Causal Game Intervention Semantics

Context

Fox2024 - Causality and Strategic Reasoning introduces causal games to answer causal queries in strategic settings. The reusable technical issue is that interventions may occur after a policy profile is fixed or before agents choose policies, in which case rational agents may adapt.

Formal Statement

For a mechanised causal game mM with rationality relations R, a conditional query over rational outcomes is:

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

For an intervention I, the interventional query is:

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

If I is fully post-policy and targets object-level variables, then the rational outcomes remain those of the original game. If I is pre-policy and targets mechanism variables such as decision-rule variables, the intervention changes the game and rational outcomes are recomputed.

For a structural causal game, counterfactuals use actual and counterfactual rational outcomes:

$$P{r}^R(x_I\mid z)=\left\{\int P{r}^{{\pi }^{\prime }}(x_I\mid e,e^{\ast })Pr(e^{\ast })P{r}^{\pi }(e\mid z)d{e}^{\prime }:(\pi ,{\pi }^{\prime })\in R(m{M}_I\mid z)\right\}.$$

Here pi is an actual rational outcome, pi' is a counterfactual rational outcome, e are original exogenous variables, and e* are fresh exogenous variables introduced by soft interventions.

Derivation / Construction

Fox decomposes interventions in causal games into four primitive operation types:

• Fix an object-level variable by replacing its conditional distribution, including hard interventions such as do(X=x).
• Fix a mechanism variable by replacing a parameter mechanism or a decision-rule rationality relation.
• Add a new object-level variable with a new conditional distribution.
• Remove an existing object-level variable and update the remaining parent sets.

Primitive interventions are claimed to be sound and complete for causal interventions in the framework. More complex intervention sets can be decomposed into primitive sequences while tracking which interventions are visible to which agents.

Counterfactuals follow Pearl’s abduction-action-prediction template but add equilibrium dependence:

1. Condition the actual model on the observation z for each actual rational outcome.
2. Apply the intervention and recompute rational responses where appropriate.
3. Return the induced counterfactual distributions for the admissible actual-counterfactual policy pairs.

The subtle step is choosing which decision-rule variables remain invariant across the actual and counterfactual worlds. Post-policy interventions keep all decision rules invariant; pre-policy counterfactuals can use either a simplicity principle that recomputes all decision rules or a closest-possible-world principle that preserves as many unaffected decision rules as justified by the relevance graph.

Implications

• In strategic systems, do(X=x) is underspecified unless one says whether agents know about the intervention before choosing policies.
• Causal-game queries often return sets, bounds, or quantified statements over equilibria rather than a single probability.
• Mechanism-level interventions are essential for questions about policy rules, incentives, signaling systems, and institutional redesign.
• The semantics explains why standard single-agent intervention reasoning can fail when other agents’ rational responses are part of the causal pathway.

Links

• Causal Games
• Multi-Agent Influence Diagrams
• Structural Equation Models
• Causal Decision Making
• Fox2024 - Causality and Strategic Reasoning