Existence and Computation of Equilibria in Games and Economies

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 legislative bargaining model that studies weighted voting games with players that are potentially heterogeneous in probabilities of being the proposer, discount factors and voting weights. Eraslan-McLennan characterize 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 Scarf’s algorithm to execute this observation. These algorithms have implications for computing equilibria of dynamic models and should be useful in other applied work.