Mixed Strategy Nash Equilibrium Research Articles

Disequilibrium Play in Tennis

Are the serves of the world’s best tennis pros consistent with the theoretical prediction of Nash equilibrium in mixed strategies? We analyze their serve direction choices (to the returner’s left, right or body) with data from an online database called the Match Charting Project. Using a new methodology, we test and decisively reject a key implication of a mixed strategy Nash equilibrium, namely, that the probability of winning a service game is the same for all serve directions. We also use dynamic programming (DP) to numerically solve for the best-response serve strategies to probability models of service game outcomes estimated for individual server-returner pairs, such as Novak Djokovic serving to Rafael Nadal. We show that for most elite pro servers, the DP serve strategy significantly increases their service game win probability compared to the mixed strategies they actually use, which we estimate using flexible reduced-form logit models. Stochastic simulations verify that our results are robust to estimation error. Classification- C61, C73, L21

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.

Implicit coordination in an <inline-formula><tex-math id="M1">$ n $</tex-math></inline-formula>-player target selection game

We show that a class of target selection games has a unique symmetric mixed strategy Nash equilibrium $ x^\ast $. We also present a simple recursion that computes $ x^\ast $ within a finite number of iterations. Along the way, we introduce the concept of value compactness that points directly to the properties of the unique symmetric mixed strategy equilibrium, including the fact that the unique equilibrium does not force any player to make an arbitrary decision about associating positive probability with subsets of targets having a particular utility– a property that we refer to as implicit coordination.

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.

Rational Inattention in the Infield

This paper provides evidence of rational inattention by experienced professionals in strategic interactions. We add rational inattention to a game of matching pennies with state-dependent payoffs. Unlike the full-information, mixed-strategy Nash equilibrium, payoffs of different actions need not be equated state by state. Moreover, players respond partially to payoff differences, this responsiveness is stronger when attention costs are lower, strategies converge to full-information Nash as stakes increase, and average payoffs across all states are approximately equal across actions. We test these predictions using data on millions of pitches from Major League Baseball, where we observe strategies, payoffs, and proxies for attention costs. (JEL C72, D83, D91, L83, Z21)

Penalty kicks as cross-fertilization: On the economic psychology of sports

We review some topics in which soccer penalty kicks are related to phenomena in game theory, decision making and psychology. In particular, we discuss the game theoretic analysis of the kicker's and goalkeeper's behavior and its relation to the concept of mixed strategy Nash Equilibrium. The main idea is that both players should not follow a predictable strategy, and therefore should choose different strategies over time. We also review the action bias of goalkeepers in penalty kicks, which results in them almost always jumping to one of the sides although staying in the goal's center is in fact a good strategy given the empirical distribution of kicks. We then turn to the order effect in penalty shootouts, and the debate to what extent kicking first gives the team an advantage in the shootout. These topics illustrate the opportunities offered by interdisciplinary research that combines sports with social sciences such as economics and psychology. While this interdisciplinary literature has grown significantly in recent years, research opportunities in this area abound and we hope that this article will encourage the readers to contribute to it.

Intention Prediction and Mixed Strategy Nash Equilibrium-Based Decision-Making Framework for Autonomous Driving in Uncontrolled Intersection

Decision-making in uncontrolled intersection is one of the main challenges in urban autonomous driving. This paper proposed a new decision-making framework in uncontrolled intersection based on the intention prediction method and Mixed Strategy Nash Equilibrium theory. The framework is a three-stage method: target vehicle motion prediction, driving mode decision, and motion planning. The driving intention (left turn, right turn, or go straight) of the target vehicle at the intersection can be predicted using the combination algorithm of GMM-HMM and SVM. According to the driving intention and road structure, the trajectory fitting module would use the Bezier curve to fit the predicted trajectory of a target vehicle. Furthermore, combined with trajectories of the ego vehicle, the S-T diagram is used to judge whether there is a spatio-temporal conflict point of the target vehicle and ego vehicle. If there is a conflict point of the target vehicle and ego vehicle, the driving mode (‘yield’ or ‘cross’) of the ego vehicle is selected by using the Mixed Strategy Nash Equilibrium theory. This method can not only avoid premature or unnecessary deceleration due to conservation but also avoid a collision or violent deceleration due to greed. According to the driving mode, the planning module uses the model predictive control algorithm to determine the optimal acceleration strategy. Have been verified by the vehicle test, it indicates that the proposed decision-making framework can make ego vehicles pass through the intersection safely and comfortably.

Stochastic Demand Response Management Using Mixed-Strategy Stackelberg Game

Stochastic solutions to the demand response (DR) problem enable utility companies to address the uncertainties in customer demand. In another words, stochastic DR solutions are recently developed considering storage systems, renewable resources, and the electricity market. However, a DR approach for uncertain response of consumers to an incentive-based DR problem has not been studied before. This article presents a new prospective stochastic DR approach by using a mixed-strategy Stackelberg game (MSG) between an aggregator and residential customers during peak load periods. We use real dataset collected by a local utility. We cluster customers based on customer baseline load (CBL), select one reference customer for each cluster, and model the MSG between that customer and the aggregator. We map the MSG to subgames and prove the equilibrium using mixed-strategy Nash Equilibrium. Our results show that the peak load can be shifted to off-peak hours for almost 40 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> of load, and players gain monetary rewards in return.

A Multi-Class Channel Access Scheme for Cognitive Edge Computing-Based Internet of Things Networks

Edge computing-based framework is capable of improving users’ quality of experience in cognitive Internet of Things (IoT) networks. To explore the advantages of this edge computing-based framework, possible offloading and processing delay resulting from computation bottlenecks, and the offloading latency caused due to inter-cell interference must be properly considered. This paper thus considered a multi-class channel access mechanism for cognitive edge computing-based IoT networks where IoT users were categorized based on their quality of experience requirements. Essential IoT devices are permitted to offload to the edge server at any time following the hybrid channel access model, while delay-tolerant IoT devices are only permitted to offload to the server when the channel is idle following the overlay channel access model. Analyses were obtained for transmission rate and offloading delay to demonstrate the performance of the proposed mechanism, while important metrics such as total offloading latency and total offloading cost were investigated. The total offloading costs were formulated through the mixed strategy Nash equilibrium method. The proposed mechanism achieves lower offloading latencies and costs for both type 1 and type 2 CUs when compared with existing methods. The obtained results showed that multi-class channel access mechanisms can reduce packet offloading delay in cognitive edge computing-based IoT networks.

Game Attack–Defense Graph Approach for Modeling and Analysis of Cyberattacks and Defenses in Local Metering System

We propose a game attack–defense graph (GADG) approach that integrates the attack–defense graph and the game theory to model and analyze cyberattacks and defenses in the local metering system (LMS). Different from previous studies concentrating on static analyses of cybervulnerabilities, the GADG method considers correlations among these vulnerabilities. Besides, to avoid the uncertainty brought by unilateral analysis, we introduce the game theory for the interaction analysis between the attacker and the defender. The mixed-strategy Nash equilibrium that they finally reach can serve as the input for the inference of system states. We also propose two eliminating algorithms to reduce the complexity of solving mixed-strategy Nash equilibrium on large-scale LMS. In the case study, our GADG model has been conducted on a real LMS, and the results prove its efficiency. This research aims to assist the power company in optimizing the allocation of limited defensive resources, especially specific in attacking steps. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article is motivated by the practical demand of protecting the cybersecurity of the local metering system (LMS), which is a critical subsystem of the smart grid and has been widely deployed to collect and transmit time-stamped information. Different from typical research works on local metering cybersecurity, which focuses on the analysis and identification of cybervulnerabilities, in this study, real experiments considering the interactive scenarios between the attacker and the defender are conducted on the LMS. Based on this, we developed the attack graph to clearly illustrate various cyberattack paths and their corresponding effects. We then applied the game theory model on the attack graph to obtain the optimal attack and defense strategy under each attack scenario. Based on this, the defender, i.e., power company, can optimally allocate limited defense resources among various attacking steps according to our model. The proposed model and method are demonstrated on one exemplar LMS used by our industrial partner China Southern Grid (CSG). Given the generalizability of the proposed model and method, they can be applied to other similar cyber–physical systems.

Market Exit and Minimax Regret

This paper shows how minimax regret sheds new light on an old economic topic, market-exit games. It focuses on wars of attrition, namely overcrowded duopoly markets where the strategic variable is the exit time. The only symmetric Nash equilibrium (NE) of the game studied is a mixed-strategy equilibrium that leads to a null expected payoff, i.e., the payoff a firm gets when it immediately exits the market. This result is not convincing, both from a behavioral and from a strategic viewpoint. The minimax regret approach that builds upon opposite regrets — exiting the market too late and exiting the market too early — is more convincing and ensures that both firms obtain a strictly positive expected payoff. Highlights Minimax regret behavior sheds new light on the optimal exit-times in overcrowded markets. Minimax regret behavior provides a new behavioral content to mixed strategies. In overcrowded duopolies, if the firms’ aim consists in minimizing regrets, then they get better payoffs than in the mixed-strategy NE.

Particle swarm optimization algorithm with proportional factor based on Nash equilibrium

Abstract Based on the global gradient-less stochastic search method of traditional particle swarm optimization algorithm, an improved particle swarm optimization algorithm for finding mixed-strategy Nash equilibrium is proposed in view of the shortcomings of prone particle overflow and unconstrained relationship between individual dimensions. In the improved evolution equation, the position function is improved to conform to the concept and properties of the combination of strategies, and the proportional factor is introduced to ensure that the numerical value of particle iteration is mapped to the solution space of normal form game. Through the test of several groups of classical examples, the results show that the algorithm not only improves efficiency and accuracy, but also solves the problems of divergence of particle and low efficiency of iteration. The experiments show that the algorithm has good practical performance.

Consensus Game: An Extension of Battle of the Sexes Game

This paper introduces “Consensus Game” as a multiplayer extension of the well-known “Battle of the Sexes” game to model situations where multiple parties (e.g., buyers–sellers) need to come to an agreement on a common issue in order for all players to be better off. We develop closed-form analytical expressions for the mixed-strategy Nash Equilibrium of a Consensus Game and thereafter, under a mildly restrictive assumption of completely mixed strategies, demonstrate the uniqueness of this equilibrium. We also show that in the constant-sum version of the Consensus Game, this unique Nash Equilibrium guarantees equal payoffs to all players. This result pertaining to equal payoffs as the only Nash Equilibrium suggests that in applications where multiple parties need to arrive at a consensus in order to share a common and fixed pool of a resource, an equal-utility distribution of the pool is the only stable solution that does not provide any party a unilateral incentive to deviate.

Corrigendum to “On the Existence of Pure and Mixed Strategy Nash Equilibrium in Discontinuous Games”

Corrigendum to “On the Existence of Pure and Mixed Strategy Nash Equilibrium in Discontinuous Games”

A Single-Adversary-Single-Detector Zero-Sum Game in Networked Control Systems*

This paper proposes a game-theoretic approach to address the problem of optimal sensor placement for detecting cyber-attacks in networked control systems. The problem is formulated as a zero-sum game with two players, namely a malicious adversary and a detector. Given a protected target vertex, the detector places a sensor at a single vertex to monitor the system and detect the presence of the adversary. On the other hand, the adversary selects a single vertex through which to conduct a cyber-attack that maximally disrupts the target vertex while remaining undetected by the detector. As our first contribution, for a given pair of attack and monitor vertices and a known target vertex, the game payoff function is defined as the output-to-output gain of the respective system. Then, the paper characterizes the set of feasible actions by the detector that ensures bounded values of the game payoff. Finally, an algebraic sufficient condition is proposed to examine whether a given vertex belongs to the set of feasible monitor vertices. The optimal sensor placement is then determined by computing the mixed-strategy Nash equilibrium of the zero-sum game through linear programming. The approach is illustrated via a numerical example of a 10-vertex networked control system with a given target vertex.

Malware propagation model for cluster-based wireless sensor networks using epidemiological theory.

Due to limited resources, wireless sensor network (WSN) nodes generally possess weak defense capabilities and are often the target of malware attacks. Attackers can capture or infect specific sensor nodes and propagate malware to other sensor nodes in WSNs through node communication. This can eventually infect an entire network system and even cause paralysis. Based on epidemiological theory, the present study proposes a malware propagation model suitable for cluster-based WSNs to analyze the propagation dynamic of malware. The model focuses on the data-transmission characteristics between different nodes in a cluster-based network and considers the actual application parameters of WSNs, such as node communication radius, node distributed density, and node death rate. In addition, an attack and defense game between malware and defending systems is also established, and the infection and recovery rates of malware propagation under the mixed strategy Nash equilibrium condition are given. In particular, the basic reproductive number, equilibrium point, and stability of the model are derived. These studies revealed that a basic reproductive number of less than 1 leads to eventual disappearance of malware, which provides significant insight into the design of defense strategies against malware threats. Numerical experiments were conducted to validate the theory proposed, and the influence of WSN parameters on malware propagation was examined.

On-ramp merging strategy for connected and automated vehicles based on complete information static game

Improper handling of vehicle on-ramp merging may hinder traffic flow and contribute to lower fuel economy, while also increasing the risk of collisions. Cooperative control for connected and automated vehicles (CAVs) has the potential to significantly reduce negative environmental impact while also improve driving safety and traffic efficiency. Therefore, in this paper, we focus on the scenario of CAVs on-ramp merging and propose a centralized control method. Merging sequence (MS) allocation and motion planning are two key issues in this process. To deal with these problems, we first propose an MS allocation method based on a complete information static game whereby the mixed-strategy Nash equilibrium is calculated for an individual vehicle to select its strategy. The on-ramp merging problem is then formulated as a bi-objective (total fuel consumption and total travel time) optimization problem, to which optimal control based on Pontryagin's minimum principle (PMP) is applied to solve the motion planning issue. To determine the proper parameters in the bi-objective optimization problem, a varying-scale grid search method is proposed to explore possible solutions at different scales. In this method, an improved quicksort algorithm is designed to search for the Pareto front, and the (approximately) unbiased Pareto solution for the bi-objective optimization problem is finally determined as the optimal solution. The proposed on-ramp merging strategy is validated via numerical simulation, and comparison with other strategies demonstrates its effectiveness in terms of fuel economy and traffic efficiency. • MS allocation is modeled as a two-player game, where Pontryagin's minimum principle is used for control law calculation. • Illustrating in detail mixed-strategy Nash equilibrium of the game for MS allocation. • Proposing a varying-scale grid search algorithm to search for candidated parameters in the cost function.

A Privacy Protection Method of Lightweight Nodes in Blockchain

Aiming at the privacy protection of lightweight nodes based on Bloom filters in blockchain, this paper proposes a new privacy protection method. Considering the superimposition effect of query information, node and Bloom filter are regarded as the two parties of the game. A privacy protection mechanism based on the mixed strategy Nash equilibrium is proposed to judge the information query. On this basis, a Bloom filter privacy protection algorithm is proposed when the probability of information query and privacy, not being leaked, is less than the node privacy protection. It is based on variable factor disturbance, adjusting the number of bits’ set to 1 in the Bloom filter to improve the privacy protection performance in different scenarios. The experiment uses Bitcoin transaction data from 2009 to 2019 as the test data to verify the effectiveness, reliability, and superiority of the method.

Frequency competition among airlines on coordinated airports network

Frequency competition is critical for a full-service airline in gaining market share, and adopting a proper strategy can improve an airline’s profits. This study proposes a new equilibrium programming model with flow balance to address frequency competition on airports network with time slot constraints. We first show that a pure-strategy Nash equilibrium may not always exist, and thus forming a pure strategy profile in frequency competition among airlines may naturally lead to deviation from current frequency. Therefore, we formulate the problem as a programming model with a mixed-strategy Nash equilibrium. To avoid shocks from dramatic frequency changes across the network, airlines tend to fine-tune frequencies on select segments during each adjustment. We propose a procedure to generate a computationally tractable amount of representative strategies from a finite set of feasible strategies to demonstrate mixed-strategy Nash equilibrium. We conduct an empirical analysis using an example in which industry profitability increased by as much as 7.89%. We then extend the model to formulate frequency competition among metal-neutral alliances. The results show that forming metal-neutral alliances can improve total industry profits by 10.59%. In particular, a sensitivity analysis with real data on the tolerance of flow imbalance demonstrates that deducting the potential costs due to the relaxation of flow balance between congested airports may earn additional total industry profits in a frequency competition.

Game-Theoretic Mode Scheduling for Dynamic TDD in 5G Systems

Dynamic time-division duplexing (TDD) enables independent uplink/downlink mode scheduling at each cell, based on the local traffic. However, this creates cross-interference among cells. Thus, the joint power allocation and scheduling problem becomes mixed-integer non-convex and turns out to be NP-hard. We propose a low-complexity and decentralized solution, where power allocation and scheduling are decoupled. First, power is allocated in a decentralized fashion, and then modes are scheduled by a non-cooperative game to achieve the mixed-strategy Nash equilibrium. We consider two possible approaches to compute the payoffs in the game, according to the cross-interference power model and the entailed communication overhead among cells. Simulation results are presented for an outdoor dense small-cell scenario, showing that our approaches outperform static TDD significantly.