Pure Strategy Nash Equilibrium Research Articles

On the existence of Pareto undominated mixed-strategy Nash equilibrium in normal-form games with infinite actions

We show that each mixed strategy of a normal-form game can be reformulated as a pure strategy of an induced Bayesian game. Moreover, a normal-form game has a Pareto undominated mixed-strategy Nash equilibrium if and only if its induced Bayesian game has a pure-strategy Bayesian Nash equilibrium. By relying on the existence result of Pareto undominated pure-strategy Nash equilibrium in Bayesian games in Fu and Yu (2015), we also show that every normal-form game has a Pareto undominated mixed-strategy Nash equilibrium.

Selfish Caching Games on Directed Graphs

Caching networks can reduce the routing costs of accessing contents by caching contents closer to users. However, cache nodes may belong to different entities and behave selfishly to maximize their own benefits, which often lead to performance degradation for the overall network. While there has been extensive literature on allocating contents to caches to maximize the social welfare, the analysis of selfish caching behaviors remains largely unexplored. In this paper, we model the selfish behaviors of cache nodes as selfish caching games on arbitrary directed graphs with heterogeneous content popularity. We study the existence of a pure strategy Nash equilibrium (PSNE) in selfish caching games, and analyze its efficiency in terms of social welfare. We show that a PSNE does not always exist in arbitrary-topology caching networks. However, if the network does not have a mixed request loop, i.e., a directed loop in which each edge is traversed by at least one content request, we show that a PSNE always exists and can be found in polynomial time. Furthermore, we can avoid mixed request loops by properly choosing request forwarding paths. We then show that the efficiency of Nash equilibria, captured by the price of anarchy (PoA), can be arbitrarily poor if we allow arbitrary content request patterns, and adding extra cache nodes can make the PoA worse, i.e., cache paradox happens. However, when cache nodes have homogeneous request patterns, we show that the PoA is bounded even allowing arbitrary topologies. We further analyze the selfish caching games for cache nodes with limited computational capabilities, and show that an approximate PSNE exists with bounded PoA in certain cases of interest. Simulation results show that increasing the cache capacity in the network improves the efficiency of Nash equilibria, while adding extra cache nodes can degrade the efficiency of Nash equilibria.

Non-cooperative queueing games on a network of single server queues

This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2times 2-grid, and (iv) 2-player games on an Atimes B-grid with small differences in the service rates.

Pricing in a competitive stochastic insurance market

This paper studies a one-period stochastic game to determine the optimal premium strategies of non-life insurers in a competitive market. Specifically, the optimal premium strategy is determined by the Nash equilibrium of an n-player game, in which each player is assumed to maximise the expected utility of terminal wealth. The terminal wealth is stochastic, since the number of policies and the size of claims are considered to be random variables. The total loss of each insurer is described by the collective risk model. The expected number of policies is affected by all the premiums in the market and further investigated by two distinct demand functions. Both models have an exponential functional form, that is characterised by market and price sensitivity parameters. The demand in the first model is zero for premiums above a given threshold, whereas the second model does not include such restriction. The pure strategy Nash equilibrium premiums are given as solutions to constrained optimisation problems. For the first model we prove the existence and uniqueness of a pure strategy Nash equilibrium, whereas for the second model we provide a formula when it exists. Two numerical examples are provided to illustrate the applicability of our findings.

Impact of Confirmation Bias on Competitive Information Spread in Social Networks

This article investigates the impact of confirmation bias on competitive information spread in the social network that comprises individuals in a social network and competitive information sources at a cyber layer. We formulate the problem of information spread as a zero-sum game, which admits a unique Nash equilibrium in pure strategies. We characterize the dependence of pure Nash equilibrium on the public's innate opinions, the social network topology, as well as the parameters of confirmation bias. We uncover that confirmation bias moves the equilibrium toward the center only when the innate opinions are not neutral, and this move does not occur for the competitive information sources simultaneously. Numerical examples in the context of well-known Krackhardt's advice network are provided to demonstrate the correctness of theoretical results.

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

In this paper, we prove the existence of pure strategy Nash Equilibrium for 2-contestant Colonel Blotto games with asymmetric values, Tullock contest success functions with r <= 1, asymmetric lobbying effectiveness and use-it-or-lose-it budgets. We show that the solution to the pure strategy Nash Equilibrium can be equivalent to the problem of solving the roots of an equation. We give the lower bound and the upper bound of these roots. We also give some sufficient conditions, with which the uniqueness of pure strategy Nash Equilibrium can be proved.

Joint Computation Offloading, Channel Access and Scheduling Optimization in UAV Swarms: A Game-Theoretic Learning Approach

Coalition-based unmanned aerial vehicle (UAV) swarms havebeen widelyused in urgent missions. To fasten the completion, mobile edge computing (MEC) has been introduced into UAV networks where coalition leaders act as servers to help members with data computing. This paper investigates a relative delay optimization in MEC-assisted UAV swarms. Considering that the scheduling methods have great impact on the delay, some theoretical analyses are made and a scheduling method based on the shortest effective job first (SEJF) is proposed. Based on the coupled relationship between scheduling and resource allocation, the computation offloading and channel access problems are then jointly optimized. To solve the problem in distributed UAV networks, the optimization problem is formulated as an offloading game. It is proved that the game is an exact potential game (EPG) and it has at least one pure strategy Nash Equilibrium (PNE). To reach the PNE, a distributed offloading algorithm based on concurrent best-better response (CBBR) is designed. Finally, the simulations show that the performance of the proposed CBBR algorithm is better than traditional algorithms. Compared with other scheduling methods, the proposed scheduling method based on SEJF reduces the delay by up to 30%.

M Equilibrium: A Theory of Beliefs and Choices in Games

We introduce a set-valued solution concept, M equilibrium, to capture empirical regularities from over half a century of game-theory experiments. We show M equilibrium serves as a meta theory for various models that hitherto were considered unrelated. M equilibrium is empirically robust and, despite being set-valued, falsifiable. We report results from a series of experiments comparing M equilibrium to leading behavioral-game-theory models and demonstrate its virtues in predicting observed choices and stated beliefs. Data from experimental games with a unique pure-strategy Nash equilibrium and multiple M equilibria exhibit coordination problems that could not be anticipated through the lens of existing models.

A Free and Fair Economy: A Game of Justice and Inclusion

Frequent violations of fair principles in real-life settings raise the fundamental question of whether such principles can guarantee the existence of a self-enforcing equilibrium in a free economy. We show that elementary principles of distributive justice guarantee that a pure-strategy Nash equilibrium exists in a finite economy where agents freely (and non-cooperatively) choose their inputs and derive utility from their pay. Chief among these principles is that: 1) your pay should not depend on your name, and 2) a more productive agent should not earn less. When these principles are violated, an equilibrium may not exist. Moreover, we uncover an intuitive condition---technological monotonicity---that guarantees equilibrium uniqueness and efficiency. We generalize our findings to economies with social justice and inclusion, implemented in the form of progressive taxation and redistribution, and guaranteeing a basic income to unproductive agents. Our analysis uncovers a new class of strategic form games by incorporating normative principles into non-cooperative game theory. Our results rely on no particular assumptions, and our setup is entirely non-parametric. Illustrations of the theory include applications to exchange economies, surplus distribution in a firm, contagion and self-enforcing lockdown in a networked economy, and bias in the academic peer-review system.

Pricing-Based Channel Selection for D2D Content Sharing in Dynamic Environments

In order to make device-to-device (D2D) content sharing give full play to its advantage of improving local area services, one of the important issues is to decide the channels that D2D pairs occupy. Most existing works study this issue in static environment, and ignore the guidance for D2D pairs to select the channel adaptively. In this paper, we investigate this issue in dynamic environment where D2D pairs’ activeness and wireless channel are dynamic. Specifically, we propose a pricing-based approach to guide D2D pairs to select different channels according to the spectrum resource states adaptively. Then, we formulate the pricing-based channel selection problem as an expected global price-to-performance ratio minimum problem. In order to solve it in a tractable manner, we make an approximately equivalent transformation to it. After that, we model the transformed problem as a stochastic game and prove it to be an exact potential game, which has at least one pure strategy Nash Equilibrium (NE) point. In order to reach the pure strategy NE points in dynamic environment, we design a channel selection learning algorithm based on stochastic learning automata, which only requires little information exchange. Simulation results show that our proposed algorithm outperforms other benchmark algorithms.

Channel coordination of a risk-averse supply chain: a mean–variance approach

In the age of rapid technological advancement and digitization, coordination strategy remains an important issue for the supply chain. Additionally, the uncertainty caused by the disruption often induces the risk aversion in the supply chain members. Motivated by this issue, here we propose a coordination mechanism for a risk-averse supply chain using mean–variance approach. Here, we consider both centralized and decentralized cases and show that our analysis holds good for a central planner as well as for a decentralized supply chain under channel coordinating contracts such as buyback and revenue-sharing schemes. With the help of theoretical and numerical analysis, we exhibit how an individual supply chain agent’s risk aversion behavior can impact the contracts selection mechanism - from the profitability perspective. We extend our analysis to a dyadic setting to a single-supplier multiple-retailer network and confirm that pure strategy Nash equilibrium exists when all the retailers are risk-averse with varying risk attitude.

A game-theoretical model of the landscape theory

In this paper we examine a game-theoretical generalization of the landscape theory introduced by Axelrod and Bennett (1993). In their two-bloc setting each player ranks the blocs on the basis of the sum of her individual evaluations of members of the group. We extend the Axelrod–Bennett setting by allowing an arbitrary number of blocs and expanding the set of possible deviations to include multi-country gradual deviations. We show that a Pareto optimal landscape equilibrium which is immune to profitable gradual deviations always exists. We also indicate that while a landscape equilibrium is a stronger concept than Nash equilibrium in pure strategies, it is weaker than strong Nash equilibrium.

Distributive politics with other‐regarding preferences

AbstractThis paper analyzes a nonsmooth model of probabilistic voting with two parties and a broad family of other‐regarding behavior, including fairness and quasi‐maximin preferences, income‐dependent altruism, and inequity aversion. The paper provides conditions for equilibrium existence and uniqueness. It also characterizes the Nash equilibrium in pure strategies when parties hold either symmetric payoffs, or minor forms of asymmetries. The characterization shows that the two parties converge to an equilibrium policy that maximizes a mixture of a “self‐regarding utilitarian” social welfare function and an aggregate of society's other‐regarding preferences. These results are shown to be applicable to other nonsmooth frameworks, such as probabilistic voting with loss averse voters. The characterization also shows that the direction and the size of the inefficiencies emerging from electoral competition depend in a subtle way on the nature of the other‐regarding preferences (and resp., loss aversion).

Hierarchical Optimal Scheduling for Air Conditioner’s Load Aggregator

In this paper, a scheduling framework based on air-conditioning load aggregator is first proposed. Then the split constant frequency air-conditioning load is optimized by minute level. The step function characteristic between the average air-conditioning load per hour and the temperature difference set at the beginning and the end of the time is obtained and linearized. A temperature difference-power model suitable for air-conditioning hourly scheduling is established. Secondly, aiming at the multiple load aggregators existing in the distribution area, this paper proposes a non-cooperative game model of air conditioning load aggregators. Considering the problem of discretization of control variables in the game revenue function, this paper introduces continuous intermediate variables, establishes a hierarchical optimization model of the game, and gives the proof that the hierarchical optimization of the game has a unique pure strategy Nash equilibrium solution and the solution method. Finally, the feasibility of the proposed method is verified by numerical simulation.

Game Theoretic Strategies for Supplier Capability Assessment and Manufacturing Order Allocation

An effective method is required to determine the amount and priority for the deployment of suppliers for multiple manufacturing processes, particularly when the available budget for each manufacturing process is limited. In this study, we propose an integrated approach for supplier assessment that consists of two game theory models which are designed to recommend manufacturers on how best to choose suppliers given budgetary limitations. In the first model, the interactive behaviors between the key factors representing the manufacturer and the supplier are modeled and analyzed as a two-player and zero-sum game, after which the Supplier Power Value (SPV) is derived from the pure or mixed strategy Nash equilibrium. In the second model, 12 SPVs are used to compute a Shapley value for each supplier, in terms of the thresholds of the majority levels in one manufacturing process. The Shapley values are then applied to create an allocated set of limited manufacturing orders for suppliers. The experimental results present that the manufacturer can use our approach to quantitatively evaluate the suppliers and easily allocate suppliers within one manufacturing process.

Pricing in a Competitive Stochastic Insurance Market

This paper studies a one-period stochastic game to determine the optimal premium strategies of non-life insurers in a competitive market. Specifically, the optimal premium strategy is determined by the Nash equilibrium of an n-player game, in which each player is assumed to maximise the expected utility of terminal wealth. The terminal wealth is stochastic, since the number of policies and the size of claims are considered to be random variables. The total loss of each insurer is described by the collective risk model. The expected number of policies is affected by the all premiums in the market and further investigated by two distinct demand functions. Both models have an exponential functional form, that is characterized by market and price sensitivity parameters. The demand in the first model is zero for premiums above a given threshold, whereas the second model does not include such restriction. The pure strategy Nash equilibrium premiums are given as solutions to constrained optimisation problems. For the first model we prove the existence and uniqueness of a pure strategy Nash equilibrium, whereas for the second model we provide a formula when it exists. Two numerical examples are provided to illustrate the applicability of our findings.

Joint access and backhaul resource allocation for D2D-assisted dense mmWave cellular networks

Device-to-Device (D2D) communication technology is a promising solution to solve the problem that millimeter-wave (mmWave) signal propagation is vulnerable to blockage and thus rapidly weakened. However, after introducing in-band D2D communication to dense mmWave cellular networks, we must consider how to reasonably share resources and effectively control interference between more types of communication links (e.g., D2D, access, backhaul), which makes the resource allocation problem more challenging. In other words, it is more challenging to investigate joint access and backhaul resource allocation problem for in-band D2D-assisted dense mmWave cellular networks. To address this challenging problem, we firstly decouple it into the two sub-problems (i.e., joint access and backhaul resource allocation sub-problem, and joint D2D access and forwarding link resource allocation sub-problem). Then, we formulate the two sub-problems as the two non-cooperative games respectively and prove that they are exact potential game and there exist feasible pure strategy Nash equilibrium under some mild conditions. Finally, based on the formulated game models, we propose a centralized algorithm and a decentralized algorithm to get resource allocation results respectively. The extensive simulation results show that the proposed algorithms can effectively mitigate the impact of blockage on network performance.

A Dynkin Game on Assets with Incomplete Information on the Return

This paper studies a two-player zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques, we first reduce the problem to a zero-sum Dynkin game on a bidimensional diffusion (X,Y). Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of (X,Y) to a set with a moving boundary. A detailed description of the stopping sets for the two players is provided along with global C1 regularity of the value function.

Contests with a constrained choice set of effort

We consider a symmetric two-player contest, in which the choice set of effort is constrained. We apply a fundamental property of the payoff function to show that, under standard assumptions, there exists a unique Nash equilibrium in pure strategies. It is shown that all equilibria are near the unconstrained equilibrium. Perhaps surprisingly, this is not the case when players have different prize evaluations.

Disclose or Exploit? A Game-Theoretic Approach to Strategic Decision Making in Cyber-Warfare

Today, countries are engaged in the cyber-arms race. With over 16 K new hardware/software vulnerabilities reported in 2018 alone, an important question confronts senior government decision makers when their cyber-warfare units discover a new vulnerability. Should they disclose the vulnerability to the vendor who produced the vulnerable product? Or should they “stockpile” the vulnerability, holding it for developing exploits (i.e., cyber-weapons) that can be targeted at an adversary? Choosing the first option may be important when the affected company is a corporation in the nation state that discovers the vulnerability and/or if that nation state would have a big exposure to that vulnerability. Choosing the second option has obvious advantages to the discovering nation's defense. We formulate the cyber-competition between countries as a repeated cyber-warfare game (RCWG), where two countries (players) compete over a series of vulnerabilities by deciding, at the time of vulnerability discovery, 1) whether to exploit or disclose it and 2) how long to exploit it if they decide to exploit. We define the equilibrium state of the RCWG as a pure strategy Nash equilibrium, and propose a learning-while-competing framework to compute the pure strategy Nash equilibrium of the formulated RCWG. Although testing our results with real data in the murky world of cyber-warfare is challenging, we were able to obtain real statistics from other sources and demonstrate the effectiveness of our proposed algorithm through a set of simulation results under different scenarios using these third-party statistics. We also report on our DiscX system that can help support government decision makers in their decision whether to disclose or exploit a vulnerability that they find.