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Subgame Credible Nash Equilibrium

Mehmet Mars Seven

TL;DR

SCNE introduces Subgame Credible Nash Equilibrium, a refinement of subgame perfect equilibrium for multi-stage games that adds an external-credibility condition across equivalent subgames to prevent self-harming punishments. It combines internal credibility—$(s^*|g)$ is a Nash equilibrium for every subgame $g$—with external credibility, requiring $u_i(s^*|g') \\ge u_i(s^*|g)$ whenever $g'$ is an off-path subgame equivalent to $g$ and the player $i$ has not deviated prior to $g$. The authors prove existence of SCNE for all multi-stage games and show uniqueness under the condition that each stage game has a unique Nash equilibrium, while contrasting SCNE with renegotiation-proof notions like WRP and SRP. The framework extends to general extensive-form games and provides a non-cooperative refinement that excludes SPNEs relying on self-punishment, offering a tractable tool for analyzing credible threat structures in dynamic games.

Abstract

We propose the Subgame Credible Nash Equilibrium (SCNE), a refinement of subgame perfect Nash equilibrium (SPNE) for multi-stage games. SCNE retains the internal credibility requirement of SPNE -- equilibrium behavior in every subgame -- and adds an external credibility requirement across equivalent subgames: whenever a player's prescribed continuation strategy differs across equivalent subgames, her own continuation payoff must not decrease. The intuition is that credible punishments or promises should not strictly harm the punisher relative to an equivalent no-punishment subgame. The SCNE eliminates equilibria sustained by self-harming punishments or promises while preserving existence. Every multi-stage game admits an SCNE, and if each stage game has a unique Nash equilibrium, the SCNE is unique.

Subgame Credible Nash Equilibrium

TL;DR

SCNE introduces Subgame Credible Nash Equilibrium, a refinement of subgame perfect equilibrium for multi-stage games that adds an external-credibility condition across equivalent subgames to prevent self-harming punishments. It combines internal credibility— is a Nash equilibrium for every subgame —with external credibility, requiring whenever is an off-path subgame equivalent to and the player has not deviated prior to . The authors prove existence of SCNE for all multi-stage games and show uniqueness under the condition that each stage game has a unique Nash equilibrium, while contrasting SCNE with renegotiation-proof notions like WRP and SRP. The framework extends to general extensive-form games and provides a non-cooperative refinement that excludes SPNEs relying on self-punishment, offering a tractable tool for analyzing credible threat structures in dynamic games.

Abstract

We propose the Subgame Credible Nash Equilibrium (SCNE), a refinement of subgame perfect Nash equilibrium (SPNE) for multi-stage games. SCNE retains the internal credibility requirement of SPNE -- equilibrium behavior in every subgame -- and adds an external credibility requirement across equivalent subgames: whenever a player's prescribed continuation strategy differs across equivalent subgames, her own continuation payoff must not decrease. The intuition is that credible punishments or promises should not strictly harm the punisher relative to an equivalent no-punishment subgame. The SCNE eliminates equilibria sustained by self-harming punishments or promises while preserving existence. Every multi-stage game admits an SCNE, and if each stage game has a unique Nash equilibrium, the SCNE is unique.
Paper Structure (2 sections, 2 theorems, 4 equations, 3 figures)

This paper contains 2 sections, 2 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction and Definition
  2. Comparison with related concepts in the literature

Key Result

Theorem 1

Every multi-stage game $G$ admits an SCNE.

Figures (3)

  • Figure 1: A two-stage game: an SPNE that is not Subgame Credible.
  • Figure 2: An SCNE that is not WRP.
  • Figure 3: A WRP equilibrium that is not an SCNE.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1: Existence
  • proof
  • Proposition 1: Uniqueness
  • proof