Higher-Order Belief in Incomplete Information MAIDs
Jack Foxabbott, Rohan Subramani, Francis Rhys Ward
TL;DR
The paper addresses the limitation of traditional MAIDs in representing heterogeneous beliefs and higher-order beliefs across agents under incomplete information. It introduces incomplete information MAIDs (II-MAIDs), defines both infinite- and finite-depth variants, and proves an equivalence to incomplete-information extensive form games (II-EFGs) without a common prior over types. This equivalence allows existing equilibrium concepts to be inherited, but the authors argue these concepts can be unrealistic when there is no common prior, motivating a more realistic recursive best-response solution concept. The work provides theoretical foundations for II-MAIDs, discusses their implications for rational behavior under diverse beliefs, and demonstrates applicability through an AI-evaluation example to illustrate how II-MAIDs can model and analyze such scenarios.
Abstract
Multi-agent influence diagrams (MAIDs) are probabilistic graphical models which represent strategic interactions between agents. MAIDs are equivalent to extensive form games (EFGs) but have a more compact and informative structure. However, MAIDs cannot, in general, represent settings of incomplete information -- wherein agents have different beliefs about the game being played, and different beliefs about each-other's beliefs. In this paper, we introduce incomplete information MAIDs (II-MAIDs). We define both infinite and finite-depth II-MAIDs and prove an equivalence relation to EFGs with incomplete information and no common prior over types. We prove that II-MAIDs inherit classical equilibria concepts via this equivalence, but note that these solution concepts are often unrealistic in the setting with no common prior because they violate common knowledge of rationality. We define a more realistic solution concept based on recursive best-response. Throughout, we describe an example with a hypothetical AI agent undergoing evaluation to illustrate the applicability of II-MAIDs.