Pure Strategy Nash Equilibrium Research Articles

Evolutionary Game of Fertility Desire of Women of Childbearing Age

In recent years, China's fertility rate has shown an obvious downward trend, which has aroused people's widespread concern. As social attitudes change, some women's concepts of childbearing also change and they dont want to have children too early. This paper constructs a game model between the government and women of childbearing age who don't want to have children and explores the effects of economic subsidies, psychological benefits, and job security brought by the government's policies to one-child or two-child families on the fertility desire. Through the study, it was found that current domestic fertility policies have not been effective in promoting higher fertility rates. The pure strategy Nash equilibrium obtained by the line drawing method is a strategy in which the government push fertility policies and women choose not to have children. In order to effectively increase the fertility rate, on the basis of increasing subsidies, the government should guide women who have not given birth to a correct view of fertility and provide women who have given birth to one child with more economic subsidies and employment guarantees.

Multi-battle contests over complementary battlefields

AbstractThis paper studies contests with complementary prizes where each agent simultaneously distributes a fixed budget over multiple battlefields. Each battlefield has a single prize which is divided among the competitors in proportion to an arbitrary power function of their investment levels. A unique pure strategy Nash equilibrium is shown to exist under arbitrarily sensitive battlefield success functions if objective functions exhibit constant subunitary elasticity of substitution between prize shares. In contrast, Blotto contests with linear objectives have only mixed strategy Nash equilibria if battlefield success functions are sufficiently sensitive to investment levels. Sufficient complementarity between prize shares allows pure strategy Nash equilibria to exist under arbitrarily sensitive battlefield success functions.

Red light game identifies ineffective criminal deterrence

AbstractThe proceeding investigates the ineffectiveness of law enforcement placement in specific circumstances from the penological framework of rational choice theory. This work employs game theory modeling to determine an individual's propensity to commit a crime. The succeeding illustrates how it is counterproductive to nudge potential criminals where socially preferable behavior corresponds to a pure strategy Nash equilibrium in a simultaneous two‐player game. The result of this analysis demonstrates an addendum to contemporary rational choice theory in criminology. The solutions of these stage games show that increasing the placement of criminal deterrents, like surveillance or fines intended to reduce the number of opportunities to commit deviant behavior, is dysfunctional under particular circumstances. This suggests that law enforcement resources should not be directed to enforcing certain crimes. The implications of this work could help optimize department assets and personnel hours.

Competition among Pairwise Lottery Contests

We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a fully polynomial-time approximation scheme to compute an approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.

Relay selection in Underwater Acoustic Sensor Networks for QoS-based cooperative communication using game theory

Underwater Acoustic Sensor Networks (UASN) plays a crucial role in monitoring and transmitting environmental information for marine resource exploration. However, underwater communication faces significant challenges such as signal attenuation, noise interference, and unstable underwater environments. Moreover, with the deployment of a large number of nodes, inefficient resource allocation strategies in cooperative communication lead to strained communication resources as well as increased energy consumption. This paper introduces a novel relay selection approach based on game theory that takes into account both competition and cooperation among nodes. By building a potential game model and a utility function, we derive the optimal relay selection strategy that can enhance system Quality of Service (QoS). We also define conditions for the feasibility of pure strategy Nash Equilibria (NE) and develop a low-complexity QoS-based Relay Selection Iterative Algorithm (QoS-RSIA) to procure feasible pure strategy NE for the designed game. Through comparative analysis with other relay selection algorithms, simulation results show that the proposed method not only enhances system throughput and average spectral efficiency by at least 8.3%, but also significantly reduces time complexity compared to exhaustive search algorithms.

Nash Equilibria in Two-Resource Congestion Games with Player-Specific Payoff Functions

In this paper, we examine the class of congestion games with player-specific payoff functions introduced by Milchtaich, I. (1996). Focusing on the special case of two resources, we give a short and simple method for identifying all Nash equilibria in pure strategies. We also provide a computation algorithm based on our theoretical analysis.

Sequential licensing with several competing technologies

AbstractWe assume a multistage oligopoly wherein a given number of innovators compete by selling their substitutive technologies. Each innovator sequentially and independently chooses how many licenses to sell, and subsequently, all licensees compete à la Cournot in the product market. We show that, in equilibrium, the total number of licensees grows exponentially with the number of innovators. In addition, this sequential outcome is also obtained as a subgame perfect Nash equilibrium in pure strategies of a game with endogenous timing. Interestingly, by extending the duopoly model of Badia et al. (Math Soc Sci 108:8–13, 2020) to the case of more than two innovators and exploring pure strategy equilibria instead of mixed strategy equilibria, we derive drastically different policy implications, in terms of patent regulations. Our results suggest that more competition in the upstream market (e.g., by relaxing patent protection against the appearance of similar technologies) tends to increase downstream competition and welfare instead of discouraging or delaying technology adoption. In addition, our analysis is extended to explore the strategic role of public investment in basic R &D.

Policy polarization, primaries, and strategic voters

We consider two-stage electoral competitions with strategic voters, investigating the effects of valence (i.e., a candidate’s personal quality) on policy polarization. In our model, two parties compete in a general election, and each party has two office-seeking candidates. Parties first hold a primary election to decide their representative, and then put forward their winning candidate to compete in the general election. Candidates are thus characterized by their promised policy and their valence. Although voters value policies differently, they value valence uniformly. We focus on a competitive regime pure strategy Nash equilibrium where the candidate with the highest valence does not necessarily win the general election. We provide the sufficient conditions under which such an equilibrium exists, as well as a novel comparative static analysis of the valence differences between candidates (or the valence advantage). Our findings demonstrate that changing the valence advantage in general and primary elections can pose differing implications for policy outcomes and policy polarization.

Distributed dynamic task allocation for unmanned aerial vehicle swarm systems: A networked evolutionary game-theoretic approach

Task allocation is a key aspect of Unmanned Aerial Vehicle (UAV) swarm collaborative operations. With an continuous increase of UAVs’ scale and the complexity and uncertainty of tasks, existing methods have poor performance in computing efficiency, robustness, and real-time allocation, and there is a lack of theoretical analysis on the convergence and optimality of the solution. This paper presents a novel intelligent framework for distributed decision-making based on the evolutionary game theory to address task allocation for a UAV swarm system in uncertain scenarios. A task allocation model is designed with the local utility of an individual and the global utility of the system. Then, the paper analytically derives a potential function in the networked evolutionary potential game and proves that the optimal solution of the task allocation problem is a pure strategy Nash equilibrium of a finite strategy game. Additionally, a PayOff-based Time-Variant Log-linear Learning Algorithm (POTVLLA) is proposed, which includes a novel learning strategy based on payoffs for an individual and a time-dependent Boltzmann parameter. The former aims to reduce the system’s computational burden and enhance the individual’s effectiveness, while the latter can ensure that the POTVLLA converges to the optimal Nash equilibrium with a probability of one. Numerical simulation results show that the approach is optimal, robust, scalable, and fast adaptable to environmental changes, even in some realistic situations where some UAVs or tasks are likely to be lost and increased, further validating the effectiveness and superiority of the proposed framework and algorithm.

On the Existence of Stationary Nash Equilibria for Mean Payoff Games on Graphs

In this paper we extend the classical concept of positional strategies for a mean payoff game to a general mixed stationary strategy approach, and prove the existence of mixed stationary Nash equilibria for an arbitrary m-player mean payoff game on graphs. Traditionally, a positional strategy represents a pure stationary strategy in a classical mean payoff game, where a Nash equilibrium in pure stationary strategies in general may not exist. Based on a constructive proof of the existence of specific equilibria for an m-player mean payoff game we propose a new approach for determining the optimal mixed stationary strategies. Additionally we characterize and extend the general problem of the existence of pure stationary Nash equilibria for some special classes of mean payoff games.

A Strategic Decision Support System Using Multiplayer Non-Cooperative Games for Resource Allocation After Natural Disasters

A severe natural disaster causes concurrent emergencies in separate geographical locations. However, tackling all the emergencies simultaneously with the limited available resources is challenging. A novel single-stage, non-cooperative multiplayer game-based solution approach for resource allocation is proposed in this paper, where the crisis locations in the demand of resources are treated as the players. The game-based decision support system (DSS) is intended to be implemented by the concerned disaster management authority to obtain a practical strategy for the allocation of indivisible and divisible resources among individual players in a limited resource environment. Any feasible allocation is associated with some cost on the basis of a non-monetary cost function. A discrete strategic game is formulated to tackle indivisible resource allocation and a continuous-kernel game for divisible resources. A mathematical analysis establishes that the game with the proposed cost function possesses at least one pure strategy Nash equilibrium (PSNE) for both types of games. Based on different selection criteria, a particular PSNE is chosen from the collection of multiple PSNEs and used as a desirable resource allocation strategy. A complexity analysis of the proposed algorithm is also carried out. Case studies are given in this paper to demonstrate the results developed. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The cost function proposed in this paper to formulate the non-cooperative game-based DSS will be useful to disaster management authorities in resource allocation problems after any natural disaster. While distributing a limited quantity of essential resources, the cost function maintains an overall balance so that none of the disaster locations is deprived or favoured unreasonably. The feasible allocations of the existing resources are determined by the solutions of a multiplayer game. The case study indicates that the disaster management authority can choose different post-disaster ground truths to compute the possible resource demands of the disaster locations. Using the demand vector (a vector comprising the demands of the players as elements) and the number of available resources as the inputs, the game-theoretic model generates the allocation vector, according to which the disaster locations can be allocated resources.

Optimal tie-breaking rules

We consider two-player contests with the possibility of ties and study the effect of different tie-breaking rules on effort. For ratio-form and difference-form contests that admit pure-strategy Nash equilibrium, we find that the effort of both players is monotone decreasing in the probability that ties are broken in favor of the stronger player. Thus, the effort-maximizing tie-breaking rule commits to breaking ties in favor of the weaker agent. With symmetric agents, we find that the equilibrium is generally symmetric and independent of the tie-breaking rule. We also study the design of random tie-breaking rules that are ex-ante fair and identify sufficient conditions under which breaking ties before the contest actually leads to greater expected effort than the more commonly observed practice of breaking ties after the contest.

Computing lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles

In 1975 the first author proved that every finite tight two-person game form g is Nash-solvable, that is, for every payoffs u and w of two players the obtained normal form game (g;u,w) has a Nash equilibrium (NE) in pure strategies. Several proofs of this theorem were obtained later. Here we strengthen the result and give a new proof, which is shorter than previous ones. We show that game (g;u,w) has two types of NE, realized by a lexicographically safe (lexsafe) strategy of one player and some special best response of the other. The proof is constructive, we obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form g is given explicitly. Yet, in applications g is frequently realized by an oracle O such that size of g is exponential in the size |O| of O. We assume that game form g=g(O) generated by O is tight and that an arbitrary ±1game(g;u0,w0) (in which payoffs u0 and w0 are zero-sum and take only values ±1) can be solved in time polynomial in |O|. These assumptions allow us to compute two (one for each player) lexsafe NE in time polynomial in |O|. These NE may coincide. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.

Towards sustainable and accountable subsidy design: Identifying effective subsidisation systems for forest stands

Governments often subsidise forest stands to ensure a sustainable supply of wood and environmental services. Governments need to design subsidy systems to effectively achieve policy goals with less government expenditure. This study explored effective subsidy systems for forest stands by comparing four subsidy systems using a simple forest resource model. Although a system formulated with minimal constraints required minimum government expenditure in all cases, the rotation age in a stand could change cyclically with the target supply. Assuming that forest owners cooperate with the government, the maximisation of a cost-effectiveness index in each stand could provide the second-lowest government expenditure for almost all cases. However, when we assumed that forest owners were economically rational and strived to maximise their land expectation values, the optimality of this system was disturbed. In this situation, alternate optimisations of rotation ages by forest owners and subsidy rates by the government converged on the subsidy system with the third-lowest government expenditure. The system adjusts the subsidy rate so that the maximum land expectation value is zero, which is the pure-strategy Nash equilibrium between the forest owners and the government.

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&amp;D technology is efficient, the sales contract emerges as the unique Pareto-inefficient Nash equilibrium. Otherwise, if the green R&amp;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.