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The concept of perfect equilibrium, formulated by Selten (Int J Game Theory 4:25–55, 1975), serves as an effective characterization of rationality in strategy perturbation. In our study, we propose a modified version of perfect equilibrium that incorporates perturbation control parameters. To match the beliefs with the equilibrium choice probabilities, the logistic quantal response equilibrium (logistic QRE) was established by McKelvey and Palfrey (Games Econ Behav 10:6–38, 1995), which is only able to select a Nash equilibrium. By introducing a linear combination between a mixed strategy profile and a given vector with positive elements, this paper develops a variant of the logistic QRE for the selection of the special version of perfect equilibrium. Expanding upon this variant, we construct an equilibrium system that incorporates an exponential function of an extra variable. Through rigorous error-bound analysis, we demonstrate that the solution set of this equilibrium system leads to a perfect equilibrium as the extra variable approaches zero. Consequently, we establish the existence of a smooth path to a perfect equilibrium and employ an exponential transformation of variables to ensure numerical stability. To make a numerical comparison, we capitalize on a variant of the square-root QRE, which yields another smooth path to a perfect equilibrium. Numerical results further verify the effectiveness and efficiency of the proposed differentiable path-following methods. [ABSTRACT FROM AUTHOR]

Reflecting the ongoing public concern about strategic behavior among colleges, we develop a game-theoretic higher education model with peer effects to examine the equilibrium market response to changes in state and federal student aid. In addition to establishing perfect equilibrium existence, we find that state and federal student aid programs are vulnerable to equilibrium effects that conflict with the intended impact of policy instruments. For instance, even when college objectives are devoid of selfish intent, strict increases in state and federal student aid can induce decreased student access to higher education or even increased financial strain from college enrollment. [ABSTRACT FROM AUTHOR]

In this paper, we apply the preplay negotiation procedure proposed by Kalai [1981] [Preplay negotiations and the prisoner's dilemma, Math. Soc. Sci. 1, 375–379] to the n -person prisoners' dilemma with n ≥ 3 and examine whether it promotes cooperation. First, we demonstrate that every perfect equilibrium is a proper equilibrium for any extensive-form game in which players have only two alternatives in each information set. We show that if the preplay is carried out once, the preplay game has a proper equilibrium that realizes cooperation. We also show that if the preplay is executed twice, the game has a proper equilibrium that realizes cooperation regardless of the entering action profile (the starting point of the preplays). Finally, we demonstrate that the preplay game consisting of at least two preplays has a proper equilibrium that realizes cooperation. [ABSTRACT FROM AUTHOR]

The linear tracing procedure plays a central role in the equilibrium selection theory of Harsanyi and Selten (1988). Nevertheless, it fails to always select a perfect equilibrium when there are more than two players. To fill this gap, we develop a variant of the linear tracing procedure by constituting a perturbed game in which each player maximizes her payoff against a linear convex combination between a totally mixed prior belief profile and a given mixed strategy profile of other players. Applying the optimality conditions to the integration of the perturbed game and a convex-quadratic-penalty game, we establish with a fixed-point argument and transformations on variables the existence of a smooth path from a unique starting point to a perfect equilibrium. Moreover, we present a variant of Harsanyi's logarithmic tracing procedure and a simplicial linear tracing procedure to select a perfect equilibrium. [ABSTRACT FROM AUTHOR]

Greenberg et al. (Econ Theory 40:91–112, 2009. https://doi.org/10.1007/s00199-008-0349-5) offered a solution concept of a "mutually acceptable course of action" (MACA) for situations in which rational individuals with different beliefs and views of the world agree to a shared course of action. In this paper we investigate epistemic conditions for MACA, in terms of information, beliefs, and rationality. Within a semantic framework, we formulate and show, using the notion of "lexicographic probability system" (LPS) introduced by Blume et al. (Econometrica 59:61–79, 1991a), that MACA is the logical consequence of "perfect" rationality, common belief in "perfect" rationality, and correct mutual belief in agreement on the shared course of action. We also demonstrate how epistemic characterizations for "perfect" versions of solution concepts can be derived from our epistemic characterization for MACA, by varying the degree of completeness of the shared course of action. [ABSTRACT FROM AUTHOR]

To extend the concept of subgame perfect equilibrium to an extensive-form game with imperfect information but perfect recall, Selten (Int J Game Theory 4:25–55, 1975) formulated the notion of perfect equilibrium and attained its existence through the agent normal-form representation of the extensive-form game. As a strict refinement of Nash equilibrium, a perfect equilibrium always yields a sequential equilibrium. The selection of a perfect equilibrium thus plays an essential role in the applications of game theory. Moreover, a different procedure may select a different perfect equilibrium. The existence of Nash equilibrium was proved by Nash (Ann Math 54:289–295, 1951) through the construction of an elegant continuous mapping and an application of Brouwer's fixed point theorem. This paper intends to enhance Nash's mapping to select a perfect equilibrium. By incorporating the complementarity condition into the equilibrium system with Nash's mapping through an artificial game, we successfully eliminate the nonnegativity constraints on a mixed strategy profile imposed by Nash's mapping. In the artificial game, each player solves against a given mixed strategy profile a strictly convex quadratic optimization problem. This enhancement enables us to derive differentiable homotopy systems from Nash's mapping and establish the existence of smooth paths for selecting a perfect equilibrium. The homotopy methods start from an arbitrary totally mixed strategy profile and numerically trace the smooth paths to a perfect equilibrium. Numerical results show that the methods are numerically stable and computationally efficient in search of a perfect equilibrium and outperform the existing differentiable homotopy method. [ABSTRACT FROM AUTHOR]

Game theory has become a standard tool for depicting and demonstrating various game‐like phenomena by providing appropriate mathematical models and for analyzing and predicting agents' behaviors and their decisions by formalizing solution concepts. The conventional game model mainly concerns ideal systems that would always guarantee optimal responses, which appears unrealistic for practical game scenarios since decision‐making usually entails resource costs. Therefore, this study considers players' decision‐making in extensive games when the computational cost of searching the strategy space is limited. We start with a new mathematical model of extensive games that features a bound on computational resources during players' decision‐making process such that they can only foresee a part of the available alternatives in the future. This model is more appropriate in predicting players' strategies than the conventional model, under which we investigate the effects of computational costs on players' strategies as well as the computational complexity. Furthermore, a simulation experiment is performed to seek the connection between the amount of resources and the goodness of the outcomes. This study is expected to provide a foundation for players' rational decision‐making with computational costs. [ABSTRACT FROM AUTHOR]

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large. [ABSTRACT FROM AUTHOR]

We study the complexity of computing or approximating refinements of Nash equilibrium for finite n -player extensive form games of perfect recall (EFGPR), n ≥ 3. Our results apply to a number of well-studied refinements, including sequential equilibrium, extensive-form perfect equilibrium, and quasi-perfect equilibrium. Informally, we show that, for all these refinements, approximating such a refined equilibrium for an n -player EFGPR is not any harder than (i.e., can be efficiently reduced to) approximating a Nash equilibrium for a 3-player normal form game. More specifically, we show that approximating such a refined equilibrium for a given EFGPR, within given desired precision, is FIXP a -complete. We also study corresponding notions of "almost" equilibrium for these refinements, and we show that computing one is PPAD -complete. (In all these cases our main results show containment in FIXP a and containment in PPAD. Hardness follows from earlier results for simpler games.) [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

As a strict refinement of Nash equilibrium, the concept of perfect equilibrium was formulated and extensively studied in the literature. To determine perfect equilibrium, this paper extends the logistic version of quantal response equilibrium (logit QRE) to a perturbed game. As a result of this extension, a smooth path is constructed for determining perfect equilibrium. The path starts from an arbitrary totally mixed strategy profile and leads to a perfect equilibrium. Numerical examples show that the extended QRE is comparable with the logit QRE and further confirm the effectiveness of the path. [ABSTRACT FROM AUTHOR]

This paper studies pure strategy perfect and proper equilibria for games with non-atomic measure spaces of players and infinitely many actions. A richness condition (nowhere equivalence) on the measure space of players is shown to be both necessary and sufficient for the existence of such equilibria. The limit admissibility of perfect and proper equilibria is also proved. [ABSTRACT FROM AUTHOR]

As a strict refinement of Nash equilibrium, the concept of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975). A well-known application of this concept is that every perfect equilibrium of the agent normal form game of an extensive form game with perfect recall yields a trembling-hand perfect equilibrium (consequently a sequential equilibrium). To compute a perfect equilibrium, this paper extends Kohlberg and Mertens's equivalent reformulation of Nash equilibrium to a perturbed game. This extension naturally leads to a homotopy mapping on the Euclidean space. With this homotopy mapping and a triangulation, we develop a simplicial homotopy method for approximating perfect equilibria. It is proved that every limit point of the simplicial path yields a perfect equilibrium. Numerical results further confirm the effectiveness of the method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that supermodular games satisfying sequential better-reply security possess a pure strategy perfect equilibrium and a strategically stable set of pure strategy equilibria. We illustrate that in continuous supermodular games, perfect equilibria may contain weakly dominated actions. Moreover, in discontinuous supermodular games satisfying sequential better-reply security, perfect equilibria may involve play of actions in the interior of the set of weakly dominated actions. We show that supermodular games satisfying sequential better-reply security possess pure strategy perfect equilibria outside the interior of the set of weakly dominated action profiles. [ABSTRACT FROM AUTHOR]

In almost common value auctions, even a small private payoff advantage is usually supposed to have an explosive effect on the outcomes in a second-price sealed-bid common value auction. According to (Bikhchandani in J. Econ. Theory 46:97-119, ) and (Klemperer in Eur. Econ. Rev. 42:757-769, ) the large set of equilibria obtained for common value auction games drastically shrinks, so that the advantaged player always wins the auction, at a price that sharply decreases the seller's payoff. Yet this result has not been observed experimentally. In this paper, we show that Bikhchandani's equilibria are not the only equilibria of the game. By introducing discontinuities in the bids, we establish a new family of perfect equilibria with interesting properties, among them: (i) the advantaged bidder does no longer win the auction regardless of her private information, (ii) she may pay a much higher price than in Bikhchandani's equilibria, (iii) there is no ex-post regret, (iv) the intersection with level-k reasoning is not empty. We also show that a private advantage limits the number of possible discontinuities: one can introduce any number of discontinuities in the common value auction, but this is not possible in presence of a private advantage. [ABSTRACT FROM AUTHOR]

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

Abstract: Simon and Stinchcombe distinguish two approaches to perfect equilibrium, the “trembling hand” approach, and the “finitistic” approach, for games with compact action spaces and continuous payoffs. We investigate relations between the different types of perfect equilibrium introduced by Simon and Stinchcombe. We also propose an improved version of the finitistic approach, and prove existence. Despite the fact that the finitistic approach appeals to basic intuition, our results—specifically Examples 3 and 4—seem to imply a severe critique of this approach. Further examples illustrate the relations between the two approaches and the relation to admissibility of strategies. [Copyright &y& Elsevier]

We provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. It is done by constructing a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by a generalization of Kakutani theorem, which is proved in this paper. This work offers a new approach to perfect equilibria, which would hopefully facilitate further study on this topic. We also hope our direct proof would be the first step toward building an algorithm to find the set of all perfect equilibria of a strategic game. [ABSTRACT FROM AUTHOR]

A game-theoretic model of territorial environmental production under Cournot competition is studied. The process is modeled as cooperative differential game. The stable distribution mechanism of the common cooperative benefit among players is proposed. We proved that the cooperative total stock of accumulated pollution is strictly less then the pollution under Nash equilibrium for the whole duration of the game. We design a stable Shapley value as a cooperative solution, which is time-consistent. The Shapley value is also strategic stable and satisfies the irrational-behavior-proofness condition. The numerical example is given. [ABSTRACT FROM AUTHOR]

New characterizations of sequential equilibrium, perfect equilibrium, and proper equilibrium are provided that use nonstandard probability. It is shown that there exists a belief system μ such that $${(\vec{\sigma},\mu)}$$ is a sequential equilibrium in an extensive game with perfect recall iff there exist an infinitesimal $${\epsilon}$$ and a completely mixed behavioral strategy profile σ′ (so that $${\sigma_i'}$$ assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from $${\vec{\sigma}}$$ such that at each information set I for player i, σ i is an $${\epsilon}$$-best response to $${\vec{\sigma}'_{-i}}$$ conditional on having reached I. Note that the characterization of sequential equilibrium does not involve belief systems. There is a similar characterization of perfect equilibrium; the only difference is that σ i must be a best response to $${\vec{\sigma}'_{-i}}$$ conditional on having reached I. Yet another variant is used to characterize proper equilibrium. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of different players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game. A suitable change in the terminology used in this paper provides (in the case of perfect altruism between generations) a new Nash equilibrium theorem for standard stochastic games with uncountable state space. [Copyright &y& Elsevier]

In this paper we discuss three examples of approval voting games. The first one illustrates that a stronger solution concept than perfection is needed for a strategic analysis of this type of games. The second example shows that sophisticated voting can imply that the Condorcet winner gets no vote. The third example shows the possibility of insincere voting being a stable equilibrium. [ABSTRACT FROM AUTHOR]

The well-known model of altruistic growth/strategic bequest is studied. A stochastic transition function is considered and fairly general sufficient conditions for the existence of Markov-stationary subgame-perfect equilibrium are given. Also some special cases in which the equilibrium policy is continuous and nondecreasing are discussed. [ABSTRACT FROM AUTHOR]

There is an implicit assumption in electronic commerce that induces the buyers to believe that their deals will be handled appropriately. However, after a seller has already committed to a buyer, he may be tempted by several requests though he will not be able to supply them all. We analyze markets in which a finite set of automated buyers interacts repeatedly with a finite set of automated sellers. These sellers can satisfy one buyer at a time, and they can be tempted to break a commitment they already have. We have found the perfect equilibria that exist in markets with a finite horizon, and with an unrestricted horizon. A significant result stemming from our study reveals that sellers are almost always tempted to breach their commitments. However, we also show that if markets' designers implement an external mechanism that restricts the automated buyers actions, then sellers will keep their commitments. [ABSTRACT FROM AUTHOR]

Copyright of OR Spectrum is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of OR Spectrum is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

If game theory is to be used as a negotiation support tool, it should be able to provide unambiguous recommendations for a target to aim at and for actions to reach this target. This need cannot be satisfied with the Nash equilibrium concept, based on the standard instrumental concept of rationality. These equilibria, as is well known. are generally multiple in a game. The concept of substantive or instrumental rationality has proved to be so pregnant, however, that researchers, instead of re-evaluating its use in game theory, have simply tried to design concepts related to the Nash equilibrium, but with the property of being unique in a game -- i.e., they have devised ways of selecting among Nash equilibria. These concepts have been labeled refined Nash equilibria. The purpose of this paper is to show the following. (a) The different types of refined Nash equilibria, based on the principle of backward induction, can lead to severe contradictions within the framework itself. This makes these concepts utterly unsatisfactory and calls for a new appraisal of the reasoning process of the players. (b) The degree of confidence in the principle of backward induction depends upon the evaluation of potential deviations with respect to the extended Nash equilibrium concept used and upon the possible interpretations of such deviations by the different players. Our goal is to show that the nature of these possible interpretations reinforces the argument that a serious conceptual reappraisal is necessary. (c) Some form of forward induction should then become the real yardstick of rationality, extending Simonian procedural rationality towards the concept of cognitive rationality. This could open the way to a renewed game theoretic approach to negotiation support systems. Such a research program, which would be a revision of the basic game theoretic concepts, is dealt with in the end of the paper. [ABSTRACT FROM AUTHOR]

An experiment designed to test whether experimenter observation influences play of the ultimatum game led to a comparison of proportions of perfect equilibria obtained under different conditions. Our analysis features a Bayesian approach to this comparison. Our method extends recent work in the nonparametric Bayesian analysis of bioassay problems when the prior constrains the form of the potency curve. Sampling-based approaches to calculating posterior features of interest are used and discussed. We conclude that experimenter observation has a small influence but not enough to account for most of the deviation from perfect-equilibrium play. [ABSTRACT FROM AUTHOR]