Existence Of Equilibria In Games Research Articles

Equilibrium existence in games with ties

We provide conditions that simplify applying Reny's (1999) better‐reply security to Bayesian games and use these conditions to prove the existence of equilibria for classes of games in which payoff discontinuities arise only at “ties.” These games include a general version of all‐pay contests, first‐prize auctions with common values, and Hotelling models with incomplete information.

Equilibrium existence in games with a concave Bayesian potential

We establish existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games that have a continuous and concave potential at all states of nature, without assuming absolute continuity of information. As an application, we show that Bayesian Nash equilibrium exists in many well-known games that have semi-quadratic payoffs (including Bertrand and Cournot oligopolies with linear demand), for general information structures.

The Gale–Nikaido–Debreu lemma with discontinuous excess demand

We provide a generalization of the Gale–Nikaido–Debreu’s lemma for discontinuous excess demand in the light of recent work on the existence of equilibria in games with discontinuous payoffs. The standard upper hemicontinuity property of the excess demand is replaced by the weaker concept of “continuous inclusion property” introduced by He and Yannelis (J Math Anal Appl 450(2):1421–1433, 2017) and we allow for the cone P of admissible prices to be general enough to cover cases for which commodities cannot be freely disposed of.

Quasi-Transfer Continuity and Nash Equilibrium

We introduce a new notion of continuity, called quasi-transfer continuity, and show that it is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. This holds true in a large class of discontinuous games. We show that our result strictly generalizes the pure strategy existence theorem of Carmona [Carmona, G. [2009] An existence result for discontinuous games, J. Econ. Theory 144, 1333–1340]. We also show that our result is neither implied by nor does it imply the existence theorems of Reny [Reny, J. P. [1999] On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056] and Baye et al. [Baye, M. R., Tian, G. and Zhou, J. [1993] Characterizations of the existence of equilibria in games with discontinuous and nonquasiconcave payoffs, Rev. Econ. Studies 60, 935–948].

Equilibrium existence in games: Slight single deviation property and Ky Fan minimax inequality

In this paper, we exhibit a relation between the Nash equilibrium existence problem in ordinal games and the Ky Fan minimax inequality. In this way, we obtain new sufficient conditions for the existence of equilibria. The main tools used in the paper are a necessary condition introduced in Nassah and Tian (2016) for normal form games and a generalization of the single deviation property. Examples compare our result with the previous ones.

Ambiguous games without a state space and full rationality

This work aims to differentiate and to better understand the assumptions that must be imposed on the structure of ambiguity and on the attitudes towards ambiguity in order to have existence of equilibria in games under ambiguous belief correspondences. In the present paper, this class of games is studied under substantially weaker assumptions on agents' preferences, as they are not required to be rational and therefore do not have any functional representation.A new approach is required to deal with preferences that are not rational, in this particular framework; in fact, the present work shows that the attitudes of agents towards the imprecision of probabilistic beliefs play a key role in the issue of equilibrium existence, whenever they are combined with some property of convexity/concavity of the ambiguous belief correspondences.The paper also studies the role played by these assumptions in different specific models (such as incomplete information games with multiple priors or games under strategic ambiguity), so as to illustrate the applicability of the results of equilibrium existence and connections with previous literature.

On Strategic Complementarities in Discontinuous Games

This paper studies the existence of a pure strategy Nash equilibrium in games with strategic complementarities where the strategy sets are totally ordered. By relaxing the conventional conditions related to upper semicontinuity and single crossing, we enlarge the class of games to which monotone techniques are applicable. The results are illustrated with a number of economics-related examples.

On the existence of equilibria in games with arbitrary strategy spaces and preferences

This paper provides necessary and sufficient conditions for the existence of pure strategy Nash equilibria by replacing the assumptions concerning continuity and quasiconcavity with a unique condition, passing strategy space from topological vector spaces to arbitrary topological spaces. Preferences may also be nontotal/nontransitive, discontinuous, nonconvex, or nonmonotonic. We define a single condition, recursive diagonal transfer continuity (RDTC) for aggregator payoff function and recursive weak transfer quasi-continuity (RWTQC) for individuals’ preferences, respectively, which establishes the existence of pure strategy Nash equilibria in games with arbitrary (topological) strategy spaces and preferences without imposing any kind of quasiconcavity-related conditions.

Existence of equilibria in constrained discontinuous games

We establish the existence of pure strategy equilibria in games with discontinuous payoffs where the set of feasible actions of each player varies, also in a discontinuous fashion, as a function of the actions of the other players. Such games are used in modeling abstract economies and other games where players share common constraints. Our approach circumvents the difficulties that arise from the presence of discontinuities by modifying the original problem and allowing the players to use strategies that possibly lie outside their feasible sets. We then show that each modified game has \(\varepsilon \)-equilibria points. Under certain conditions, and as the extent of modification becomes smaller and smaller and \(\varepsilon \) approaches zero, the \(\varepsilon \)-equilibria points of the modified games will converge to a strategy profile that is an equilibrium of the original game. Hence, we obtain a set of sufficient conditions for the existence of pure equilibria of the original game. We apply our results to a number of classic games that have discontinuous payoffs and discontinuous constraint correspondences.

Uncertainty aversion and equilibrium existence in games with incomplete information

We consider games with incomplete information à la Harsanyi, where the payoff of a player depends on an unknown state of nature as well as on the profile of chosen actions. As opposed to the standard model, playersʼ preferences over state-contingent utility vectors are represented by arbitrary functionals. The definitions of Nash and Bayes equilibria naturally extend to this generalized setting. We characterize equilibrium existence in terms of the preferences of the participating players. It turns out that, given continuity and monotonicity of the preferences, equilibrium exists in every game if and only if all players are averse to uncertainty (i.e., all the functionals are quasi-concave).

Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players’ payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of \({\epsilon}\)-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.

An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud. 53 (1986) 1–26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935–948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029–1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.

Uncertainty Aversion and Equilibrium Existence in Games with Incomplete Information

We consider games with incomplete information a la Harsanyi, where the payoff of a player depends on an unknown state of nature as well as on the profile of chosen actions. As opposed to the standard model, players' preferences over state--contingent utility vectors are represented by arbitrary functionals. The definitions of Nash and Bayes equilibria naturally extend to this generalized setting. We characterize equilibrium existence in terms of the preferences of the participating players. It turns out that, given continuity and monotonicity of the preferences, equilibrium exists in every game if and only if all players are averse to uncertainty (i.e., all the functionals are quasi--concave). We further show that if the functionals are either homogeneous or translation invariant then equilibrium existence is equivalent to concavity of the functionals.

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.

Nash equilibria for non-binary choice rules

We prove the existence of equilibria in games with players who employ abstract (non-binary) choice rules. This framework goes beyond the standard, transitive model and encompasses games where players have non-transitive preferences (e.g., skew-symmetric bilinear preferences).

The existence of equilibrium in games with randomly perturbed payoffs and applications to experimental economics

I present an equilibrium existence result for games with perturbed payoffs by extending Milgrom and Weber’s [Mathematics and Operations Research 10 (1985) 619] proof of existence of equilibrium in incomplete information games to integrable utility functions and show that their assumption of ‘Equicontinuous Payoffs’ is redundant. I apply this result to games with randomly, for example, normally, logistically, or double exponentially perturbed payoffs studied in experimental economics.

Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs

This paper characterizes pure-strategy and dominant-strategy Nash equilibrium in non-cooperative games which may have discontinuous and/or non-quasiconcave payoffs. Conditions called diagonal transfer quasiconcavity and uniform transfer quasiconcavity are shown to be necessary and, with conditions called diagonal transfer continuity and transfer upper semicontinuity, sufficient for the existence of pure-strategy and dominant-strategy Nash equilibrium, respectively. The results are used to examine the existence or non-existence of equilibrium in some well-known economic games with discontinuous and/or non-quasiconcave payoffs. For example, we show that the failure of the existence of a pure-strategy Nash equilibrium in the Hotelling model is due to the failure of an aggregator function to be diagonal transfer quasiconcave—not the failure of payoffs to be quasiconcave, as has been elsewhere conjectured.