Andrew McLennan

Consequences of the Condorcet Jury Theorem for Beneficial Information Aggregation by Rational Agents

“NaA¯ve†Condorcet Jury Theorems automatically have “sophisticated†versions as corollaries. A Condorcet Jury Theorem is a result, pertaining to an election in which the agents have common preferences but diverse information, asserting that the outcome is better, on average, than the one that would be chosen by any particular individual. Sometimes there is the additional assertion that, as the population grows, the probability of an incorrect decision goes to zero. As a consequence of simple properties of common interest games, whenever “sincere†voting leads to the conclusions of the theorem, there are Nash equilibria with these properties. In symmetric environments the equilibria may be taken to be symmetric.

SEQUENTIAL BARGAINING AS A NONCOOPERATIVE FOUNDATION FOR WALRASIAN EQUILIBRIUM

An allocation for an exchange economy with smooth preferences is shown to be Walrasian if there is a set of net trades that is closed under addition, contains the negations of net trades that would improve any agents final bundle, and is such that each agents final bundle is weakly preferred to the sum of the initial endowment and any allowed net trade. These conditions characterize the sets of net trades available in equilibria of market games in which randomly paired agents bargain repeatedly and imply that steady state equilibria are Walrasian. Copyright 1991 by The Econometric Society.

Justifiable Beliefs in Sequential Equilibrium

An action in an extensive game that has a suboptimal payoff in every sequential equilibrium is said to be useless. There are sequential equilibria in which beliefs at each information set assign positive probability only to those nodes reached by the fewest useless actions. An action is second order useless if it is not useless but is strictly suboptimal in every equilibrium satisfying this condition on beliefs, and there exist sequential equilibria in which beliefs assign positive probability only to those nodes reached by the fewest useless actions, and, in this set, only those nodes requiring the fewest second order useless actions. Higher order uselessness is defined inductively, and beliefs satisfying the associated sequence of conditions are said to be justifiable. The existence of sequential equilibria with justifiable beliefs is demonstrated.

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49-73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N R I, the expected number of Nash equilibria is approximately (R(π log N) / 2)(M-1)/R M. Letting M = N R I, the expected number of Nash equilibria is exp(NM + O(log N)), where M A 0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.

Uniqueness of Stationary Equilibrium Payoffs in Coalitional Bargaining

We study a model of sequential bargaining in which, in each period before an agreement is reached, the proposerʼs identity is randomly determined, the proposer suggests a division of a pie of size one, each other agent either approves or rejects the proposal, and the proposal is implemented if the set of approving agents is a winning coalition for the proposer. The theory of the fixed point index is used to show that stationary equilibrium expected payoffs of this coalitional bargaining game are unique. This generalizes Eraslan [34] insofar as: (a) there are no restrictions on the structure of sets of winning coalitions; (b) different proposers may have different sets of winning coalitions; (c) there may be a positive probability that no proposer is selected.

Consistent conditional in noncooperative game theory

A space ofconsistent conditional systems is obtained by combining concepts introduced by Kreps and Wilson (1982) and Myerson (1986). The sequential equilibrium concept is displayed as the set of fixed points of a best response correspondence from this space to itself. A thorough exploration of the space shows that the best response correspondence satisfies the hypotheses of the Eilenberg-Montgomery (1946) fixed point theorem, and this allows notions of stability analogous to those introduced by Kohlberg and Mertens (1986) to be defined.

Simple complexity from imitation games

We give simple proofs of refinements of the complexity results of Gilboa and Zemel (1989), and we derive additional results of this sort. Our constructions employ imitation games, which are two person games in which both players have the same sets of pure strategies and the second player wishes to play the same pure strategy as the first player.

The space of conditional systems is a ball

LetS be a nonempty finite set. The space Δ(S) of conditional systems onS, as defined by Myerson (1986), is shown to be homeomorphic to the closed unit ball in (#S-1)-dimensional Euclidean space.

Strategic candidacy for multivalued voting procedures

Dutta et al. (Econometrica 69 (2001) 1013) (Dutta, Jackson, and Le Breton-DJLeB) initiate the study of manipulation of voting procedures by a candidate who withdraws from the election. A voting procedure is candidate stable if this is never possible. We extend the DJLeB framework by allowing: (a) the outcome of the procedure to be a set of candidates; (b) some or all of the voters to have weak preference orderings of the candidates. When there are at least three candidates, any strongly candidate stable voting selection satisfying a weak unanimity condition is characterized by a serial dictatorship. This result generalizes Theorem 4 of DJLeB

Truthful implementation and preference aggregation in restricted domains

In a setting where agents have quasi-linear utilities over social alternatives and a transferable commodity, we consider three properties that a social choice function may possess: truthful implementation (in dominant strategies); monotonicity in differences; and lexicographic affine maximization. We introduce the notion of a flexible domain of preferences that allows elevation of pairs and study which of these conditions implies which others in such domain. We provide a generalization of the theorem of Roberts (1979) [36] in restricted valuation domains. Flexibility holds (and the theorem is not vacuous) if the domain of valuation profiles is restricted to the space of continuous functions defined on a compact metric space, or the space of piecewise linear functions defined on an affine space, or the space of smooth functions defined on a compact differentiable manifold. We provide applications of our results to public goods allocation settings, with finite and infinite alternative sets.