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Robust Mechanism Design∗ Robust Mechanism Design∗ Dirk Bergemann† Stephen Morris‡ First Version: December 2000 This Version: November 2001 Preliminary and incomplete Abstract The implementation literature is often criticized because the mechanisms do not seem to be robust to assumed features of the underlying environment. This paper explores one way of formalizing robustness: fixing the set of pay- off relevant types of each player, we ask whether it is possible to implement a given object on large type spaces (reflecting a lack of common knowledge the environment). We show how equilibrium implementation conditions on the universal type space translates into strong solution concepts on the original types space of payoff relevant types. Keywords: Mechanism Design, Common Knowledge, Universal Type Space, Interim Equilibrium, Ex-Post Equilibrium. Jel Classification: ∗This research is support by NSF Grant #SES-0095321. We are grateful for comments from seminar participants at the Institute for Advanced Study, the Stony Brook Game Theory Conference: Seminar on Contract Theory, and the University of Pennsylvania who heard very preliminary versions of this work. We thank Matt Jackson, Jon Levin, Zvika Neeman, Andrew Postlewaite and especially Bart Lipman for valuable discussions about the universal type space. †Department of Economics, Yale University, 28, Hillhouse Avenue, New Haven, CT 06511, [email protected]. ‡Department of Economics, Yale University, 30, Hillhouse Avenue, New Haven, CT 06511, [email protected]. “Game theory has a great advantage in explicitly analyzing the con- sequences of trading rules that presumably are really common knowl- edge; it is deficient to the extent it assumes other features to be com- mon knowledge, such as one agent’s probability assessment about an- other’s preferences or information. I foresee the progress of game theory as depending on successive re- ductions in the base of common knowledge required to conduct use- ful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality.” Wilson (1987) 1. Introduction Two different strands of thought motivate this research. First, the implementation literature is often thought to be fragile because optimal mechanisms seem to be too sensitive to the assumed structure of the environment. This leads researchers to look for mechanisms that are insensitive to certain features of the environment. However, there is not a clear theoretical rationale for the notions of robustness used. We would like to develop a better language for discussing the robustness of mechanisms. Second, the work of Harsanyi (1967/1968) and Mertens and Zamir (1985), establishes that we can, without loss of generality, model environments with in- complete information by a Bayesian game. That is, without loss of generality, one can assume that there is common knowledge among the players of each player’s type spaces and each type’s beliefs over the types of other players. However, as a practical matter, applied economic analysis tends to assume much smaller type spaces than the universal type space, and yet maintain the assumption that there is common knowledge among the players of each player’s type spaces and each type’s beliefs over the types of other players. Of course, the latter assump- tion is certainly not without loss of generality. It is then useful to check if and when the results of applied economic analysis are robust to the inclusion of a rich type spaces with little common knowledge. A recent important paper by Neeman (1999) suggests that the small type space assumption may be especially problematic in the context of mechanism design. We would like to examine the robustness of the implementation literature to higher order uncertainty. Bringing these two strands together, we propose to introduce rich type spaces (with non-trivial higher order beliefs) into the implementation literature; in doing so, we hope to provide a natural language for talking about robustness. More precisely, our approach to robustness is to weaken the base of common knowledge by allowing agents to have richer type spaces and in particular richer higher- order beliefs. This effectively reduces the common knowledge among agents and designer. In consequence, our investigation encompasses, but is not restricted to earlier notions of robustness. In this context, we may mention briefly two of the 2 most prominent positions taken in the literature. Wilson (1985) states that a desirable property of a trading rule is that it “does not rely on features of the agents’ common knowledge, such as their probability assessments.” By gradually extending the type space, and asking whether a social choice function can still be implemented, we obtain a precise notion to what extent a trading rule relies on common knowledge among the agents. Similarly, Maskin and Dasgupta (2000) “seek auction rules that are independent of the details - such as functional forms or distribution of signals - of any particular application and that work well in a broad range of circumstances”. Finally, a frequently raised criticism to the mechanism design literature is that the assumption that the designer shares the common prior with the agents. While this is certainly one direction in which common knowledge can be weakened and it is a special case of our analysis, it is perhaps the easiest one to solve, as any information which is non-exclusive in the sense of Postlewaite and Schmeidler (1986), can always be truthfully elicited from the agents in an interim equilibrium. We proceed as follows. First, we can fixasetofpayoff-relevant types for each player and social choice function mapping profiles of payoff-relevant types to outcomes. The objective of the mechanism designer will be to implement this social choice function. We will say that a social choice function is truth- fully implemented if a truthtelling equilibrium in a direct mechanism yields the outcome prescribed by the social choice function. The social choice function is simply implemented if there exists a mechanism under which the prescription is always followed under every equilibrium. While holding fixed this environment, we can construct many type spaces, where a player’s type specifies both his payoff- relevant type and his belief about other players’ types. Crucially, there may be manytypesofaplayerwiththesamepayoff relevant type. The larger the type space we construct, the harder it will be to implement the social choice function, and so the more “robust” the resulting mechanism will be. Thesmallesttypespacewecanworkwithisthe“naivetypespace,”where we set the possible types of each player equal to the set of payoff-relevant types, and assume a common knowledge prior over this type space. This is the usual exercise performed in the mechanism design literature. The largest type space we can work with is the “universal type space,” where we allow players to have any beliefs or higher order beliefs about other players’ payoff relevant types. This space can be constructed from the players’ payoff relevant types along the lines of Mertens and Zamir (1985). There are many interesting type spaces in between the naive type space and the universal type space that are also interesting to study. For example, we can look at the union of all naive type spaces (so that the players have common knowledge of a prior over payoff relevant types but the mechanism designer does not); and we can look at the subset of types in the universal type space for whom the common prior assumption holds. In the current work, we present three benchmark results. 1. We show that requiring equilibrium implementation on the universal type 3 space is equivalent to requiring ex post equilibrium implementation on the naive type space. 2. We provide a characterization of when the social choice function can be equilibrium implemented on an arbitrary finite type space satisfying the common prior assumption. The condition shows that it is enough to check if equilibrium implementation on the naive type space would have been possible for certain independent types. 3. We show that requiring full equilibrium implementation on the universal type space is equivalent to requiring an incomplete information version of dominance solvability on the naive type space. In the concluding section, we discuss other issues we hope to address using this framework. 2. Framework We introduce the payoff relevant environment in Subsection 2.1, then define a variety type spaces in Subsection 2.2 and finally discuss and solution and imple- mentationconceptsinSubsection2.3. 2.1. The Payoff Relevant Environment We consider a finite set of agents = 1, 2, ..., I . Agent i’s payoff relevant type is I { } θi Θi, where Θi is a finite set. We write θ Θ = Θ1 ... ΘI . A social outcome ∈ ∈ × × is denoted x X, where X is a finite set. A vector of transfers is denoted y RI . ∈ ∈ Every agent has a utility function ui : X Θ R,defined on social outcomes and a vector of payoff relevant types. Each× agent→ has quasi-linear utility, so that his utility if the outcome is x,thepayoff relevant type vector is θ and his transfer is yi is ui (x, θ)+yi. 2.2. Type Spaces We are interested in studying type spaces that are richer than Θ,thesetofpayoff relevant types. This is important because we want to allow for the possibility that two types of a player may be identical from a payoff relevant perspective, but they have different beliefs about, say, the payoff relevant types of other players. In addition, we will want to allow for interim type spaces, where there are no restrictions on a type’s interim belief about other player’s types. Requiring that types’ interim beliefs be derived from some prior probability distribution on the type space will then represent an important special case. Agent i’s type is ti Ti. A type of agent i must include a description of his ∈ payoff relevant type.
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