While game theory has been transformative for decision-making, the assumptions made can be overly restrictive in certain instances. In this work, we investigate some of the underlying assumptions of rationality, such as mutual consistency and best response, and consider ways to relax these assumptions using concepts from level-$k$ reasoning and quantal response equilibrium (QRE) respectively. Specifically, we propose an information-theoretic two-parameter model called the Quantal Hierarchy model, which can relax both mutual consistency and best response while still approximating level-$k$, QRE, or typical Nash equilibrium behaviour in the limiting cases. The model is based on a recursive form of the variational free energy principle, representing higher-order reasoning as (pseudo) sequential decision-making in extensive-form game tree. This representation enables us to treat simultaneous games in a similar manner to sequential games, where reasoning resources deplete throughout the game-tree. Bounds in player processing abilities are captured as information costs, where future branches of reasoning are discounted, implying a hierarchy of players where lower-level players have fewer processing resources. We demonstrate the effectiveness of the Quantal Hierarchy model in several canonical economic games, {both simultaneous and sequential}, using out-of-sample modelling.

翻译：尽管博弈论对决策制定产生了变革性影响，但其假设在某些情境下可能过于严格。本研究探讨了理性的一些基本假设，例如相互一致性和最优响应，并考虑分别利用层级-k推理和量子响应均衡（QRE）中的概念来放松这些假设。具体而言，我们提出了一种基于信息论的双参数模型，称为量子层级模型，该模型能够同时放松相互一致性和最优响应，同时在极限情况下仍能近似层级-k、QRE或典型纳什均衡行为。该模型基于变分自由能原理的递归形式，将高阶推理表示为扩展形式博弈树中的（伪）序贯决策制定。这一表示使我们能够以类似序贯博弈的方式处理同时博弈，其中推理资源在博弈树中逐渐消耗。玩家处理能力的界限被捕捉为信息成本，未来的推理分支被折现，从而形成一个层级结构，其中低层级玩家拥有较少的处理资源。我们通过样本外建模在多个经典经济博弈（包括同时博弈和序贯博弈）中展示了量子层级模型的有效性。