Declining valuations in sequential auctions

We analyze an independent private values model where a number of objects are sold in sequential first- and second-price auctions. Bidders have unit demand and their valuation for an object is decreasing in the rank number of the auction in which it is sold. We derive efficient equilibria if prices are announced after each auction or if no information is given to bidders. We show that the sequence of prices constitutes a supermartingale. Even if we correct for the decrease in valuations for objects sold in later auctions we find that average prices are declining.Received June 2004We are grateful to Christian Groh, Wolfgang Köhler and Benny Moldovanu for helpful suggestions. Financial support from the German Science Foundation through SFB 504 and SFB/TR 15 at the University of Mannheim and the University of Bonn is gratefully acknowledged.     

An axiomatization of the consistent non-transferable utility value

The Maschler–Owen consistent value for non-transferable utility games is axiomatized, by means of a marginality axiom.Previous versions: October 2003 (Center for the Study of Rationality DP-337), December 2004. Research partially supported by a grant of the Israel Academy of Sciences and Humanities. The author thanks Andreu Mas-Colell, Bezalel Peleg, Peter Sudhölter, the referees, and the editor for their comments and suggestions.     

On the geometry of Nash equilibria and correlated equilibria

It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.We are grateful to Francoise Forges, Dan Arce, the editors, and several anonymous referees for helpful comments. This research was supported by the National Science Foundation under grant 98–09225 and by the Fuqua School of Business.The use of correlated mixed strategies in 2-player games was discussed by Raiffa (1951), who noted: it is a useful concept since it serves to convexify certain regions [of expected payoffs] in the Euclidean plane. (p. 8)Received: April 2002 / Revised: November 2003     

A dynamic location problem with maximum decreasing capacities

In this paper a capacitated dynamic location problem with opening, closure and reopening of facilities is formulated and a primal-dual heuristic that can solve this problem is described. The problem formulated considers the situation where a facility is open (or reopens) with a certain maximum capacity that decreases as clients are assigned to that facility during its operating periods. This problem is NP-hard. Computational results are presented and discussed. This research was partially supported by research project POCTI/ISFL-1/152 and POCTI/MAT/139/2001.     

Correlated equilibrium and concave games

This paper shows that if a game satisfies the sufficient condition for the existence and uniqueness of a pure-strategy Nash equilibrium provided by Rosen (Econometrica 33:520, 1965), then the game has a unique correlated equilibrium, which places probability one on the unique pure-strategy Nash equilibrium. In addition, it shows that a weaker condition suffices for the uniqueness of a correlated equilibrium. The condition generalizes the sufficient condition for the uniqueness of a correlated equilibrium provided by Neyman (Int J Game Theory 26:223, 1997) for a potential game with a strictly concave potential function. I thank the editor, an associate editor, and an anonymous referee for detailed comments and suggestions, which have substantially improved this paper. Special thanks are due to the referee for pointing out Lemmas 4 and 5. I acknowledge financial support by The Japan Economic Research Foundation and by MEXT, Grant-in-Aid for Scientific Research. All remaining errors are mine.     

Farsighted Stability in an n-Person Prisoner’s Dilemma

Farsighted stability is examined in an n-person prisoner’s dilemma. It is shown that any individually rational and Pareto efficient outcome is a farsighted stable set and no other farsighted stable set exists; it is further shown that the largest consistent set consists of all individually rational outcomes.     

Axiomatizing the Harsanyi solution,the symmetric egalitarian solution and the consistent solution for NTU-games

The validity of the axiomatization of the Harsanyi solution for NTU-games by Hart (1985) is shown to depend on the regularity conditions imposed on games. Following this observation, we propose two related axiomatic characterizations, one of the symmetric egalitarian solution (Kalai and Samet, 1985) and one of the consistent solution (Maschler and Owen, 1992). The three axiomatic results are studied, evaluated and compared in detail.Revised October 2004We thank an anonymous referee and an associate editor for their helpful comments. Geoffroy de Clippel also thanks Professors Sergiu Hart, Jean-François Mertens and Enrico Minelli. Horst Zank thanks the Dutch Science Foundation NWO and the British Council for support under the UK-Netherlands Partnership Programme in Science (PPS 706). The usual disclaimer applies.     

Resource-monotonicity for house allocation problems

We study a simple model of assigning indivisible objects (e.g., houses, jobs, offices, etc.) to agents. Each agent receives at most one object and monetary compensations are not possible. We completely describe all rules satisfying efficiency, independence of irrelevant objects, and resource-monotonicity. The characterized rules assign the objects in a sequence of steps such that at each step there is either a dictator or two agents trade objects from their hierarchically specified endowments.We wish to thank Szilvia Pápai, William Thomson, and two anonymous referees for their useful comments.L. Ehlers gratefully acknowledges financial support from the SSHRC (Canada) and FQRSC (Québec).B. Klauss research is supported by a Ramón y Cajal contract of the Spanish Ministerio de Ciencia y Tecnología. B. Klaus is also partially supported by Research Grant BEC2002-02130 from the Spanish Ministerio de Ciencia y Tecnología and by the Barcelona Economics Program of CREA.     

Bargaining sets in exchange economies with indivisibilities and money

This paper introduces and investigates the semireactive bargaining set (Sudhölter and Potters (2001)) and the reactive bargaining set (Granot (1994)), which are originally solution concepts for TU-games, in economies in which agents exchange indivisible goods and one perfectly divisible good (money). Under the assumptions that the preferences of the agents are quasi-linear and the endowments satisfy the Total Abundance condition, a condition on the amounts of money agents initially have, it is shown that the (semi)reactive bargaining set is nonempty. In addition, we prove that in such an economy the (semi)reactive bargaining set and the (strong) core coincide if and only if the (semi)reactive bargaining set and the core of the underlying TU-game coincide.The authors are grateful to Peter Sudhölter for his helpful suggestions that greatly improved the paper     

On stochastic games in economics

In this paper we show that many results on equilibria in stochastic games arising from economic theory can be deduced from the theorem on the existence of a correlated equilibrium due to Nowak and Raghavan. Some new classes of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria are also presented. Three nontrivial examples of dynamic stochastic games arising from economic theory are given closed form solutions. Research partially supported by MNSW grant 1 P03A 01030.     

Fuzzy play,matching devices and coordination failures

We revisit n-player coordination games with Pareto-ranked Nash equilibria. As a novelty, we introduce fuzzy play and a matching device. By fuzzy play we mean that each player does not choose which pure strategy to play, but instead chooses a nonempty subset of his strategy set that he submits to the matching device. The matching device is a very simple one. It randomly selects a match if possible, and it selects randomly some strategy belonging to the strategy set sent by each player otherwise. That is, it does not impose that the best alternatives are matched. Using the concepts of perfect Nash equilibrium and of trembling-hand perfect rationalizability, we show that players coordinate directly on the Pareto optimal outcome. This implies that they neither use the option of fuzzy play, nor make use of the matching device.We thank an anonymous referee and an Associate Editor for valuable comments. Jean-Jacques Herings would like to thank the Netherlands Organisation for Scientific Research (NWO) for financial support. Vincent Vannetelbosch is Chercheur Qualifié at the Fonds National de la Recherche Scientifique. The research of Ana Mauleon has been made possible by a fellowship of the Fonds Européen du Développement Economique Régional (FEDER). Financial support from the Belgian French Communitys program Action de Recherches Concertée 99/04-235 (IRES, Université catholique de Louvain) is gratefully acknowledged.     

Two-stage stochastic hierarchical multiple risk problems: models and algorithms

In this paper, we consider a class of two-stage stochastic risk management problems, which may be stated as follows. A decision-maker determines a set of binary first-stage decisions, after which a random event from a finite set of possible outcomes is realized. Depending on the realization of this outcome, a set of continuous second-stage decisions must then be made that attempt to minimize some risk function. We consider a hierarchy of multiple risk levels along with associated penalties for each possible scenario. The overall objective function thus depends on the cost of the first-stage decisions, plus the expected second-stage risk penalties. We develop a mixed-integer 0–1 programming model and adopt an automatic convexification procedure using the reformulation–linearization technique to recast the problem into a form that is amenable to applying Benders’ partitioning approach. As a principal computational expedient, we show how the reformulated higher-dimensional Benders’ subproblems can be efficiently solved via certain reduced-sized linear programs in the original variable space. In addition, we explore several key ingredients in our proposed procedure to enhance the tightness of the prescribed Benders’ cuts and the efficiency with which they are generated. Finally, we demonstrate the computational efficacy of our approaches on a set of realistic test problems. Dr. H. D. Sherali acknowledges the support of the National Science Foundation under Grant No. DMI-0552676. Dr. J. C. Smith acknowledges the support of the Air Force Office of Scientific Research under Grant No. AFOSR/MURI F49620-03-1-0477.     

A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approaches

The paper investigates model reduction techniques that are based on a nonlocal quasi-continuum-like approach. These techniques reduce a large optimization problem to either a system of nonlinear equations or another optimization problem that are expressed in a smaller number of degrees of freedom. The reduction is based on the observation that many of the components of the solution of the original optimization problem are well approximated by certain interpolation operators with respect to a restricted set of representative components. Under certain assumptions, the “optimize and interpolate” and the “interpolate and optimize” approaches result in a regular nonlinear equation and an optimization problem whose solutions are close to the solution of the original problem, respectively. The validity of these assumptions is investigated by using examples from potential-based and electronic structure-based calculations in Materials Science models. A methodology is presented for using quasi-continuum-like model reduction for real-space DFT computations in the absence of periodic boundary conditions. The methodology is illustrated using a basic Thomas–Fermi–Dirac case study.     

The gamma-core and coalition formation

This paper reinterprets the γ-core (Chander and Tulkens in Int Tax Pub Financ 2:279–293, 1995; in Int J Game Theory 26:379–401, 1997) and justifies it as well as its prediction that the efficient coalition structure is stable in terms of the coalition formation theory. The problem of coalition formation is formulated as an infinitely repeated game in which the players must choose whether to cooperate or not. It is shown that a certain equilibrium of this game corresponds to the γ-core assumption that when a coalition forms the remaining players form singletons, and that the grand coalition is an equilibrium coalition structure. An earlier version of this paper was presented at the conference on Game Theory and Its Applications held in Mumbai in 2003 and was subsequently circulated as CORE Discussion Paper 2003/46.     

An exotic sphere with positive curvature almost everywhere

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere. In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.     

Submodularity and the evolution of Walrasian behavior

Vega-Redondo (1997) showed that imitation leads to the Walrasian outcome in Cournot Oligopoly. We generalize his result to aggregative quasi-submodular games. Examples are the Cournot Oligopoly, Bertrand games with differentiated complementary products, Common-Pool Resource games, Rent-Seeking games and generalized Nash-Demand games.I thank Jörg Oechssler, Rolf Tisljar, the editor, and an anonymous referee for helpful comments and suggestions. Financial support by the DFG is gratefully acknowledged.Submitted: September 2001 / Revised: January 2004     

An isolic generalization of Cauchy's theorem for finite groups

In his note [5] Hausner states a simple combinatorial principle, namely:

Linear programming in the foreign exchange markets

This article demonstrates the use of a linear programming model to achieve an optimal allocation ot liquid funds among various currencies in different countries. The model takes into account interest rates, projected changes in currency values, relative risk and corporate policies and safeguards. Currency trading has reached unprecedented proportions 1. 5 trillion dollars are traded daily and the volume keeps increasing. World trade in goods, for comparison, amounts to $ 4 trillion per year.     

The money-burning refinement: With an application to a political signalling game

We define an equilibrium refinement in signalling games by allowing agents to refine the game by burning money whenever they can act. We apply the refinement in a model where the public is unsure about the ability of an agent, say a government, to foresee the effects of long-term decisions. A government with ample information about the consequences of decisions should invest either immediately or not at all. Poorly informed agents should wait for better information. We identify pooling equilibria in which excessive rush or waiting occurs. The money-burning refinement may eliminate both rash and waiting pooling equilibria, but it involves wasting money and, for high discount factors, a decrease in welfare.First version August 1997/This version March 2004For valuable suggestions and comments I would like to thank Hans-Jörg Beilharz, Eric van Damme, Ulrich Erlenmaier, Ami Glazer, Paul Grout, Volker Hahn, Hans Haller, Majia Halonen, Martin Hellwig, Verena Liessem, Till Requate, Gisèle Umbhauer, seminar participants in Bristol, Heidelberg and Mannheim and at the Conference of the European Public Choice Society 1998 in Goeteborg, two anonymous referees, and the associate editor.     

A branch-and-price algorithm for the two-dimensional level strip packing problem

The two-dimensional level strip packing problem (2LSPP) consists in packing rectangular items of given size into a strip of given width divided into levels. Items packed into the same level cannot be put on top of one another and their overall width cannot exceed the width of the strip. The objective is to accommodate all the items while minimizing the overall height of the strip. The problem is -hard and arises from applications in logistics and transportation. We present a set covering formulation of the 2LSPP suitable for a column generation approach, where each column corresponds to a feasible combination of items inserted into the same level. For the exact optimization of the 2LSPP we present a branch-and-price algorithm, in which the pricing problem is a penalized knapsack problem. Computational results are reported for benchmark instances with some hundreds items.