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。这类情况可分为两个领域：（1）博弈论，决策者被称为参与者;（2）数学规划，参与者的目标是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着显著的差异。一般来说，文献集中在游戏中，其中：（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，即所谓的正规形式游戏。** 我们研究的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计划，仅举几例。通过研究计算博弈均衡的算法的发展，人们可以：（1）对问题的复杂性进行分类，从而建立关于博弈者采用均衡的现实性的假设;（2）将博弈论的结果推广到更广泛的博弈类别，同时，推进数学规划工具;* （3）通过重新设计具有更多“方便均衡”的博弈，提出政策改变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，研究动态和不完全信息博弈。这样的调查有可能建议更透明的政策，通过适当的竞争游戏规则激励进步，并实现社会效益。

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.