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之间的互动。这种情况 分为两个领域：（1）博弈论，决策者被称为参与者; 和（2）数学规划，玩家旨在优化他们的个人 收益（Payoff）。整数规划 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： 每个玩家的目标都是通过数学优化程序来描述的 他们的收益取决于竞争对手此外，该框架允许一个 对实际问题中固有的离散决策进行编码。 IPG和大多数IPG之间存在显著差异， 博弈论文献中。一般来说，文献集中在以下博弈中：（1）每个局中人的优化问题 是凸的，或者（2）所有可能的游戏结果都明确列举在 输入，即所谓的正规形式博弈。 我们研究的目的是 处理不合作的IPG。具体来说，我们的目标是确定什么是最重要的 参与者可以选择的理性策略，即所谓的均衡 战略布局 了解整数规划 游戏是至关重要的，因为这个类使我们能够正确地反映 不同的自利实体相互作用的实际情况。为 例如，国土安全背景下的防御者-攻击者交互，移植 计划（市场），旨在优化他们的利益， 自己的病人，生产计划的不同公司在同一竞争 市场，仅举几例。 通过调查 计算博弈均衡的算法可以：（1）将问题分类 复杂性，从而建立假设的现实主义的球员 采用均衡;（2）将博弈论结果推广到更广泛的 类游戏，同时，推进数学规划工具; (3)建议修改政策，通过重新设计游戏来修复正在进行的游戏， 更方便的均衡。 这个项目的最终目标 研究是超越静态游戏，走向动态， 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.