Game theory for decision-making

许多现实世界的决定 流程涉及相互竞争的决策者之间的互动。这样的情况 属于两个领域：(1)博弈论，决策者被称为玩家； 和(2)数学规划，玩家的目标是优化他们的个人 结果(回报)。整数规划 游戏(IPGs)是最近定义的一类游戏，它将这两个领域结合在一起：它模拟情况 其中每个球员的目标通过数学优化程序来描述 谁的收益取决于竞争对手。此外，该框架还允许 对实际问题所固有的离散决策进行编码。 IPG和MOST之间有一个显著的区别 博弈论文献。一般说来，文献主要集中在游戏中：(1)每个玩家的最优化问题 是凸的，或者(2)所有可能的博弈结果都在 输入，也就是所谓的范式博弈。 我们研究的目标是 解决不合作的IPG问题。具体地说，我们的目标是确定哪些是最 玩家可以选择的理性策略，即所谓的均衡 战略。 了解整数规划 游戏是至关重要的，因为这门课使我们能够正确地反映 不同的自利实体相互作用的实际情况。为 例如，在国土安全的背景下，防御者和攻击者的互动，移植 跨国家/地区的计划(市场)，旨在优化其 自己的病人，不同企业竞争的生产计划相同 市场，仅举几例。 通过调查发展情况 计算博弈均衡的算法可以：(1)对问题进行分类 复杂性，并因此建立了关于玩家现实主义的假设 采用均衡；(2)将博弈论结果推广到这个更广泛的领域 游戏类，同时，在数学编程工具方面取得进展； (3)通过重新设计游戏，提出修复正在进行的游戏的政策更改 更多的“便利均衡”。 这样做的最终目的是 研究是超越静态游戏，向动态和 不完全信息博弈。这样的调查有可能表明 更透明的政策，通过适当的游戏规则激励进步 竞争，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.