The lexicographic composition of abstract games

The lexicographic composition of reflexive and complete binary relations (which are often called abstract games) is studied. The necessary and sufficient conditions are obtained for a lexicographic composition of quasi transitive relations to be quasi transitive.The case of acyclicity of lexicographic composition of two relations is also investigated. Received: September 25, 2000; revised version: November 28, 2000     

Dynamic R&D competition with memory

We consider a differential game of R&D competition and explore the impact of rivalry on the firms' investment behavior over time. Using closed-loop strategies and hence allowing for strategic interactions among rival firms we show that R&D spending by the individual competitor is increased due to competition in the race for priority. This leads us to argue that competitive encounters enhance R&D activities at the same time as increasing efficiency in the race for a technological breakthrough.     

Collusion in one-shot second-price auctions

Summary. Most of the literature on collusive behavior in auctions ignores two important issues that make collusion difficult to sustain at least in one-shot interactions: the detection of cheating and the verification of bids. Colluding bidders may deceive each other by using shill bidders. Also, if the identities of the bidders and their bids are not published then it would be difficult to verify the bid of a colluding bidder. This paper addresses these problems in one shot second price auctions where one bidder offers another bidder a side payment in exchange for not participating in the auction, while the number of other bidders is stochastic. In spite of the barriers to collusion mentioned above, a simple side payment mechanism which depends only on the auction price is introduced. It induces a successful collusion, eliminates the verification problem, provides no incentive for the use of shill bidders and guarantees that the proponent obtains ex-post non-negative payoff. The colluding bidders are ex-ante strictly better off compared with the competitive case, irrespective of their types.Received: 27 November 2002, Revised: 28 January 2005, JEL Classification Numbers: C72, D44, D82.Yair Tauman: Correspondence toWe would like to thank an anonymous referee for very valuable comments and suggestions that significantly improved the paper. We thank Shmuel Zamir for a helpful discussion.     

Finding a Nash equilibrium in spatial games is an NP-complete problem

Summary. We consider the class of (finite) spatial games. We show that the problem of determining whether there exists a Nash equilibrium in which each player has a payoff of at least k is NP-complete as a function of the number of players.Received: 15 September 2002, Revised: 9 March 2003, JEL Classification Numbers: C72.Correspondence to: H. HallerWe thank a referee for helpful comments. The hospitality of the Institute of Economics, University of Copenhagen, and the Institute for Advanced Studies, Vienna, is gratefully acknowledged by the third author.     

Trees and decisions

Summary. The traditional model of sequential decision making, for instance, in extensive form games, is a tree. Most texts define a tree as a connected directed graph without loops and a distinguished node, called the root. But an abstract graph is not a domain for decision theory. Decision theory perceives of acts as functions from states to consequences. Sequential decisions, accordingly, get conceptualized by mappings from sets of states to sets of consequences. Thus, the question arises whether a natural definition of a tree can be given, where nodes are sets of states. We show that, indeed, trees can be defined as specific collections of sets. Without loss of generality the elements of these sets can be interpreted as representing plays. Therefore, the elements can serve as states and consequences at the same time.Received: 23 January 2003, Revised: 2 November 2003, JEL Classification Numbers: C72, D70. Correspondence to: Klaus RitzbergerWe are grateful to Larry Blume, Ariel Rubinstein, Jörgen Weibull, an anonymous referee, and seminar participants at the universities of Vienna, Salamanca, and Heidelberg for helpful comments. Financial support from the Austrian Science Fund (FWF) under project P15281 is also gratefully acknowledged.     

Nonpoint Source Pollution Taxes and Excessive Tax Burden

If a regulator is unable to measure firms’ individual emissions, an ambient tax can be used to achieve the socially desired level of pollution. With this tax, each firm pays a unit tax on aggregate emissions. In order for the tax to be effective, firms must recognize that their decisions affect aggregate emissions. When firms behave strategically with respect to the tax-setting regulator, under plausible circumstances their tax burden is lower under an ambient tax, relative to the tax which charges firms on the basis of individual emissions. Firms may prefer the case where the regulator is unable to observe individual firm emissions, even if this asymmetric information causes the regulator to tax each firm on the basis of aggregate emissions.     

Absorbing Team Games

A team is a group of people having the same motives but possibly different available actions. A team game is a game where two teams face each other. An absorbing game is a repeated game where some of the entries are absorbing, in the sense that once they are chosen the play terminates, and all future payoffs are equal to the payoff at the stage of termination. We prove that every absorbing team game has an equilibrium payoff and that there are -equilibrium profiles with cyclic structure. Journal of Economic Literature Classification Numbers: C72, C73.     

A theory of sequential reciprocity

Many experimental studies indicate that people are motivated by reciprocity. Rabin [Amer. Econ. Rev. 83 (1993) 1281] develops techniques for incorporating such concerns into game theory and economics. His theory is developed for normal form games, and he abstracts from information about the sequential structure of a strategic situation. We develop a theory of reciprocity for extensive games in which the sequential structure of a strategic situation is made explicit, and propose a new solution concept—sequential reciprocity equilibrium—for which we prove an equilibrium existence result. The model is applied in several examples, and it is shown that it captures very well the intuitive meaning of reciprocity as well as certain qualitative features of experimental evidence.     

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games.     

Communication equilibrium payoffs in repeated games with imperfect monitoring

We characterize the set of communication equilibrium payoffs of any undiscounted repeated matrix-game with imperfect monitoring and complete information. For two-player games, a characterization is provided by Mertens, Sorin, and Zamir (Repeated games, Part A (1994) CORE DP 9420), mainly using Lehrer's (Math. Operations Res. (1992) 175) result for correlated equilibria. The main result of this paper is to extend this characterization to the n-player case. The proof of the characterization relies on an analogy with an auxiliary 2-player repeated game with incomplete information and imperfect monitoring. We use Kohlberg's (Int. J. Game Theory (1975) 7) result to construct explicitly a canonical communication device for each communication equilibrium payoff.