Computation of Nash equilibria in finite games: introduction to the symposium

This is a collection of ten invited articles on the computation of Nash equilibria in finite games. Three (by Ruchira Datta, Jean-Jacques Herings and Ronald Peeters, and Tim Roughgarden) are longer surveys, the others present new research. All contributions attempt to make the material accessible and interesting to the non-specialist. The collection reflects in equal measure the state of research, as well as providing an exposition of the main ideas and questions of the field. The articles are also bridged by a unity of ideas that appear in many places. The topic of this collection is a computational problem: given a game in some finite description, find its Nash equilibria; or, a more modest task: find one Nash equilibrium of the game. These questions pose highly interesting challenges for economics, mathematics, and computer science. For the applied economist, noncooperative games are a modeling tool, with the Nash equilibrium as the central solution concept. Interpreting and discussing a Nash equilibrium is essential for the model, but the equilibrium has to be found first, which often takes a substantial amount of the analysis. A computer program for this analysis should save time and allow for more detailed modeling. Why should the applied economist, as a modeler and user of game theory, care about algorithms? It seems easy just to convert the equilibrium problem to a suitable optimization problem, run somemathematical software on it, and use the solution. The problem is that such a “black box algorithm” is unlikely to work on all but the smallest cases. Any algorithm that “scales” reasonably well in allowing for more detailed games as input, and still finishes in reasonable time, must exploit the mathematical structure of the problem.