Quasi-perfect Equilibrium Research Articles

Backward induction in games without perfect recall

The equilibrium concepts that we now think of as various forms of backwards induction, namely, subgame perfect equilibrium (Selten, 1965), perfect equilibrium (Selten, 1975), sequential equilibrium (Kreps and Wilson, 1982), and quasi-perfect equilibrium (van Damme, 1984), while formally well defined in a wider class of games, are explicitly restricted to games with perfect recall.We argue that in games without perfect recall the original definitions are inappropriate. Our reading of the original papers is that the authors were aware that their definitions did not formally require the assumption of perfect recall but they were also aware that without the assumption of perfect recall the definitions they gave were not the “correct” ones. We give definitions of these concepts that identify the same outcomes in games with perfect recall and behave well in linear games without perfect recall.

A characterization of quasi-perfect equilibria

We provide a characterization of quasi-perfect equilibria in n-player games, showing that any quasi-perfect equilibrium can be obtained as limit point of a sequence of Nash equilibria of a certain class of perturbed games in sequence form, and any limit point of a sequence of Nash equilibria of these perturbed games is a quasi-perfect equilibrium. We prove that, in games with three or more players, we need trembles defined as rational functions of the perturbation magnitude ε, whereas, in two-player games with nature, trembles expressed in terms of polynomial functions of ε suffice. Exploiting the relationship between sequence form and extensive form, we also provide a similar characterization in terms of perturbed games in extensive form, though not compliant with Selten's definition of perturbed game.

Quasi-Perfect Stackelberg Equilibrium

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasiperfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings—that is, settings where a leader can commit to a strategy—which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes.

The Real Computational Complexity of Minmax Value and Equilibrium Refinements in Multi-player Games

We show that for several solution concepts for finite n-player games, where n ≥ 3, the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the reals. This holds for trembling hand perfect equilibrium, proper equilibrium, and CURB sets in strategic form games and for (the strategy part of) sequential equilibrium, trembling hand perfect equilibrium, and quasi-perfect equilibrium in extensive form games of perfect recall. For obtaining these results we first show that the decision problem for the minmax value in n-player games, where n ≥ 3, is also equivalent to the decision problem for the existential theory of the reals. Our results thus improve previous results of NP-hardness as well as Sqrt-Sum-hardness of the decision problems to completeness for ${\exists {\mathbb {R}}}$ , the complexity class corresponding to the decision problem of the existential theory of the reals. As a byproduct we also obtain a simpler proof of a result by Schaefer and Stefankovic giving ${\exists {\mathbb {R}}}$ -completeness for the problem of deciding existence of a probability constrained Nash equilibrium.

On the Characterization of Quasi-Perfect Equilibria

Van Damme [1984] introduces the concept of quasi-perfect equilibrium, which re-fines sequential equilibrium as well as normal-form perfect equilibrium. It has been argued by Mertens [1995] that quasi-perfection is conceptually superior to extensive-form perfection, since quasi-perfection guarantees normal-form perfection, which for two-player games is equivalent to admissibility. On the other hand, while extensive-form perfect equilibria are defined as limit points of sequences of Nash equilibria of a general class of perturbed games in extensive form, till now, to the best of our knowledge, there is no characterization of quasi-perfect equilibria in terms of limit points of equilibria of perturbed games. The only known result is Lemma 1 by Miltersen and Sorensen [2010], showing that limit points of sequences of Nash equilibria of a particular class of perturbed games in sequence form are quasi-perfect equilibria of the original, unperturbed game in extensive form. However, as the authors point out, their main result only proves that a subset of the quasi-perfect equilibria can be obtained as limit points of equilibria of their class of perturbed games, and, thus, their paper provides no characterization of quasi-perfect equilibria in terms of perturbed games. The present paper fills this gap providing such characterization, showing that any quasi-perfect equilibrium can be obtained as limit point of a sequence of Nash equilibria of a certain class of perturbed games in sequence form, at least for the case of two-player games with nature. This result shows that the sequence form is not merely a computationally efficient representation, but it also captures game features that other forms are not able to effectively express.

Further Results on Verification Problems in Extensive-Form Games

The computational study of games is receiving increasing attention both in game theory and computer science. The challenge is distinguishing computationally tractable problems (also said easy), admitting polynomial–time algorithms, from the intractable ones (also said hard). In this paper, we focus on extensive form games, as the computational problems defined on such games are largely unexplored. We study the problem (aka verification problem) of certifying that a solution given in input is an equilibrium according to different refinements for extensive form games as the input change. We show that, when the input is a realization plan strategy profi (i.e., strategies for the sequence form representation), deciding whether the input is a Subgame Perfect Equilibrium or is a part of a Sequential Equilibrium is NP-hard even in two-player games (we conjecture the same holds also for Quasi Perfect Equilibrium). This means that there is no polynomial–time algorithm unless P = NP, but it is commonly believed that P ≠ NP. Subsequently, we show that in two-player games, when the input is a behavioral strategy profile there is a polynomial-time algorithm deciding whether the input is a Quasi-Perfect Equilibrium, and a simple variation of the algorithm decides whether the input is part of some Sequential Equilibrium (in games with three or more players, the problem is known to be NP-hard for both Quasi-Perfect Equilibrium and Sequential Equilibrium). Finally, we show that, when the input is an assessment, there is a polynomial-time algorithm deciding whether the input is a Sequential Equilibrium regardless the number of players.

Characterizing solution concepts in terms of common knowledge of rationality

Characterizations of Nash equilibrium, correlated equilibrium, and rationalizability in terms of common knowledge of rationality are well known. Analogous characterizations of sequential equilibrium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using earlier results of Halpern characterizing these solution concepts using non-Archimedean fields.

On the equivalence between (quasi-)perfect and sequential equilibria

We prove the generic equivalence between quasi-perfect equilibrium and sequential equilibrium. Combining this result with Blume and Zame (Econometrica 62:783–794, 1994) shows that perfect, quasi-perfect and sequential equilibrium coincide in generic games.

On the Equivalence between (Quasi-)Perfect and Sequential Equilibria

We prove the generic equivalence between quasi-perfect equilibrium and sequential equilibrium. Combining this result with Blume and Zame (1994) shows that perfect, quasi-perfect and sequential equilibrium coincide in generic games.

Axiomatic Equilibrium Selection for Generic Two-Player Games

We impose three conditions on refinements of the Nash equilibria of finite games with perfect recall that select closed connected subsets, called solutions. A. Each equilibrium in a solution uses undominated strategies; B. Each solution contains a quasi-perfect equilibrium; C. The solutions of a game map to the solutions of an embedded game, where a game is embedded if each player’s feasible strategies and payoffs are preserved by a multilinear map. We prove for games with two players and generic payoffs that these conditions characterize each solution as an essential component of equilibria in undominated strategies, and thus a stable set as defined by Mertens (1989).

Computing a quasi-perfect equilibrium of a two-player game

Refining an algorithm due to Koller, Megiddo and von Stengel, we show how to apply Lemke’s algorithm for solving linear complementarity programs to compute a quasi-perfect equilibrium in behavior strategies of a given two-player extensive-form game of perfect recall. A quasi-perfect equilibrium is known to be sequential, and our algorithm thus resolves a conjecture of McKelvey and McLennan in the positive. A quasi-perfect equilibrium is also known to be normal-form perfect and our algorithm thus provides an alternative to an algorithm by von Stengel, van den Elzen and Talman. For the case of a zero-sum game, we devise variants of the algorithm that rely on linear programming rather than linear complementarity programming and use the simplex algorithm or other algorithms for linear programming rather than Lemke’s algorithm. We argue that these latter algorithms are relevant for recent applications of equilibrium computation to artificial intelligence.

On Forward Induction

We examine Hillas and Kohlberg's conjecture that invariance to the addition of payoff-redundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply forward induction. Although it suffices in simple games to interpret backward induction as a subgame-perfect or sequential equilibrium, to obtain general theorems we use a quasi-perfect equilibrium, viz. a sequential equilibrium in strategies that are admissible continuations from each information set. Using this version of backward induction, we prove the Hillas-Kohlberg conjecture for two-player extensive-form games with perfect recall. We also prove an analogous theorem for general games by interpreting backward induction as a proper equilibrium, since a proper equilibrium is equivalent to a quasi-perfect equilibrium of each extensive form with the same normal form, provided beliefs are justifed by perturbations invariant to inessential transformations of the extensive form. For a two-player game we prove that if a set of equilibria includes a proper equilibrium of every game with the same reduced normal form then it satisfies forward induction, i.e. it includes a proper equilibrium of the game after deleting strategies that are inferior replies to all equilibria in the set. We invoke slightly stronger versions of invariance and properness to handle nonlinearities in an N-player game.

Justification of Stable Equilibria

Two assumptions are used to justify selection of equilibria in stable sets. One assumption requires that a selected set is invariant to addition of redundant strategies. The other is a strong version of backward induction. Backward induction is interpreted as the requirement that behavior strategies in an extensive-form game are sequentially rational and conditionally admissible at every information set; viz., quasi-perfect as defined by van Damme. The strong version requires 'truly' quasi-perfect, in that every action perturbation selects a quasi-perfect equilibrium in the set. For two-player games we also provide an exact characterization of stable sets.

Sequential and quasi-perfect rationalizability in extensive games

Within an epistemic model for two-player extensive games, we formalize the event that each player believes that his opponent chooses rationally at all information sets. Letting this event be common certain belief yields the concept of sequential rationalizability. Adding preference for cautious behavior to this event likewise yields the concept of quasi-perfect rationalizability. These concepts are shown to (a) imply backward induction in generic perfect information games, and (b) be non-equilibrium analogues to sequential and quasi-perfect equilibrium, leading to epistemic characterizations of the latter concepts. Conditional beliefs are described by the novel concept of a system of conditional lexicographic probabilities.

A relation between perfect equilibria in extensive form games and proper equilibria in normal form games

The concept of quasi-perfect equilibria for games in extensive form is introduced. It is shown that a proper equilibrium of a normal form game induces a quasi-perfect equilibrium in every extensive form game having this normal form.