Pure Strategy Nash Equilibrium Research Articles

Pure strategy Nash Equilibrium in 2-contestant generalized lottery Colonel Blotto games

We consider a large class of 2-contestant Colonel Blotto games, for which the budget and valuation are both asymmetric between players and the contest success functions are in Tullock form with battle-specific discriminatory power in (0,1] and battle-and-contestant-specific lobbying effectiveness. We prove the existence of pure strategy Nash Equilibrium of the game through a direct equilibrium characterization approach. We examine some important equilibrium properties. Among them, the rate of equilibrium rent dissipation ratio between the two contestants is constant across all the battles, and the payoff dominance relationship is impossible between any two distinct equilibria. Furthermore, we provide two sets of sufficient conditions, one for symmetric valuations and the other for asymmetric valuations, under each of which the equilibrium of the game is unique. Our study contributes to a better understanding regarding the existence, uniqueness, characterization and properties of equilibria in Blotto games.

Pricing Game of Smart Charging Services for Risk-Averse Users in the Smart Grid

Electric vehicles play a key role in the transition to an environmental-friendly transportation system and can meanwhile enhance the power system’s evolution to the smart grid. With the adoption of dynamic pricing and usage scheduling enabled by the smart grid equipment, a variety of smart charging strategies have been designed to make the most of flexibility contained in their considerable electricity demand, whereas less effort is devoted to users’ willingness to participate. In this paper, we model a noncooperative pricing game between two types of charging stations. One offers conventional fast charging and the other uses the electric vehicles’ onboard batteries to provide regulation service to the grid. With drivers’ risk attitudes and bounded rationality taken into consideration, we design a prospect theory-based decision model to calculate the proportion of users that would go for the regulation-providing charging option. The decision model of the customer base is a critical determinant of profitability and it enables two competitors to strategically set their prices that optimally balance between gaining in market share and growing in profit per client. We prove the existence of a pure strategy Nash equilibrium for the game proposed and compute the equilibrium prices in different circumstances with respect to market settings and user segments. A comprehensive analysis of the results gives insights into the key factors at play and provides the grid operators with indications of how to increase the penetration of electric vehicles in the ancillary service market.

Competition and coordination in public transport: A mode choice experiment

Public transport plays an important role in sustainable transportation by reducing traffic congestion and greenhouse gas emissions. This study investigated the competition and coordination behavior in public transport mode choice with both positive and negative externalities. Subjects participated in a mode-choice game in which they were required to choose between bus and metro. The externalities in one public transport mode were determined by its characteristics and how many individuals chose it. We examined two treatments with different amounts of information: less information and more information, corresponding to non-implementing and implementing advanced traveler information systems, respectively. We derived three different theoretical solutions (i.e., the pure-strategy Nash equilibrium, the mixed-strategy equilibrium, and the fair-reference point) as benchmarks to compare with the experimental outcomes, finding that the fair-reference point was achieved when more information was provided, whereas the pure-strategy Nash equilibrium was better to predict the choice behavior when providing less information. Furthermore, the comparison of the two treatments revealed the information paradox, indicating that information affected the public transport mode choice behavior, and more information might lead to worse outcomes. The spontaneous collective actions induced by less information and risk aversion could exert the benefits from positive externalities, thereby reducing travel costs compared with more information. Also, we adopted an adaptive learning model to reproduce the main findings in the experiment and exhibit simulated results under different levels of information provision, providing a parsimonious explanation for the information paradox.

Three-player game-theoretic allocation of indivisible resources during natural disasters

Resource allocation is an integral part of disaster management. After a natural disaster, multiple concurrent emergencies in distinct locations often make resource allocation challenging for the disaster management authority. This article focuses on disaster resource allocation based on a novel three-player, non-cooperative, single-stage, strategic game where the emergency locations are considered as the players. The decision-making authority solves a game-theoretic algorithm to determine a suitable strategy for allocating indivisible resources among three disaster locations where the available resources are insufficient to satisfy all the players simultaneously. Based on a unique non-monetary cost function, each player incurs a penalty for any possible allocation strategy. Mathematical analysis shows that, for the proposed game, at least one pure strategy Nash equilibrium (PSNE) always exists, which can be a desirable allocation to the players. Payoff dominance and utopia-point-based solutions are used to select a single PSNE from a set of multiple PSNEs.

Multi-Agent Reinforcement Learning for a Random Access Game

This work investigates a random access (RA) game for a time-slotted RA system, 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> players choose a set of slots of a frame and each frame consists of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> multiple time slots. We obtain the pure strategy Nash equilibria (PNEs) of this RA game, where slots are fully utilized as in the centralized scheduling. As an algorithm to realize a PNE (Pure strategy Nash Equilibrium), we propose an Exponential-weight algorithm for Exploration and Exploitation (EXP3)-based multi-agent (MA) learning algorithm, which has the computational complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(N N_{\max }^{2} T)$</tex-math></inline-formula> . EXP3 is a bandit algorithm designed to find an optimal strategy in a multi-armed bandit (MAB) problem that users do not know the expected payoff of each strategy. Our simulation results show that the proposed algorithm can achieve PNEs. Moreover, it can adapt to time-varying environments, where the number of players varies over time.

Uncommon Knowledge in Multiparty Auctions

The pure strategy Nash equilibrium (PSNE) solution to multiparty auctions makes the strong but unrealistic assumption that all participants share the same beliefs about the type distributions of the others, and that all know that this information is mutually known. This paper proposes two alternative analyses of such auction problems that do not make that presupposition. The first is based on a solution concept similar to the PSNE. The second is employs the level k thinking solution concept.

Optimal strategies and cost-benefit analysis of the {varvec{n}}-player weightlifting game

The study of cooperation has been extensively studied in game theory. Especially, two-player two-strategy games have been categorized according to their equilibrium strategies and fully analysed. Recently, a grand unified game covering all types of two-player two-strategy games, i.e., the weightlifting game, was proposed. In the present study, we extend this two-player weightlifting game into an n-player game. We investigate the conditions for pure strategy Nash equilibria and for Pareto optimal strategies, expressed in terms of the success probability and benefit-to-cost ratio of the weightlifting game. We also present a general characterization of n-player games in terms of the proposed game. In terms of a concrete example, we present diagrams showing how the game category varies depending on the benefit-to-cost ratio. As a general rule, cooperation becomes difficult to achieve as group size increases because the success probability of weightlifting saturates towards unity. The present study provides insights into achieving behavioural cooperation in a large group by means of a cost–benefit analysis.

Anti-jamming channel access in 5G ultra-dense networks: a game-theoretic learning approach

This paper investigates the Quality of Experience (QoE) oriented channel access anti-jamming problem in 5th Generation Mobile Communication (5G) ultra-dense networks. Firstly, considering that the 5G base station adopts beamforming technology, an anti-jamming model under Space Division Multiple Access (SDMA) conditions is proposed. Secondly, the confrontational relationship between users and the jammer is formulated as a Stackelberg game. Besides, to achieve global optimization, we design a local cooperation mechanism for users and formulate the cooperation and competition among users as a local altruistic game. By proving that the local altruistic game is an Exact Potential Game (EPG), we further prove the existence of pure strategy Nash Equilibrium (NE) among users and Stackelberg Equilibrium (SE) between users and jammer. Thirdly, to obtain the equilibrium solutions of the proposed games, we propose an anti-jamming channel selection algorithm and improve its convergence speed through heterogeneous learning parameters. The simulation results validate the convergence and effectiveness of the proposed algorithm. Compared with the throughput optimization scheme, our proposed scheme obtain a greater network satisfaction rate. Finally, we also analyze user fairness changes during the algorithm convergence process and get some interesting conclusions.

Commodity tax competition and cross-border shopping in a tripoint model

This paper extends commodity tax competition to cross-border shopping in a two-country model of three countries with a tripoint border and investigates whether a pure-strategy Nash equilibrium of tax competition exists. First, we find that unlike existing studies on the two-country model, there is no pure-strategy Nash equilibrium for tax competition in the three-country tripoint model. Second, we consider partial tax coordination between two countries and investigate how the population size of the countries affects the Nash equilibrium tax rates and tax revenues under tax coordination. Our results suggest that the existence of a pure-strategy Nash equilibrium is limited to a specific case of only two countries. When three countries with a tripoint engage in tax competition, we cannot predict what happens in the fiscal game between governments by the Nash equilibrium. Furthermore, the Appendix focuses on the marginally deviation-proof sets of tax rates instead of the Nash equilibrium and compares tax rates and tax revenues among the two sets of tax rates.

Joint Task Offloading, D2D Pairing, and Resource Allocation in Device-Enhanced MEC: A Potential Game Approach

The emergence of intelligent applications produces the demand for computing. How to reduce the computation pressure in mobile-edge computing (MEC) under massive computation demand is an urgent problem to solve. Specifically, the allocation of heterogeneous resources, including communication resources and computing resources, needs to be optimized simultaneously. From the perspective of joint optimization of channel allocation, device-to-device (D2D) pairing, and offloading mode, this article studies the multiuser computing task offloading problem in device-enhanced MEC. The objective is maximizing the aggregate offloading benefits, i.e., the tradeoff between delay and energy consumption, of all compute-intensive users in the network. By introducing game theory, the problem is modeled as a multiuser computation task offloading game, which is proved to be an exact potential game (EPG) with at least one pure-strategy Nash equilibrium (NE) solution. In order to find a desirable solution, this article proposes a better reply-based distributed multiuser computation task offloading algorithm (BR-DMCTO). Simulation results show that the proposed offloading mechanism can improve the benefit of users, and verify the effectiveness and convergence of the proposed algorithm.

Competition and equilibrium effort choice

We examine optimal effort choice in a competition model where the agents are averse to low relative status and to exerting excessive effort. The game has a unique pure strategy Nash equilibrium. When the agents are homogeneous, a stronger competition incentive induces higher effort levels and results in lower utilities. When agents are heterogeneous, the model predicts that (i) all else equal, initial losers exert more effort but cannot become final winners, (ii) changes in effort choices of a subgroup of agents have a spillover effect, and (iii) agents with competitive advantages may prefer not to compete if the advantages are not significant enough. Some extensions and variants of the baseline model are also examined. Overall, the model’s theoretical predictions are consistent with some competition-related findings documented in the empirical literature.

Expectations-based loss aversion in contests

This paper studies a multi-player lottery contest in which heterogeneous contestants exhibit reference-dependent loss aversion à la Kőszegi and Rabin (2006, 2007). We verify the existence and uniqueness of pure-strategy choice-acclimating personal Nash equilibrium (CPNE) under moderate loss aversion and fully characterize the equilibrium. The equilibrium sharply contrasts with that in the two-player risk-neutral symmetric case. Loss aversion can lead contestants' individual efforts to change nonmonotonically, while the total effort strictly decreases. Further, it always leads to a more elitist distributional outcome, in the sense that a smaller set of contestants remain active in the competition and stronger contestants' equilibrium winning probabilities increase. We demonstrate that loss aversion generates a fundamentally different decision problem than risk aversion and develop a rationale that explains the contrasting predictions from the two frameworks. Finally, our results are robust under the alternative equilibrium concept of preferred personal Nash equilibrium (PPNE).

Game theory without theory: Interactive choice in pigeons, humans and machines

A unique approach to the experimental analysis of interactive choice behavior is described. Pairs of subjects were coupled in such a manner that each subject’s choice determined either the probability of reinforcement or the points awarded to the other subject’s choice. In this manner, several of game theory’s familiar games, like Prisoner’s Dilemma, were created. With pigeons as subjects, with humans playing against pigeons or humans playing each other (either in the laboratory or on the Internet), and even with a simple computer simulation, the results were clear. In every instance, players’ choices gravitated to the game’s pure strategy Nash equilibrium in games that have them and to a near equality of responding in the game Rock-Paper-Scissors, which does not. The principle of positive reinforcement entirely accounts for the results. There is no need for appeal to complex mathematical models or higher-order concepts of strategy, optimality, rational expectations and the like.

On a Fixed Point Theorem for General Multivalued Mappings on Finite Sets with Applications in Game Theory

We propose a new fixed point theorem that completely characterizes the existence of fixed points for multivalued maps on finite sets. Our result can be seen as a generalization of Abian’s fixed point theorem. In the context of finite games, our result can be used to characterize the existence of a Nash equilibrium in pure strategies and can therefore distinguish pure strategy equilibria from mixed strategy equilibria in the celebrated Nash theorem.

Online-Offline Competition with Heterogeneous Consumers: An Example for No Existence of Pure Strategy Nash Equilibrium

Existing literature on competition between online and offline firms has focused on market conditions that guarantee the existence of a pure strategy Nash equilibrium. In this note, by constructing a concrete example, we provide a first attempt to show that the equilibrium existence result does not necessarily hold when consumers’ preferences are heterogeneous. Specifically, we consider the competition between one online firm and several offline firms in a market organized as a Salop model, where consumers’ preferences have a binary distribution. We identify a boundary scenario where the type distribution is binary with one type of consumer loyal to online shopping and the other type loyal to offline shopping. We show that there is no pure strategy Nash equilibrium for this boundary scenario, which indicates that the market may not be stable under such conditions. Our study contributes to a better understanding of the equilibrium existence conditions for the online versus offline retail competition.

Pure Strategy Nash Equilibrium in 2-Contestant Generalized Lottery Colonel Blotto Games

Pure Strategy Nash Equilibrium in 2-Contestant Generalized Lottery Colonel Blotto Games

Distribution Games: A New Class of Games With Application to User Provided Networks

User Provided Network (UPN) is a promising solution for sharing the limited network resources by utilizing user capabilities as a part of the communication infrastructure. In UPNs, it is an important problem to decide how to share the resources among multiple clients in decentralized manner. Motivated by this problem, we introduce a new class of games termed distribution games that can be used to distribute efficiently and fairly the bandwidth capacity among users. We show that every distribution game has at least one pure strategy Nash equilibrium (NE) and any best response dynamics always converges to such an equilibrium.We consider social welfare functions that are weighted sums of bandwidths allocated to clients.We present tight upper bounds for the price of anarchy and price of stability of these games provided that they satisfy some reasonable assumptions. We define two specific practical instances of distribution games that fit these assumptions.We conduct experiments on one of these instances and demonstrate that in most of the settings the social welfare obtained by the best response dynamics is very close to the optimum. Simulations show that this game also leads to a fair distribution of the bandwidth.

Utility Design for Infinite Distributed Welfare Games

A distributed welfare game is a non-cooperative game-theoretic model for a resource allocation problem that maximizes welfare. In this paper, two utility function designs for resource allocation problems with infinite resources are considered. One is called wonderful life utility, which was proposed for the problem. The second one employs a solution concept of the cooperative game, Egalitarian Non-Separable Contribution (ENSC). The main contributions of the paper are two holds. The first contribution is to clarify the existence of a pure strategy Nash equilibrium in the games defined via the above utility functions when the welfare function is monotone, continuously twice differentiable, and diminishing return (DR) submodular. The second is to derive lower bounds of the Price of Anarchy (PoA) for both games. The PoA is the worst-case ratio between the optimal value and values at the pure strategy Nash equilibria.

Existence of Pure Strategy Nash Equilibrium in Contests: Arbitrary Grouping and Asymmetric Valuations

Existence of Pure Strategy Nash Equilibrium in Contests: Arbitrary Grouping and Asymmetric Valuations

A tractable multi-leader multi-follower peak-load-pricing model with strategic interaction

While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems.