Dries Vermeulen

The private value single item bisection auction

In this paper we present a new iterative auction, the bisection auction, that can be used for the sale of a single indivisible object. The bisection auction has fewer rounds than the classical English auction and causes less information to be revealed than the Vickrey auction. Still, it preserves all characteristics the English auction shares with the Vickrey auction: there exists an equilibrium in weakly dominant strategies in which everyone behaves truthfully, the object is allocated in accordance with efficiency requirements to the buyer who has the highest valuation, and the price paid by the winner of the object equals the second-highest valuation

The structure of the set of equilibria for two person multicriteria games

In this paper the structure of the set of equilibria for two person multicriteria games is analysed. It turns out that the classical result for the set of equilibria for bimatrix games, that it is a finite union of polytopes, is only valid for multicriteria games if one of the players only has two pure strategies. A full polyhedral description of these polytopes can be derived when the player with an arbitrary number of pure strategies has one criterion.

An axiomatic approach to egalitarianism in TU-games

A core concept is a solution concept on the class of balanced games that exclusively selects core allocations. We show that every continuous core concept that satisfies both the equal treatment property and a new property called independence of irrelevant core allocations (IIC) necessarily selects egalitarian allocations. IIC requires that, if the core concept selects a certain core allocation for a given game, and this allocation is still a core allocation for a new game with a core that is contained in the core of the first game, then the core concept also chooses this allocation as the solution to the new game. When we replace the continuity requirement by a weak version of additivity we obtain an axiomatization of the egalitarian solution concept that assigns to each balanced game the core allocation minimizing the Euclidean distance to the equal share allocation.

Some characterizations of egalitarian solutions on classes of TU-games

Abstract In this paper we derive characterizations of egalitarian solutions on two subclasses of the class of balanced games. Firstly we show that the Dutta–Ray solution is the only solution that satisfies symmetry, independence of irrelevant core allocations, and continuity on the class of convex games. Secondly, together with the other two requirements, a strengthening of continuity to monotonicity in the value of the grand coalition turns out to be sufficient for the characterization of the lexicographically maximal solution on the class of large core games.

WPO, COV and IIA bargaining solutions for non-convex bargaining problems

We characterize all n-person multi-valued bargaining solutions, defined on the domain of all finite bargaining problems, and satisfying Weak Pareto Optimality (WPO), Covariance (COV), and Independence of Irrelevant Alternatives (IIA). We show that these solutions are obtained by iteratively maximizing nonsymmetric Nash products and determining the final set of points by so-called LDR decompositions. If, next, we assume the (set-theoretic) Axiom of Determinacy, then this class coincides with the class of iterated Nash bargaining solutions; but if we assume the Axiom of Choice then we are able to construct an additional large set of discontinuous and even nonmeasurable solutions. We show however that none of these nonmeasurable solutions can be defined in terms of set theoretic formulae. We next show that a number of existing results in the literature as well as some new results are implied by our approach. These include a characterization of all WPO, COV and IIA solutions—including single-valued ones—on the domain of all compact bargaining problems, and an extension of a theorem of Birkhoff characterizing translation invariant and homogeneous orderings.

Convergence of Bayesian learning to general equilibrium in mis-specified models

Abstract A central unanswered question in economic theory is that of price formation in disequilibrium. This paper lays the groundwork for a model that has been suggested as an answer to this question in, particularly, Arrow [Toward a theory of price adjustment, in: M. Abramovitz, et al. (Ed.), The Allocation of Economic Resources, Stanford University Press, Stanford, 1959], Fisher [Disequilibrium Foundations of Equilibrium Economics, Cambridge University Press, Cambridge, 1983] and Hahn [Information dynamics and equilibrium, in: F. Hahn (Ed.), The Economics of Missing Markets, Information, and Games, Clarendon Press, Oxford, 1989]. We consider sellers that monopolistically compete in prices but have incomplete information about the structure of the market they face. They each entertain a simple demand conjecture in which sales are perceived to depend on the own price only, and set prices to maximize expected profits. Prior beliefs on the parameters of conjectured demand are updated into posterior beliefs upon each observation of sales at proposed prices, using Bayes’ rule. The rational learning process, thus, constructed drives the price dynamics of the model. Its properties are analysed. Moreover, a sufficient condition is provided, relating objectively possible events and subjective beliefs, under which the price process is globally stable on a conjectural equilibrium for almost all objectively possible developments of history.

The communication complexity of private value single-item auctions

This paper is concerned with information revelation in single-item auctions. We compute how much data needs to be transmitted in three strategically equivalent auctions-the Vickrey auction, the English auction and the recently proposed bisection auction-and show that in the truth-telling equilibrium the bisection auction is the best performer.

On the nucleolus of neighbor games

The class of neighbour games is the intersection of the class of assignment games (cf. Shapley and Shubik (1972)) and the class of component additive games (cf. Curiel et al. (1994)). For assignment games and component additive games there exist polynomially bounded algorithms of order p4 for calculating the nucleolus, where p is the number of players. In this paper we present a polynomially bounded algorithm of order p2 for calculating the nucleolus of neighbour games.

Perfect equilibrium in games with compact action spaces

Simon and Stinchcombe distinguish two approaches to perfect equilibrium, the “trembling hand” approach, and the “finitistic” approach, for games with compact action spaces and continuous payoffs. We investigate relations between the different types of perfect equilibrium introduced by Simon and Stinchcombe. We also propose an improved version of the finitistic approach, and prove existence.

A generalization of the Shapley–Ichiishi result

The Shapley–Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper, we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley–Ichiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.