Game theory relies heavily on the availability of cardinal utility functions, but in fields such as matching markets, only ordinal preferences are typically elicited. The literature focuses on mechanisms with simple dominant strategies, but many real-world applications lack dominant strategies, making the intensity of preferences between outcomes important for determining strategies. Even though precise information about cardinal utilities is not available, some data about the likelihood of utility functions is often accessible. We propose to use Bayesian games to formalize uncertainty about the decision-makers' utilities by viewing them as a collection of normal-form games. Instead of searching for the Bayes-Nash equilibrium, we study how uncertainty in utilities is reflected in uncertainty of strategic play. To do this, we introduce a novel solution concept called $\alpha$-Rank-collections, which extends $\alpha$-Rank to Bayesian games. This allows us to analyze strategic play in, for example, non-strategyproof matching markets, for which appropriate solution concepts are currently lacking. $\alpha$-Rank-collections characterize the expected probability of encountering a certain strategy profile under replicator dynamics in the long run, rather than predicting a specific equilibrium strategy profile. We experimentally evaluate $\alpha$-Rank-collections using instances of the Boston mechanism, finding that our solution concept provides more nuanced predictions compared to Bayes-Nash equilibria. Additionally, we prove that $\alpha$-Rank-collections are invariant to positive affine transformations, a standard property for a solution concept, and are efficient to approximate.

翻译：博弈论在很大程度上依赖于基数效用函数的可用性，但在匹配市场等领域，通常只能获取序数偏好。现有文献主要关注具有简单占优策略的机制，然而许多实际应用场景缺乏占优策略，此时结果间偏好的强度对策略确定至关重要。尽管无法获得基数效用的精确信息，但关于效用函数可能性的某些数据往往可以获取。我们提出使用贝叶斯博弈来形式化决策者效用的不确定性，将其视为一系列标准形式博弈的集合。不同于寻找贝叶斯-纳什均衡，我们研究效用不确定性如何反映在策略博弈的不确定性中。为此，我们引入一种称为α-Rank集合的新解概念，将α-Rank扩展至贝叶斯博弈。这使得我们能够分析例如非策略证明匹配市场中的策略博弈，这类市场目前缺乏合适的解概念。α-Rank集合刻画了在复制动力学下长期遇到特定策略剖面的期望概率，而非预测特定的均衡策略剖面。我们通过波士顿机制实例对α-Rank集合进行实验评估，发现相较于贝叶斯-纳什均衡，我们的解概念能提供更精细的预测。此外，我们证明α-Rank集合具有正仿射变换不变性——这是解概念的标准性质，并且能够高效近似计算。