Homotopy Methods to Compute Equilibria in Game Theory

This paper presents a complete survey of the use of homotopy methods in game theory.Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by variousselection theories. We present all relevant techniques underlying homotopy algorithms.We give detailed expositions of the Lemke-Howson algorithm and the Van den Elzen-Talman algorithm to compute Nash equilibria in 2-person games, and the Herings-Vanden Elzen, Herings-Peeters, and McKelvey-Palfrey algorithms to compute Nash equilibriain general n-person games.operations research and management science;

Competition against peer-to-peer networks

In this paper, we consider the competition of providers of information products against P2P networks that offer illegal versions of the information products. Depending on the generic cost factor of downloading—incorporating factors including, among other things, the degree of legal enforcement of intellectual property rights—we find that the firm may employ pricing strategies to either deter the entry of a network or to accommodate it. In the latter case, we find that the equilibrium price moves in the opposite direction of the generic cost factor of downloading. This counter-intuitive result corresponds to a very subtle form of platform competition between the firm and the network. Furthermore, profits for the firm ambiguously decrease when the generic cost factor of downloading declines, whereas total welfare unambiguously increases. This implies that it may well be welfare enhancing to relax the legal enforcements of intellectual property rights.Strategy;

Piracy on the internet: Accommodate it or fight it? A dynamic approach

This paper uses a dynamic stochastic model to solve for the optimal pricing policy of themusic recording companies in the presence of P2P file-sharing networks eroding their CDsales. We employ a policy iteration algorithm on a discretized state space to numericallycompute the optimal price policy. The realistically calibrated model reflects the real-worldfigures we observe and provides estimates of figures we can not observe, such as changesin total welfare. The results suggest that, thanks to the existence of P2P networks, totalwelfare in 2008 in the U.S. is about $25.6 billion more per annum than in 1999 before P2Pwas introduced. Moreover, the results predict that the current trend of decreasing CDsales will continue until around the year 2020 when it will stabilize at around 231.2 millioncopies per year, comparing to the industry all-time high of 938.9 million in 1999. Thecomparative static analysis shows that full enforcement of intellectual property rights,although helpful for the industrial profit, may have adverse effect on total welfare.microeconomics ;

A Globally Convergent Algorithm to Compute All Nash Equilibria of n-Person Games

In this paper an algorithm is presented to compute all Nash equilibria for games in normal form on the only premises that the number of Nash equilibria is finite. The algorithm relies on decomposing the game by means of support-sets. For each support-set, the set of totally mixed equilibria of the support-restricted game that results by restricting the players to strategies in the support-set can be characterized by a system of polynomial equations and inequalities. By solving those systems for each support-set, all equilibria are found. The algorithm belongs to the class of homotopy-methods and is implementable. Finally, several techniques to speed up computations are proposed.Economics ;

Equilibrium Selection in Stochastic Games

Game theory, stochastic games, Equilibrium selection, Linear tracing procedure, Correlated beliefs