as action s, namely (r, z), and/or can be choosing an action s̃ that leads
to a contract (r̃ , z̃), which is different from (r, z).
For v ∈ [0, v̄] let m(v) ∈ [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 different from (r, z). We assume that m is a measurable function of v, with m(v) ∈ [0, 1] for all v ∈ [0, v̄]. Now, for simplicity suppose that other than s,
there is only one more action that leads to (r, z), call it ŝ.22 A fraction β(v) of m(v) type v
chooses action s and a fraction 1 − β(v) of m(v) chooses action ŝ. We assume that β is a
measurable function of v, with β(v) ∈ [0, 1] for all v ∈ [0, v̄]. Thus, the seller’s posteriors at
22. The case where more than two actions lead to (r, z) can be handled analogously.
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the beginning
the final period of the game after observing
 %of
 % s s, and respectively ŝ, are given
s
 %0v̄ β(t)m(t)dF(t) , for v ∈ [0, v̄]
 %0v̄ (1−β(t))m(t)dF(t) , for v ∈ [0, v̄]
β(t)m(t)dF(t)
by F2 (s) = 0
, and F̂2 (s) = 0 (1−β(t))m(t)dF(t)
.


0,
otherwise
0,
otherwise
% v̄
Since both actions are chosen with strictly positive probability, we have that 0 β(t)m(t)dF(t) >
% v̄
0 and 0 (1 − β(t))m(t)dF(t) > 0. When all types that choose either s or ŝ, all coordinate on
m
one
 % s of the two, say s, the posterior after the seller observes s at t = 1 is given by F2 (s) =
 %0v̄ m(t)dF(t) for v ∈ [0, v̄]

0
m(t)dF(t)
0,
otherwise.
Let z 2 (F2 ), respectively, z 2 ( F̂2 ) and z 2 (F2m ), solve (2) given posterior F2 , respectively, F̂2
and F2m . In terms of t = 1 allocation, actions s and ŝ are equivalent since they lead to the same
contract (r, z), but they may lead to different t = 2 options since it is possible that z 2 (F2 ) .=
z 2 ( F̂2 ).
In the first step of our proof we show that this is impossible: in order for both actions s and
ŝ to be chosen with strictly positive probability at t = 1, they must lead to the same t = 2 options,
namely z 2 (F2 ) = z 2 ( F̂2 ). We also show that z 2 (F2m ) = z 2 ( F̂2 ) = z 2 (F2 ). In the second and final
step of our proof, we show that z 2 (F2 ) ≤ z 2 (v̄). The details can be found in the Appendix. 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 1. 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 Propositions 7 and 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 benefit. The potential benefit from offering 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 revenuemaximizing 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 ≥ RNC (δ), where RC and RNC denote the highest revenue that the seller can achieve with and
without commitment, respectively. The difference in RC and RNC (δ) depends on the discount
factor. When the buyer and the seller are very patient (in this model the buyer and the seller have
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REVIEW OF ECONOMIC STUDIES
the same discount factor), the seller will find it beneficial to postpone all trade to the last period
of the game where she has commitment power. It follows that when δ = 1 expected revenue with
and without commitment coincide. The same is true for δ = 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 RNC (0) = RC . For intermediate values of the discount factor it holds
that RNC < RC . To get some idea about the magnitude of the difference 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
corresponding expected revenue is RC = 0·25. Now let us determine RNC (δ). Our theorem tells us
that the seller will maximize revenue by posting a price in each period. Let v̄ denote the valuation
of the buyer who is indifferent between accepting the price at t = 1, denoted by z 1 , and accepting
2 (v̄)
the price at t = 2, denoted by z 2 (v̄), that is, v̄ − z 1 = δ v̄ −δz 2 (v̄) or v̄ = z1 −δz
1−δ
/ v . For the assumed
for v ∈ [0, v̄]
v̄ ,
prior we have that, if the buyer rejects the price offer at t = 1, then F2 (v) =
,
0, otherwise
and the price posted at t = 2 is given by z 2 (v̄) = v̄2 . Substituting this expression of z 2 (v̄) into v̄
we get that z 1 = v̄(1 − 0·5δ). Given the above relationship between z 1 , v̄, and z 2 (v̄) the seller
will pick
0
1
v̄ v̄
v̄ ∈ arg max(1 − v̄)v̄(1 − 0· 5δ) + δ v̄ −
.
2 2
The following table gives the solution for different values of the discount factor.
Discount factor δ
0
0·3
0·5
0·7
0·9
1
Price at t = 1, z 1
0·5
0·46
0·45
0·44
0·46
0·5
Price at t = 2, z 2 (v̄)
0·5
0·27
0·3
0·34
0·42
0·5
v̄
RNC
0·5
0·54
0·6
0·68
0·84
1
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 PBE, in a game where the seller
can choose a different mechanism in each period. The set of PBE-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 finally (iv) on the length of the time horizon. Regarding the generality of the
strategy that the buyer may be 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 definition of mechanisms we have assumed that a mechanism
consists of a deterministic game form. The same is true in Bester and Strausz (2001)23 and in the
earlier literature from Hart and Tirole (1988) to Rey and Salanie (1996).
23. In Bester and Strausz (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.
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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) and by the
earlier literature. In this case, the assumption that game forms are deterministic is without any
loss.24 A recent paper by 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 Bester and Strausz (2004), the message that the agent reports), increases
the commitment power of the mechanism designer.25 Bester and Strausz (2004) show that in this
case, the principal can do better compared to the case where she observes s. In Skreta (2005c)
we establish that our result is robust to the case that the seller does not observe s, as in Bester
and Strausz (2004), but in addition she does not observe g(s),26 and observes only whether trade
took place or not. In this case, the mechanism designer has more commitment than Bester and
Strausz (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.27
Finally, in Skreta (2005c) one can also find the complete analysis for T > 2. The main
structure of the proof for the T = 2 case extends to longer finite 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 find 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 of
the seller is to find what price to post. We provide a reason for the seller’s choice to post a price,
even though she can use infinitely 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 infinite-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.
24. Suppose that given an action s, g(s) is random, with probability π the outcome is (r, z), and with probability
(1 − π) the outcome is (r̂ , ẑ). 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 (r̂ , ẑ)—in other words, the second period price will
be simply a function of s, call it z 2 (s). Types below z 2 (s) reject this price at t = 2, hence for those types the expected
outcome from choosing s is p = πr + (1 − π)r̂ and x = π z + (1 − π)ẑ. Now types above z 2 (s) accept the price at t = 2
and for those types the expected outcome from choosing s is given by p = π(r + (1 −r )δ) + (1 − π)(r̂ + (1 − r̂ )δ) and x =
π(z +(1−r )δz 2 (s))+(1−π)(ẑ +(1− r̂ )δz 2 (s)), which can be also rewritten as p = πr +(1−π)r̂ + [1−πr −(1−π)r̂ ]δ
and x = π z + (1 − π)ẑ+ [1 − πr − (1 − π)r̂ ]δz 2 (s). These outcomes can arise by a deterministic mapping that maps s to
r̃ = πr + (1 − π)r̂ and z̃ = π z + (1 − π)ẑ.
25. Recall that if the mechanism designer can commit not to observe anything, the commitment solution is sequentially rational.
26. Recall that g(s) is a probability of trade r and an expected payment z.
27. The result is also robust to the case where the mechanism is defined 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 are available from the author upon request.
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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 defined on V , whereas a solution of Program B is an allocation and a payment rule defined on [0, 1]. We first
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 defining 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]\V is an element of {pA (v), xA (v)}v∈V , it is fine for our purposes. The “extended” allocation and payment rule still
have to satisfy IC, PC, 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 0 on [0, 1]\V , we have that
$1
0
xA (v)dF(v) =
$
xA (v)dF(v).
V
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, PC, 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 ).
We will be done if we establish that
R( pA ) ≤ R( pB ).
(8)
R( pA ) > R( pB ).
(9)
We argue by contradiction. Suppose not, then
Now consider the allocation and payment rule defined as follows. For v ∈ V we set
p̄A (v) = pA (v), and x̄A (v) = xA (v).
Before we describe what happens for v ∈ [0, 1]\V , note that it is without loss to take V to be closed. Otherwise, we can
define 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 . Now, suppose that (v L , v H ) is an interval in [0, 1]\V and let
v̂ ∈ (v L , v H ) denote the type that is indifferent between choosing the actions specified for v L and the actions specified
for v H , that is
(10)
pA (v H )v̂ − xA (v H ) = pA (v L )v̂ − xA (v L ).
Such a type exists, since for v L we have that
pA (v L )v L − xA (v L ) ≥ pA (v H )v L − xA (v H )
and for v H we have that
pA (v H )v H − xA (v H ) ≥ pA (v L )v H − xA (v L );
then by continuity, there exists v̂ ∈ (v L , v H ) such that (10) holds.
On (v L , v H ), p̄A and x̄A are defined as
p̄A (v) = pA (v L ) and x̄A (v) = xA (v L )
p̄A (v) = pA (v H ) and x̄A (v) = xA (v H )
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if v ∈ (v L , v̂)
if v ∈ [v̂, v H ).
(11)
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Proceed similarly for all subintervals of types that are not contained in V . Now since dF(v) = 0 for v ∈ [0, 1]\V we
have that
(12)
R( p̄A ) = R( pA ).
From (9) and (12) we obtain that
R( p̄A ) > R( pB ).
We establish that p̄A and x̄A 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 {p̄A (v), x̄A (v)}v∈[0,1] is exactly the same as
{pA (v), xA (v)}v∈V . 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 p̄A , x̄A satisfy IC on V . Now to check IC on [0, 1], consider types in (v L , v H ).
From the fact that pA , xA satisfy IC on V , we have that pA (v H ) ≥ pA (v L ). Then we know from (10) that all types
below v̂ prefer ( pA (v L ), xA (v L )) and all types above v̂ prefer ( pA (v H ), xA (v H )). By the definition of p̄A , x̄A it then
follows that deviations within (v L , v H ) are not profitable. Now from the fact that pA and xA satisfy IC on V we can
conclude that type v L prefers outcome ( pA (v L ), xA (v L )) to all other options in {pA (v), xA (v)}v∈V and type v H prefers
outcome ( pA (v H ), xA (v H )) to all other options in {pA (v), xA (v)}v∈V . Then using (10) it is very easy to see that no
type in v ∈ (v L , v H ) can profitably deviate by behaving like a type in [0, 1]\(v L , v H ). Similarly, we can check that IC is
satisfied for all v ∈ [0, 1]\V . It is also immediate that p̄A and x̄A satisfy voluntary participation constraints and resource
constraints.
Hence p̄A , x̄A 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 PBE assessment (σ, µ) and let ( p, x) denote the allocation and payment rules
implemented by it. Point (i) is obvious. All PBEs are BNEs 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 PBE 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, v̄], with 0 ≤ v̄, 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 z 2 (F2 ) and define
v L = inf{v ∈ Y s.t. v accepts z 2 (F2 ) at t = 2}.
First we show that v L = z 2 (F2 ), then we establish that for v ∈ (v L , v̄) we have that p(v) = r + (1 −r )δ. Finally, we show
that for v ∈ [0, v L ) we have that p(v) = r .
Step 1. We show that v L = z 2 (F2 ). At a PBE, the buyer’s strategy must be a best response to the seller’s strategy.
This implies that
[r + (1 − r )δ]v L − [z + (1 − r )δz 2 (F2 )] ≥ r v L − z.
We now show that this inequality must hold with equality. We argue by contradiction. Suppose that [r + (1 − r )δ]v L −
[z + (1 − r )δz 2 (F2 )] > r v L − z, then the seller can increase z 2 (F2 ) by #z such that
[r + (1 − r )δ]v L − [z + (1 − r )δz 2 (F2 )] − δ#z = r v L − z
and raise higher revenue at the continuation game that starts at t = 2. All types v ≥ v L still prefer to choose (1, z 2 (F2 )) at
t = 2 then to choose (0, 0). Hence the seller has a profitable deviation at t = 2, contradicting the fact that we are looking
at a PBE. At a PBE we therefore have that
[r + (1 − r )δ]v L − [z + (1 − r )δz 2 (F2 )] = r v L − z,
(13)
from which it is immediate that v L = z 2 (F2 ).
Step 2. We claim that for v ∈ (z 2 (F2 ), v̄), where z 2 (F2 ) < v̄, we have that p(v) = r + (1 − r )δ. We will establish
this result by observing that if a v ∈ (z 2 (F2 ), v̄) is choosing a sequence of actions with positive probability, then it must
be that the expected discounted probability from these alternative actions, p̂, is such that p̂ = r + (1 − r )δ. Otherwise
depending on whether p̂ < r + (1 − r )δ or p̂ ≥ r + (1 − r )δ, either a type smaller than v, say z 2 (F2 ), or greater than v,
say v̄, has a profitable deviation. Take, for instance, the case that p̂ > r + (1 − r )δ. Then, for v it must hold p̂v − x̂ ≥
[r + (1 − r )δ]v − [z + (1 − r )δz 2 (F2 )], which implies together with (13), that all types in (v, v̄) strictly prefer an action
different from s, contradicting the fact that [0, v̄] is the convex hull of the set of types that choose s.
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REVIEW OF ECONOMIC STUDIES
Step 3. For v ∈ [0, z 2 (F2 )), with z 2 (F2 ) .= 0 we have that p(v) = r . Again if a v ∈ (0, z 2 (F2 )) is choosing a sequence of actions with positive probability, then it must be that the expected discounted probability from these alternative
actions p̂, is such that p̂ = r . Otherwise depending on whether p̂ < r or p̂ > r either a type smaller than v, say 0, or
greater than v, say z 2 (F2 ), has a profitable deviation. Suppose that p̂ < r , then for type v it must be p̂v − x̂ ≥ r v − z, but
then 0 has a profitable deviation since p̂0 − x̂ > r 0 − 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 < z 2 (F2 ) < v̄, from Step 2 we have that p(v) = r + (1 − r )δ for v ∈ (z 2 (F2 ), v̄), and from Step 3
we have that p(v) = r , for v ∈ [0, z 2 (F2 )). Now if z 2 (F2 ) = v̄, then p(v̄) ∈ [r, 1] (since p(z 2 (F2 )) ∈ [r,r + (1 − r )δ] and
p(v̄) ∈ [r + (1 − r )δ, 1]). It follows that if an allocation rule is implemented by an assessment that is a PBE 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
R( p) =
$z 2
0
r sdF(s) −
$z 2
0
r [1 − F(s)]ds
+
$v̄
[r + (1 − r )δ]sdF(s) −
+
$1
sdF(s) −
z2
v̄
$1
$v̄
z2
[r + (1 − r )δ][1 − F(s)]ds
[1 − F(s)]ds.
v̄
In order to find an optimal allocation rule out of P ∗ we have to choose (i) z 2 , (ii) r , and (iii) v̄ optimally.
Step 1.
At an optimum z 2 = z 2 (v̄).
From the definition of an allocation rule in P ∗ we have that z 2 must be less or equal to z 2 (v̄). We now establish that
at an optimum of P ∗ it will be z 2 = z 2 (v̄). Since z 2 (v̄) solves (2) given beliefs (5) we have that for all z 2 < z 2 (v̄)
z$2 (v̄)
z2
which since F(v̄) > 0 implies that
z$2 (v̄)
sdF(s) −
z2
z$2 (v̄)2
dF(s)
−
F(v̄)
s
z2
1−
3
F(s)
ds < 0,
F(v̄)
z$2 (v̄)
[F(v̄) − F(s)]ds < 0,
z2
from which it is immediate that
z$2 (v̄)
z2
[r + (1 − r )δ]sdF(s) −
z$2 (v̄)
z2
[r + (1 − r )δ][1 − F(s)]ds < 0.
It follows that at an optimum z 2 = z 2 (v̄).
Step 2.
At an optimum r = 0 or r = 1.
Differentiating with respect to r we get that
∂ R( p)
=
∂r
z$2 (v̄)
0
+
sdF(s) −
$v̄
z 2 (v̄)
z$2 (v̄)
0
[1 − F(s)]ds
(1 − δ)sdF(s) −
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$v̄
z 2 (v̄)
(1 − δ)[1 − F(s)]ds,
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SEQUENTIALLY OPTIMAL MECHANISMS
1107
p)
∂ R( p)
which does not depend on r . If ∂ R(
∂r ≥ 0, then at an optimum it must be r = 1 for all v ∈ [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 v̄ determines optimal sequence of prices.
From Steps 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, z 1 ), where z 1 denotes the price that the seller must post
z −δz (v̄)
at t = 1 to keep type v̄, indifferent between z 1 at t = 1 and z 2 (v̄) at t = 2, that is v̄ − z 1 = δ(v̄ − z 2 (v̄)) or v̄ = 1 1−δ2 .
v̄
∈
[0,
1]
determines
z 1 at
Types [0, v̄] of the buyer choose (0, 0) at t = 1, whereas types (v̄, 1] choose& (1, z 1 ). Each
'
z −δz (v̄)
F(z 2 )
and the optimal price at t = 2, z 2 (v̄), via maxz 2 ∈[0,v̄] 1 − F(v̄)
z 2 . In order to find an optimal
t = 1 via v̄ = 1 1−δ2
allocation rule out of P ∗ , the seller chooses v̄ optimally. It is very easy to see that this is a PBE 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 v̂ > v̄ but z 2 (v̂) < z 2 (v̄), then
[F(v̂) − F(v̄)]z 2 (v̄) > [F(v̂) − F(v̄)]z 2 (v̂).
(14)
&
'
F(z 2 )
Since z 2 (v̄) is the optimal price given beliefs (5) it solves z 2 (v̄) maxz 2 ∈[0,v̄] 1 − F(v̄)
z 2 . It follows that
&
'
'
&
F(z 2 (v̄))
F(z 2 (v̂))
1 − F(v̄) z 2 (v̄) ≥ 1 − F(v̄) z 2 (v̂) or
[F(v̄) − F(z 2 (v̄))]z 2 (v̄) ≥ [F(v̄) − F(z 2 (v̂))]z 2 (v̂).
(15)
Multiplying (14) and (15) by F(1v̂) and adding them up we get
1
[(F(v̂) − F(v̄))z 2 (v̄) + (F(v̄) − F(z 2 (v̄)))z 2 (v̄)]
F(v̂)
>
(16)
1
[(F(v̂) − F(v̄))z 2 (v̂) + (F(v̄) − F(z 2 (v̂)))z 2 (v̂)].
F(v̂)
Now (16) can be rewritten as,
1
1
[F(v̂) − F(z 2 (v̄))z 2 (v̄)] >
[F(v̂) − F(z 2 (v̂))z 2 (v̂)],
F(v̂)
F(v̂)
contradicting the definition of z 2 (v̂).
-
Proof of Proposition 8.
Step 1.
Step 1.1.
We establish z 2 (F2m ) = z 2 ( F̂2 ) = z 2 (F2 ).
First we show that z 2 (F2 ) = z 2 ( F̂2 ).
Suppose not, and without any loss, assume that z 2 (F2 ) < z 2 ( F̂2 ). Then for all v ∈ V we have that
[r + (1 − r )δ]v − [z + (1 − r )δz 2 (F2 )] > [r + (1 − r )δ]v − [z + (1 − r )δz 2 ( F̂2 )],
hence for all v ≥ z 2 (F2 ) the buyer strictly prefers s to ŝ. But then when the seller sees ŝ, she can infer that the valuation
of the buyer is below z 2 (F2 ), which in turn implies that a price of z 2 ( F̂2 ) > z 2 (F2 ) cannot be optimal. Contradiction.
This completes the proof of Step 1.1.
Step 1.2.
This step establishes that z 2 (F2m ) = z 2 ( F̂2 ) = z 2 (F2 ).
Assume wlog that z 2 (F2m ) < z 2 (F2 ). By the definition of z 2 (F2m ) (given by (2), with F replaced by F2m (v)), it
follows that
z 2$(F2 )
z 2$(F2 ) 2
3
$ s
m (t) ds ≥ 0.
sdFm
(s)
−
dF
(17)
1
−
2
2
z 2 (F2m )
z 2 (F2m )
0
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REVIEW OF ECONOMIC STUDIES
% z (F )
% z (F ) 4
%
5
But since z 2 (F2m ) is not a maximizer for beliefs F2 we have that z 2(F m2 ) sdF2 (s) − z 2(F m2 ) 1 − 0s dF2 (t) ds < 0, or
2 2
2 2
.
6
&
'
1 % z 2 (F2 ) sβ(s)m(s)dF(s) − % z 2 (F2 ) % v̄ β(t)m(t)dF(t) − % s β(t)m(t)dF(t) ds < 0, where 1 ≡
1
,
% v̄
0
A
A
z (F m )
z (F m ) 0
2
2
2
which in turn implies that
z 2$(F2 )
z 2 (F2m )
0 β(t)m(t)dF2 (t)
2
sβ(s)m(s)dF(s) −

z 2$(F2 ) $v̄
z 2 (F2m )


0
β(t)m(t)dF(t) −
$s
0


β(t)m(t)dF(t) ds < 0.
(18)
'
%
% z ( F̂ ) &
% z ( F̂ )
Similarly, since z 2 (F2m ) is not a maximizer for beliefs F̂2 we have that z 2(F m2 ) sd F̂2 (s) − z 2(F m2 ) 1 − 0s d F̂2 (t) ds <
2 2
2 2
0, which implies
z 2$( F̂2 )
z 2 (F2m )
s(1 − β(s))m(s)dF(s) −

z 2$( F̂2 ) $v̄
z 2 (F2m )

 (1 − β(t))m(t)dF(t) −
0
$s
0

(1 − β(t))m(t)dF(t) ds < 0
adding (18) and (19) and recalling that z 2 (F2 ) = z 2 ( F̂2 ) we obtain that
z 2$( F̂2 )
z 2 (F2m )
sm(s)dF(s) −

z 2$( F̂2 ) $v̄
z 2 (F2m )
%
which since 0v̄ m(t)dF(t) > 0 also implies that


0
m(t)dF(t) −

$s
0

(19)


m(t)dF(t) ds < 0,

%s
m(t)dF(t) 


m(s)dF(s)
1 − 0
 ds < 0, or
s
−


v̄
v̄
%
%


m(t)dF(t) z 2 (F2m )
m(t)dF(t)
z 2 (F2m )
z 2$( F̂2 )
z 2$( F̂2 ) 
0
z 2$( F̂2 )
z 2 (F2m )
0
sdFm
2 (s) −
z 2$( F̂2 ) 
z 2 (F2m )
1 −
$s
0


dFm
2 (t) ds < 0.
(20)
But (20) contradicts (17).
Now we argue that z 2 (F2m ) > z 2 (F2 ) is not possible either. To see this simply, reverse the roles of z 2 (F2m ) and
z 2 (F2 ), z 2 ( F̂2 ) and replace in the above inequalities “≥” with “<” and “<” with “≥”. It follows then that z 2 (F2m ) =
z 2 ( F̂2 ) = z 2 (F2 ). This completes the proof of Step 1.2.
So far we have established that z 2 (F2m ) = z 2 ( F̂2 ) = z 2 (F2 ). Now, we move on to compare z 2 (F2m ) and z 2 (v̄).
Step 2.
We establish that z 2 (F2 ) ≤ z 2 (v̄).
First, we find that only types above z 2 (F2 ) can be choosing with strictly positive probability actions that lead to
contracts other than (r, z). Then m(v) = 1 for all v ∈ [0, z 2 (F2 )).
Step 2.1.
We show that m(v) = 1 for all v ∈ [0, z 2 (F2 )).
We argue by contradiction. Suppose that there exists ṽ ∈ [0, z 2 (F2 )) that is choosing with strictly positive probability
an action s̃ that leads to a contract (r̃ , z̃) for which either r̃ .= r and/or z̃ .= z. Let z̃ 2 denote the price that the seller will
post at t = 2 after the history that the buyer choose s̃ at t = 1. Now, from Proposition 4 we know that for v ∈ [0, z 2 (F2 ))
we have that p(v) = r , therefore it must be the case that either (a) r = r̃ or (b) r = r̃ + (1 − r̃ )δ.
We show that case (a) is impossible, since r̃ = r implies z̃ = z, which implies in turn, that (r̃ , z̃) is the same contract as
(r, z). To see this we argue by contradiction. Suppose not, and wlog let z̃ > z. Depending on whether z + (1 −r )δz 2 (F2 ) <
z̃ +(1− r̃ )δ z̃ 2 or z +(1−r )δz 2 (F2 ) ≥ z̃ +(1− r̃ )δ z̃ 2 , there are two cases to consider. If z +(1−r )δz 2 (F2 ) < z̃ +(1− r̃ )δ z̃ 2
then, since we also have that z < z̃, all types of the buyer prefer (r, z) to (r̃ , z̃). The reason is that the probability of
c 2006 The Review of Economic Studies Limited
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SKRETA
SEQUENTIALLY OPTIMAL MECHANISMS
1109
obtaining the object is always the same under these two options, r = r̃ and of course r + (1 − r )δ = r̃ + (1 − r̃ )δ, but
payments are always lower at contract (r, z). But then the types that are choosing with strictly positive probability action
s̃, which leads to (r̃ , z̃), are not best-responding, contradicting the supposition that we are looking at a PBE. Now in
the case that z + (1 − r )δz 2 (F2 ) ≥ z̃ + (1 − r̃ )δ z̃ 2 , then all types above z 2 (F2 ) prefer (r̃ , z̃) over (r, z). This implies that
the seller posts at t = 2 a price z̃ 2 strictly lower than z 2 (F2 ) given a posterior with support [z 2 (F2 ), v̄]. This cannot be
optimal, which contradicts the fact that we are looking at PBE. Hence r̃ = r implies that z̃ = z.
We now show that (b) is impossible. First if r = r̃ + (1 − r̃ )δ then it must be the case that z = z̃ + (1 − r̃ )δ ẑ 2 . If
z > z̃ + (1 − r̃ )δ z̃ 2 than no type below z 2 (F2 ) would choose s, contradicting the fact that [0, v̄] is the convex hull of types
that choose s. If z < z̃ + (1 − r̃ )δ z̃ 2 , than no type would choose the sequence of actions that leading to r̃ + (1 − r̃ )δ, and
z̃ + (1 − r̃ )δ z̃ 2 . Hence, it must be z = z̃ + (1 − r̃ )δ z̃ 2 . Now since ṽ is choosing (r̃ , z̃) at t = 1 and accepts (1, z̃ 2 ) at t = 2,
it must be the case that z̃ 2 ≤ ṽ. For type z̃ 2 it holds that r̃ z̃ 2 − z̃ = [r̃ + (1 − r̃ )δ]z̃ 2 − [z̃ + (1 − r̃ )δ z̃ 2 ] = r z̃ 2 − z. But since
0 ≤ z̃ 2 and r̃ ≤ r we have that r̃ 0 − z̃ ≥ r 0 − z contradicting the definition 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 ∈ [0, z 2 (F2 )). Then, it is immediate that
F2m (v) =

F(v)
,
%



F(z 2 (F2 ))+ zv̄ (F ) m(s)dF(s)


2 2
v ∈ [0, z 2 (F2 ))
(21)
,
%

F(z 2 (F2 ))+ zv (F ) m(s)dF(s)



,
% v̄2 2

v ∈ [z 2 (F2 ), v̄]
F(z 2 (F2 ))+ z (F ) m(s)dF(s)
2 2
%
where F(z 2 (F2 )) + zv̄ (F ) m(s)dF(s) > 0 since action s is chosen by strictly positive probability.
2 2
Now that we have more information about F2m , and we know that z 2 (F2 ) = z 2 (F2m ), we investigate whether it is
possible to have z 2 (F2 ) > z 2 (v̄).
Step 2.2.
Case 1.
We show that z 2 (F2 ) ≤ z 2 (v̄).
z 2 (F2 ) = 0
It is then immediate that z 2 (F2 ) ≤ z 2 (v̄).
Case 2.
z 2 (F2 ) > 0
Since z 2 (v̄) is the optimal price at T = 2, it solves (2) for beliefs given by (5). From the previous step we have that
z 2 (F2 ) = z 2 (F2m ), where z 2 (F2m ) solves (2), for (21). The relevant expressions are28


>
1
$ṽ
$ṽ
> F
1 

φ v, ṽ >>
= F(v̄)
 sdF(s) − (F(v̄) − F(t))dt  ,
F(v̄)
F(v̄)
0
and
v


φ(v, ṽ|F2m ) =  F(z 2 (F2 )) +
$v̄
z 2 (F2 )
(22)
v


$ṽ
$ṽ


m
m(t)dF(t)  sdFm
2 (s) − [1 − F2 (s)]ds  .
v
v
Since m(v) = 1 for v, ṽ ∈ [0, z 2 (F2 )), we obtain that


φ(v, ṽ|F2m ) = 
$ṽ
v
tdF(t) −
$ṽ
v


 F(z 2 (F2 )) +
$v̄
z 2 (F2 )


 
m(s)dF(s) − F(t) dt  .
%
28. Multiplying by a constant, F(v̄), in the case of (5) and by F(z 2 ) + zv̄ m(t)dF(t) in the case of (21), does not
2
change anything, because we care about the sign of the integral.
c 2006 The Review of Economic Studies Limited
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1110
REVIEW OF ECONOMIC STUDIES
>
E
F
>
To make φ v, ṽ > F(Fv̄) more easily comparable with φ(v, ṽ|F2m ), let us do some rewriting. By adding and subtracting
>
E
F
% ṽ
> F
v F(z 2 (F2 ))dt to φ v, ṽ > F(v̄) we get that
>
0
1 $ṽ
$ṽ
> F
φ v, ṽ >>
= tdF(t) − (F(z 2 (F2 )) + F(v̄) − F(z 2 (F2 )) − F(t))dt.
F(v̄)
v
v
>
F
%
>
Hence for v, ṽ ∈ [0, z 2 (F2 )) φ(v, ṽ|F2m ) and φ v, ṽ > F(Fv̄) differ by a constant zv̄ (F ) m(s)dF(s) in the case of
2 2
>
E
F
>
φ(v, ṽ|F2m ), versus F(v̄) − F(z 2 (F2 )) in the case of φ v, ṽ > F(Fv̄) . Now because m(v) ∈ [0, 1] and F ≥ 0, we have that
%
F(z 2 (F2 )) + F(v̄) − F(z 2 (F2 )) ≥ F(z 2 (F2 )) + zv̄ (F ) m(t)dF(t) which implies that
E
2
2
>
0
1
> F
φ v, ṽ >>
≤ φ(v, ṽ|FTm ),
F(v̄)
for all v, ṽ ∈ [0, z 2 (F2 )).
(23)
Suppose that z 2 (v̄) < z 2 (F2 ), then by the definition of z 2 (v̄), it follows that
>
0
1
> F
φ z 2 (v̄), ṽ >>
≥ 0, for all ṽ ∈ [z 2 (v̄), z 2 (F2 )],
F(v̄)
which together with (23) implies that
φ(z 2 (v̄), ṽ|F2m ) ≥ 0,
for all ṽ ∈ [z 2 (v̄), z 2 (F2 )],
(24)
φ(z 2 (F2 ), ṽ|F2m ) ≥ 0,
for all ṽ ∈ [z 2 (F2 ), v̄],
(25)
φ(z 2 (v̄), ṽ|F2m ) ≥ 0,
for all ṽ ∈ [z 2 (v̄), v̄],
and by the definition of z 2 (F2 ) we have that
but (24) and (25) imply
contradicting the definition of z 2 (F2 ). Hence, we have shown that z 2 (v̄) ≥ z 2 (F2 ).
The reason why this is true follows from the fact that only types greater than z 2 (F2 ) may 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, v̄] compared to (5). This completes the proof of Step 2.2. Acknowledgements. 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 significantly 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 financial support. A significant 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.
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