Existence and Computation of Equilibria in Games and Economies

In this dissertation we study the existence and computation of equilibria in games and economies. The results in this dissertation have been presented in the form of several papers, one of which have already been published. The first paper is entitled “Computation of Stationary Equilibrium Payoffs in Coalitional Bargaining.” In this paper, we provide an algorithm to compute the equilibrium payoffs in the coalitional bargaining model of Eraslan-McLennan (Journal of Economic Theory, 2013) by using recent developments in methods of numerical algebraic geometry. The Eraslan-McLennan model is a legisla- tive bargaining model which studies weighted voting games with players that are heterogeneous in their discount factors, voting weights and in terms of the probabilities of being selected as the proposer. Eraslan-McLennan characterizes the equilibria as fixed points of a set-valued function. In this paper, we show that the equilibria of the game can be characterized by solutions to a system of polynomial equations and provide an algorithm to compute the equilibrium payoffs. As an alternative approach, we show that all equilibria of such games can be characterized by fixed points of a continuous function, and use a variety of fixed point algorithms to execute this observation. These algorithms have implications for computing equilibria of dynamic models and should be useful in other applied work. The second paper is entitled “On the Nonemptiness of the α-core of Dis- continuous Games: Transferable and Nontransferable Utilities” (published in Journal of Economic Theory, 2015). The nonemptiness of the α-core of games with continuous payoff functions was proved by Scarf (1971) for nontransfer- able utilities and by Zhao (1999a) for transferable utilities. In this paper we present generalizations of their results to games with possibly discontinuous payoff functions. Our handling of discontinuity is based on Reny’s (1999) better- reply-security concept. We present examples to show that our generalizations are nonvacuous. The third paper is entitled “On the Nonemptiness of the Transferable Utility β-core of Discontinuous Games.” Zhao (1999) proved the nonemptiness of the transferable utility β-core of games with continuous payoff functions. In this paper we present generalizations of his result to games with possibly discontinuous payoff functions by applying the concepts and methods we introduced in the second paper. We rely on Reny’s (1999) concept of “better-reply-security” to handle any discontinuities that may arise. We present applications to show that our generalizations are nonvacuous. The fourth paper is entitled “On the Existence of Equilibrium with Discon- tinuous Preferences: Games and Economies.” In this paper, we present three fixed point theorems that can be seen as generalizations of the earlier works of Browder and Fan-Glicksberg and illustrate the use of these theorems as a ‘methodological toolkit’ for existence issues in a variety of economic settings. In particular, we present a synthetic treatment of the problem of the existence of an equilibrium in games and economies when the preferences of the individuals are not necessarily continuous or ordered. We also relate our results to those available in the antecedent literature