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Robust Equilibria in Generic Extensive form Games

Lucas Pahl, Carlos Pimienta

TL;DR

The article proves that, for generic two-player extensive-form games, hyperstable equilibrium components coincide with components whose index is nonzero, providing an operational, index-theoretic characterization of Kohlberg–Mertens stability. By introducing excluded games and using fixed-point index theory, the authors show that zero-index components fail to be hyperstable, while nonzero-index components are robust to payoff perturbations and invariant across equivalent representations. The main theorem is established via a three-step perturbation argument that leverages the product structure of indices across players and an explicit construction of equivalent perturbations that eliminate equilibria near zero-index components. The approach yields a practical method to identify hyperstable outcomes in economically relevant examples and clarifies how hyperstability relates to other refinement concepts, such as sequential stability and NWBR-type criteria, within extensional-form settings. Overall, the work provides a rigorous, computable bridge between topological index theory and strategic stability in generic two-player extensive-form games, with clear implications for predicting equilibrium outcomes under perturbations and equivalence transformations.

Abstract

We prove the 2-player, generic extensive-form case of the conjecture of Govindan and Wilson (1997a,b) and Hauk and Hurkens (2002) stating that an equilibrium component is essential in every equivalent game if and only if the index of the component is nonzero. This provides an index-theoretic characterization of the concept of hyperstable components of equilibria in generic extensive-form games, first formulated by Kohlberg and Mertens (1986). We also illustrate how to compute hyperstable equilibria in multiple economically relevant examples and show how the predictions of hyperstability compare with other solution concepts.

Robust Equilibria in Generic Extensive form Games

TL;DR

The article proves that, for generic two-player extensive-form games, hyperstable equilibrium components coincide with components whose index is nonzero, providing an operational, index-theoretic characterization of Kohlberg–Mertens stability. By introducing excluded games and using fixed-point index theory, the authors show that zero-index components fail to be hyperstable, while nonzero-index components are robust to payoff perturbations and invariant across equivalent representations. The main theorem is established via a three-step perturbation argument that leverages the product structure of indices across players and an explicit construction of equivalent perturbations that eliminate equilibria near zero-index components. The approach yields a practical method to identify hyperstable outcomes in economically relevant examples and clarifies how hyperstability relates to other refinement concepts, such as sequential stability and NWBR-type criteria, within extensional-form settings. Overall, the work provides a rigorous, computable bridge between topological index theory and strategic stability in generic two-player extensive-form games, with clear implications for predicting equilibrium outcomes under perturbations and equivalence transformations.

Abstract

We prove the 2-player, generic extensive-form case of the conjecture of Govindan and Wilson (1997a,b) and Hauk and Hurkens (2002) stating that an equilibrium component is essential in every equivalent game if and only if the index of the component is nonzero. This provides an index-theoretic characterization of the concept of hyperstable components of equilibria in generic extensive-form games, first formulated by Kohlberg and Mertens (1986). We also illustrate how to compute hyperstable equilibria in multiple economically relevant examples and show how the predictions of hyperstability compare with other solution concepts.

Paper Structure

This paper contains 20 sections, 20 theorems, 54 equations, 15 figures.

Table of Contents

  1. Introduction
  2. An example
  3. Preliminaries
  4. Equivalent strategies, equivalent normal-form games
  5. Extensive-form games
  6. Index Theory
  7. Main Result
  8. Excluded Games
  9. Genericity Assumptions
  10. Proof of Theorem \ref{['mainthm']}
  11. Step 1: For some player n, the supporting polytope has index 0 in Gn
  12. Step 2: Perturbing the excluded game so that it has no equilibria in the supporting polytope
  13. Step 3: The equivalent game and its perturbation
  14. Applications
  15. Proof of Proposition \ref{['nonzeroindex']}
  16. ...and 5 more sections

Key Result

Proposition 2.3

If an equilibrium outcome $Q$ induced by a component of equilibria in mixed strategies $K$ has full support, then $K$ has non-zero index.

Figures (15)

  • Figure 1: Entry game
  • Figure 2: Modifying function $f$ to eliminate two fixed points with indexes adding up to zero.
  • Figure 3: A game equivalent to that in Figure \ref{['Entry']} when $\varepsilon=0$
  • Figure 4: Game G
  • Figure 5: Game ${G}^{\varepsilon_1, \varepsilon_2}$
  • ...and 10 more figures

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Definition 3.2: Excluded game
  • Remark 3.3
  • Example 3.4
  • ...and 53 more