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

1984 ◽  
Vol 13 (1) ◽  
pp. 1-13 ◽  
Author(s):  
E. van Damme
Keyword(s):  
Normal Form ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Normal Form Games ◽  
Perfect Equilibria

scholarly journals Non-altruistic Equilibria

2019 ◽  
Vol 67 (3-4) ◽  
pp. 185-195
Author(s):  
Kazuhiro Ohnishi
Keyword(s):  
Normal Form ◽  
Cooperative Games ◽  
The Other ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Normal Form Games ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria ◽  
Non Cooperative Games

Which choice will a player make if he can make one of two choices in which his own payoffs are equal, but his rival’s payoffs are not equal, that is, one with a large payoff for his rival and the other with a small payoff for his rival? This paper introduces non-altruistic equilibria for normal-form games and extensive-form non-altruistic equilibria for extensive-form games as equilibrium concepts of non-cooperative games by discussing such a problem and examines the connections between their equilibrium concepts and Nash and subgame perfect equilibria that are important and frequently encountered equilibrium concepts.

scholarly journals The Refined Best Reply Correspondence and Backward Induction

2019 ◽  
Vol 20 (1) ◽  
pp. 52-66
Author(s):  
Dieter Balkenborg ◽  
Christoph Kuzmics ◽  
Josef Hofbauer
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
The Other ◽  
Backward Induction ◽  
Perfect Recall ◽  
Extensive Form ◽  
Remarkable Property ◽  
Extensive Form Games ◽  
Other Hand ◽  
Perfect Equilibria

Abstract Fixed points of the (most) refined best reply correspondence, introduced in Balkenborg et al. (2013), in the agent normal form of extensive form games with perfect recall have a remarkable property. They induce fixed points of the same correspondence in the agent normal form of every subgame. Furthermore, in a well-defined sense, fixed points of this correspondence refine even trembling hand perfect equilibria, while, on the other hand, reasonable equilibria that are not weak perfect Bayesian equilibria are fixed points of this correspondence.

scholarly journals Decentralized No-regret Learning Algorithms for Extensive-form Correlated Equilibria (Extended Abstract)

2021 ◽  
Author(s):  
Andrea Celli ◽  
Alberto Marchesi ◽  
Gabriele Farina ◽  
Nicola Gatti
Keyword(s):  
Normal Form ◽  
Private Information ◽  
Imperfect Information ◽  
Research Question ◽  
Correlated Equilibrium ◽  
Extensive Form ◽  
Learning Dynamics ◽  
Extensive Form Games ◽  
Normal Form Games ◽  
Correlated Equilibria

The existence of uncoupled no-regret learning dynamics converging to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and the presence of private information, correlation in extensive-form games possesses significantly different properties than in normal-form games. The extensive-form correlated equilibrium (EFCE) is the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device ({\em a.k.a.} mediator) must take into account the evolution of beliefs of each player as they make observations throughout the game. Due to this additional complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics which provably converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play after T game repetitions is guaranteed to be a O(T^-1/2)-approximate EFCE with high probability, and an EFCE almost surely in the limit.

scholarly journals Coarse Correlation in Extensive-Form Games

2020 ◽  
Vol 34 (02) ◽  
pp. 1934-1941
Author(s):  
Gabriele Farina ◽  
Tommaso Bianchi ◽  
Tuomas Sandholm
Keyword(s):  
Normal Form ◽  
Saddle Points ◽  
Solution Concept ◽  
Correlated Equilibrium ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Rational Agents ◽  
Correlation Models ◽  
Correlated Equilibria ◽  
Special Case

Coarse correlation models strategic interactions of rational agents complemented by a correlation device which is a mediator that can recommend behavior but not enforce it. Despite being a classical concept in the theory of normal-form games since 1978, not much is known about the merits of coarse correlation in extensive-form settings. In this paper, we consider two instantiations of the idea of coarse correlation in extensive-form games: normal-form coarse-correlated equilibrium (NFCCE), already defined in the literature, and extensive-form coarse-correlated equilibrium (EFCCE), a new solution concept that we introduce. We show that EFCCEs are a subset of NFCCEs and a superset of the related extensive-form correlated equilibria. We also show that, in n-player extensive-form games, social-welfare-maximizing EFCCEs and NFCCEs are bilinear saddle points, and give new efficient algorithms for the special case of two-player games with no chance moves. Experimentally, our proposed algorithm for NFCCE is two to four orders of magnitude faster than the prior state of the art.

scholarly journals Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games

2019 ◽  
Vol 33 ◽  
pp. 1917-1925
Author(s):  
Gabriele Farina ◽  
Christian Kroer ◽  
Tuomas Sandholm
Keyword(s):  
Nash Equilibrium ◽  
Convex Sets ◽  
Convex Compact ◽  
Sequential Decision ◽  
Quantal Response ◽  
Decision Point ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Regret Minimization ◽  
Perfect Equilibria

Regret minimization is a powerful tool for solving large-scale extensive-form games. State-of-the-art methods rely on minimizing regret locally at each decision point. In this work we derive a new framework for regret minimization on sequential decision problems and extensive-form games with general compact convex sets at each decision point and general convex losses, as opposed to prior work which has been for simplex decision points and linear losses. We call our framework laminar regret decomposition. It generalizes the CFR algorithm to this more general setting. Furthermore, our framework enables a new proof of CFR even in the known setting, which is derived from a perspective of decomposing polytope regret, thereby leading to an arguably simpler interpretation of the algorithm. Our generalization to convex compact sets and convex losses allows us to develop new algorithms for several problems: regularized sequential decision making, regularized Nash equilibria in zero-sum extensive-form games, and computing approximate extensive-form perfect equilibria. Our generalization also leads to the first regret-minimization algorithm for computing reduced-normal-form quantal response equilibria based on minimizing local regrets. Experiments show that our framework leads to algorithms that scale at a rate comparable to the fastest variants of counterfactual regret minimization for computing Nash equilibrium, and therefore our approach leads to the first algorithm for computing quantal response equilibria in extremely large games. Our algorithms for (quadratically) regularized equilibrium finding are orders of magnitude faster than the fastest algorithms for Nash equilibrium finding; this suggests regret-minimization algorithms based on decreasing regularization for Nash equilibrium finding as future work. Finally we show that our framework enables a new kind of scalable opponent exploitation approach.

Extensive Form Rationalizability

2014 ◽  
Author(s):  
Herbert Gintis
Keyword(s):  
Normal Form ◽  
Nash Equilibria ◽  
Common Knowledge ◽  
Backward Induction ◽  
Forward Induction ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Dominated Strategies ◽  
Extensive Form Game ◽  
Common Knowledge Of Rationality

The extensive form of a game is informationally richer than the normal form since players gather information that allows them to update their subjective priors as the game progresses. For this reason, the study of rationalizability in extensive form games is more complex than the corresponding study in normal form games. There are two ways to use the added information to eliminate strategies that would not be chosen by a rational agent: backward induction and forward induction. The latter is relatively exotic (although more defensible). Backward induction, by far the most popular technique, employs the iterated elimination of weakly dominated strategies, arriving at the subgame perfect Nash equilibria—the equilibria that remain Nash equilibria in all subgames. An extensive form game is considered generic if it has a unique subgame perfect Nash equilibrium. This chapter develops the tools of modal logic and presents Robert Aumann's famous proof that common knowledge of rationality (CKR) implies backward induction. It concludes that Aumann is perfectly correct, and the real culprit is CKR itself. CKR is in fact self-contradictory when applied to extensive form games.

scholarly journals Extensive Form Reasoning in Normal Form Games

Econometrica ◽  
1993 ◽  
Vol 61 (2) ◽  
pp. 273 ◽  
Author(s):  
George J. Mailath ◽  
Larry Samuelson ◽  
Jeroen M. Swinkels
Keyword(s):  
Normal Form ◽  
Extensive Form ◽  
Normal Form Games

scholarly journals Normal Form Structures in Extensive Form Games

1994 ◽  
Vol 64 (2) ◽  
pp. 325-371 ◽  
Author(s):  
George J. Mailath ◽  
Larry Samuelson ◽  
Jeroen M. Swinkels
Keyword(s):  
Normal Form ◽  
Extensive Form ◽  
Extensive Form Games
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