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：在演算游戏理论($AGT$)中,设计计算纳什平衡的高效算法提出了相当大的挑战。我们在实地取得了进展,并重新展示了算法游戏理论和整数编程之间的交叉点。我们引入了以ZERO$(美元)为折算、计算和选择纯纳什平衡法(PNE$)的通用平面算法(PNE$),这是同时和不合作游戏的类别。我们为我们的算法推理提供了一个理论基础,并为纯纳什平衡游戏和网络构建游戏提供了综合特征。我们引入了以美元平价美元为折合金的概念,并设计了以美元为折价的折价美元为单位的折价平面平面法,将非平衡法与美元分开。我们测试了两种游戏的范式类:Knapsack游戏和网络构建游戏,这是在$GUDA的相关应用中成功展示了稳定的游戏。