Pure Strategy Nash Equilibrium Research Articles

A traveler-centric mobility game: Efficiency and stability under rationality and prospect theory.

In this paper, we study a routing and travel-mode choice problem for mobility systems with a multimodal transportation network as a "mobility game" with coupled action sets. We formulate an atomic routing game to focus on the travelers' preferences and study the impact on the efficiency of the travelers' behavioral decision-making under rationality and prospect theory. To control the innate inefficiencies, we introduce a mobility "pricing mechanism," in which we model traffic congestion using linear cost functions while also considering the waiting times at different transport hubs. We show that the travelers' selfish actions lead to a pure-strategy Nash equilibrium. We then perform a Price of Anarchy and Price of Stability analysis to establish that the mobility system's inefficiencies remain relatively low and the social welfare at a NE remains close to the social optimum as the number of travelers increases. We deviate from the standard game-theoretic analysis of decision-making by extending our mobility game to capture the subjective behavior of travelers using prospect theory. Finally, we provide a detailed discussion of implementing our proposed mobility game.

A game theoretic approach to balance privacy risks and familial benefits

As recreational genomics continues to grow in its popularity, many people are afforded the opportunity to share their genomes in exchange for various services, including third-party interpretation (TPI) tools, to understand their predisposition to health problems and, based on genome similarity, to find extended family members. At the same time, these services have increasingly been reused by law enforcement to track down potential criminals through family members who disclose their genomic information. While it has been observed that many potential users shy away from such data sharing when they learn that their privacy cannot be assured, it remains unclear how potential users’ valuations of the service will affect a population’s behavior. In this paper, we present a game theoretic framework to model interdependent privacy challenges in genomic data sharing online. Through simulations, we find that in addition to the boundary cases when (1) no player and (2) every player joins, there exist pure-strategy Nash equilibria when a relatively small portion of players choose to join the genomic database. The result is consistent under different parametric settings. We further examine the stability of Nash equilibria and illustrate that the only equilibrium that is resistant to a random dropping of players is when all players join the genomic database. Finally, we show that when players consider the impact that their data sharing may have on their relatives, the only pure strategy Nash equilibria are when either no player or every player shares their genomic data.

Pure-strategy equilibrium in Bayesian potential games with absolutely continuous information

We prove the existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games with absolutely continuous information and a Bayesian potential that is upper semi-continuous in actions for any realization of the players' types. In particular, all Bayesian potential games with finitely many actions and absolutely continuous information possess a pure-strategy Bayesian Nash equilibrium.

Contests between two groups for a group-specific public-good/bad prize

We study contests between groups for a group-specific public-good/bad prize in which the contest success function for a group can be represented by a continuous function in each group’s effort level, where each group’s effort level equals the sum of effort levels that the individual players in that group expend. The players expend their effort simultaneously and independently to win the prize or not to win it (or both). Obtaining the pure-strategy Nash equilibrium, we establish that, in each group, only the player with the highest valuation and the player with the lowest valuation may be active. We further establish that there are only two active players, either in the same group or in different groups, and the rest expend zero effort.

Environmental delegation versus sales delegation: a game-theoretic analysis

AbstractRecently, in their 2019 paper, Poyago-Theotoky and Yong consider a managerial Cournot duopoly with pollution externalities and emission taxes and propose an explicit environmental incentive in a managerial compensation contract. The authors compare several exogenous equilibria emerging in the symmetric sub-games in which the owner offers either the environmental delegation contract or the standard sales delegation contract: abatement and social welfare (resp. emission taxes) under environmental delegation are higher (resp. lower) than under sales delegation. The present work extends their model using a game-theoretic approach to analyse the asymmetric sub-games, in which only one firm adopts the environmental contract, and adds the contract decision stage. Results show that the environmental contract never emerges as the unique sub-game perfect Nash equilibrium of this non-cooperative managerial decision game. Indeed, if the green R&D technology is efficient, the sales contract emerges as the unique Pareto-inefficient Nash equilibrium. Otherwise, if the green R&D technology is inefficient, multiple Nash equilibria in pure strategies exist (coordination game). Our findings offer direct policy implications.

Competitive information disclosure in random search markets

We analyze the role of competition in information provision in random search markets. Multiple symmetric senders compete for the receiver's investment by disclosing information about their respective project qualities, and the receiver conducts random search to learn about the qualities of the projects. We show that in any symmetric pure strategy Nash equilibrium, each sender chooses a strategy with the lowest possible reservation value. There is no active search, and the receiver does not benefit from the competition of the senders.

Market-dependent preferences, positive and negative network effects and welfare

This article specifies the conditions on the utility function and on the cost function for which congestion or production overcapacity emerge in a market when information is perfect and prices are flexible. Consumers are of two types: Some appreciate a large production market, some do not. A typology underlines the shape of utility functions and cost functions which generate an optimal production overcapacity or an optimal congestion at the pure strategy Nash equilibrium. We show that the sign of the consumption externalities is endogenous. The result is independent of the nature (positive or negative) of the consumption externalities that affect the consumers’ preferences. A welfare analysis is performed. We underline the conditions under which the social planner chooses whether to clear the market or not. Application to a network economy is developed. Extensions to the following three cases are provided: (1) unsold stocks of goods with or without warehouse inventories, (2) quality of goods and (3) software markets.

Equilibria in bottleneck games

This paper introduces a bottleneck game with finite sets of commuters and departing time slots as an extension of congestion games of Konishi et al. (J Econ Theory 72:225–237, 1997a). After characterizing Nash equilibrium of the game, we provide sufficient conditions for which the equivalence between Nash and strong equilibria holds. Somewhat surprisingly, unlike in congestion games, a Nash equilibrium in pure strategies may often fail to exist, even when players are homogeneous. In contrast, when there is a continuum of atomless players, the existence of a Nash equilibrium and the equivalence between the set of Nash and strong equilibria hold as in congestion games (Konishi et al. 1997a).

On the solution of games with arbitrary payoffs: An application to an over‐the‐counter financial market

AbstractThis paper defines a variety of game theoretic solution concepts in the language of soft set theory. We begin by defining the Nash equilibrium in pure strategies. We assume that the gains of the players are totally ordered and non‐desirable alternatives are absent. Moreover, we introduce the notions of strong and semi‐strong utility. These two completely new notions, serve as a mechanism for converting non‐ordered gains into totally ordered ones. We define the Nash equilibrium in mixed strategies in a general framework by introducing the notion of an extended game and strategy space. We finally define the Nash solution to cooperative bargaining games within the framework of soft set theory, illustrate a practical application to an over‐the‐counter financial market, and provide a detailed numerical example.

Nash equilibrium realization of population games based on social learning processes.

In the two-population game model, we assume the players have certain imitative learning abilities. To simulate the learning process of the game players, we propose a new swarm intelligence algorithm by combining the particle swarm optimization algorithm, where each player can be considered a particle. We conduct simulations for three typical games: the prisoner's dilemma game (with only one pure-strategy Nash equilibrium), the coin-flip game (with only one fully-mixed Nash equilibrium), and the coordination game (with two pure-strategy Nash equilibria and one fully-mixed Nash equilibrium). The results show that when the game has a pure strategy Nash equilibrium, the algorithm converges to that equilibrium. However, if the game does not have a pure strategy Nash equilibrium, it exhibits periodic convergence to the only mixed-strategy Nash equilibrium. Furthermore, the magnitude of the periodical convergence is inversely proportional to the introspection rate. After conducting experiments, our algorithm outperforms the Meta Equilibrium Q-learning algorithm in realizing mixed-strategy Nash equilibrium.

Secure Downlink Transmission Strategies against Active Eavesdropping in燦OMA Systems: A Zero-Sum Game Approach

Non-orthogonal multiple access technology (NOMA), as a potentially promising technology in the 5G/B5G era, suffers from ubiquitous security threats due to the broadcast nature of the wireless medium. In this paper, we focus on artificial-signal-assisted and relay-assisted secure downlink transmission schemes against external eavesdropping in the context of physical layer security, respectively. To characterize the non-cooperative confrontation around the secrecy rate between the legitimate communication party and the eavesdropper, their interactions are modeled as a two-person zero-sum game. The existence of the Nash equilibrium of the proposed game models is proved, and the pure strategy Nash equilibrium and mixed-strategy Nash equilibrium profiles in the two schemes are solved and analyzed, respectively. The numerical simulations are conducted to validate the analytical results, and show that the two schemes improve the secrecy rate and further enhance the physical layer security performance of NOMA systems.

Non-Cooperative Games for Privacy-Preserving and Cost-Efficient Smart Grid Energy Management

In this paper, we design privacy-preserving and cost-efficient energy management strategies for smart grid users that are equipped with renewable energy sources. The adversary is assumed to employ a factorial hidden Markov model based inference for load disaggregation, and the corresponding joint log-likelihood of the model is utilized as the privacy measure. The studied dynamic pricing model is applicable to a commodity-limited market, where the price of unit amount of energy is determined by the users’ aggregated power request. The users’ energy management strategies are designed under a non-cooperative game framework, where each user aims to optimize a weighted sum objective of both privacy measure and energy cost saving. The users’ non-cooperative game is shown to admit a unique pure strategy Nash equilibrium. As an extension, a computational-efficient distributed Nash equilibrium energy management strategy seeking method is proposed, which also avoids the privacy leakage due to the sharing of payoff functions between users. The performance of practical designs of the energy management strategies in the equilibrium is finally illustrated by numerical experiments.

Decentralized payment clearing using blockchain and optimal bidding

In this paper, we construct a decentralized clearing mechanism which endogenously and automatically provides a claims resolution procedure. This mechanism can be used to clear a network of obligations through blockchain. In particular, we investigate default contagion in a network of smart contracts cleared through blockchain. In so doing, we provide an algorithm which constructs the blockchain so as to guarantee the payments can be verified and the miners earn a fee. We, additionally, consider the special case in which the blocks have unbounded capacity to provide a simple equilibrium clearing condition for the terminal net worths; existence and uniqueness are proven for this system. Finally, we consider the optimal bidding strategies for each firm in the network so that all firms are utility maximizers with respect to their terminal wealths. We show that a pure strategy Nash equilibrium need not exist, and then look for mixed Nash equilibrium bidding strategies. The implications of these strategies, and more broadly blockchain, on systemic risk are considered.

Efficient and Robust Equilibrium Strategies of Utilities in Day-Ahead Market With Load Uncertainty

We consider the scenario where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> utilities strategically bid for electricity in the day-ahead market and balance the mismatch between the committed supply and actual demand in the real-time market, with uncertainty in demand and local renewable generation in consideration. We model the interactions among utilities as a noncooperative game, in which each utility aims atminimizing its per-unit electricity cost. We investigate utilities’ optimal bidding strategies and show that all utilities bidding according to (net load) prediction is a unique pure strategy Nash equilibrium with two salient properties. First, it incurs no loss of efficiency; hence, competition among utilities does not increase the social cost. Second, it is robust and (0, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N-1$</tex-math></inline-formula> ) fault immune. That is, fault behaviors of irrational utilities only help to reduce other rational utilities’ costs. The expected market supply–demand mismatch is minimized simultaneously, which improves the planning and supply-and-demand matching efficiency of the electricity supply chain. We prove the results hold under the settings of correlated prediction errors and a general class of real-time spot pricing models, which capture the relationship between the spot price, the day-ahead clearing price, and the market-level mismatch. Simulations based on real-world traces corroborate our theoretical findings. Our article adds new insights to market mechanism design. In particular, we derive a set of fairly general sufficient conditions for the market operator to design real-time pricing schemes so that the interactions among utilities admit the desired equilibrium.

Existence and uniqueness of Nash equilibrium in discontinuous Bertrand games: a complete characterization

Since (Reny in Econometrica 67:1029–1056, 1999) a substantial body of research has considered what conditions are sufficient for the existence of a pure strategy Nash equilibrium in games with discontinuous payoffs. This work analyzes a general Bertrand game, with convex costs and an arbitrary sharing rule at price ties, in which tied payoffs may be greater than non-tied payoffs when both are positive. On this domain, necessary and sufficient conditions for (i) the existence of equilibrium (ii) the uniqueness of equilibrium are presented. The conditions are intuitively easy to understand and centre around the relationships between intervals of real numbers determined by the primitives of the model.

Dynamic supplier competition and cooperation for buyer loyalty on service

A firm’s service-driven always-a-share behavior may have a significant effect on the competitive and cooperative inventory policies of its suppliers. To explore this effect, we consider a model of a repeat buyer (she) sharing her patronage among two heterogeneous newsvendor-type suppliers over an infinite horizon. To enjoy the best service advantage, the buyer plays one supplier (him) against the other by rewarding product availability with repurchase (loyalty) and punishing stockouts with switching (disloyalty) in the next period. Faced with this behavior, the optimal ordering policy of each supplier is a basestock policy with a non-negative “active” basestock level when the supplier has the buyer’s loyalty and a zero basestock level when he does not. Under competition, the optimal active basestock level of each supplier is greater than his myopic basestock level and increases in the other supplier’s active basestock level. Under a mild condition, the active basestock levels of both suppliers have at least one pure-strategy Nash equilibrium solution. If the suppliers cooperate, the optimal active basestock level of the supplier with the highest/lowest myopic profit is greater/smaller than his myopic basestock level. To better comprehend these results, we apply them to the case where the buyer’s demand is exponentially distributed. This allows us to obtain exact expressions for the optimal active basestock levels and payoff functions, which we then use in a numerical sensitivity analysis. We conclude with a discussion of the extension of the results to more than two suppliers.

Exploiting Non-Cooperative Game Against Cache Pollution Attack in Vehicular Content Centric Network

Vehicular Content Centric Network (VCCN) has been proposed to address the issues of user mobility and sporadic connectivity in Vehicular Ad hoc Network (VANET). The in-network caching of VCCN can indeed improve the performance of content distribution in terms of cache hit and delivery latency by making each node store frequently accessed data. However, data caching unfortunately suffers from Cache Pollution Attack (CPA) that sends out fabricated requests to pollute the data cache. To prevent the degradation of network performance caused by such an attack, we propose a detection and defense scheme for CPA by adopting game theory. We model the attack scenario into a non-cooperative game model. First, we prove the non-existence of pure strategy Nash Equilibrium (NE) in the attacker and defender game, and propose a normal form game model with complete information and an extensive Bayesian form game model with incomplete information. Under the guidance of NE, we propose a punishment strategy to prevent the conspiracy attack and make the attackers behave normally. Moreover, we propose a cooperative detection scheme to give the Road Side Unit (RSU) the final decision on CPA based on the direct and indirect suspicious lists which are generated by each node according to the observed traffic pattern. Simulation evaluations demonstrate that our scheme outperforms state-of-the-art schemes in terms of cache hit, detecting ratio, and cache accuracy for detecting and defending CPA.

Rational Play in Extensive-Form Games

We argue in favor of a departure from the equilibrium approach in game theory towards the less ambitious goal of describing only the actual behavior of rational players. The notions of Nash equilibrium and its refinements require a specification of the players’ choices and beliefs not only along the equilibrium play but also at counterfactual histories. We discuss an alternative—counterfactual-free—approach that focuses on choices and beliefs along the actual play, while being silent on choices and beliefs at unreached histories. Such an approach was introduced in an earlier paper that considered only perfect-information games. Here we extend the analysis to general extensive-form games (allowing for imperfect information) and put forward a behavioral notion of self-confirming play, which is close in spirit to the literature on self-confirming equilibrium. We also extend, to general extensive-form games, the characterization of rational play that is compatible with pure-strategy Nash equilibrium.

Random encounters and information diffusion about product quality

AbstractThis paper explores how social interactions among consumers shape markets. In a two‐country model, consumers meet and exchange information about the quality of the goods. As information spreads, demand evolves, affecting the prices and quantities manufactured by profit‐maximizing firms. We show that market prices with informational frictions reach the duopoly price with full information at the limit. However, this convergence can take different paths depending on the size asymmetry between countries. In particular, when the country producing the low‐quality good is relatively large, the single market does not immediately turn into a duopoly and can be temporarily trapped in a situation of price instability where no Nash equilibrium in pure (but only in mixed) strategies exists and prices can fluctuate between their monopoly and duopoly levels. It follows that the classical price‐reducing effects of international trade may take longer to appear. In view of an intense globalization process, understanding how social meetings affect market outcomes is critical for understanding the performance of international economic integration.

Equilibrium characterization and shock propagation in conflict networks

We establish the general properties of equilibria (existence and uniqueness) in a very general model of interconnected multiplayer conflicts. In particular, under mild conditions on the cost function and the contest technology, we show that a pure-strategy Nash equilibrium always exists and the set of Nash equilibria is convex. Furthermore, under the strong monotonicity of the cost function, the equilibrium is unique, regardless of the conflict structure. To establish these properties of equilibria, we use variational inequality techniques to obtain equivalent equilibrium characterizations. After conducting comparative statics analysis with respect to the model primitives, we explore, in several simple examples of conflict structures, how shocks that originated locally propagate to the rest of the conflict network. Our general framework subsumes a number of models studied in the contest literature, as we are able to generalize many results obtained in these papers.