In Algorithmic Game Theory (AGT), designing efficient algorithms to compute Nash equilibria poses considerable challenges. We make progress in the field and shed new light on the intersection between Algorithmic Game Theory and Integer Programming. We introduce ZERO Regrets, a general cutting plane algorithm to compute, enumerate, and select Pure Nash Equilibria (PNEs) in Integer Programming Games, a class of simultaneous and non-cooperative games. We present a theoretical foundation for our algorithmic reasoning and provide a polyhedral characterization of the convex hull of the Pure Nash Equilibria. We introduce the concept of equilibrium inequality and devise an equilibrium separation oracle to separate non-equilibrium strategies from PNEs. We test ZERO Regrets on two paradigmatic classes of games: the Knapsack Game and the Network Formation Game, a well-studied game in AGT. Our algorithm successfully solves relevant instances of both games and shows promising applications for equilibria selection.
翻译:在算法游戏理论(AGT)中,设计计算纳什平衡的高效算法(AGT)带来了相当大的挑战。我们在实地取得了进展,并重新揭示了算法游戏理论和整数程序之间的交叉点。我们引入了ZERO Risquet(ZERO Risquet),这是用来计算、计算和在整数编程游戏中选择Pure Nash Equilibria(PNEE)的通用切割平面算法(PNAGT),这是一个同时和不合作游戏的类别。我们为我们的算法推理提供了理论基础,并为纯纳什平衡结构提供了综合特征。我们引入了均衡不平等的概念,并设计了平衡分离法,将非平衡战略与PNEOs分开。我们测试了ZERO DRisquen,这是AGT中经过良好研究的游戏。我们的算法成功地解决了两个游戏的有关实例,并展示了对等平衡选择的有希望的应用。