Dynamic Mechanism Design: Revenue Equivalence,Pro…t Maximization and Information Disclosure Preliminary and incomplete.
Alessandro Pavan Northwestern UNiversity最胆大的人体模特摄影
Ilya Segal
宁姓氏读音
Stanford University
Juuso Toikka
Stanford University June2,2008
Abstract
人艺术照This paper examines the problem of how to design incentive-compatible mechanisms in environments in which the agents’private information evolves stochastically over time and in
手指滑板动画片which decisions have to be made in each period.The environments we consider are fairly general
in that the agents’types are allowed to evolve in a non-Markov way,decisions are allowed to a¤ect
the type distributions and payo¤s are not restricted to be parable over time.Our…rst result is
the characterization of a dynamic formula for(the derivative of)the agents’equilibrium payo¤s
in an incentive-compatible mechanism.The formula summarizes all local…rst-order conditions
taking into account how current types a¤ect the dynamics of expected payo¤s.The formula
generalizes the familiar envelope condition from static mechanism design:the key di¤erence is
that a variation in the current types now impacts payo¤s in all subquent periods both directly
and through the e¤ect on the distributions of future types.We…rst identify assumptions on the
primitive environment that guarantee that our dynamic payo¤formula is a necessary condition
溃退的意思
for incentive compatibility.Next,we specialize this formula to quasi-linear environments and
u it to establish a dynamic revenue-equivalence result.Lastly,we turn to the characterization关于爱情的词语
of su¢cient conditions for incentive compatibility.We then apply the results to study the
properties of revenue-maximizing mechanisms in a variety of applications that include dynamic
auctions with AR(k)values and the provision of experience goods.
JEL Classi…cation Numbers:D82,C73,L1.
Keywords:dynamic mechanisms,asymmetric information,stochastic process,long-term con-
tracting,incentives
This paper supercedes previous working papers“Revenue Equivalence,Pro…t Maximization,and Transparency in Dynamic Mechanisms”by Segal and Toikka and“Long-Term Contracting in a Changing World”by Pavan.Ac-knowledgements to be added.
1Introduction
We consider the problem of how to design incentive-compatible mechanisms in a dynamic environ-ment in which agents receive private information over time and decisions may be made over time. The model allows for rial correlation of the agents’private information as well as the dependence of information on past allocations.For example,it covers as special cas such problems as the alloc
ation of resources to agents who valuations follow a stochastic process,the procedures for lling new experience goods who value is re…ned by the buyers upon consumption,or the design of multiperiod procurement auctions for bidders who cost parameters evolve stochastically over time and may exhibit learning-by-doing e¤ects.
The fundamental di¤erence between dynamic and static mechanism design is that in the former, an agent has access to a lot more potential deviations.Namely,instead of a simple misreprentation of his true type,the agent can make this reprentation conditional on the information he has obrved in the mechanism,in particular on his past types,his past reports(which need not have been truthful),and what he inferred about the other agents’types in the cour of the mechanism.Despite the resulting complications,we deliver some general necessary conditions for incentive compatibility and some su¢cient conditions,and u them to characterize pro…t-maximizing mechanisms in veral applications.
未来地球The cornerstone of our analysis is the derivation of a formula for the derivative of an agent’s expected payo¤in an incentive-compatible mechanism with respect to his private information. Similarly to Mirrlees’s…rst-order approach for static environments(Mirrlees,1971),our formula (hereafter referred to as dynamic payo¤formula)provides an envelope-theorem condition sum-marizi
ng local incentive compatibility constraints.In contrast to the static model,however,the derivation of this formula relies on incentive compatibility in all the future periods,not just in one given period.Furthermore,unlike some of the earlier papers about dynamic mechanism design, we identify conditions on the primitive environment for which the dynamic payo¤formula is a necessary condition for any incentive-compatible mechanism(not just for“well-behaved”ones). In addition to carrying over the usual static assumptions of“smoothness”of the agent’s payo¤function in his type and connectedness of the type space(,Milgrom and Segal,2002),the dynamic tting requires additional assumptions on the stochastic process governing the evolution of each agent’s information.Intuitively,our dynamic payo¤formula reprents the impact of an (in…nitesimal)change in the agent’s current type on his equilibrium expected payo¤.This change can be decompod into two parts.The…rst one is the familiar e¤ect of the current type on the agent’s expected utility,as in static mechanism design.The cond part captures the indirect e¤ect of the current type on the expected utility through its impact on the type distributions in each of the subquent periods.Note that in general the current type may a¤ect the future type distributions
directly as well as indirectly through its impact on the type distributions in intermediate periods. All changes in the type distributions are then evaluated by looking at their ultimate impact on the agent’s
utility,holding constant the agent’s messages to the mechanism(by the usual envelope theorem logic).
The dynamic payo¤formula can be established either by iterating backward the local incentive-compatibility conditions or by using the quantile function theorem(e, e.g.,Angus,1994)to reprent the agents’types as the result of independent innovations(shocks).While the two ap-proaches lead to the same formula,the conditions on the primitive environment that validate this formula a necessary condition for incentive compatibility are somewhat di¤erent.In this n the two approaches are complementary(e also Eso and Szentes,2007,for a similar approach in a two-period-one-decision model).
To ea the exposition,in the…rst part of the paper(Section3)we consider an environment with a single agent who obrves all the relevant history of the mechanism.There we derive the envelope formula that determines the agent’s equilibrium payo¤in a incentive-compatible mechanism.In Section4we then show how to adapt the envelope formula to a multi-agent environment.The key di¤erence between the two ttings is that in the latter an agent obrves only a part the entire history generated by the mechanism:an agent must thus form beliefs about the unobrved types of the other agents as well as the decisions that the mechanism has induces with the agents. We sh
ow that the derivation for the single-agent ca extends to multi-agent mechanisms provided that the stochastic process governing the evolution of the agents’types are independent of one another,except through their e¤ect on the decisions that are obrved by the agents.In other words,we show how the familiar“Independent Types”assumption for static mechanism design should be properly adjusted to a dynamic tting to guarantee that the agents’equilibrium payo¤s can still be pinned down by an envelope formula.
For the special ca of quasilinear environments,we…rst u the dynamic envelope formula to establish a dynamic“revenue equivalence theorem”that links the payment rules in any two Bayesian incentive-compatible mechanisms that implement the same allocation rule.In particular, if we have a single agent who participates in a deterministic mechanism,this theorem pins down,in each state,the total payment that is necessary to implement a given allocation rule,up to a scalar that does not depend on the state.With many agents,or with a stochastic mechanism,the theorem pins down the expected payments as function of each agent’s type history,where the expectation is with respect to the other agents’types and/or the stochastic decisions taken by the mechanism. However,if one requires a strong form of“robustness”—according to which the mechanism must remain incentive-compatible even if an agent is shown at the very beginning of the game all the other agents’(future)types—then the theorem again pins down the total payments for each state.
Next,we u the dynamic envelope formula to express the expected pro…ts in an incentive-compatible and individually rational mechanism as the expected“virtual surplus,”appropriately de…ned for the dynamic tting.This derivation us only the agents’local incentive constraints, and only the participation constraints of the lowest-types in the initial period.Ignoring all the other incentive and participation constraints yields a dynamic“Relaxed Program,”which is in general a dynamic programming problem.In particular,the Relaxed Program gives us a simple intuition for the optimal distortions introduced by a pro…t-maximizing principal:Since only the …rst-period participation constraints bind(this is due to the unlimited bonding possibilities in the quasilinear environment with unbounded transfers),the distortions are created to balance the rent-extraction versus e¢ciency trade-o¤,as perceived from the perspective of period one.However, due to informational linkages in the stochastic type process,the principal will not only distort the agent’s consumption in period one but also in any subquent period whenever his type in period t is informative about the…rst-period type.The informativeness is here measured by an “information index”that captures all the direct and indirect e¤ects of the…rst-period type on the type distributions in all subquent periods.
It turns out that when an agent’s type in period t>1hits its highest or lowest possible value,the informational linkage disappears and the principal implements the e¢cient level of consumption in th
at period(provided that payo¤s are additively time-parable).However,for intermediate types in period t,the optimal mechanism entails distortions(for example,when types are positively correlated over time in the n of First-Order Stochastic Dominance,and the agent’s payo¤s satisfy the single-crossing property,the optimal mechanism entails downward distortions).Thus, in contrast to the static model,with a continuous but bounded type space,distortions in each period t>1are never monotonic in the agent’s type.This is also in contrast with the results of Battaglini(2005)for the ca of a Markov process with only two types in each period.
Studying the Relaxed Program is not fully satisfactory unless one also provides su¢cient con-ditions for its solution to satisfy all of the remaining incentive and participation constraints.We are indeed able to provide some such conditions.In particular,we show that in the ca where the agents’types follow a Markov process and their payo¤s are Markovian in their types(so that it is enough to check one-stage deviations from truthtelling),a su¢cient condition for an allocation rule to be implementable is that the partial derivative of the agent’s expected utility with respect to his current type when he misreports be nondecreasing in the report.One can then u the dynamic payo¤formula to calculate this partial derivative—the condition is fairly easy to check. (Unfortunately,this condition is not necessary for incentive-compatibility—a tight characterization is evasive becau of the multidimensi
onal decision space of the problem.)This su¢cient condi-tion also turns uful when checking incentive compatibility is some non-Markov ttings that are
su¢ciently“parable.”
2000年属什么In some standard ttings we can actually state an even simpler su¢cient condition for incentive compatibility,which also ensures that incentive compatibility is robust to an agent learning in advance all of the other agents’types(and therefore to any weaker form of information leakage in the mechanism).This condition is that the transitions that describe the evolution of the agents’private information are monotone in the n of First-Order Dominance,the payo¤s satisfy a single-crossing property,and the allocation rule is“strongly monotonic”in the n that the consumption of a given agent in any period is nondecreasing in each of the agent’s type reports,for any given pro…le of reports by the other agents.
In Section5,we apply the general results to a few simple,yet illuminating,applications.The analysis proves especially simple when the agents’types follow an autoregressive stochastic process of degree k(AR(k)).If we assume in addition that each agent’s payo¤is a¢ne in his types (but not necessarily in his consumption),then the principal’s Relaxed Program turns out to be very similar to t
he expected social surplus maximization program,the only di¤erence being that the agents’true values in each period are replaced by their corresponding“virtual values.”In the AR(k)ca,the di¤erence between an agent’s true value and his virtual value in period t, which can be called his“handicap”in period t,is determined by the agent’s…rst-period type, the hazard rate of the…rst period type’s distribution,and the“impul respon coe¢cient”of the AR(k)process.1Intuitively,the impul respon coe¢cient determines the informational link between period t and period1,while the…rst-period hazard rate captures the importance that the principal assigns to the trade-o¤between e¢ciency and rent-extraction as perceived from period one’s perspective(just as in the static model).Importantly,since the handicaps depend only on the…rst-period type reports,the Relaxed Program at any period t 2can be solved by running an e¢,expected surplus-maximizing)mechanism on the handicapped values.Thus,while building an e¢cient mechanism may in general require solving an involved dynamic programming problem(due to possible intertemporal payo¤interactions),once a solution is found it can be easily adapted to obtain a solution to the Relaxed Program.We also u the fact that the solution to the Relaxed Program looks“quasi-e¢cient”from period2onward to show that it can be implemented in a mechanism that is incentive compatible from period2onward(following truthtelling in period one).This can be done for example using the“Team Mechanism”payments propod by Athey and Segal(2007)to implement e¢cient allo
cation rules.As for verifying incentives in period1,we have only been able to do it in a few special ttings.
We also consider two other applications.The…rst one is the designing of quential auctions for environments in which the agents’payo¤s are time-parable while their private types follow an 1The term“handicapped auction”was…rst ud in Eso and Szentes(2007).
