We introduce a new paradigm for game theory -- Bayesian satisfaction. This novel approach is a synthesis of the idea of Bayesian rationality introduced by Aumann, and satisfaction games. The concept of Bayesian rationality for which, in part, Robert Aumann was awarded the Nobel Prize in 2005, is concerned with players in a game acting in their own best interest given a subjective knowledge of the other players' behaviours as represented by a probability distribution. Satisfaction games have emerged in the engineering literature as a way of modelling competitive interactions in resource allocation problems where players seek to attain a specified level of utility, rather than trying to maximise utility. In this paper, we explore the relationship between optimality in Aumann's sense (correlated equilibria), and satisfaction in games. We show that correlated equilibria in a satisfaction game represent stable outcomes in which no player can increase their probability of satisfaction by unilateral deviation from the specified behaviour. Thus, we propose a whole new class of equilibrium outcomes in satisfaction games which include existing notions of equilibria in such games. Iterative algorithms for computing such equilibria based on the existing ideas of regret matching are presented and interpreted within the satisfaction framework. Numerical examples of resource allocation are presented to illustrate the behaviour of these algorithms. A notable feature of these algorithms is that they almost always find equilibrium outcomes whereas existing approaches in satisfaction games may not.

翻译：我们提出了一种新的博弈论范式——贝叶斯满足。这一新颖方法综合了奥曼提出的贝叶斯理性思想与满意度博弈。贝叶斯理性概念（罗伯特·奥曼因在此领域的贡献荣获2005年诺贝尔奖）关注博弈中玩家基于对其他玩家行为（由概率分布表征）的主观认知，做出符合自身最大利益的选择。满意度博弈则源于工程文献，是一种在资源分配问题中对竞争性互动进行建模的方式，其中玩家寻求达到特定的效用水平，而非追求效用最大化。本文探索了奥曼意义下的最优性（关联均衡）与博弈中满足感之间的关系。我们证明，在满意度博弈中，关联均衡表征了稳定结果：任何玩家都无法通过单方面偏离指定行为来提高其满足概率。因此，我们提出了一类全新的满意度博弈均衡结果，它包含了此类博弈中已有的均衡概念。我们基于现有的遗憾匹配思想，提出了计算此类均衡的迭代算法，并在满意度框架下进行了解读。通过资源分配数值示例，展示了这些算法的行为特征。这些算法的显著特点是几乎总能找到均衡结果，而满意度博弈的现有方法可能无法做到。