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3. Implementable Collective Choice
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Chapter 3 Implementable Collective Choice The Gibbard-Satterthwaite Theorem (Theorem 2.1) says that if we ask people to tell us their preferences to arrive at some social choice, we cannot be sure that they will tell us the truth. But this result does not say that it is generally impossible to elicit the truthful revelation of preferences. Indeed, given a (possibly set-valued) collective choice rule ep, the primary concern is in selecting the "right" outcomes ep(p) ~ X for any preference profile, p. Consequently, if it is possible to insure an outcome in ep(p) for any truthful profile p, then whether or not we can identify the details of the profile itself is immaterial. Exploring this possibility is known as the implementation problem. Suppose a society has agreed upon some collective choice rule to be used for making compromises among the constituent individuals'pr eferences and suppose, further, that application of the rule in any particular choice is the responsibility of a distinct agent, a "planner"; for example, a government agency or a judge. Although the concerned individuals know the true preference profile over the set of alternatives, it is assumed the planner does not. This is reasonable if the application is, for instance, to a small group of signatories to a contract and the planner is a judge; it is somewhat less plausible if the application is to a large and disparate electorate where the planner is an election board. Nevertheless, when individuals are presumed to know at most their own preferences, the problem is more subtle but qualitatively similar which, at least in part, legitimates a working assumption that the preference profile is known to all but the planner. Given the assumption, the problem confronting a planner faced with such limited knowledge and an 69 70 CHAPTER 3. IMPLEMENTABLE COLLECTIVE CHOICE interest in respecting the collective choice rule is to design an institution or a set of rules with the property that, when acting under these rules, strategically rational individuals induce outcomes that would be chosen under the collective choice rule were the true preference profile known to the planner. Whether institutions with the requisite properties exist for any collective choice rule turns out to depend in general on the assumed strategic theory of individual choice. The concept of strategy-proofness reflects a particularly appealing hypothesis: if an individual can never do better than report his or her true preferences, then we expect the individual to reveal these preferences . But when reporting true preferences is not the best an individual could do, the minimal behavioral assumption of strategy-proofness makes no prediction and a stronger model of strategic decision-making is necessary. Although there are several possibilities, the most important such model and the central concept for much of this and subsequent chapters is Nash equilibrium . Loosely speaking, the idea here is to look for institutions under which every individual does the best they can by revealing their true preferences , given that everyone else is likewise revealing their true preferences. Thus, Nash behavior generalizes strategy-proof behavior in that, if telling the truth is the best anyone can do whatever others are doing (strategyproof ), then it is also the best anyone can do when others are telling the truth (Nash); the converse, however, is not true. So Nash equilibrium exists and yields predictions in cases where strategy-proofness is silent; the price of such general existence, however, is that there typically exist many Nash equilibria in any given setting, not all of which involve mutual truth-telling. 3.1 Mechanisms and equilibria The idea of implementation is to design a mechanism, or set of rules, with the property that strategic individuals making choices under these rules end up with outcomes that would be chosen were we to apply the relevant collective choice rule to their true preference profile directly. If we can do this, then the fact that a given collective choice rule is not strategy-proof becomes unimportant because individuals' efforts to manipulate the mechanism turn out to be mutually offsetting. The first thing to do, therefore, is make precise what is meant by a "mechanism" and by "mutually offsetting" strategic behavior. This we now do. Assume the set of alternatives, X, is finite. For any individual i E N, let Mi denote a set of messages or, more generally, actions that an individual has available. For example, a message could be a preference 3.1. MECHANISMS AND EQUILIBRIA 71 ordering over the set...