Uncommon Belief in Rationality (2412.09407v1)

Abstract: Common knowledge/belief in rationality is the traditional standard assumption in analysing interaction among agents. This paper proposes a graph-based language for capturing significantly more complicated structures of higher-order beliefs that agents might have about the rationality of the other agents. The two main contributions are a solution concept that captures the reasoning process based on a given belief structure and an efficient algorithm for compressing any belief structure into a unique minimal form.

• The paper introduces a novel doxastic rationalisability concept to model complex higher-order beliefs among agents.
• It employs a directed, labeled graph model and a partition refinement algorithm to derive minimal belief structures.
• The approach provides actionable insights for multiagent systems by transcending traditional common belief assumptions.

Overview of Uncommon Belief in Rationality

This paper presents a rigorous investigation into the complex structures of higher-order beliefs in rationality among agents, challenging the conventional assumption of common knowledge or belief in game-theoretic contexts. The authors introduce a graph-based formalism for modeling these intricate belief structures and propose a novel solution concept named doxastic rationalisability, along with an efficient algorithm for minimizing belief structures to a unique canonical form.

Contributions

1. Graph-Based Language for Belief Modelling: The paper puts forward a directed, labeled graph model to capture uncommon rationality and beliefs in rationality (RBR) among various agent types. Such a model accommodates more complexity than the traditional RCBR models by permitting uncommon belief hierarchies in rationality networks.
2. Solution Concept - Doxastic Rationalisability: A new solution concept, doxastic rationalisability, is devised for these uncommon RBR systems. This concept entails an iterative rationalisation process to predict strategically rationalisable behaviors of agents based on nuanced belief hierarchies that may not follow the RCBR assumption.
3. Graph Minimisation Algorithm: The authors devise an efficient algorithm, based on partition refinement, to condense any belief graph into its minimal equivalent form. The reasoning is inspired by techniques from automata theory, particularly Myhill-Nerode theorem and Hopcroft's algorithm. The time complexity of the algorithm is $O(|\mathcal{A}|\cdot|N|^2\cdot\log|N|)$, where $|\mathcal{A}|$ is the number of agents and $|N|$ is the number of nodes in the graph.

Key Results & Implications

• Iterative Rationalisation: By exploring a simplified "guess 2/3 of the average" game, the paper illustrates how rationalisation depends on agents' belief sequences. Agents' beliefs trigger iterative steps of strategy elimination until equilibrium strategies emerge, highlighting the interactive depth of rational decision making beyond common belief assumptions.
• Equivalence in Beliefs and Graphs: A decisive condition for doxastic equivalence of nodes is established: they must share identical belief hierarchies. This theoretical insight extends to equivalence of entire RBR graphs, allowing reliable simplification and understanding of multiagent systems based on minimal equivalent canonical forms.
• Doxastic Representation of Reality: The model deftly represents doxastic aspects, where beliefs entail entities not existing in reality but which influence perceived equilibria. Thus, the framework can model bounded rationality and real-world cognitive limitations in agents.

Theoretical and Practical Implications

The theoretical framing of beliefs in this paper offers a more flexible yet precise tool for analyzing multiagent interactions, bypassing the limitations imposed by rigid common belief structures. By embracing uncommon belief hierarchies, the framework better reflects nuanced real-world interactions—where perfect knowledge assumptions rarely hold. Practically, the algorithm's extraction of minimal, canonical belief structures facilitates computational tractability and clarity in predicting agent behavior in complex systems.

Future Directions

Future developments could extend this model to incorporate probabilistic belief structures, blending dynamic uncertainty with strategic rationalisability. Moreover, empirical validation in settings like autonomous systems or market strategies, where agent beliefs significantly diverge, could enrich the model's applicability and practical influence.

In conclusion, this paper opens avenues for nuanced game-theoretic modeling, enabling researchers and practitioners to explore sophisticated belief frameworks where traditional assumptions fall short. The marriage of epistemic game theory with efficient computational algorithms presents a promising leap in the analysis and optimisation of multiagent systems.