"Efficient and Durable Decision Rules with Incomplete Information", by Bengt Holmström and Roger B. Myerson

Transcription

1 April 15, 2015 "Efficient and Durable Decision Rules with Incomplete Information", by Bengt Holmström and Roger B. Myerson Econometrica, Vol. 51, No. 6 (Nov., 1983), pp Stable URL: This paper investigates Pareto efficiency when agents have private information that affect their utilities. The notion of Pareto dominance has both normative and positive uses in economic theory: normative: The consulting economist may suggest a change in policy to a group of agents that makes none of them worse off while strictly improving the welfare of at least one of them. positive: If a Pareto improvement exists for a group of agents, then they would presumably agree to change the policy to achieve the improvement. Pareto efficiency therefore characterizes equilibrium. ThisisthelogicthatunderliestheCoaseTheorem. Both the normative and the positive uses of Pareto dominance become more complicated if the agents have private information that affect their utilities. Specifically, how can the consulting economist suggest a Pareto improvement if he does not fully know the preferences of the agents? And can the agents identify and agree to a Pareto improvement if they do not know each other s preferences? In situations with incomplete information, the normative and the positive implications of Pareto improvements can be different. For instance, the agents may be aware of an improvement that the consulting economist is unaware of, given his ignorance of their preferences. This paper formally sorts out the impact of private information upon Pareto efficiency. The greatest impact of this paper is its taxonomy or classification of different notions of Pareto efficiency as distinguished by the state of knowledge and the potential for strategic behavior by the agents. The taxonomy provides a precise language for discussing efficiency in settings with incomplete information. You should learn this language and use it. The Model There are agents. For each agent, let denote his finite set of possible types. While the definitions in this discussion extend to infinite type sets, the use of finite type sets keeps the discussion simple (e.g., it avoids the distraction of dealing with issues of measurability). The set of information states is = 1 Agent s beliefs are given by a probability measure ( ) on. The paper does not require the common prior assumption (i.e., ( ) = ( ) for all 1 and ) that is so commonly assumed in the theory of Bayesian games. The group of agents can select a choice 0. The paper allows a randomized selection of a group choice and so it considers the set of all probability measures on 0. The group choice could be an allocation of goods, the choice of a leader, a level of a public good, etc.. It is assumed that the preferences of each agent are given by a utility function : R. An agent s utility can depend upon the types of other agents. The utility functions of the agents are fixed and assumed to be common knowledge. The possibility of making the group choice contingent on the information state leads us to consider decision rules :. We do not need to consider randomized decision rules because randomization is built into. Let denote the set of all possible decision rules. We use the following notation and terminology for an agent s expected payoff given the decision rule : 122

2 agent s ex ante expected utility is ( ) = [ ( ( ) )] agent s interim expected utility conditional on his type is agent s ex post utility in the state is ( )= [ ( ( ) ) ] ( ) = ( ( ) ) This corresponds to the increasing revelation of information: the ex ante stage is when an agent has beliefs but has not yet observed his own type, the interim stage is when he has observed his type but has beliefs about the types of the other agents, and the ex post stage is when the information state is fully realized and known commonly among the agents (but not necessarily to an external consultant). There is an old adage that "timing is everything in comedy." It turns out that "the timing of constraints (ex ante, interim or ex post) is everything in Bayesian mechanism design," as will be discussed below. The presence of private information creates the possibility that an agent may not reveal his type honestly. A decision rule is incentive compatible if honest revelation defines a Bayesian Nash equilibrium, ( )= [ ( ( ) ) ] [ ( ( ) ) ] for all 1 and,. Let denote the set of incentive compatible decision rules. The revelation principle can be invoked to justify restricting attention to revelation mechanisms. Pareto Dominance and Efficiency We have three notions of Pareto dominance in comparing two decision rules and, distinguishedbythe timing of the welfare comparison: ex ante dominates if ( ) ( ) for =1 with strict inequality for at least one. This means that each agent weakly prefers to before learning his type, with at least one agent strictly preferring to. interim dominates if ( ) ( ) for =1 and all, with strict inequality for at least one and type. This means that each agent weakly prefers to after learning his type, whatever value of type he observes, with at least one agent strictly preferring to upon learning some particular value of his type. ex post dominates if ( ) ( ) for =1 and all, with strict inequality for at least one and information state. This means that each agent weakly prefers to in each information state, for all possible information states, with at least one agent strictly preferring to in at least one information state. The burden of evaluation for an external consultant multiplies as one passes from the ex ante to the interim to the ex post stages if he is to be certain that a change in the decision rule from to is a Pareto improvement for the welfare of the agents: at the ex ante stage, he must make comparisons of expected utility values; at the interim stage, he must make X =1 123

3 comparisons; at the ex post stage, he must make comparisons. This is all because the consultant does not privately observe the types of the agents. Ex ante dominance respects the welfare of each of the agents. Interim dominance respects the welfare of each type of each agent: it treats each type of each agent as an individual whose welfare is to be respected. Ex post dominance respects the welfare of each individual in each possible information state: it treates an agent in a particular state as a distinct individual whose welfare must be respected. Our three notions of Pareto dominance combine with our two sets of feasible decision rules ( and ) to produce six notions of Pareto efficiency. The adjective "classical" is used by Holmström and Myerson to indicate Pareto efficiency in, ignoring the constraint of incentive compatibility. The adjective "incentive" is used to indicate Pareto efficiency in, wherein comparisons are only made among incentive compatible decision rules. The six sets of Pareto efficient decision rules are:, the ex ante classically efficient decision rules;, the interim classically efficient decision rules;, the ex post classically efficient decision rules;, the ex ante Incentive efficient decision rules;, the interim incentive efficient decision rules;,theexpostincentiveefficient decision rules. It can be shown that and are each closed and convex. Consider the decision rules that maximize the welfare measure X X ( ) = ( ) ( ) ( ( ) ) (29) =1 where each ( ) maps into R +. The function ( ) is a welfare weight. The efficient decision rules maximize ( ) over or for some choice of the welfare weights. Consideration of arbitrary weights ( ) produces the ex post efficient decision rules., consideration of welfare weights such that each ( ) produces the interim efficient allocation rules, and consideration of constant welfare weights R + produces the ex ante efficient decision rules. This implies that and Also, and Holmström and Myerson (p. 1807) appraise these six notions of Pareto efficiency as follows: All notions, with the exception of,havebeenusedearlierintheliterature. However, in our view, only three of these notions (one for each evaluation stage) are relevant: ex ante incentive efficiency, interim incentive efficiency, and ex post classical efficiency. If the entire information state were to become publicly known before the decision in is chosen, then there would be no incentive problems and would be the right efficiency concept to use. If the decision rule must be selected when each individual knows only his own type, then the incentive constraints (3.1) apply, and is the right efficiency concept. If the decision rule can be selected before individuals learn their types, but if the individuals cannot commit themselves ex ante to honestly report their types after they learn them, then is the appropriate efficiency concept. Question: Is strategic misrepresentation possible in situations of complete information? In the years following this paper, has become relevant in the literature using a different notion of incentive compatibility than the interim, Bayesian-Nash solution concept that defines. A person can act strategically and lie even when everyone knows that the person is lying. An alternative notice of incentive compatibility is ex post Nash equilibrium. Here, this means that truthful revelation ex post is a best response to truthful revelation by everyone else, i.e., ( ) = ( ( ) ) ( ( ) ) for all, 1 124

4 This is an appropriate notion of incentive compatibility for the ex post stage. Finally, interim incentive efficiency is commonly referred to as incentive efficiency, andex post classical efficiency is commonly shortened to simply ex post efficiency. Notice that the focus here is very much upon the effect of private information: no one yet has private information at the ex ante stage, each agent has private information at the interim stage, and no one has private information at the ex post stage. The distinction between the interim and ex post stages most directly isolates the impact of private information on group decisions. The Usefulness of Interim Incentive Efficiency Holmström and Myerson identify ex ante incentive efficiency, interim incentive efficiency, and ex post classical efficiency as the three useful notions of Pareto efficiency for situations with incomplete information. The incentive constraint is applied at the ex ante and the interim stages in which private information may permit an agent to affect the group decision through misrepresentation. Of these three notions of efficiency, interim incentive efficiency is the hardest to understand. Most papers on Bayesian mechanism design compare ex ante incentive efficiency to ex post classical efficiency as a way of identifying the costs to agents of private information. What is the usefulness of interim incentive efficiency? One use arises from the theorem proven above: if the agents have adopted an interim incentive efficient decision rule, then it cannot be common knowledge at the interim stage that the agents should switch to some other decision rule. There is therefore "friction" or difficulty in switching to a better decision rule, given the realization of the agents types. This provides a positive justification of interim incentive efficient decision rules: they may be likely to persist because there is an inherent obstacle to switching to something better. We may therefore be interested in the interim incentive efficient decision rules because they represent what we may expect to see in practice. This viewpoint is pursued by Robert Wilson in his paper, "Incentive Efficiency of Double Auctions," 8 which shows that equilibria in a particular kind of trading game are interim incentive efficient but not ex post classical efficient. A goal of the paper is to explain why an equilibrium in a particular trading institution might persist even though it fails to achieve all possible gains from trade in every information state. Alternative justifications are provided in two papers by Ledyard and Palfrey. 9 Thepointcanbemade here using the simplest model of the two papers. A group of agents must decide whether or not to produce a discrete public good and who should pay for it. A feasible allocation for this problem is any vector ( 1 ) R {0 1} such that X =1 The value of =1indicates that the public good is provided while =0indicates that it is not. The value of is the tax on agent and is a fixed constant. Ledyard and Palfrey also consider a variety of bounds on as part of feasibility but we will not need this for our discussion. Agent s utility is = (30) where is his type. The value of determines his marginal cost of taxation. Notice that a more standard approach is = (31) where his type =1 is the value that he places on the public good. The two approaches are equivalent. The simplest Ledyard-Palfrey model considersthecaseoftwopossiblevaluesof, The types of the agents are i.i.d.. Let denote the probability that any agent s type is 1. An allocation rule specifies the probability that the public good is provided and expected taxes for each of the agents for each vector of their reported types. Assume symmetry of the allocation rule in its treatment of the agents, i.e., two agents whose announce the same type are taxed the same amount. Let 8 Econometrica, Vol. 53, No. 5 (Sep., 1985), pp "Voting and Lottery Drafts as Efficient Public Goods Mechanisms," The Review of Economic Studies, Vol. 61, No. 2 (Apr., 1994), pp , and "Characterization of Interim Efficiency with Public Goods," Econometrica, Vol. 67, No. 2 (Mar., 1999), pp

5 denote the conditional probability that the public good is provided and the expected tax of an agent given that his type is. For a welfare weight 0, Ledyard and Palfrey consider ( ) that maximize the welfare objective ( )+(1 )( ) Multiplying this by to account for all of the agents produces the welfare objective (29) in which the type 1 is assigned the weight and the type 2 is assigned the weight 1. We can dispense with the constant, which does not affect the maximization of this objective subject to incentive compatibility, feasibility and wealth constraints, and participation constraints. Notice that different individuals of the same type are weighted equally. This is mainly for reasons of tractibility, but it might also be argued on the grounds that is reasonable to treat people differently based upon the pain they feel from taxation (their types) but not something arbitrary such as their numerical index. We do wish to argue here that interim incentive efficiency is the correct efficiency standard here if thewelfareofpeopleofdifferent types should be weighed differently. A person of lower type 1, for instance, is hurt less by a dollar of taxation that a person of high type 2. It might therefore be argued that 1 is appropriate, i.e., greater weight should be attached to the 2 -individuals who suffer the most from taxation. It might also be the case that 1 corresponds to rich individuals and 2 to poor individuals (we haven t specified the wealth endowments in this discussion). It is common in the discussion of social welfare to place greater weight on the welfare of the poor than the rich. Selecting welfare weights is a question of values and judgment. The point here is that interim incentive efficiency corresponds to weighing the interests of different types of individuals differently. Ex ante incentive efficiency is simply the case of =1, so that each type of each agent is weighed equally (again, we have assumed equal treatment across agents). Ledyard and Palfrey make a second argument in favor of interim incentive efficiency in this model. Consider the case of = 2 1, so that the objective reduces to µ µ (1 )( ) 1 Now multiply the objective by 1 2 to obtain µ µ µ 1 1 +(1 ) µ This is the objective for ex ante incentive efficiency in the model in which 1 is an agent s reservation value for the public good. The solution to maximizing this objective is therefore the ex ante efficient decision rule for a different normalization of the utility functions in the model. A quotation from Ledyard and Palfrey (1994, p. 333) summarizes the point here: Actually, in our problem, the ex ante optimal allocation rule is simply a particular point on the interim efficiency frontier, corresponding to =1. In fact, each of the other points on the frontier ( 6= 1) corresponds to an ex ante optimal allocation rule for a different normalization of the utility functions. In this sense, one can view our approach of characterizing the symmetric interim efficient frontier as equivalent to characterizing the set of allocation rules that are ex ante efficient for some normalization of the utility functions. Effectively, interim incentive efficiency is in their paper is a way to generalize ex ante incentive efficiency to all possible normalizations of the utility functions in their model. 126