Computing an Extensive-Form Correlated Equilibrium in Polynomial Time

Abstract

This paper defines the extensive-form correlated equilibrium (EFCE) for extensive games with perfect recall. The EFCE concept extends Aumann's strategic-form correlated equilibrium (CE). Before the game starts, a correlation device generates a move for each information set. This move is recommended to the player only when the player reaches the information set. In two-player perfect-recall extensive games without chance moves, the set of EFCE can be described by a polynomial number of consistency and incentive constraints. Assuming P is not equal to NP, this is not possible for the set of CE, or if the game has chance moves.Key words: correlated equilibrium; extensive game; polynomial-time computable MSC2000 subject classification: Primary: 91A18; secondary: 91A05, 91A28, 68Q17 OR/MS subject classification: Primary: noncooperative games; secondary: computational complexity History: Received March 22, 2006; revised September 19, 2007, and March 20, 2008. 1. Introduction. Aumann [1] defined the concept of correlated equilibrium (abbreviated as CE, also for the plural equilibria) for games in strategic form. Before the game starts, a device selects private signals from a joint probability distribution and sends them to the players. In the canonical representation of a CE, these signals are strategies that players are recommended to play. This paper proposes a new concept of correlated equilibrium for extensive games, called extensive-form correlated equilibrium or EFCE. Like in a CE (which is defined in terms of the strategic form), the recommendations to the players are moves that are generated before the game starts. However, each recommended move is assumed to be in a "sealed envelope" and is only revealed to a player when he reaches the information set where he can make that move.As recommendations become local in this way, players know less. Consequently, the set of EFCE outcomes is larger than the set of CE outcomes. However, an EFCE is more restrictive than an agent-form correlated equilibrium (AFCE). In the agent form of the game, moves are chosen by a separate agent for each information set of the player. In an EFCE, players remain in control of their future actions, which is important when they consider deviating from their recommended moves.The EFCE is a natural definition of correlated equilibrium for extensive games with perfect recall as defined by Kuhn [25]. Earlier extensions of Aumann's concept applied only to multistage games, including Bayesian games and stochastic games, which have a special time and information structure. These earlier approaches are discussed in §2.4.The main motivation for the EFCE concept is computational. The algorithmic input is some description of the extensive game with its game tree, information sets, moves, chance probabilities and payoffs. Polynomial (or linear or exponential) size and time always refer to the size of this description. The strategic form of the extensive game has typically exponential size. Hence, there are also exponentially many linear constraint...

No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

“…The problem of computing an optimal EFCE in extensive-form games with more than two players and/or chance moves is known to be NP-hard (von Stengel and Forges, 2008). However, Huang and von Stengel (2008) show that the problem of finding one EFCE can be solved in polynomial time via a variation of the Ellipsoid Against Hope algorithm (Papadimitriou and Jiang and Leyton-Brown, 2015). This holds for arbitrary EFGs with multiple players and/or chance moves.…”

“…Original contributions We focus on general-sum sequential games with an arbitrary number of players (including the chance player). In this setting, the problem of computing a feasible EFCE (and, therefore, a feasible EFCCE) can be solved in polynomial time in the size of the game tree (Huang and von Stengel, 2008) via a variation of the Ellipsoid Against Hope algorithm (Papadimitriou and Jiang and Leyton-Brown, 2015). However, in practice, this approach cannot scale beyond toy problems.…”

Computing Optimal Coarse Correlated Equilibria in Sequential Games

“…An optimal EFCE can be found efficiently in two-player games without Chance moves but, in games with three or more players (including Chance), finding an EFCE (or an AFCE) is NPhard (von Stengel and Forges 2008). The only positive result for multi-player games is a polynomial-time algorithm to find an EFCE (Huang and von Stengel 2008).…”

The Principle of Maximum Causal Entropy for Estimating Interacting Processes

“…Efficient approximation approaches have been employed [38], [33], but tractable applicability has been limited to small games [33]. For the far more modest goal of finding an arbitrary CE in a range of compact games, algorithms that are polynomial in the number of agents have been developed [45], [24] and extended to sequential games [20].…”