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The literature on strategic ambiguity in classical games provides generalized notions of equilibrium in which each player best responds to ambiguous or imprecise beliefs about his opponents' strategic choices. In a recent paper, strategic ambiguity has been extended to psychological games, by taking into account ambiguous hierarchies of beliefs and max–min preferences. Given that this kind of preference seems too restrictive as a general method to evaluate decisions, in this paper we extend the analysis by taking into account α -max–min preferences in which decisions are evaluated by a convex combination of the worst-case (with weight α) and the best-case (with weight 1 − α ) scenarios. We define the α -max–min psychological Nash equilibrium; an illustrative example shows that the set of equilibria is affected by the parameter α and the larger the ambiguity, the greater the effect. We also provide a result of stability of the equilibria with respect to perturbations that involve the attitudes toward ambiguity, the structure of ambiguity, and the payoff functions: converging sequences of equilibria of perturbed games converge to equilibria of the unperturbed game as the perturbation vanishes. Surprisingly, a final example shows that the existence of equilibria is not guaranteed for every value of α. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions on the primitives of a class of discontinuous Bayesian games such that all games in the class share equilibria. If a Bayesian game in the class also satisfies a weak efficiency condition, then we show its normal form is better-reply secure. The invariance property then provides an existence result for all Bayesian games in the class. Results are shown for both pure strategy and behavioral strategy equilibrium. We illustrate the application of the results with an example of a class of contests with bid caps. [ABSTRACT FROM AUTHOR]

This paper presents various examples of two‐player submodular or supermodular contest games. Emphasizing the three main elements of a contest model, our examples revolve around situations where (i) contest success function allows for a draw, (ii) winning prize is not exogenously given but rather jointly produced, or (iii) individual effort cost also depends on the rival's effort. We then illustrate how submodularity and supermodularity can be used to study the existence of equilibrium, the order structure of the equilibrium set, and monotone comparative statics in such contest game examples. [ABSTRACT FROM AUTHOR]

The literature on strategic ambiguity in classical games provides generalized notions of equilibrium in which each player best responds to ambiguous or imprecise beliefs about his opponents’ strategic choices. In a recent paper, strategic ambiguity has been extended to psychological games, by taking into account ambiguous hierarchies of beliefs and max–min preferences. Given that this kind of preference seems too restrictive as a general method to evaluate decisions, in this paper we extend the analysis by taking into account α-max–min preferences in which decisions are evaluated by a convex combination of the worst-case (with weight α) and the best-case (with weight 1−α) scenarios. We define the α-max–min psychological Nash equilibrium; an illustrative example shows that the set of equilibria is affected by the parameter α and the larger the ambiguity, the greater the effect. We also provide a result of stability of the equilibria with respect to perturbations that involve the attitudes toward ambiguity, the structure of ambiguity, and the payoff functions: converging sequences of equilibria of perturbed games converge to equilibria of the unperturbed game as the perturbation vanishes. Surprisingly, a final example shows that the existence of equilibria is not guaranteed for every value of α.

We provide sufficient conditions on the primitives of a class of discontinuous Bayesian games such that all games in the class share equilibria. If a Bayesian game in the class also satisfies a weak efficiency condition, then we show its normal form is better-reply secure. The invariance property then provides an existence result for all Bayesian games in the class. Results are shown for both pure strategy and behavioral strategy equilibrium. We illustrate the application of the results with an example of a class of contests with bid caps.

During a pandemic people have to find a trade-off between meeting others and staying safely at home. While meeting others is pleasant, it also increases the risk of infection. We consider this dilemma by introducing a game-theoretic network creation model in which selfish agents can form bilateral connections. They benefit from network neighbors, but at the same time, they want to maximize their distance to all other agents. This models the inherent conflict that social distancing rules impose on the behavior of selfish agents in a social network. Besides addressing this familiar issue, our model can be seen as the inverse to the well-studied Network Creation Game by Fabrikant et al. (in: PODC 2003, pp 347–351, 2003. https://doi.org/10.1145/872035.872088), where agents aim at being as central as possible in the created network. We look at two variants of network creation governed by social distancing. Firstly, a variant without connection restrictions, where we characterize optimal and equilibrium networks, and derive asymptotically tight bounds on the Price of Anarchy and Price of Stability. The second variant allows connection restrictions. As our main result, we prove that Swap-Maximal Routing-Cost Spanning Trees, an efficiently computable weaker variant of Maximum Routing-Cost Spanning Trees, actually resemble equilibria for a significant range of the parameter space. Moreover, we give almost tight bounds on the Price of Anarchy and Price of Stability. These results imply that under social distancing the agents' selfishness has a strong impact on the quality of the equilibria. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Psychological games aim to represent situations in which players may have belief-dependent motivations. In this setting, utility functions are directly dependent on the entire hierarchy of beliefs of each player. On the other hand, the literature on strategic ambiguity in classical games highlights that players may have ambiguous (or imprecise) beliefs about opponents' strategy choices. In this paper, we look at the issue of ambiguity in the framework of simultaneous psychological games by taking into account ambiguous hierarchies of beliefs and study a natural generalization of the psychological Nash equilibrium concept to this framework. We give an existence result for this new concept of equilibrium and provide examples that show that even an infinitesimal amount of ambiguity may alter significantly the equilibria of the game or can work as an equilibrium selection device. Finally, we look at the problem of stability of psychological equilibria with respect to ambiguous trembles on the entire hierarchy of correct beliefs and we provide a limit result that gives conditions so that sequences of psychological equilibria under ambiguous perturbation converge to psychological equilibria of the unperturbed game. • The present paper aims to jointly take into account two issues that arise from different strands of literature: Players' preferences might depend on the entire hierarchy of beliefs; Beliefs might be ambiguous (or imprecise) in equilibrium. • We study a natural generalization of the psychological Nash equilibrium concept in simultaneous psychological games allowing for ambiguous hierarchies of beliefs. • We give an existence result for this new concept of equilibrium and provide examples that show many different effects of ambiguity. • In the examples, we show that an infinitesimal amount of ambiguity can work as an equilibrium selection device. • We provide a limit result that gives conditions so that sequences of psychological equilibria under ambiguous perturbation converge to psychological equilibria of the unperturbed psychological game. [ABSTRACT FROM AUTHOR]

In this study, I analyze games in which the functions mapping a vector of efforts into each player's share of the prize and its value exhibit an arbitrary degree of homogeneity. I present a simple way to compute the equilibrium strategy and sufficient conditions for a unique interior symmetric pure‐strategy Nash equilibrium. The setup nests Malueg and Yates (2006), who exploit homogeneity for rent‐seeking contests with exogenous prize valuation, and shows that homogeneity can be used to solve (i) a wider range of rent‐seeking contests and (ii) other classes of games, like Cournot games with nonlinear inverse demand and possibly non homogeneous goods. [ABSTRACT FROM AUTHOR]

The solution concept of a Bayesian equilibrium of a Bayesian game is inherently an interim concept. The corresponding ex ante solution concept has been termed a Harsányi equilibrium; examples have appeared in the literature showing that there are Bayesian games with uncountable state spaces that have no Bayesian approximate equilibria but do admit a Harsányi approximate equilibrium, thus exhibiting divergent behaviour in the ex ante and interim stages. Smoothness, a concept from descriptive set theory, has been shown in previous works to guarantee the existence of Bayesian equilibria. We show here that higher rungs in the countable Borel equivalence relation hierarchy can also shed light on equilibrium existence. In particular, hyperfiniteness, the next step above smoothness, is a sufficient condition for the existence of Harsányi approximate equilibria in purely atomic Bayesian games. [ABSTRACT FROM AUTHOR]

Traffic incidents are recognized as a key contributor to non-recurrent congestion, which causes many negative effects in economy, environment, health and lifestyle. In this article, we investigate an incident management policy considering both signal control and route choice, which presents a real-time systematic effort to provide a rapid recovery from an incident and mitigate incident-related congestion according to different incident effects. Firstly, we introduce a route choice method on a multiple-route urban road network with consideration of bottleneck delays. Then, we analyze the route travel costs under incident effects and give the equilibrium existence condition after the occurrence of an incident. Furthermore, combining with the route choice method, a novel traffic signal control policy is proposed and the condition for equilibrium existence is given with the consideration of dynamic signal control and route choice simultaneously. Sufficient conditions for the dynamic road system to be stable are also derived and validated by using Lyapunov stability theorem. The analytical results indicate that opposite signal control policies should be applied in road networks under different incident circumstances and the proposed control policy can achieve the improved recovery rate and system stability than existing control policies in terms of dynamic incident effects in road networks. Finally, numerical results have been conducted to demonstrate the effectiveness of our proposed incident control policy and confirm the conditions for road system stability when different incident circumstances had been identified. [ABSTRACT FROM AUTHOR]

We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from (Vasquez (2017)), but is more direct as it does not make use of countably additive representations of finitely additive measures by (Yosida and Hewitt (1952)). [ABSTRACT FROM AUTHOR]

This paper shows that continuous-time stochastic games of fixed duration need not possess equilibria in Markov strategies. The example requires payoffs and transitions to depend on time in a continuous but irregular (almost nowhere almost differentiable) way. This example offers a correction to the erroneous construction presented previously in Levy (Dyn Games Appl 3(2):279–312, 2013. https://doi.org/10.1007/s13235-012-0067-2). [ABSTRACT FROM AUTHOR]

A finite number of vertically differentiated firms simultaneously compete for and screen agents with private information about their payoffs. In equilibrium, higher firms serve higher types. Each firm distorts the allocation downward from the efficient level on types below a threshold, but upward above. While payoffs in this game are neither quasi‐concave nor continuous, if firms are sufficiently differentiated, then any strategy profile that satisfies a simple set of necessary conditions is a pure‐stategy equilibrium, and an equilibrium exists. A mixed‐strategy equilibrium exists even when firms are less differentiated. The welfare effects of private information are drastically different than under monopoly. The equilibrium approaches the competitive limit quickly as entry costs grow small. We solve the problem of a multi‐plant firm facing a type‐dependent outside option and use this to study the effect of mergers. [ABSTRACT FROM AUTHOR]

We prove the existence of a behavioral-strategy Bayesian Nash equilibrium in contests where each contestant's probability to win is continuous in efforts outside the zero-effort profile, monotone in his own effort, and greater that 1/2 if that contestant is the only one exerting positive effort. General type spaces, and in particular a continuum of information types, are allowed. As a corollary, the existence of a pure-strategy Bayesian Nash equilibrium is established in generalized Tullock contests, where the probability to win is strictly concave in one's own effort for any non-zero effort profile of other players. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of coalitional equilibria for coalitional abstract economies with nonordered preferences. Accordingly, the authors give a fuzzy extension of the coalitional equilibrium existence theorem. The main result is illustrated with applications to coalitional abstract economies with payoff functions, strong Nash equilibria and absolute optimal solutions. [ABSTRACT FROM AUTHOR]

We introduce a novel method of proving existence of rational expectations equilibria (REE) in multi-dimensional CARA-Gaussian environments. Our approach is to construct a mapping from agents' initial beliefs (which are characterized by a positive semidefinite matrix), to their updated beliefs, after reaching and observing equilibrium; we then show Brouwer's fixed point theorem applies. We apply our approach to a finite-market version of Admati (1985), which is a multi-asset noisy REE asset pricing model with dispersed information. We present an algorithm to numerically solve for equilibrium of the finite model, as well as several examples illustrating the difference in equilibrium behavior between the finite and infinite models. Our method can be applied to any multi-dimensional REE model with Gaussian uncertainty and behavior that is linear in agents' information. [ABSTRACT FROM AUTHOR]

An Equivalence Theorem between geometric structures and utility functions allows new methods for understanding preferences. Our classification of valuations into "Demand Types" incorporates existing definitions (substitutes, complements, "strong substitutes," etc.) and permits new ones. Our Unimodularity Theorem generalizes previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a "demand type." Contrary to popular belief, equilibrium is guaranteed for more classes of purely‐complements than of purely‐substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include matching and coalition‐formation, and the "Product‐Mix Auction" introduced by the Bank of England in response to the financial crisis. [ABSTRACT FROM AUTHOR]

In the classic Hotelling-Downs model of political competition, no pure strategy equilibrium with three or more strategic candidates exists when the distribution of voters’ preferred policies is unimodal. I study the effect of introducing two idealist candidates to the model who are non-strategic (i.e., fixed to their policy platforms), while allowing for an unlimited number of strategic candidates. Doing so, I show that equilibrium is restored for a non-degenerate set of unimodal distributions. In addition, the equilibria have the following features: (1) the left-most and right-most candidates (i.e., extremists) are idealists; (2) strategic candidates never share their policy platforms, which instead are spread out across the policy space; and (3) if more than one strategic candidate enters, the distribution of voter preferences must be asymmetric. I also show that equilibria can accommodate idealist fringes of candidates toward the extremes of the political spectrum. [ABSTRACT FROM AUTHOR]

We discuss how linear equilibrium pricing in certain competitive market structures may represent nonlinear equilibrium pricing of Aliprantis et al. (J Econ Theory 100:22-72, 2001, J Econ Theory 121:51-74, 2005). Their work extends the theory of value beyond the scope of the Walrasian single market linear price model. Our arguments include a new and general result on the existence of linear price equilibrium with multiple markets. Each market has its own price vector (linear functional), and agents’ involvement in various markets is heterogeneous. As a result, price differences across markets may prevail in equilibrium. We present an example in which single market linear price equilibrium does not exist, but certain corresponding equilibrium with two markets does. This example is a particular instance of a prevalent nonexistence problem in atomless economies with differential information. Bypassing the nonexistence problem is one of the achievements of the nonlinear equilibrium theory. Our equilibrium with multiple markets, on the other hand, offers a solution with a more standard economic interpretation. Besides, our general framework is a model of multiple markets in their own right, and our results are related to the role of economic intermediation and bilateral trade. [ABSTRACT FROM AUTHOR]

A non-cooperative solution, the Equilibrium in Secure Strategies (EinSS), is defined as an extension of the Nash equilibrium in pure strategies, and is meant to solve games where players are “cautious,” i.e. , looking for secure positions and avoiding threats. This concept abstracts and unifies ad hoc solutions already formulated in various applied economic games that have been discussed extensively in the literature. A general existence theorem is provided and then applied to the price-setting game in the Hotelling location model, to Tullock's rent-seeking contests, and to Bertrand–Edgeworth duopoly. Finally, competition in the insurance market game is re-examined and the Rothschild–Stiglitz–Wilson contract is shown to be an EinSS even when the Nash equilibrium breaks down. [ABSTRACT FROM AUTHOR]

The previous literature on general equilibrium has assumed that all traders belong to a single market. However, traders often participate in more than one exchange to diversify their portfolios. Moreover, there is evidence that the security listings of exchanges overlap. Our model captures these facts: there are multiple exchanges, the same security is listed in different exchanges, and traders can belong to more than one exchange. We show that, in the presence of convex transaction costs, there exists an equilibrium, and that individual demand is a continuously differentiable function in prices. [ABSTRACT FROM AUTHOR]

It is known that multidimensional Downsian competition fails to admit an equilibrium in pure strategies unless very stringent conditions on the distribution of voters' bliss points are imposed ( Plott, 1967 ). This paper revisits this problem considering that the two vote share maximizing candidates are differentiated . That is, candidates strategically decide positions only in some of the n dimensions while in the rest of them their positions are assumed to be fixed. These fixed dimensions may be viewed as candidates' immutable characteristics (race, religion, culture, etc.). We find that for any distribution of voters' bliss points , a unique Nash equilibrium in pure strategies is guaranteed to exist if candidates are sufficiently differentiated –if in the fixed dimensions their positions are sufficiently different. This is true even if there exists a unique fixed dimension and candidates are flexible in all other n − 1 dimensions. [ABSTRACT FROM AUTHOR]

We study non-stationary markets in which traders are randomly matched to bargain over the price of a heterogeneous good or the terms of a partnership. The economy consists of a continuum of players drawn from a finite set of types. Players exogenously enter the market over time and exit upon trading. At every date, matching probabilities for each pair of types are determined by the endogenous distribution of trader types in the market. The balance of bargaining power at any stage depends on variations in potential gains from trade, the inflows of new traders, the structure of agreements at future dates, and the induced frequency of trading opportunities. We establish that an equilibrium always exists. Moreover, all equilibria that lead to the same evolution of the economy are payoff equivalent. However, we show that multiple self-fulfilling expectations about the trajectory of the economy, generating distinct equilibrium dynamics and payoffs, may coexist. [ABSTRACT FROM AUTHOR]

This paper addresses the question of existence of stationary Markov perfect equilibria in a class of dynamic games that includes many known bargaining models and models of coalition formation. General sufficient conditions for existence of equilibria are currently lacking in a number of interesting environments, e.g., models with non-convexities, consumption lower bounds, or an evolving state variable. The main result establishes existence of equilibrium under compactness and continuity conditions, without the structure of convexity or strict comprehensiveness used in the extant literature. The proof requires a precise selection of voting equilibria following different proposals using a generalization of Fatou's lemma. [ABSTRACT FROM AUTHOR]

The paper proves the existence of equilibrium in non-renewable resource markets when extraction costs are non-convex and resource storage is possible. Inventories flatten the consumption path and eliminate price jumps at the end of the extraction period, so that market equilibrium becomes possible. We distinguish between two types of solutions, one with immediate and one with delayed buildup of inventories. For both cases, we do not only characterize potential optimal paths but also show that equilibria actually exist under fairly general conditions. It is found that optimum resource extraction involves increasing quantities over a period of time. What is generally interpreted as an indicator of increasing resource abundance is thus perfectly compatible with constant resource stocks. [ABSTRACT FROM AUTHOR]

What is the minimal structure that is needed to perform equilibrium analysis in large extensive form games? To answer this question, this paper provides conditions that are simultaneously necessary and sufficient for the existence of a subgame perfect equilibrium in any well-behaved perfect information game defined on a large game tree. In particular, the set of plays needs to be endowed with a topology satisfying two conditions. (a) Nodes are closed as sets of plays; and (b) the immediate predecessor function is an open map. [ABSTRACT FROM AUTHOR]

For a broad definition of time-discrete stochastic games, their zero-sum varieties have values. But the existence of $$\epsilon $$ -equilibrium for the corresponding non-zero-sum games has proven elusive. We present the problems associated with $$\epsilon $$ -equilibria in non-zero-sum stochastic games, from both the perspectives of proving existence and demonstrating a counter-example. [ABSTRACT FROM AUTHOR]