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On the Outcome Equivalence of Extensive-Form and Behavioral Correlated Equilibria

Brian Hu Zhang, Tuomas Sandholm

TL;DR

This paper proves that extensive-form correlated equilibria (EFCE) and behavioral correlated equilibria (BCE) are outcome-equivalent: any $\\varepsilon$-EFCE can be transformed into an $\\varepsilon$-BCE that induces the same distribution over terminal nodes. It provides a constructive, polynomial-time method to perform this transformation, enabling the first polynomial-time algorithm for computing a BCE in extensive-form games. The authors introduce counterfactual-regret-based BCE deviations, define a transformation via counterfactual best-response strategies $x_i^{Ia}$, and show the resulting profile is a BCE with the same outcome distribution as the original EFCE. They also discuss implications for optimizing equilibria, learning dynamics, and the limits of stronger notions of outcome equivalence, outlining future directions in polynomial-time computation and dynamics for BCE.

Abstract

We investigate two notions of correlated equilibrium for extensive-form games: extensive-form correlated equilibrium (EFCE) and behavioral correlated equilibrium (BCE). We show that the two are outcome-equivalent, in the sense that every outcome distribution achievable under one notion is achievable under the other. Our result implies, to our knowledge, the first polynomial-time algorithm for computing a BCE.

On the Outcome Equivalence of Extensive-Form and Behavioral Correlated Equilibria

TL;DR

This paper proves that extensive-form correlated equilibria (EFCE) and behavioral correlated equilibria (BCE) are outcome-equivalent: any -EFCE can be transformed into an -BCE that induces the same distribution over terminal nodes. It provides a constructive, polynomial-time method to perform this transformation, enabling the first polynomial-time algorithm for computing a BCE in extensive-form games. The authors introduce counterfactual-regret-based BCE deviations, define a transformation via counterfactual best-response strategies , and show the resulting profile is a BCE with the same outcome distribution as the original EFCE. They also discuss implications for optimizing equilibria, learning dynamics, and the limits of stronger notions of outcome equivalence, outlining future directions in polynomial-time computation and dynamics for BCE.

Abstract

We investigate two notions of correlated equilibrium for extensive-form games: extensive-form correlated equilibrium (EFCE) and behavioral correlated equilibrium (BCE). We show that the two are outcome-equivalent, in the sense that every outcome distribution achievable under one notion is achievable under the other. Our result implies, to our knowledge, the first polynomial-time algorithm for computing a BCE.
Paper Structure (14 sections, 5 theorems, 11 equations, 2 figures)

This paper contains 14 sections, 5 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extensive-Form Games
  4. Extensive-Form and Behavioral Correlated Equilibria
  5. Main Result Statement and Examples
  6. Proof of Main Result
  7. Algorithm for Converting from EFCE to BCE
  8. Discussion
  9. Optimal Equilibria
  10. Hindsight Rationality and No-Regret Learning
  11. Stronger Notions of Outcome Equivalence
  12. Surjectivity
  13. Counterfactual Regret and the Definition of BCE
  14. Conclusions and Future Research

Key Result

Theorem 3.2

Every $\varepsilon$-EFCE is outcome-equivalent to an $\varepsilon$-BCE.

Figures (2)

  • Figure 1: Left: The extended battle of the sexes game in \ref{['ex:morrill']}. The players are $\blacktriangle$ (P1) and $\blacktriangledown$ (P2). Infosets are connected by dotted lines. Player 1 first chooses whether or not to upgrade ($U$). Then, both players simultaneously choose an event ($X$ or $Y$) to attend. Player 1 prefers $X$, while player 2 prefers $Y$. If the players attend different events, they are unhappy and get utility $0$. If the players attend the same event, the player attending its preferred event gets $2$, and the other player gets $1$. Upgrading gives an extra point of utility if the players attend the same event. Right: The game used in \ref{['ex:counterfactual']} illustrating that use counterfactual regret is also significant.
  • Figure 2: A game showing that the EFCE-BCE map in this paper is not surjective (for any tiebreaking method). The root node is a nature node; nature moves uniformly at random. The MP subtree is the matching pennies game; the Coord subtree is a coordination game, but P2 has a strictly dominant Exit action.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2: Morrill20:Hindsight
  • Definition 2.3: Stengel08:ExtensiveMorrill21:Efficient
  • Definition 2.4: Morrill21:Efficient
  • Definition 3.1
  • Theorem 3.2: Main result
  • Example 3.3: Morrill20:Hindsight
  • Example 3.4
  • Definition 5.1
  • Lemma 5.2
  • ...and 5 more