scholarly journals The implications of finite‐order reasoning

2021 ◽  
Vol 16 (4) ◽  
pp. 1605-1654
Author(s):  
Adam Brandenburger ◽  
Alexander Danieli ◽  
Amanda Friedenberg
Keyword(s):  
Experimental Data ◽  
Finite Order ◽  
Prisoner’S Dilemma ◽  
Solution Concept ◽  
Response Sequence ◽  
Forward Induction ◽  
Extensive Form ◽  
Strong Belief ◽  
Response Sets ◽  
Best Response

The epistemic conditions of rationality and mth‐order strong belief of rationality (R mSBR; Battigalli and Siniscalchi, 2002) formalize the idea that players engage in contextualized forward‐induction reasoning. This paper characterizes the behavior consistent with R mSBR across all type structures. In particular, in a class of generic games, R( m − 1)SBR is characterized by a new solution concept we call an m‐best response sequence ( m‐BRS). Such sequences are an iterative version of extensive‐form best response sets (Battigalli and Friedenberg, 2012). The strategies that survive m rounds of extensive‐form rationalizability are consistent with an m‐BRS, but there are m‐BRS's that are disjoint from the former set. As such, there is behavior that is consistent with R( m − 1)SBR but inconsistent with m rounds of extensive‐form rationalizability. We use our characterization to draw implications for the interpretation of experimental data. Specifically, we show that the implications are nontrivial in the three‐repeated Prisoner's Dilemma and Centipede games.

Incomplete information and iterated strict dominance

2020 ◽  
Author(s):  
Christian W Bach ◽  
Andrés Perea
Keyword(s):  
Incomplete Information ◽  
Solution Concept ◽  
Complete Information ◽  
Common Belief ◽  
Person Perspective ◽  
Response Sets ◽  
Best Response ◽  
Necessary And Sufficient

Abstract The solution concept of iterated strict dominance for static games with complete information recursively deletes choices that are inferior. Here, we devise such an algorithm for the more general case of incomplete information. The ensuing solution concept of generalized iterated strict dominance is characterized in terms of common belief in rationality as well as in terms of best response sets. Besides, we provide doxastic conditions that are necessary and sufficient for modelling complete information from a one-person perspective.

PLAUSIBILITY ORDERINGS IN DYNAMIC GAMES

2014 ◽  
Vol 30 (3) ◽  
pp. 331-364 ◽  
Author(s):  
Andrés Perea
Keyword(s):  
Belief Revision ◽  
Dynamic Games ◽  
Backward Induction ◽  
Common Belief ◽  
Revision Theory ◽  
Forward Induction ◽  
Strong Belief ◽  
Game Theoretic ◽  
Plausibility Ordering

In this paper we explore game-theoretic reasoning in dynamic games within the framework of belief revision theory. More precisely, we focus on the forward induction concept of ‘common strong belief in rationality’ (Battigalli and Siniscalchi (2002) and the backward induction concept of ‘common belief in future rationality’ (Baltag et al. 2009; Perea 2014). For both concepts we investigate whether the entire collection of selected belief revision policies for a player can be characterized by a unique plausibility ordering. We find that this is indeed possible for ‘common strong belief in rationality’, whereas this may be impossible in some games for ‘common belief in future rationality’.

scholarly journals Solving Large Extensive-Form Games with Strategy Constraints

2019 ◽  
Vol 33 ◽  
pp. 1861-1868
Author(s):  
Trevor Davis ◽  
Kevin Waugh ◽  
Michael Bowling
Keyword(s):  
Private Information ◽  
Imperfect Information ◽  
Risk Mitigation ◽  
Solution Concept ◽  
Optimal Strategies ◽  
Linear Constraints ◽  
Convex Constraints ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Large Extensive Form Games

Extensive-form games are a common model for multiagent interactions with imperfect information. In two-player zerosum games, the typical solution concept is a Nash equilibrium over the unconstrained strategy set for each player. In many situations, however, we would like to constrain the set of possible strategies. For example, constraints are a natural way to model limited resources, risk mitigation, safety, consistency with past observations of behavior, or other secondary objectives for an agent. In small games, optimal strategies under linear constraints can be found by solving a linear program; however, state-of-the-art algorithms for solving large games cannot handle general constraints. In this work we introduce a generalized form of Counterfactual Regret Minimization that provably finds optimal strategies under any feasible set of convex constraints. We demonstrate the effectiveness of our algorithm for finding strategies that mitigate risk in security games, and for opponent modeling in poker games when given only partial observations of private information.

scholarly journals Win-Stay-Lose-Shift as a self-confirming equilibrium in the iterated Prisoner’s Dilemma

2021 ◽  
Vol 288 (1953) ◽  
pp. 20211021
Author(s):  
Minjae Kim ◽  
Jung-Kyoo Choi ◽  
Seung Ki Baek
Keyword(s):  
Observational Learning ◽  
Prisoner's Dilemma ◽  
Evolutionary Game ◽  
Prisoner’S Dilemma ◽  
Large Uncertainty ◽  
Iterated Prisoner's Dilemma ◽  
Best Response ◽  
The Cost ◽  
With Memory ◽  
Iterated Prisoner’S Dilemma

Evolutionary game theory assumes that players replicate a highly scored player’s strategy through genetic inheritance. However, when learning occurs culturally, it is often difficult to recognize someone’s strategy just by observing the behaviour. In this work, we consider players with memory-one stochastic strategies in the iterated Prisoner’s Dilemma, with an assumption that they cannot directly access each other’s strategy but only observe the actual moves for a certain number of rounds. Based on the observation, the observer has to infer the resident strategy in a Bayesian way and chooses his or her own strategy accordingly. By examining the best-response relations, we argue that players can escape from full defection into a cooperative equilibrium supported by Win-Stay-Lose-Shift in a self-confirming manner, provided that the cost of cooperation is low and the observational learning supplies sufficiently large uncertainty.

scholarly journals Smoothing Method for Approximate Extensive-Form Perfect Equilibrium

2017 ◽  
Author(s):  
Christian Kroer ◽  
Gabriele Farina ◽  
Tuomas Sandholm
Keyword(s):  
Nash Equilibrium ◽  
Imperfect Information ◽  
Large Scale ◽  
Nash Equilibria ◽  
Convex Polytope ◽  
Solution Concept ◽  
Extensive Form ◽  
Game Tree ◽  
Equilibrium Refinements ◽  
Perfect Equilibria

Nash equilibrium is a popular solution concept for solving imperfect-information games in practice. However, it has a major drawback: it does not preclude suboptimal play in branches of the game tree that are not reached in equilibrium. Equilibrium refinements can mend this issue, but have experienced little practical adoption. This is largely due to a lack of scalable algorithms.Sparse iterative methods, in particular first-order methods, are known to be among the most effective algorithms for computing Nash equilibria in large-scale two-player zero-sum extensive-form games. In this paper, we provide, to our knowledge, the first extension of these methods to equilibrium refinements. We develop a smoothing approach for behavioral perturbations of the convex polytope that encompasses the strategy spaces of players in an extensive-form game. This enables one to compute an approximate variant of extensive-form perfect equilibria. Experiments show that our smoothing approach leads to solutions with dramatically stronger strategies at information sets that are reached with low probability in approximate Nash equilibria, while retaining the overall convergence rate associated with fast algorithms for Nash equilibrium. This has benefits both in approximate equilibrium finding (such approximation is necessary in practice in large games) where some probabilities are low while possibly heading toward zero in the limit, and exact equilibrium computation where the low probabilities are actually zero.

scholarly journals A NOTE ON THE EQUIVALENCE OF RATIONALIZABILITY CONCEPTS IN GENERALIZED NICE GAMES

2006 ◽  
Vol 08 (04) ◽  
pp. 669-674
Author(s):  
ALEXANDER ZIMPER
Keyword(s):  
Real Line ◽  
Solution Concept ◽  
The Real ◽  
Individual Strategy ◽  
Dominated Strategies ◽  
Best Response ◽  
Strategic Solution ◽  
The Individual ◽  
Best Responses ◽  
Surprising Finding

Moulin (1984) describes the class of nice games for which the solution concept of point-rationalizability coincides with iterated elimination of strictly dominated strategies. As a consequence nice games have the desirable property that all rationalizability concepts determine the same strategic solution. However, nice games are characterized by rather strong assumptions. For example, only single-valued best responses are admitted and the individual strategy sets have to be convex and compact subsets of the real line ℝ. This note shows that equivalence of all rationalizability concepts can be extended to multi-valued best response correspondences. The surprising finding is that equivalence does not hold for individual strategy sets that are compact and convex subsets of ℝn with n ≥ 2.

Strong Forward Induction

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Bingyong Zheng
Keyword(s):  
Solution Concept ◽  
Forward Induction ◽  
Equilibrium Outcome ◽  
Dominance Condition ◽  
Local Dominance

AbstractForward induction, as defined by Govindan and Wilson (2009. “On Forward Induction.” Econometrica 77:1–28), places a local dominance condition on off-equilibrium beliefs that restricts relevant strategy profiles for an equilibrium outcome to be infinitely more likely than profiles that include irrelevant strategies. Meanwhile, it places no global dominance restrictions and thus leaves open the possibility that a dominated strategy is deemed more likely than strategies dominating it. This paper defines strong forward induction, which improves upon forward induction. We also develop a solution concept called strong forward induction equilibrium that is obtained from iterative application of the strong forward induction criterion.

scholarly journals The consequences of switching strategies in a two-player iterated survival game

2021 ◽  
Vol 82 (3) ◽  
Author(s):  
Olivier Salagnac ◽  
John Wakeley
Keyword(s):  
Survival Probability ◽  
Cooperative Behavior ◽  
Prisoner’S Dilemma ◽  
Optimal Number ◽  
Single Step ◽  
Critical Values ◽  
Survival Probabilities ◽  
Best Response ◽  
Cooperative Partner ◽  
Switching Strategies

AbstractWe consider two-player iterated survival games in which players are able to switch from a more cooperative behavior to a less cooperative one at some step of an n-step game. Payoffs are survival probabilities and lone individuals have to finish the game on their own. We explore the potential of these games to support cooperation, focusing on the case in which each single step is a Prisoner’s Dilemma. We find that incentives for or against cooperation depend on the number of defections at the end of the game, as opposed to the number of steps in the game. Broadly, cooperation is supported when the survival prospects of lone individuals are relatively bleak. Specifically, we find three critical values or cutoffs for the loner survival probability which, in concert with other survival parameters, determine the incentives for or against cooperation. One cutoff determines the existence of an optimal number of defections against a fully cooperative partner, one determines whether additional defections eventually become disfavored as the number of defections by the partner increases, and one determines whether additional cooperations eventually become favored as the number of defections by the partner increases. We obtain expressions for these switch-points and for optimal numbers of defections against partners with various strategies. These typically involve small numbers of defections even in very long games. We show that potentially long stretches of equilibria may exist, in which there is no incentive to defect more or cooperate more. We describe how individuals find equilibria in best-response walks among n-step strategies.

scholarly journals When to Follow the Tip: Security Games with Strategic Informants

2020 ◽  
Author(s):  
Weiran Shen ◽  
Weizhe Chen ◽  
Taoan Huang ◽  
Rohit Singh ◽  
Fei Fang
Keyword(s):  
Solution Concept ◽  
Extensive Form ◽  
Research Attention ◽  
The Past ◽  
Bayesian Equilibrium ◽  
Strategic Behaviors ◽  
Security Games ◽  
Extensive Form Game ◽  
Patrol Strategies ◽  
Security Game

Although security games have attracted intensive research attention over the past years, few existing works consider how information from local communities would affect the game. In this paper, we introduce a new player -- a strategic informant, who can observe and report upcoming attacks -- to the defender-attacker security game setting. Characterized by a private type, the informant has his utility structure that leads to his strategic behaviors. We model the game as a 3-player extensive-form game and propose a novel solution concept of Strong Stackelberg-perfect Bayesian equilibrium. To compute the optimal defender strategy, we first show that although the informant can have infinitely many types in general, the optimal defense plan can only include a finite (exponential) number of different patrol strategies. We then prove that there exists a defense plan with only a linear number of patrol strategies that achieve the optimal defender's utility, which significantly reduces the computational burden and allows us to solve the game in polynomial time using linear programming. Finally, we conduct extensive experiments to show the effect of the strategic informant and demonstrate the effectiveness of our algorithm.

scholarly journals On the Determinants of Cooperation in Infinitely Repeated Games: A Survey

2018 ◽  
Vol 56 (1) ◽  
pp. 60-114 ◽  
Author(s):  
Pedro Dal Bó ◽  
Guillaume R. Fréchette
Keyword(s):  
Experimental Data ◽  
Repeated Games ◽  
Multiple Equilibria ◽  
Prisoner’S Dilemma ◽  
Personal Characteristics ◽  
Repeated Game ◽  
Experimental Literature ◽  
Imperfect Monitoring ◽  
Strategic Uncertainty ◽  
Infinitely Repeated Games

A growing experimental literature studies the determinants of cooperation in infinitely repeated games, tests different predictions of the theory, and suggests an empirical solution to the problem of multiple equilibria. To provide a robust description of the literature's findings, we gather and analyze a metadata set of experiments on infinitely repeated prisoner's dilemma games. The experimental data show that cooperation is affected by infinite repetition and is more likely to arise when it can be supported in equilibrium. However, the fact that cooperation can be supported in equilibrium does not imply that most subjects will cooperate. High cooperation rates will emerge only when the parameters of the repeated game are such that cooperation is very robust to strategic uncertainty. We also review the results regarding the effect of imperfect monitoring, changing partners, and personal characteristics on cooperation and the strategies used to support it. (JEL C71, C73, D81, D83)

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