Super-modularity: Applications to convex games and to the greedy algorithm for LP

L. S. Shapley (Internat. J. Game Theory1 (1971), 11–26) showed that if a game is convex, then all the marginal worth vectors are members of the core. J. Edmonds (in “Combinatorial Structures and Their Applications” (R. Guy et al., Eds.), pp. 69–87, Gordon & Breach, New York, 1970) considered a class of linear programming problems for which the greedy algorithm works. The present paper unifies these studies, and establishes the converse of each theorem.     

A contraction principle for finite global games

I provide a new proof of uniqueness of equilibrium in a wide class of global games. I show that the joint best-response in these games is a contraction. The uniqueness result then follows as a corollary of the contraction principle. Furthermore, the contraction-mapping approach provides an intuition for why uniqueness arises: complementarities in games generate multiplicity of equilibria, but the global-games structure dampens complementarities so that only one equilibrium exists.     

A spectral decomposition for structural VAR models

Based on structural VARs, this paper proposes a spectral decomposition which allows to infer the effects of changes in one variable on the other variables in the frequency domain. It is shown that there is a close relationship between this concept and conventional forecast error variance decomposition techniques for VARs. An empirical example demonstrates the usefulness of this additional tool in analyzing the relationships among time series.     

Markets for contracts: experiments exploring the compatibility of games and markets for games

Summmary. The research explores the relationship between games and the economic environment in which the games might be embedded. The focus is on a market institution in which agents buy and sell rights to participate in a follow-on stage of strategic interaction. The central question posed concerns how two different types of processes, the game and the market, interact. The market converges to a competitive equilibrium that is consistent with the Nash equilibrium that obtains in the game, and the convergence of the market to a competitive equilibrium lags the convergence of behaviors in the game to a Nash equilibrium.     

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.     

Utility functions for simple games

This note characterizes the Shapley-Shubik index and the Banzhaf index on the class of simple games as utility functions which reflect different postures toward risk.     

A cooperative study of one-commodity market games

A competitive model is attached to a particular market where several agents trade a same commodity. We focus on the stability of the price systems and define a solution concept, akin to the core, proving an existence result. The author wishes to thank the Editor and an anonymous referee for useful comments and remarks.     

Global Newton Method for stochastic games

The Global Newton Method for games in normal form and in extensive form is shown to have a natural extension to computing Markov-perfect equilibria of stochastic games.     

A decomposition of the Harberger expression for tax incidence

In this paper we propose a decomposition of the Harberger expression for the incidence of the corporate income tax into ‘commodity price distortion’ and ‘factor price distortion’ effects. By applying Harberger's estimates of the parameters for the U.S. economy, we will demonstrate that the factor price distortion effect, which has an extremely simple formal expression, explains 115 percent of the total effect of the tax change on the rate of return. This suggests that the commodity price distortion effect may be relatively unimportant in explaining the tax incidence. The difference between our two effects and Mieszkowski's (1967) output and factor substitution effects is also noted.     

A strategic implementation of the Average Tree solution for cycle-free graph games

In this note we provide a strategic implementation of the Average Tree solution for zero-monotonic cycle-free graph games. That is, we propose a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the average hierarchical outcome of the game. This mechanism takes into account that a player is only able to communicate with other players (i.e., to make proposals about a division of the surplus of cooperation) when they are connected in the graph.     

An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.     

A decomposition of training probabilities

The determinants of workers' training probabilities are analysed. A distinction is made between the unconditional probability that a worker is employed by a firm that provides any training opportunities at all, and the conditional probability that the worker receives training given that s/he works for a firm that provides training. For this analysis a censored version of the bivariate probit model is applied. The results indicate that establishment size, industry dummies, type of schooling, age and experience affect only the unconditional probability, whereas the type of labour contract, working hours and job level affect only the conditional probability. Years of formal schooling affect both probabilities; more highly educated workers are both more likely to work for training-providing firms and, given that they work for such a firm, to be selected for training programmes.     

A remark on the measurability of large games

A game with a continuum of players is described by a function assigning payoff functions to players and satisfying some measurability properties. In this note we establish the equivalence between several measurability assumptions that have been made in the literature. I thank an anonymous referee for very helpful comments and John Huffstot for editorial assistance. Financial support from a Nova Fórum grant is gratefully acknowledged.     

Enumeration of Nash equilibria for two-player games

This paper describes algorithms for finding all Nash equilibria of a two-player game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method lrsnash is based on the known vertex enumeration program lrs, for “lexicographic reverse search”. It enumerates the vertices of only one best-response polytope, and the vertices of the complementary faces that correspond to these vertices (if they are not empty) in the other polytope. The second method is a modification of the known EEE algorithm, for “enumeration of extreme equilibria”. We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of lrsnash and EEE, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses.     

Stable lexicographic rules for shortest path games

For the class of shortest path games, we propose a family of new cost sharing rules satisfying core selection. These rules allocate shares according to some lexicographic preference relation. A computational procedure is provided. Our results relate to those of Tijs et al. (2011).     

An equilibrium closure result for discontinuous games

For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.     

Consistent solutions for cooperative games with externalities

In order to calculate the worth of a coalition of players, the coalition needs to predict the actions of outsiders. We propose that, for a given solution concept, such predictions should be made by applying the solution concept to the “reduced society” consisting of the non-members. We illustrate by computing the r-core for the case of Bertrand competition with differentiated commodities.