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The Theory of Mechanism Design: an Overview

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The Theory of Mechanism Design: an Overview COMMENTARY voters are free to reveal any ranking, they The Theory of Mechanism can also misrepresent their true ranking. Clearly their report will depend on how Design: An Overview they believe others will vote. A natural question is whether a voting rule can be designed where voters can be induced to Arunava Sen reveal their true ranking irrespective of their beliefs regarding the voting behaviour Three US­based economists have he Nobel Prize in economics for of others. Unfortunately this does not hold been awarded the Nobel Prize 2007 was awarded to Leonid for either the Plurality Rule or rules based for economics for 2007 for laying THurwicz (University of Minnesota, on the Majority Principle. US), Eric Maskin (Institute of Advanced To see this consider (for simplicity) the the foundations of mechanism Studies, Princeton, US) and Roger Myerson case where there are are three voters 1, 2 design theory. A description (University of Chicago, US) for “having and 3 and exactly three proposals a, b and and discussion of this theory, laid the foundations of mechanism design c. Suppose that voter 1’s true ranking is a its importance and the work of theory”. What is this theory and why is it better than b better than c, 2’s true rank­ important? What are the contributions ing is b better than c better than a while 3’s the award­winning economists. of the recipients of this year’s prize? true ranking is c better than a better than Mechanism design could help This article briefly attempts to address b. If these rankings were announced, a tie policymaking in a number of these questions. would result according to both the Plurality areas, one potential area is in Rule (since there is exactly one voter for 1 Introduction whom each of a, b and c is first ranked) drawing up the rules for land Consider the following example on the de­ and any rule based on the Majority Principle acquisition. sign of voting rules, which is a particular (since a majority of voters prefer a to b, b aspect of the theory of mechanism design. to c and c to a). Assume without any loss of Suppose that there are N voters in a com­ generality that the outcome in this situa­ mittee who have to choose one of K tion is a. Then voter 2 who believes that 1 proposals. Each voter has an opinion on and 3 will vote truthfully, will be better the proposals, which can be expressed in off by lying and reporting c better than b the form of a ranking of the proposals. For better than a, thereby obtaining c (for instance, voter j may prefer proposal a to both the Plurality Rule or a rule based on proposal b to proposal c and so on. A “voting the Majority Principle) which is better rule” is a method for aggregating individual than a, the outcome that would occur if opinions into an outcome. It determines she voted truthfully, according to her the chosen proposal at any given profile of true ranking. Note that lying leads to rankings of voters – the chosen proposal non­optimal outcomes. can be thought of as the “the optimal” pro­ As a result, a host of questions arise posal in that situation. An example of a which are addressed by mechanism design voting rule is the familiar “Plurality Rule” theory. Are there other rules where voters where the proposal selected is the first­ are induced to vote truthfully? Can the ranked candidate for the largest number of truth­telling requirement be weakened voters. Another well known class of voting sensibly? What happens if monetary com­ rules is based on the “Majority Principle” pensation or randomisation is permitted? which picks the proposal that beats all other What if voters had better information proposals in pairwise majority contests about the true rankings of other voters? whenever such a proposal exists.1 Is there any benefit in designing more complex communication systems than Revealing Information simply requiring voters to reveal their A natural assumption in a setting where rankings directly? I am grateful to Parikshit Ghosh and Debasis individuals vote is that each voter’s opinion Mishra for their comments. is known only to the voter. Moreover vot­ Devising a Mechanism ing is the act of revealing this information. In general terms, mechanism design theory Arunava Sen ([email protected]) is at the Indian If voter opinions were commonly known, is concerned with resource allocation Statistical Institute, New Delhi. voting itself would be superfluous. Since (understood very broadly) in multi­agent 8 december 8, 2007 Economic & Political Weekly COMMENTARY environments. The key feature of the represents the collective will of all the a game of complete information and a so­ problem is that the determination of an agents. The goals of the planner are cap­ lution concept such as Nash equilibrium, “optimal” allocation depends on informa­ tured by a “Social Choice Correspondence” is appropriate.5 In the incomplete informa- tion which agents possess privately. In (SCC) F which is a mapping from the set Θ tion setting, each agent has only partial order to achieve an optimal allocation, to the set of non­empty subsets of A. Thus, information regarding the state. this private information must therefore be for any θ∈Θ, F(θ)⊂A are the alternatives Typically, θ comprises an N tuple (θ1,.., θN) elicited from the agents. However, agents which are socially optimal in the state θ. where θi is the private information of are sophisticated in the sense that they The crux of the problem is that the plan­ agent i. In general θ1,.., θN are random vari­ recognise that they may (depending on ner does not know the realisation of the ables whose joint distribution is commonly beliefs that they have about the information state of the world, i e, does not know known (by all agents and the planner) but revealed by the other agents) be served which θ∈Θ has occurred. Agents however θi is observed only by agent i. In the pri- better by lying rather than by telling the have better information about the state vate values model (to which I shall mainly truth. Computing the optimal allocation and in particular, jointly know the θ. In confine my attention), θi determines i’s from incorrect information may entail se­ other words, if their information is pooled, preferences Ri(θi). Here, (G,θ) is a game of rious errors; hence the challenge is to de­ then it identifies θ exactly (I shall try and incomplete information and there are two vise a mechanism or a procedure for com­ make these notions more precise very solution concepts which are natural. The municating the information of agents such shortly). The planner must induce the first is dominant strategy equilibrium where that the outcome is an optimal allocation agents to reveal their information. She each agent has an optimal message which even when these agents behave strategi­ does so by constructing a mechanism G gives him a better outcome no matter what 6 cally. Mechanism design theory can there­ which consists of a message set Mi for messages are sent by the other agents. fore be thought of as a theory of the design each agent i and an outcome function The second is the weaker requirement of institutions or the design of the rules of g:M1×M2×...×MN → A. of Bayes­Nash equilibrium which is a strat­ interactions amongst fully strategic agents Each agent i sends a message mi from egy N­tuple from which no agent, regard­ in order to achieve desirable outcomes. his message set; the outcome function less of the information he receives, can A critical step in the development of a then specifies an outcome g(m1,..,mN) as a deviate unilaterally and obtain a higher theory is the formulation of a conceptual function of the messages received. The expected pay­off. A more complicated framework that allows consideration of rele­ mechanism is commonly known to all the model is the interdependent values model vant issues at a high degree of generality. agents. A pair (G,θ), θ∈Θ therefore defines in the incomplete information setting, For instance, progress in game theory was a game. Consider a solution concept E ap­ where each agent i observes θi but whose possible only after von Neumann and propriate for this game and let E(G,θ)⊂A preferences depend not just on but also on Morgenstern introduced the idea of normal denote the set of equilibrium outcomes ac­ θi the entire vector θ, i e, we have Ri(θi). and extensive form representations. This cording to E. We say that the mechanism G I now give a number of applications which permitted a unified analysis ofall strategic implements F (according to E) if there ex­ illustrate the general framework outlined interaction between players including in its ists a mechanism G such that E(G,θ)=F(θ) above and describe some important results. ambit for instance, interaction as disparate for all θ∈Θ.4 Moreover, the SCC F is imple- as that in chess and that in oil cartels. In mentable if there exists a mechanism 3 Complete Information this respect, the foundations of mechanism which implements it. If a SCC is imple­ Adjudication Problems: The most cele­ design theory were laid by Hurwicz (1960, mentable, the planner can put the mecha­ brated example is the so­called King 1972). The basic model ,which I outline nism which implements it into operation Solomon problem from the Book of Kings below, is both simple and versatile.
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