[Paper Review] Rationality and Escalation in Infinite Extensive Games

Pierre Lescanne|arXiv (Cornell University)|Dec 6, 2011

Computability, Logic, AI Algorithms48 references4 citations

TL;DR

This paper demonstrates that escalation in infinite extensive games—such as speculative market crashes or endless bidding—can be rationally justified when resources are assumed to be infinite. Using coinductive reasoning, the authors formalize infinite games and strategy profiles in Coq, proving that behaviors deemed 'madness' in finite models become rational under infinite resource assumptions, challenging conventional game-theoretic intuitions.

ABSTRACT

The aim of this of this paper is to study infinite games and to prove formally some properties in this framework. As a consequence we show that the behavior (the madness) of people which leads to speculative crashes or escalation can be fully rational. Indeed it proceeds from the statement that resources are infinite. The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by economic agents unknowingly. When used consciously, this concept is not as simple as induction and we could paraphrase Newton: "Modeling the madness of people is more difficult than modeling the motion of planets".

Motivation & Objective

• To formalize infinite extensive games using coinductive methods from computer science.
• To challenge the conventional view that escalation in games is irrational by showing it can be logically rational under infinite resource assumptions.
• To demonstrate that equilibria in finite games do not necessarily extend to infinite games, revealing new equilibrium types in infinite settings.
• To provide a formal proof framework using Coq for reasoning about infinite strategy profiles and game behaviors.
• To bridge insights from logic, computer science (coinduction), and game theory to resolve paradoxes in infinite strategic interactions.

Proposed method

• The paper models infinite games using coinductive definitions, representing games as recursive structures with infinite depth.
• It defines strategy profiles coinductively, allowing for infinite sequences of decisions without requiring a terminal node.
• A formal implementation in the Coq proof assistant ensures correctness of reasoning over infinite structures.
• The authors use backward coinduction to prove invariants and equilibrium properties in infinite games.
• They introduce a generalized utility framework with preorders to handle payoffs and preferences in infinite contexts.
• The $0,1$ game example is used to illustrate how infinite recursion leads to rational equilibria distinct from finite approximations.

Experimental results

Research questions

• RQ1 Can escalation in games be rationally justified when resources are infinite?
• RQ2 Why do standard game-theoretic equilibria in finite games fail to extend to infinite games?
• RQ3 How can coinductive reasoning be applied to formalize infinite strategy profiles and prove their properties?
• RQ4 What is the role of infinite resource assumptions in making seemingly irrational behaviors rational?
• RQ5 How do infinite games differ from their finite approximations in terms of equilibrium existence and structure?

Key findings

• Escalation in infinite games is not irrational but logically consistent when resources are assumed to be infinite.
• The $0,1$ game exhibits a rational equilibrium under coinductive reasoning that does not exist in finite approximations.
• Finite-game equilibria do not converge to infinite-game equilibria, indicating a fundamental break between finite and infinite models.
• Coinduction provides a formal framework to reason about infinite games and strategy profiles, correcting flawed extrapolations from finite cases.
• The paper shows that Weierstrass-type phenomena—where infinite limits behave differently from finite approximations—also occur in game theory.
• Using Coq, the authors formally verify that infinite strategy profiles can support rational equilibria, validating the coinductive approach.