This work contains the mathematical exploration of a few prototypical games in which central concepts from statistics and probability theory naturally emerge. The first two kinds of games are termed Fisher and Bayesian games, which are connected to Frequentist and Bayesian statistics, respectively. Later, a more general type of game is introduced, termed Statistical game, in which a further parameter, the players' relative risk aversion, can be set. In this work, we show that Fisher and Bayesian games can be viewed as limiting cases of Statistical games. Therefore, Statistical games can be viewed as a unified framework, incorporating both Frequentist and Bayesian statistics. Furthermore, a philosophical framework is (re-)presented -- often referred to as minimax regret criterion -- as a general approach to decision making. The main motivation for this work was to embed Bayesian statistics into a broader decision-making framework, where, based on collected data, actions with consequences have to be made, which can be translated to utilities (or rewards/losses) of the decision-maker. The work starts with the simplest possible toy model, related to hypothesis testing and statistical inference. This choice has two main benefits: i.) it allows us to determine (conjecture) the behaviour of the equilibrium strategies in various limiting cases ii.) this way, we can introduce Statistical games without requiring additional stochastic parameters. The work contains game theoretical methods related to two-player, non-cooperative games to determine and prove equilibrium strategies of Fisher, Bayesian and Statistical games. It also relies on analytical tools for derivations concerning various limiting cases.

翻译：本文对几个原型博弈进行了数学探索，在其中，统计学和概率论的核心概念自然涌现。前两类博弈分别称为费希尔博弈和贝叶斯博弈，它们分别与频率学派统计学和贝叶斯统计学相关联。随后，引入了一种更一般的博弈类型，称为统计博弈，其中可以设定一个额外参数——参与者的相对风险厌恶程度。本文表明，费希尔博弈和贝叶斯博弈可被视为统计博弈的极限情况。因此，统计博弈可被视为一个统一框架，同时涵盖了频率学派和贝叶斯统计学。此外，本文（重新）阐述了一个哲学框架——常被称为最小最大遗憾准则——作为决策制定的一般方法。本研究的主要动机是将贝叶斯统计学嵌入一个更广泛的决策框架中，在该框架下，基于收集到的数据，必须做出具有后果的行动，这些后果可转化为决策者的效用（或奖励/损失）。本文从最简单的玩具模型开始，该模型与假设检验和统计推断相关。这一选择有两大优势：i.) 它使我们能够确定（推测）各种极限情况下均衡策略的行为；ii.) 通过这种方式，我们可以在不引入额外随机参数的情况下介绍统计博弈。本文运用与双人非合作博弈相关的博弈论方法，来确定并证明费希尔博弈、贝叶斯博弈和统计博弈的均衡策略，并借助分析工具推导各种极限情况。