# Bounded rationality for relaxing best response and mutual consistency: The Quantal Hierarchy model of decision-making

arXiv: https://arxiv.org/abs/2106.15844 · HTML: https://mlsanity.com/paper/2106.15844/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 120 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [Sophisticated Inference](https://mlsanity.com/paper/2006.04120/) (2020)
- [Modelling Bounded Rationality in Multi-Agent Interactions by Generalized Recursive Reasoning](https://mlsanity.com/paper/1901.09216/) (2019)
- [Information-Theoretic Bounded Rationality](https://mlsanity.com/paper/1512.06789/) (2015)
- [Generalized Thompson Sampling for Sequential Decision-Making and Causal Inference](https://mlsanity.com/paper/1303.4431/) (2013)
- [A maximum entropy model of bounded rational decision-making with prior beliefs and market feedback](https://mlsanity.com/paper/2102.09180/) (2021)
- [Human Decision-Making under Limited Time](https://mlsanity.com/paper/1610.01698/) (2016)
- [Bounded rational decision-making from elementary computations that reduce uncertainty](https://mlsanity.com/paper/1904.03964/) (2019)
- [The impact of social influence in Australian real-estate: market forecasting with a spatial agent-based model](https://mlsanity.com/paper/2009.06914/) (2020)
- [Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos](https://mlsanity.com/paper/1711.02456/) (2017)

## Built on by (most cited first, 3 of 3)

- [Solving Dynamic Principal-Agent Problems with a Rationally Inattentive Principal](https://mlsanity.com/paper/2202.01691/) (2022)
- [Convergent Q-Learning for Infinite-Horizon General-Sum Markov Games through Behavioral Economics](https://mlsanity.com/paper/2508.08669/) (2025)
- [Beyond Rational Illusion: Behaviorally Realistic Strategic Classification](https://mlsanity.com/paper/2605.19674/) (2026)

## Abstract (arXiv)

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.