Game theory for decision-making

许多现实世界的决策过程涉及相互竞争的决策者的互动。这种情况分为两个领域：（1）博弈论，决策者被称为参与者;（2）数学规划，参与者旨在优化他们的个人结果（收益）。可编程游戏（IPG）是最近定义的一类游戏，它将这两个领域结合在一起：它模拟了每个玩家的目标通过数学优化程序描述的情况，其收益取决于竞争对手。此外，这个框架允许人们对实际问题所固有的离散决策进行编码。IPG与大多数博弈论文献之间存在显著差异。一般来说，文献集中在游戏中：（1）每个玩家的优化问题是凸的，或者（2）所有可能的游戏结果都在输入中显式枚举，所谓的正规形式游戏。 我们的研究目标是解决不合作的IPG。具体来说，我们的目标是确定参与者可以选择的最合理的策略，即所谓的均衡策略。 理解整数规划游戏是至关重要的，因为这个类使我们能够正确地反映不同的自我利益实体相互作用的实际情况。例如，国土安全背景下的防御者-攻击者互动，旨在优化本国患者利益的跨国家移植计划（市场），在同一市场竞争的不同公司的生产规划，仅举几例。通过研究博弈均衡计算算法的发展，可以：（1）对问题的复杂性进行分类，从而建立关于局中人采用均衡的现实性的假设;（2）将博弈论的结果推广到这一更广泛的博弈类，同时推进数学规划工具的发展;（3）通过重新设计具有更多“方便均衡”的博弈，提出政策变化以修复正在进行的博弈。本研究的最终目标是超越静态博弈，向动态和不完全信息博弈迈进。这种调查有可能提出更透明的政策，通过适当的竞争游戏规则激励进步，实现社会效益。

Many real-world decision processes involve the interaction of competing decision makers. Such situations fall in two fields: (1) Game Theory, with decision makers being called players; and (2) Mathematical Programming, with players aiming to optimize their individual outcome (payoff). Integer programming games (IPGs) is a recently defined class of games that brings together these two fields: it models situations where each player's goal is described through a mathematical optimization program whose payoff depends on the competitors. Furthermore, this framework allows one to encode discrete decisions that are inherent to practical problems. There is a striking difference between IPGs and most of the game theory literature. In general, the literature focus in games where: (1) each player's optimization problem is convex, or (2) all possible game outcomes are explicitly enumerated in the input, the so-called normal-form games. The goal of our research is to tackle non-cooperative IPGs. In specific, we aim to determine what are the most rational strategies that the players can select, the so-called equilibrium strategies. Understanding integer programming games is of crucial importance as this class enables us to properly mirror practical situations where different self-interested entities interact. For example, defender-attacker interactions in the context of homeland security, transplant programs (markets) across countries that aim to optimize the benefit of their own patients, production planning of different firms competing in the same market, to name a few. By investigating the development of algorithms to compute the games' equilibria one can: (1) classify the problem complexity and, consequently, establish hypothesis on the realism of players adopting an equilibrium; (2) generalize game theory results to this broader class of games and, simultaneously, advance on mathematical programming tools; (3) propose policy changes to repair ongoing games, by re-designing games with more "convenient equilibria". The ultimate goal of this research is to study beyond static games and move forward to dynamic and incomplete-information games. Such investigation has the potential to suggest more transparent policies, incentive progress through adequate game rules for competition, and accomplish social benefits.