Payoff Matrix Research Articles

General features of Nash equilibria in combinations of elementary interactions in symmetric two-person games

Two-person games are used in many multi-agent mathematical models to describe pair interactions. The type (pure or mixed) and the number of Nash equilibria affect fundamentally the macroscopic behavior of these systems. In this paper, the general features of Nash equilibria are investigated systematically within the framework of matrix decomposition for n strategies. This approach distinguishes four types of elementary interactions that each possess fundamentally different characteristics. The possible Nash equilibria are discussed separately for different types of interactions and also for their combinations. A relation is established between the existence of infinitely many mixed Nash equilibria and the zero-eigenvalue eigenvectors of the payoff matrix.

Research on Optimization of Array Honeypot Defense Strategies Based on Evolutionary Game Theory

Honeypot has been regarded as an active defense technology that can deceive attackers by simulating real systems. However, honeypot is actually a static network trap with fixed disposition, which is easily identified by anti-honeypot technology. Thus, honeypot is a “passive” active defense technology. Dynamic honeypot makes up for the shortcomings of honeypot, which dynamically adjusts defense strategies with the attack of hackers. Therefore, the confrontation between defenders and attackers is a strategic game. This paper focuses on the non-cooperative evolutionary game mechanism of bounded rationality, aiming to improve the security of the array honeypot system through the evolutionarily stable strategies derived from the evolutionary game model. First, we construct a three-party evolutionary game model of array honeypot, which is composed of defenders, attackers and legitimate users. Secondly, we formally describe the strategies and revenues of players in the game, and build the three-party game payoff matrices. Then the evolutionarily stable strategy is obtained by analyzing the Replicator Dynamics of various parties. In addition, we discuss the equilibrium condition to get the influence of the number of servers N on the stability of strategy evolution. MATLAB and Gambit simulation experiment results show that deduced evolutionarily stable strategies are valid in resisting attackers.

Role of synergy and antagonism in designing multidrug adaptive chemotherapy schedules.

Chemotherapeutic resistance via the mechanism of competitive release of resistant tumor cell subpopulations is a major problem associated with cancer treatments and one of the main causes of tumor recurrence. Often, chemoresistance is mitigated by using multidrug schedules (two or more combination therapies) that can act synergistically, additively, or antagonistically on the heterogeneous population of cells as they evolve. In this paper, we develop a three-component evolutionary game theory model to design two-drug adaptive schedules that mitigate chemoresistance and delay tumor recurrence in an evolving collection of tumor cells with two resistant subpopulations and one chemosensitive population that has a higher baseline fitness but is not resistant to either drug. Using the nonlinear replicator dynamical system with a payoff matrix of Prisoner's Dilemma (PD) type (enforcing a cost to resistance), we investigate the nonlinear dynamics of this three-component system along with an additional tumor growth model whose growth rate is a function of the fitness landscape of the tumor cell populations. A key parameter determines whether the two drugs interact synergistically, additively, or antagonistically. We show that antagonistic drug interactions generally result in slower rates of adaptation of the resistant cells than synergistic ones, making them more effective in combating the evolution of resistance. We then design evolutionary cycles (closed loops) in the three-component phase space by shaping the fitness landscape of the cell populations (i.e., altering the evolutionary stable states of the game) using appropriately designed time-dependent schedules (adaptive therapy), altering the dosages and timing of the two drugs. We describe two key bifurcations associated with our drug interaction parameter which help explain why antagonistic interactions are more effective at controlling competitive release of the resistant population than synergistic interactions in the context of an evolving tumor.

Evolutionary Dynamics of Zero-sum Games with Degenerate Payoff Matrix and Bisexual Population

In this paper we consider the quadratic stochastic operators describing evolution of a bisexual population. We establish correlation between such operators and evolutionary games, namely demonstrate that Volterra quadratic stochastic operator with degenerate payoff matrix is non-ergodic and corresponding evolutionary game is rock-paper-scissors game. To prove this statements we study the asymptotic behavior of trajectories of the Volterra quadratic stochastic operators with the non-hyperbolic fixed points.

Game dynamics of emotion evolution based on the Moran process.

In this paper, emotions are classified into four types, namely, respect for the strong, envying the strong, sympathy for the weak, and bullying the weak. The corresponding relationship between the four emotion types and the two behaviors of competition and cooperation is then defined. The payoff matrices of the game based on emotions are obtained and the evolutionary dynamics of the four emotion types in a finite population based on the Moran process are studied. Next, we derive the absorption probabilities of a 4×4 symmetric evolutionary game of the population. The influence of the payoff parameters and the natural selection intensity on the result of the group evolution are then analyzed. The calculations indicate that there are differences in the absorption probabilities of the four absorption states of the system. At a steady state, individuals of the types envying the strong and bullying the weak have the highest probability of occupying the entire population, and individuals of the type respect for the strong and sympathy for the weak have the lowest one. By comparing the level of cooperation and average payoffs at a steady state, we observe that the level of cooperation and average payoffs based on the proposed model are better than those of the prisoner's dilemma game with two behaviors. Therefore, emotional evolution can promote cooperation and achieve better group fitness.

Effects of the quarantine on the individuals’ risk of Covid-19 infection: Game theoretical approach

In this study, we analyze the general or self-quarantine effects to the spread of the first wave of Covid-19 pandemic in the view of the game-theoretical approach. As in some other applications of game theory in different aspects of the literature, we focus on only the application of game theory to present the effects of quarantine during the three different stages -the start, the spread, the end- of the pandemic. We first choose three countries such as South Korea for self-quarantine, Italy, and Turkey for general quarantine during the analysis of the different stages of the spread. Then, we present a formula that will be an important tool for the creation of the payoff matrices and give the general procedure for the creation of the payoff matrix for each stage of the pandemic process. After that, we generate the payoff bimatrix for each stage of the pandemic by using the average of the daily diagnosis number/number of tests for each country. Moreover, we try to find the optimal strategy of the game. Additionally, to determine the necessity of the continuity of the quarantine, we use the repeated game approach in our analysis, as well. Therefore, we convert the game only for the spread stage to the repeated game for each country. Finally, we obtain the Nash equilibrium of all games for each level of the pandemic. The results show that the quarantine has important effects to be infected or not, and the spread of the pandemic at each level. In addition to these analysis results, we compare the death rates of the considered countries and show that the results are almost parallel to that are obtained for the quarantine requirement of each country by game-theoretical approaches.

A method to solve two-player zero-sum matrix games in chaotic environment

This research article proposes a method for solving the two-player zero-sum matrix games in chaotic environment. In a fast growing world, the real life situations are characterized by their chaotic behaviors and chaotic processes. The chaos variables are defined to study such type of problems. Classical mathematics deals with the numbers as static and less value-added, while the chaos mathematics deals with it as dynamic evolutionary, and comparatively more value-added. In this research article, the payoff is characterized by chaos numbers. While the chaos payoff matrix converted into the corresponding static payoff matrix. An approach for determining the chaotic optimal strategy is developed. In the last, one solved example is provided to explain the utility, effectiveness and applicability of the approach for the problem.Abbreviations: DM= Decision Maker; MCDM = Multiple Criteria Decision Making; LPP = Linear Programming Problem; GAMS= General Algebraic Modeling System.

A model for joint decision between production rate and clinker export proportion using cooperative game theory approach

This study aims to provide a decision making model to determine clinker export proportion. Cooperative game theory approach used to build the model. Players in this game are manager of production & material planning, and manager of cement production. Payoff matrix of the game composed using Mixed Integer Linear Programming (MILP) based equation. Expected contribution of this study is to provide a decision making model in determining acceptable proportion between clinker amount to export, and clinker amount to be grinded. Furthermore, this study expected to support revenue maximizing in the company.

A Collaborative Multiagent Reinforcement Learning Method Based on Policy Gradient Potential.

Gradient-based method has been extensively used in today's multiagent reinforcement learning (MARL). In a gradient-based MARL algorithm, each agent updates its parameterized strategy in the direction of the gradient of some performance index. However, studies on the convergence of the existing gradient-based MARL algorithms for identical interest games are quite few. In this article, we propose a policy gradient potential (PGP) algorithm that takes PGP as the source of information for guiding the strategy update, as opposed to the gradient itself, to learn the optimal joint strategy that has a maximal global reward. Since the payoff matrix and the joint strategy are often unavailable to the learning agents in reality, we consider the probability of obtaining the maximal reward as the performance index. Theoretical analysis of the PGP algorithm on the continuous model involving an identical interest repeated game shows that if the component action of every optimal joint action is unique, the critical points corresponding to all optimal joint actions are asymptotically stable. The PGP algorithm is experimentally studied and compared against other MARL algorithms on two commonly used collaborative tasks-the robots leaving a room task and the distributed sensor network task, as well as a real-world minefield navigation problem where only local state and local reward information are available. The results show that the PGP algorithm outperforms the other algorithms in terms of the cumulative reward and the number of time steps used in an episode.

Maximizing cooperation in the prisoner's dilemma evolutionary game via optimal control.

The prisoner's dilemma (PD) game offers a simple paradigm of competition between two players who can either cooperate or defect. Since defection is a strict Nash equilibrium, it is an asymptotically stable state of the replicator dynamical system that uses the PD payoff matrix to define the fitness landscape of two interacting evolving populations. The dilemma arises from the fact that the average payoff of this asymptotically stable state is suboptimal. Coaxing the players to cooperate would result in a higher payoff for both. Here we develop an optimal control theory for the prisoner's dilemma evolutionary game in order to maximize cooperation (minimize the defector population) over a given cycle time T, subject to constraints. Our two time-dependent controllers are applied to the off-diagonal elements of the payoff matrix in a bang-bang sequence that dynamically changes the game being played by dynamically adjusting the payoffs, with optimal timing that depends on the initial population distributions. Over multiple cycles nT (n>1), the method is adaptive as it uses the defector population at the end of the nth cycle to calculate the optimal schedule over the n+1st cycle. The control method, based on Pontryagin's maximum principle, can be viewed as determining the optimal way to dynamically alter incentives and penalties in order to maximize the probability of cooperation in settings that track dynamic changes in the frequency of strategists, with potential applications in evolutionary biology, economics, theoretical ecology, social sciences, reinforcement learning, and other fields where the replicator system is used.

Quadratically Optimal Bi-Matrix Games

In this paper, we introduce the class of quadratically optimal (bi-matrix) games, which are bi-matrix games whose set of equilibrium points contain all pairs of probability vectors which maximize the expected pay-off of some pay-off matrix. We call the equilibrium points obtained in this way, quadratically optimal equilibrium points. We prove the existence of quadratically optimal equilibrium points of identical bi-matrix games, i.e. bi-matrix games for which the two pay-off matrices are equal, from which it easily follows that weakly potential bi-matrix games (a generalization of potential bimatrix games) are quadratically optimal. We also show that those weakly potential square bi-matrix games which have potential matrices that are two-way matrices are quadratically and symmetrically solvable games (i.e. there exists a square pay-off matrix whose expected pay-off maximizing probability vectors subject to the requirement that the two probability vectors (row probability vector and column probability vector) being equal) are equilibrium points of the bi-matrix game. We show by means of an example of a 2×2 identical symmetric potential bi-matrix game that for every potential matrix of the game, the set of pairs of probability distributions that maximizes the expected pay-off of the potential matrix is a strict subset of the set of equilibrium points of the potential game.

Payoff Matrix’s Effect on Trust Game Paradigm

This report first sorted out the definition of trust and found that researchers did not have a unified definition of trust. But trust includes two core characteristics: dependence and risk. Then we introduce the measurement methods of trust, including the trust game paradigm and its variants, in which the binary trust game paradigm and the three parameters in the payoff matrix of trust game paradigm are introduced; those three parameters are cost, benefit and temptation. Since previous studies exploring the influence of these three parameters on people’s trust behavior in the trust game paradigm have inconsistent results, some researchers reported that it is because the absolute values of these three parameters are used when analyzing the results. Therefore, based on these three parameters, some researchers proposed a risk index, a temptation index and a cooperation index, and found that the three standardized parameters can predict people’s trust behavior better.

On solving a fuzzy game problem using hexagonal fuzzy numbers

Problems with game theory can be found in a variety of fields, including economics, business administration, sociology, political science, military operations, and so on. Fuzzy set theory is an excellent foundation for researching the uncertainty in games. Payoffs are represented by fuzzy shapes in this type of game. Consider the payoff matrix in this research paper as a two-player zero-sum game with infinite values. Fuzzy Hexagonal Numbers are said to be the source of all imprecise values. An approach to solving issues Pascal's triangular graded mean integration was considered to find a solution to the dilemma of the Fuzzy game. By applying Pascal's triangle graded mean integration, we can convert the fuzzy valued game problem into a crisp valued game problem that can be solved using the conventional approach by changing the payoffs.

Choosing a servicing company’s strategy while interacting with freight owners at the road transport market

Purpose. To develop the methodology for choosing the strategies of freight forwarding companies in the situation of interaction with freight owners as customers of forwarding services. Methodology. The game-theoretical approach is used to formalize the conflict situation between a freight forwarding company and a cargo owner. A set of services proposed by forwarders is used as the base in order to represent possible strategies of a forwarder as the vector of probabilities that the respective services are provided for a client. The strategies of the cargo owner are represented as a binary variable that shows whether the client uses the provided services or not. The payoff function for a forwarder is defined as the companys profit and the clients payoff function as fee paid for forwarding services. To determine the influence of the demand parameters on the forwarders optimal strategies, the demand for transport services is represented as a flow of requests characterized by two numeric parameters delivery distance and consignment weight. Findings. The conducted experimental studies have shown that as a result of the use of the proposed methodology, the optimal strategy of a forwarding company can always be determined from the payoff matrix. The performed simulation experiment allowed us to state that in most cases the forwarders optimal strategy is mixed (the technological and commercial services should be provided with the given probability while servicing the flow of requests from freight owners). Originality. The studies on the influence of the request flow parameters on the probabilities of choosing the elementary strategies are carried in the paper for the first time. Practical value. The proposed methodology can be used as the basic tool for supporting decisions of freight forwarders while servicing the cargo owners at the market of road freight transportation.

Disturbance Decoupling for Gradient-Based Multi-Agent Learning With Quadratic Costs

Motivated by applications of multi-agent learning in noisy environments, this paper studies the robustness of gradient-based learning dynamics with respect to disturbances. While disturbances injected along a coordinate corresponding to any individual player's actions can always affect the overall learning dynamics, a subset of players can be disturbance decoupled---i.e., such players' actions are completely unaffected by the injected disturbance. We provide necessary and sufficient conditions to guarantee this property for games with quadratic cost functions, which encompass quadratic one-shot continuous games, finite-horizon linear quadratic (LQ) dynamic games, and bilinear games. Specifically, disturbance decoupling is characterized by both algebraic and graph-theoretic conditions on the learning dynamics, the latter is obtained by constructing a game graph based on gradients of players' costs. For LQ games, we show that disturbance decoupling imposes constraints on the controllable and unobservable subspaces of players. For two player bilinear games, we show that disturbance decoupling within a player's action coordinates imposes constraints on the payoff matrices. Illustrative numerical examples are provided.

Archetypal games generate diverse models of power, conflict, and cooperation

Interdependence takes many forms. We show how three patterns of power generate diverse models for understanding dynamics and transformations in social-ecological systems. Archetypal games trace pathways that go beyond a focus on a few social dilemmas to recognize and understand diversity and complexity in a landscape of social situations, including families of coordination and defection problems. We apply the extended topology of two-person two-choice (2 × 2) games to derive simple archetypes of interdependence that generate models with overlapping opportunities and challenges for collective action. Simplifying payoff matrices by equalizing outcome ranks (making ties to show indifference among outcomes) yields three archetypal games that are ordinally equivalent to payoff structures for independence, coordination, and exchange, as identified by interdependence theory in social psychology. These three symmetric patterns of power combine to make an asymmetric archetype for zero-sum conflict and further structures of power and dependence. Differentiating the ranking of outcomes (breaking ties) transforms these primal archetypes into more complex configurations, including intermediate archetypes for synergy, compromise, convention, rivalry, and advantage. Archetypal models of interdependence, and the pathways through which they generate diverse situations, could help to understand institutional diversity and potential transformations in social-ecological systems, to distinguish between convergent and divergent collective action problems for organizations, and to clarify elementary patterns of power in governance.

Evolutionary Game of Emergency Evacuation After an Earthquake at a University: How to Promote Orderly Evacuation

Safe and effective evacuation is very important to reduce casualties after earthquakes. Given that evacuation is a checking process rather than a preselection process, evolutionary game theory can be used to better understand the crowd dynamics by analyzing the imitation effects in terms of individual interactions during evacuation. The key purpose of this paper was to construct an evolutionary game model of emergency evacuation of a teaching building at a university after an earthquake to search for the specific constraints necessary to reach the expected collective behaviors for orderly evacuation. We first analyzed the evacuation scenarios, including the environment of the teaching building, emergency degree of the earthquake and behaviors of the crowd. The related game players in the evacuation process were divided into three types, impulsive evacuees, calm evacuees and university staffs, and their payoff matrix was constructed according to the evacuation scenarios. Then, an evolutionary game model via the replicated dynamic system was created, and stable strategies were discussed. Next, we simulated eighteen evacuation scenarios with adjustment of the guiding authoritativeness, emergency degree, and group-oriented coefficient; the simulation results showed that the worst stable strategies can be avoided and the expected stable strategies can be obtained by adjusting the constraints, which are closely related to the corresponding management measures. Finally, the performance of the proposed model was verified by comparisons with the real videos and the conventional strategies.

Virtuality as a precondition and necessity for the mankind development

Objective of the study: The research aims at exploring the way of interaction between virtual and physical reality and defining the approaches to analyze consequences of the virtuality penetration to the real world or vice versa.Methodology / approach: We apply a game theoretic method based on payoff matrix to show the essence of interaction between virtual and physical worlds; an internalization approach for solving optimization problem; graphical visualization of the interaction by using cones movement approach and analysis of cost functions to develop an equation of the city’s economy and to rebuild a model of optimal city size.Originality / Relevance: The majority of economic approaches have a linear-minded nature and only some take into account technological aspect. We state that the solution lies in consideration the costs as function of positive and negative exponential externalities. The proposed cost-based approach allows finding the optimal distribution of the time staying in the virtual and physical space.Main results: Obtained results of the research proved the following assumptions: interaction with new reality has changed us; a human is irrational in absolute majority; virtuality is a threat and an opportunity to the human. We confirmed that the nature of interaction between physical and virtual reality should takes the form of cooperation to ensure the mankind development.Theoretical / methodological contributions: It is one of the first attempts to develop economic theories and methods by extending them with variables of virtual reality.Social / management contributions: The study presents a broader view on human who exists on the fringes of physical and virtual reality.

IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations.

Evolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix. We present a software package to aid in the construction, analysis and visualization of three-strategy matrix games. The IsoMaTrix package computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations on a lattice grid. IsoMaTrix computes fixed points, phase flow, trajectories, (sub)velocities and uncertainty quantification for stochastic effects in spatial matrix games. We describe a result obtained via IsoMaTrix's spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor. The code is available at: https://github.com/mathonco/isomatrix. Supplementary data are available at Bioinformatics online.

Cooperative decision-making on fiscal and monetary policy in Iraq using the prisoner’s dilemma

This paper investigates the interaction between fiscal and monetary policy in Iraq after 2003 using the prisoner’s dilemma.The paper aims to determine the best form of coordination between these policies to achieve their goals; payoff matrix for both policies was constructed. To achieve the purpose, the quantitative approach was applied using several methods, including regression, building payoff matrices and decision analysis using a number of software.The results of the monetary policy payment function show that inflation rate has an inverse relationship with the auctions of selling foreign currency and a positive relationship with the government’s activity, while the fiscal policy function shows that real growth is positively related to price levels (the inverted Phillips curve) and correlates with the government’s activity. After using the Gambit Solution to determine the Nash balance, which is achieved through the expansion strategies of both policies to confirm the results, the Promethee-Gaia method was used for multi-criteria decision making. When the two policies interact with similar forces (50% each), the best decision is one of the expansionary strategies that help achieve their main objectives in the short and long term, represented by price stability and economic growth.The main conclusion is that the best way to achieve the goals of economic policy in Iraq is that the coordination of procedures between the two policies should be expansionary, since the Iraqi economy needs to be stimulated due to the under-exploitation of many its sectors, such as agriculture and industry.