Causal Games

Definition

Causal Games are graphical models for strategic settings where agents’ choices and the environment’s causal mechanisms must both be represented. In Fox’s formulation, a causal game is a Multi-Agent Influence Diagrams model whose induced distribution is a causal Bayesian network once the agents’ policy profile is fixed.

Why It Matters

Many intervention questions in multi-agent systems ask what happens when other agents can observe the intervention and adapt. A model that only fixes a policy misses this strategic response, while a game model without causal semantics cannot cleanly distinguish prediction, intervention, and counterfactual queries.

Formalism / Key Objects

• A causal game M=(G,theta) is a MAID such that, for every policy profile pi, the induced model with joint distribution Pr^pi(V) is a causal Bayesian network.
• A structural causal game adds structural equations and exogenous variables so counterfactual queries can be evaluated.
• Mechanised causal games add mechanism variables: Pi_D for decision rules and Theta_V for non-decision parameters.
• Conditional queries quantify over rational outcomes:

$$P{r}^R(x\mid z)=\{P{r}^{\pi }(x\mid z):\pi \in R(mM\mid z)\}.$$

• Interventional queries similarly range over rational outcomes in the intervened game:

$$P{r}^R(x_I)=\{P{r}^{\pi }(x_I):\pi \in R(m{M}_I)\}.$$

• Causal Game Intervention Semantics records the pre-policy/post-policy distinction and the counterfactual extension.

Connections

• Multi-Agent Influence Diagrams provide the base graphical-game language.
• Structural Equation Models provide the intervention and counterfactual semantics that causal games lift into strategic settings.
• Causal Decision Making is adjacent when policy choice is evaluated by intervention effects; causal games add multiple rational agents and equilibrium dependence.
• Strategic Reasoning supplies the equilibrium, subgame, rationality, and information-structure questions.
• Stochastic Games are a different multi-agent sequential model; causal games emphasize graphical conditional independence and causal query semantics.

Common Confusions

• A causal game is not just a game drawn as a graph; the graph must support causal interventions once policies are fixed.
• A post-policy intervention is not the same as a pre-policy intervention. The latter lets agents adapt their decision rules to the changed game.
• Causal games do not pick a unique equilibrium by default; queries often return sets or quantified claims over rational outcomes.

Key Sources

• Fox2024 - Causality and Strategic Reasoning