Causality and Strategic Reasoning

Summary

Fox’s thesis develops Causal Games as a unifying formalism for reasoning about causal questions in strategic multi-agent settings. The central idea is to combine Pearl-style causal modeling with Multi-Agent Influence Diagrams so that a model can represent both environmental causal structure and the way rational agents adapt their policies to interventions.

This partial ingest reviewed the abstract, introduction, thesis overview, mechanised-MAID definitions, subgame and equilibrium-refinement chapters, imperfect-recall results, causal-game query semantics, and conclusion. The thesis is broad and proof-heavy, so this note records the reusable formal spine rather than every theorem.

Key Claims

• Causal questions in games require tracking both environmental mechanisms and dependencies among agents’ decision rules.
• Causal games can be read two ways: as MAIDs lifted from the associational level to Pearl’s interventional and counterfactual levels, or as causal models extended with decision variables and utility variables for multiple agents.
• Mechanised MAIDs expose mechanism variables for decision rules and non-decision parameters, making dependencies among policies, parameters, and rational responses explicit.
• R-relevance generalizes strategic relevance: for a chosen rationality relation, the graph can identify which mechanisms are relevant to a decision rule.
• MAID subgames can be richer than extensive-form-game subgames because DAG structure exposes conditional independencies that a tree representation can hide.
• Subgame perfectness and trembling-hand perfectness can be defined for MAIDs, with equivalence or one-way preservation results relative to extensive-form games.
• Imperfect recall in MAIDs splits into forgetfulness and absent-mindedness; either can destroy Nash equilibrium existence in behavioral policies, while mixed policies restore general existence.
• Causal games distinguish post-policy interventions, where policies are fixed before the intervention, from pre-policy interventions, where agents observe the changed game and may rationally adapt.

Methods / Formalism

• A mechanised MAID starts from a MAID M=(G,theta) and adds mechanism variables: decision-rule variables Pi_D for decisions and parameter variables Theta_V for non-decision variables.
• A rationality relation for decision D is a serial relation

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

A policy profile is R-rational when every decision rule is a rational response under its corresponding relation.

• For best-response rationality, a decision rule satisfies:

$${\pi }_D\in \arg \underset{{\hat{\pi }}_D\in dom({\Pi }_D)}{\max }\sum _{U\in U^i}{E}_{({\hat{\pi }}_D,{\pi }_{-D})}[U].$$

• A causal game is a MAID whose induced model is a causal Bayesian network for every fixed policy profile pi. Structural causal games add structural equations so counterfactuals can be evaluated.
• The thesis defines prediction, intervention, and counterfactual queries over sets of rational outcomes, not just one fixed policy. See Causal Game Intervention Semantics for the compact intervention and counterfactual schema.
• A post-policy intervention on object-level variables leaves rational outcomes fixed; a pre-policy intervention on mechanism variables changes the game and recomputes the rational outcomes.
• The intervention semantics is decomposed into primitive operations: fixing object-level variables, fixing mechanism variables, adding object-level variables, and removing object-level variables.

Evidence / Experiments

• The evidence is mostly formal: definitions, equivalence results, graphical criteria, existence/non-existence results, complexity classifications, and proof sketches.
• The subgame chapter reports computational experiments using PyCID in a class of matching-pennies-like games, where MAID subgames allow equilibrium computation to scale much better than the corresponding extensive-form-game representation.
• The imperfect-recall chapter gives counterexamples showing that forgetfulness and absent-mindedness can prevent behavioral-policy Nash equilibria, then records existence guarantees for pure, behavioral, mixed, and correlated policies under sufficient information or sufficient recall.
• The causal-games chapter works through a job-market signaling example to show how association, intervention, and counterfactual queries differ under pre-policy versus post-policy readings.

Connections

• Seeds Causal Games as the wiki page for game-theoretic causal models.
• Seeds Multi-Agent Influence Diagrams as the background graphical-game representation on which causal games build.
• Creates Causal Game Intervention Semantics for the formal query layer: rational-outcome sets, primitive interventions, visibility, and counterfactual rational outcomes.
• Extends Causal Decision Making from single-agent or logged-policy intervention effects to strategic settings where other agents may adapt their policies.
• Extends Structural Equation Models by showing how structural causal reasoning changes when some mechanisms are rationality relations rather than ordinary functions.
• Belongs in Strategic Reasoning because equilibrium refinements, subgames, recall assumptions, and correlated equilibria are central to the thesis.
• Connects to Decision Theory through policy choice under uncertainty, equilibrium selection, rational response, and counterfactual evaluation of interventions.

Open Questions

• Should the wiki later split out separate concept pages for R-relevance, MAID subgames, and imperfect recall in graphical games?
• How should causal games connect to later AI-safety work on deception, incentives, agent discovery, and causal influence diagrams?
• Which parts of the thesis are most useful for safe-MARL modeling versus economic mechanism-design applications?
• Should PyCID become an artifact or tool note if future sources rely on it computationally?

Citation

Fox, J. (2024). Causality and Strategic Reasoning (DPhil thesis, University of Oxford).