Multi-Agent Influence Diagrams

Definition

Multi-Agent Influence Diagrams are DAG-based graphical models for finite strategic decision problems. They represent chance variables, decision variables, utility variables, information available to each decision, and the way a policy profile induces expected utilities for multiple agents.

Why It Matters

Compared with extensive-form games, MAIDs can expose conditional independence structure compactly. That structure supports faster equilibrium analysis in some games, richer subgame discovery, and a natural bridge to causal models when the graph is interpreted causally.

Formalism / Key Objects

• A MAID is commonly written M=(G,theta), where G contains chance, decision, and utility nodes and theta parameterizes non-decision mechanisms.
• A decision rule pi_D maps each decision context pa_D to a distribution over actions:

$${\pi }_D(\cdot \mid p{a}_D)\in \Delta (dom(D)).$$

• A policy profile pi induces a joint distribution over the ordinary variables and expected utilities for each agent.
• A mechanised MAID adds mechanism variables Pi_D for decision rules and Theta_V for non-decision parameters.
• A rationality relation is a serial relation

$$r_D\subseteq dom(P{a}_{{\Pi }_D})\times dom({\Pi }_D),$$

so rational outcomes are policy profiles whose decision rules satisfy their relevant rationality relations.

• R-relevance asks which mechanism variables can change a rational response for a decision rule.

Connections

• Causal Games lift MAIDs from associational graphical games to intervention and counterfactual query models.
• Strategic Reasoning uses MAIDs for equilibrium, subgame, relevance, recall, and correlation questions.
• Decision Theory supplies the expected-utility and rational-response background.
• Structural Equation Models become relevant when mechanism variables are interpreted as causal mechanisms.
• Stochastic Games are sequential and state-based; MAIDs are graphical and can sometimes represent the same strategic dependencies more compactly.

Common Confusions

• MAIDs are not automatically causal models; causal games add the causal interpretation needed for interventions.
• Behavioral policies and mixed policies can diverge in imperfect-recall settings.
• A MAID subgame need not match an extensive-form-game subgame exactly; the DAG can reveal additional conditional independence.

Key Sources

• Fox2024 - Causality and Strategic Reasoning