Acyclicity of Preferences, Nash Equilibria, and Subgame Perfect Equilibria: a Formal and Constructive Equivalence

TL;DR

This paper generalizes sequential game theory by replacing real-valued payoffs with abstract outcomes and preferences, proving the equivalence of acyclicity, Nash equilibria, and subgame perfect equilibria, all formalized in Coq.

Contribution

It introduces a formal, abstract framework for sequential games with arbitrary preferences and proves key equivalences, fully verified in Coq.

Findings

Preferences over outcomes are acyclic if and only if every game has a Nash equilibrium.

Preferences being acyclic is equivalent to the existence of subgame perfect equilibria.

The formal proof ensures correctness and clarifies the main concepts involved.

Abstract

In 1953, Kuhn showed that every sequential game has a Nash equilibrium by showing that a procedure, named ``backward induction'' in game theory, yields a Nash equilibrium. It actually yields Nash equilibria that define a proper subclass of Nash equilibria. In 1965, Selten named this proper subclass subgame perfect equilibria. In game theory, payoffs are rewards usually granted at the end of a game. Although traditional game theory mainly focuses on real-valued payoffs that are implicitly ordered by the usual total order over the reals, works of Simon or Blackwell already involved partially ordered payoffs. This paper generalises the notion of sequential game by replacing real-valued payoff functions with abstract atomic objects, called outcomes, and by replacing the usual total order over the reals with arbitrary binary relations over outcomes, called preferences. This introduces a general abstract formalism where Nash equilibrium, subgame perfect equilibrium, and ``backward induction'' can still be defined. This paper proves that the following three propositions are equivalent: 1) Preferences over the outcomes are acyclic. 2) Every sequential game has a Nash equilibrium. 3) Every sequential game has a subgame perfect equilibrium. The result is fully computer-certified using Coq. Beside the additional guarantee of correctness, the activity of formalisation using Coq also helps clearly identify the useful definitions and the main articulations of the proof.