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Mechanism_Design_Theory.pdf

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MECHANISM_DESIGN_THEORY
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15 October 2007 ____________________________________________________________________________________________________ Information Department, Box 50005, SE-104 05 Stockholm, Sweden Phone: +46 8 673 95 00, Fax: +46 8 15 56 70, E-mail: info@kva.se, Website: www.kva.se Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2007 Mechanism Design Theory Compiled by the Prize Committee of the Royal Swedish Academy of Sciences 经济学家-http://bbs.jjxj.org 经济学家-http://bbs.jjxj.org1 Introduction Economic transactions take place in markets, within ?rms and under a host of other institutional arrangements. Some markets are free of government intervention while others are regulated. Within ?rms, some transactions are guided by market prices, some are negotiated, and yet others are dictated by management. Mechanism design theory provides a coherent framework for analyzing this great variety of institutions, or “allocation mechanisms”, with a focus on the problems associated with incentives and private information. Markets, or market-like institutions, often allocate goods and services e ?ciently. Long ago, economists theoretically proved this e ?ciency under fairly stringent assump- tions concerning, among other things, the nature of the goods to be produced and traded, participants’ information about these, and the degree of competition. Mecha- nism design theory allows researchers to systematically analyze and compare a broad variety of institutions under less stringent assumptions. By using game theory, mech- anism design can go beyond the classical approach, and, for example, explicitly model how prices are set. In some cases, the game-theoretic approach has led to a new appre- ciation of the market mechanism. The theory shows, for example, that so-called double auctions (where buyers and sellers post their bid- and ask-prices) can be e ?cient trad- ing institutions when each trader has private information about his or her valuations of the goods traded. As the number of traders increases, the double-auction mechanism will more and more e ?ciently aggregate privately held information, and eventually all information is re?ected by the equilibriumprices (Wilson, 1985). These results support Friedrich Hayek’s (1945) argument that markets e ?ciently aggregate relevant private information. Mechanism design theory shows which mechanisms are optimal for di ?erent partic- ipants, say sellers or buyers (e.g. Samuelson, 1984). Such insights have been used to better understand market mechanisms that we frequently observe. For example, the 1 经济学家-http://bbs.jjxj.org 经济学家-http://bbs.jjxj.orgtheory has been used to identify conditions under which commonly observed auction forms maximize the seller’s expected revenue (Harris and Raviv, 1981; Myerson, 1981; Riley and Samuelson, 1981). The theory also admits detailed characterizations of opti- malauctionformswhentheseconditionsdonothold(Myerson, 1981; MaskinandRiley, 1984a). Likewise, mechanism design theory has enabled economists to ?nd solutions to the monopoly pricing problem, showing, for example, how the price should depend on quality and quantity so as to maximize the seller’s expected revenue (Maskin and Riley, 1984b). Again, the theoretical solution squares well with observed practice. In some cases, no market mechanism can ensure a fully e ?cient allocation of re- sources. In such cases, mechanism design theory can be used to identify other, more e ?cient institutions. A classic example concerns public goods, such as clean air or na- tional security. Paul Samuelson (1954) conjectured that no resource allocation mecha- nismcanensureafullye ?cientlevelofpublicgoods, because“itisinthesel?shinterest of each person to give false signals, to pretend to have less interest in a given collective activity than he really has...” (page 388 op. cit.). Mechanism design theory permits a precise analysis of Samuelson’s conjecture. More generally, the theory can be used to analyze the economic e ?ciency of alternative institutions for the provision of public goods, rangingfrommarketsandconsensualcollectivedecision-makingthroughmajori- tariandecisionrulesallthewaytodictatorship. Animportantinsightisthatconsensual decision-making is frequently incompatible with economic e ?ciency. The theory thus helps to justify governmental ?nancing of public goods through taxation. Applications of mechanism design theory have led to breakthroughs in a number of other areas of economics as well, including regulation, corporate ?nance, and the theory of taxation. The development of mechanism design theory began with the work of Leonid Hur- wicz (1960). He de?ned a mechanism as a communication system in which participants send messages to each other and/or to a “message center,” and where a pre-speci?ed rule assigns an outcome (such as an allocation of goods and services) for every collec- tion of received messages. Within this framework, markets and market-like institutions could be compared with a vast array of alternative institutions. Initially, much of the interest focussed on the informational and computational costs of mechanisms, while abstracting from the problem of incentives. An important contribution was Marshak and Radner’s (1972) theory of teams, which inspired much subsequent literature (e.g. Groves, 1973). However, in many situations, providing incentives to the participating agents is an important part of the problem. Mechanism design theory became relevant for a wide variety of applications only after Hurwicz (1972) introduced the key notion 2 经济学家-http://bbs.jjxj.org 经济学家-http://bbs.jjxj.orgof incentive-compatibility, which allows the analysis to incorporate the incentives of self-interested participants. In particular, it enables a rigorous analysis of economies where agents are self-interested and have relevant private information. In the 1970s, the formulation of the so-called revelation principle and the devel- opment of implementation theory led to great advances in the theory of mechanism design. The revelation principle is an insight that greatly simpli?es the analysis of mechanism design problems. In force of this principle, the researcher, when searching for the best possible mechanism to solve a given allocation problem, can restrict at- tention to a small subclass of mechanisms, so-called direct mechanisms. While direct mechanismsarenotintendedasdescriptionsofreal-worldinstitutions, theirmathemat- ical structure makes them relatively easy to analyze. Optimization over the set of all direct mechanisms for a given allocation problem is a well-de?ned mathematical task, and once an optimal direct mechanism has been found, the researcher can “translate back” that mechanism to a more realistic mechanism. By this seemingly roundabout method, researchers have been able to solve problems of institutional design that would otherwise have been e ?ectively intractable. The ?rst version of the revelation principle was formulated by Gibbard (1973). Several researchers independently extended it to the general notion of Bayesian Nash equilibrium (Dasgupta, Hammond and Maskin, 1979, Harris and Townsend, 1981, Holmstrom, 1977, Myerson, 1979, Rosenthal, 1978). Roger Myerson (1979, 1982, 1986) developed the principle in its greatest generality and pioneered its application to important areas such as regulation (Baron and Myerson, 1982) and auction theory (Myerson, 1981). The revelation principle is extremely useful. However, it does not address the issue of multiple equilibria. That is, although an optimal outcome may be achieved in one equilibrium, other, sub-optimal, equilibria may also exist. There is, then, the danger that the participants might end up playing such a sub-optimal equilibrium. Can a mechanism be designed so that all its equilibria are optimal? The ?rst general solution to this problem was given by Eric Maskin (1977). The resulting theory, known as implementation theory, is a key part of modern mechanism design. Theremainderofthissurveyisorganizedasfollows. Section2presentskeyconcepts and results, Section 3 discusses applications, and Section 4 concludes. 3 经济学家-http://bbs.jjxj.org 经济学家-http://bbs.jjxj.org2 Key concepts and insights We begin by describing incentive compatibility and the revelation principle. We then discuss some results obtained for two main solution concepts, dominant-strategy equi- librium and Bayesian Nash equilibrium, respectively. We consider, in particular, the classicallocationproblemofoptimalprovisionofpublicgoods. Wealsodiscussasimple example of bilateral trade. We conclude by discussing the implementation problem. 2.1 Incentive compatibility and the revelation principle The seminal work of Leonid Hurwicz (1960,1972) marks the birth of mechanism design theory. In Hurwicz’s formulation, a mechanism is a communication system in which participants exchange messages with each other, messages that jointly determine the outcome. Thesemessagesmaycontainprivateinformation,suchasanindividual’s(true or pretended) willingness to pay for a public good. The mechanism is like a machine that compiles and processes the received messages, thereby aggregating (true or false) private information provided by many agents. Each agent strives to maximize his or her expected payo ? (utility or pro?t), and may decide to withhold disadvantageous information or send false information (hoping to pay less for a public good, say). This leads to the notion of “implementing” outcomes as equilibria of message games, where the mechanism de?nes the “rules” of the message game. The comparison of alternative mechanisms is then cast as a comparison of the equilibria of the associated message games. To identify an optimal mechanism, for a given goal function (such as pro?ttoa given seller or social welfare), the researcher must ?rst delineate the set of feasible mechanisms, and then specify the equilibrium criterion that will be used to predict the participants’ behavior. Suppose we focus on the set of “direct mechanisms”, where the agents report their private information (for example, their willingness to pay for a public good). There is no presumption that the agents will tell the truth; they will be truthful only if it is in their self-interest. Based on all these individual reports, the direct mechanism assigns an outcome (for example, the amount provided of the public good and fees for its ?nancing). Suppose we use the notion of dominant strategy equi- librium as our behavioral criterion. 1 Hurwicz’s (1972) notion of incentive-compatibility can now be expressed as follows: the mechanism is incentive-compatible if it is a dom- 1 A strategy is dominant if it is a agent’s optimal choice, irrespective of what other agents do. 4 经济学家-http://bbs.jjxj.org 经济学家-http://bbs.jjxj.orginant strategy for each participant to report his private information truthfully. In addition, we may want to impose a participation constraint: no agent should be made worse o ? by participating in the mechanism. Under some weak assumptions on tech- nology and taste, Hurwicz (1972) proved the following negative result: in a standard exchangeeconomy, noincentive-compatiblemechanismwhichsatis?estheparticipation constraint can produce Pareto-optimal outcomes. In other words, private information precludes full e ?ciency. A natural question emanating from Hurwicz’s (1972) classic work thus is: Can Pareto optimality be attained if we consider a wider class of mechanisms and/or a less demanding equilibrium concept than dominant-strategy equilibrium, such as Nash equilibrium or Bayesian Nash equilibrium? 2 If not, then we would like to know how largetheunavoidablesocialwelfarelossesare,andwhattheappropriatestandardof e ?ciencyshouldbe. Moregenerally,wewouldliketoknowwhatkindofmechanismwill maximize a given goal function, such as pro?t or social welfare (whether this outcome is fully e ?cient or not). In the literature that followed Hurwicz (1972), these questions have been answered. Much of the success of this research program can be attributed to the discovery of the revelation principle. The revelation principle states that any equilibrium outcome of an arbitrary mech- anism can be replicated by an incentive-compatible direct mechanism. In its most general version, developed by Myerson (1979, 1982, 1986), the revelation principle is valid not only when agents have private information but also when they take unob- served actions (so-called moral hazard), as well as when mechanisms have multiple stages. Although the set of all possible mechanisms is huge, the revelation principle impliesthatanoptimal mechanismcanalwaysbefoundwithinthewell-structuredsub- class consisting of direct mechanisms. Accordingly, much of the literature has focussed on the well-de?ned mathematical task of ?nding a direct mechanism that maximizes the goal function, subject to the incentive-compatibility (IC) constraint (and, where appropriate, also the participation constraint). A rough proof of the revelation principle for the case with no moral hazard goes as follows. First, ?x an equilibrium of any given mechanism. An agent’s private infor- mation is said to be his “type”. Suppose that an agent of type t sends the message m( t) in this equilibrium. Now consider the associated direct mechanism in which each 2 In a Nash equilibrium, each agent’s strategy is a best response to the other agents’ strategies. A Bayesian Nash equilibrium is a Nash equilibrium of a game of incomplete information, as de?ned by Harsanyi (1967-8). 5 经济学家-http://bbs.jjxj.org 经济学家-http://bbs.jjxj.orgagent simply reports a type t 0 ,where t 0 may be his true type t or any other type. The reported type t 0 is his message in the direct mechanism, and the outcome is de?ned to be the same as when the agent sends the message m( t 0 ) in the equilibrium of the original mechanism. By hypothesis, an agent of type t preferred to send message m( t) intheoriginal mechanism(theagentcouldnotgainbyunilaterallydeviatingtoanother message). In particular, the agent preferred sending the message m( t) to sending the message m( t 0 ), for any for t 0 6= t. Therefore, he also prefers reporting his true type t in the direct mechanism, rather than falsely reporting any other type t 0 . So the direct mechanismis incentive compatible: noagent has anincentive tomisreport his type. By construction, the direct mechanism produces the same outcome as the original mecha- nism. Thus, any (arbitrary) equilibrium can be replicated by an incentive-compatible direct mechanism. 3 As discussed below, the revelation principle can be used to generalize Hurwicz’s (1972) impossibility result to the case of Bayesian Nash equilibrium. Thus, in settings where participants have private information, Pareto optimality in the classical sense is in general not attainable, and we need a new standard of e ?ciency which takes incen- tives into account. A direct mechanism is said to be incentiv
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