(Mechanical) Reasoning on Infinite Extensive Games

In order to better understand reasoning involved in analyzing infinite games in extensive form, we performed experiments in the proof assistant Coq that are reported here.

💡 Research Summary

The paper “Mechanical Reasoning on Infinite Extensive Games” investigates how to formalize and mechanically verify reasoning about infinite extensive‑form games using the Coq proof assistant. The authors adopt a radical stance: instead of modeling human agents with psychological constraints, they treat agents as idealized entities possessing unlimited deductive power. Their goal is to capture what a fully rational, logically perfect reasoning process would look like, and to implement this process on a computer so that every inference step can be checked by a machine.

The work begins with a concise overview of constructive logic and the Curry‑Howard correspondence, emphasizing that in this setting proofs are programs and propositions are types. Natural deduction is presented as the underlying logical system, with sequents (Γ ⊢ φ) and introduction/elimination rules for logical connectives. The authors explain how Coq’s type system mirrors these rules, allowing them to encode logical derivations as Coq terms.

A central technical contribution is the systematic use of induction for finite structures and co‑induction for infinite ones. Inductive definitions in Coq correspond to least fixed points, suitable for finite games and strategies. Co‑inductive definitions correspond to greatest fixed points, which are required to model infinite game trees, infinite strategies, and the associated equilibrium concepts. The paper discusses how temporal notions such as “always” and “eventually” can be expressed co‑inductively.

For the finite case, the authors define binary extensive‑form games (FinGame) as an inductive datatype with two constructors: a leaf carrying a utility function and an internal node containing an agent and two sub‑games. Strategies (FinStrategy) have the same shape but include a choice label (left/right) at each internal node. A recursive function f2u extracts the utility associated with a given strategy. They introduce an equivalence relation called a‑convertibility («‑a‑»), which captures the idea that two strategies are identical except possibly for the choices of a particular agent. Using this relation they define a Nash equilibrium predicate (FinNashEq) that requires no a‑convertible deviation to improve the agent’s utility.

Backward induction (BI) is formalized as an inductive predicate with three constructors: a leaf case, a left‑choice case, and a right‑choice case. The BI predicate encodes the standard textbook backward‑induction algorithm for finite horizon games. The authors prove, within Coq, that any strategy satisfying BI also satisfies FinNashEq (the theorem “BI is FinNashEq”). The proof hinges on the observation that BI characterizes the least fixed point of a set of equations, while Nash equilibrium is another fixed point; showing that the Nash fixed point is above the BI fixed point yields the result. Three auxiliary lemmas establish that the base leaf case and both inductive steps of Nash equilibrium are compatible with the corresponding BI constructors.

The infinite case is handled by defining a special class of infinite binary games, called “centipedes”, where each node consists of an agent, an infinite left sub‑game (again a centipede) and a finite right sub‑game. These games are encoded as a CoInductive datatype (InfGame). Infinite strategies (InfStrategy) are defined analogously, and a co‑inductive utility function i2u maps an infinite strategy to a utility. The key equilibrium concept for infinite games is Sub‑Game Perfect Equilibrium (SGPE), expressed as a co‑inductive predicate (SGPE) that checks, along the infinite path, that each agent’s choice is optimal given the continuation. Temporal predicates “always” and “eventually” are also defined co‑inductively to reason about properties that must hold throughout the infinite play.

Throughout the paper the authors provide Coq scripts and proof files on a public website, encouraging readers to download and run the developments. They argue that this mechanical approach reveals hidden logical dependencies (e.g., use of the law of excluded middle, double negation) that are often glossed over in informal human reasoning, especially in the context of infinite games where intuition can be misleading.

In summary, the paper makes three main contributions: (1) a complete formalization of both finite and a specific class of infinite extensive‑form games in Coq; (2) a demonstration that classic solution concepts—backward induction, Nash equilibrium for finite games, and sub‑game perfect equilibrium for infinite games—can be expressed and mechanically verified using induction and co‑induction; (3) an empirical observation that mechanical reasoning uncovers logical subtleties absent from informal analyses, offering a new perspective on what “perfect rationality” entails. Limitations include the focus on binary trees and centipede‑like infinite structures; extending the framework to n‑ary choices, imperfect information, or more general infinite horizons remains an open direction for future work.