Payoff Matrix Research Articles

Design and implementation of a game-based activity to promote learning of game theory

Game theory, an interdisciplinary field that explores interactions among entities in strategic scenarios, is paramount for understanding how individuals make decisions in interactive situations. This paper outlines the development and evaluation of a playful exercise to facilitate understanding fundamental concepts such as dominant and dominated strategies and Nash equilibrium, employing a 'design thinking' methodology. An empirical evaluation was conducted with 76 Industrial Engineering students from the Universidad Industrial de Santander, dividing them into four teams representing companies vying for customers. The teams assessed payoff matrices and made marketing decisions, utilizing a colored die to eliminate non-advantageous strategies. Findings indicate a significant increase in interest in game theory, with 86% of participants expressing a preference for this method compared to conventional classes. Furthermore, 81% desired to incorporate this activity into their regular courses, while 76% applied previous knowledge during the training. An average improvement of 17% in correct responses suggests that the playful exercise effectively facilitates the assimilation of key concepts in game theory among university students. This study underscores the potential of integrating active activities based on game theory to enrich the learning process in an academic context.

Research of the problem of fair distribution of fishing quotas by the methods of game theory

Game theory emerged as a science in the second half of the 20th century. It managed to prove itself well in the analysis of economic situations involving several subjects of economic activity (players), whose interests are completely or partially opposite. At the same time, in a number of cases, the solution of the game satisfied all players, but was not the most profitable (there was a Nash equilibrium), and in a number of other cases, it was possible to take into account the interests of all parties to the maximum (there was a Pareto optimal solution). The transfer of the principles of game theory to other areas turned out to have a number of difficulties associated, among other things, with the correct interpretation of strategies and gains of the parties in a conflict situation. For this reason, despite the obvious benefit from the possible application of game theory methods to problems of a fair distribution of quotas for catching fish and other seafood, this step has not been taken until recently. In this paper, we consider a scheme for applying the algorithms of the theory of bimatrix and cooperative games on the example of solving the problem of finding the percentage of the allowable catch of the black halibut in the Barents Sea for two countries participating in the catch and give a meaningful interpretation of the results. The basis for the calculations was real data collected by the Russian-Norwegian Fisheries Commission in recent decades to determine the proportions of the catch of the indicated fish species in the respective sea zones. Since not all components of the payoff matrices of the players are uniquely determined, it became possible to perform a parametric analysis of the mathematical model of the conflict situation both in the search for an equilibrium solution and in the implementation of the arbitration scheme. The work is an extended and supplemented version of the report [2].

Spatial Decision-Making under Uncertainties for Supporting the Prospection of Sites for Hybrid Renewable Energy Generation Systems

This research aims at developing methodological tools for decision makers (DMs) to determine the locations for generating sites for hybrid renewable energy systems, considering complementarity characteristics of the corresponding sources and their seasonal availability. The decision process on new sites starts using Geographic Information Systems for modeling relevant spatial criteria. The problem solution is associated with using the Slide-OWA operator, which permits one to control the inter-criteria compensation levels by adjusting pessimism and optimism parameters. After constructing multicriteria estimates for generating sites, the general scheme of multi-objective decision-making is applied in conditions of uncertainty since these estimates are subject to significant levels of uncertainty. Such a scheme is based on the possibilistic approach, which involves the construction of payoff matrices. At this point, it is possible that some alternatives cannot be distinguished based only on the criteria considered so far, or the DM may want to evaluate the alternatives from the point of view of additional considerations. For this purpose, the framework of multi-attribute decision-making models is applied by employing methods for preference modeling in a fuzzy environment. The present work results are applied to a case study of the Minas Gerais State in Brazil. These results play a strategic role for governments and investors in their decisions while considering the ability to meet a wide range of criteria.

Evolutionary Game Analysis of Low-Carbon Transformation of Construction Enterprises Under the "Dual Carbon" Target

Purpose: This study aims to investigate the impacts of different choices made by construction enterprises, government, and the public on the low-carbon transformation of Chinese construction enterprises in the context of the global climate crisis. Theoretical framework: The main theoretical foundation of this research is the theory of dynamic evolutionary games, it can aptly reflects the dynamic evolutionary process of strategy choices by various stakeholders during the promotion of low-carbon construction. Method/design/approach: This study adopts a mixed-method approach involving literature review, qualitative analysis, and quantitative analysis. Firstly, a payoff matrix is constructed for the three stakeholders under different strategy choices. Then, the conditions under which different strategies tend to reach equilibrium are discussed. Finally, the stability of equilibrium solutions in evolutionary game is analyzed. Results and conclusion: There are 7 equilibrium solutions that can be evolutionary stable strategies in specific situations. It is challenging for the system to achieve a progressively stable strategy when only one or two entities participate in low-carbon transformation. Research implications: This study focuses on the background of the Chinese government's "dual carbon" target. It takes an innovative approach by considering the stakeholders in the low-carbon building supply chain, including the government and the general public, and develops a tripartite evolutionary game model for the low-carbon transformation of construction enterprises. This research makes a significant contribution to the future development of low-carbon buildings in China. Originality/value:The results obtained in this study are unprecedented, innovative and relevant to the scientific community, in the context of sustainable development of low-carbon buildings.

Cultural tightness, trust, and power in enforcing tax compliance

The so-called Cultural Tightness–Looseness (CTL) concept intends to describe the socio-cultural foundation of societies by the strictness of social norms. In this paper, CTL is used to investigate its relevance to enforcing tax compliance. The CTL concept is formalised by structural payoff matrices for the confrontation between citizens and officials. Social norms and their impacts on the behaviour of the parties in a conflict are used to define several variants of tight and loose societies. The payoff matrices are employed to classify respective forms of societies. For a dynamic analysis, ‘power’ and ‘trust’, as policy measures of behaviour control, are introduced to determine which instrument or combination of instruments emerges as the most relevant and effective option in the respective societal setting. The result is that ‘trust’ emerges in tight societies as the more important instrument of behaviour control, whereas ‘power’ appears as the more crucial tool in loose societies. Therefore, it is concluded that power or the ‘cops and robbers’ approach is usually applied in loose societies to enforce tax compliance. By contrast, ‘trust’ via the provision of information and support for paying taxes can be more often used in tight societies.

A New Decomposition Linear Programming Model to Solve Zero Sum Two Person Matrix Game in Fully Fuzzy Trapezoidal Environment

This article targets to unriddle the problem of a non-cooperative fully fuzzified ’Zero Sum Two Person Matrix Game’ (ZSTPMG) with payoff matrix equipped with Trapezoidal fuzzy numbers (TrFNs). To achieve the target a unique and novel decomposition technique has been introduced. First, we develop two auxiliaries fully fuzzified linear programming problem (FFLPP) models for both the players and then we decompose these two FFLPP models into four linear programming (LP) models each, for both the players. These eight LP models are then solved by using the software TORA-2.0. The solutions of these eight LP models ascertain the optimal strategies and the optimal value of the fully fuzzified ZSTPMG for both the players. Our technique has an advantage over the existing ones as it can solve fully fuzzified ZSTPMG with all kind of TrFNs such as symmetric, asymmetric, positive or negative TrFNs. To establish this fact, the proposed methodology has been illustrated by taking three numericals equipped with various kinds of TrFNs.

Inhibition and activation of interactions in networked weak prisoner's dilemma.

In the framework of the coevolution dynamics of the weak prisoner's dilemma, inspired by prior empirical research, we present a coevolutionary model with local network dynamics in a static network framework. Viewing the edges of the network as social interactions between individuals, when individuals play the weak prisoner's dilemma game, they accumulate both payoffs and social interaction willingness based on a payoff matrix of the social interaction willingness we constructed. The edges are then inhibiting or activating based on the social interaction willingness of the two individuals, and individuals only interact with others through activated edges, resulting in local network dynamics in a static network framework. Individuals who receive more cooperation will be more likely to activate the edges around them, meaning they will participate in more social interactions. Conversely, individuals who receive more defects will do the opposite. Specifically, we investigate the evolutionary dynamics of cooperation under different levels of sensitivity to social interaction willingness and the temptation to defect. Through the simulation, we find that sparse cooperator clusters can expand greatly when social interaction sensitivity and temptation to defect are low. In contrast, dense cooperator clusters form rapidly in a high social interaction sensitivity, which protects the cooperation from high temptation.

The Evolution of Cooperation in Multigames with Uniform Random Hypergraphs

How to explain the emergence of cooperative behavior remains a significant problem. As players may hold diverse perceptions on a particular dilemma, the concept of multigames has been introduced. Therefore, a multigame is studied within various binary networks. Since group structures are common in human society and a person can participate in multiple groups, this paper studies an evolutionary multigame with high-order interaction properties. For this purpose, a uniform random hypergraph is adopted as the network structure, allowing players to interact with all nodes in the same hyperedge. First, we investigate the effect of the multigame payoff matrix differences on the evolution of cooperation and find that increasing the differences in the payoff matrix promotes cooperation on the hypergraph network. Second, we discover that an increase in the average hyperdegree of the hypergraph network promotes network reciprocity, wherein high-hyperdegree nodes influence surrounding nodes to form a cooperator cluster. Conversely, groups with a low hyperdegree are more susceptible to betrayal, leading to a decline in cooperation.

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

Abstract 3164: Disulfiram re-sensitizes ER+ breast cancer cells resistant to chemotherapy through an evolutionary double-bind

Abstract Most ER+ metastatic breast cancer patients develop resistance to endocrine therapy and are also typically cross-resistant to chemotherapy, resulting in disease progression. However, chemotherapy resistant lines have been shown to acquire cross-sensitivity to therapies such as disulfiram, a drug used to treat alcoholism. Because of this cross-sensitivity, we hypothesized the existence of an “evolutionary double-bind”, a situation where treatment with one drug improves the response to a different agent. This can occur when resistance mechanisms rewire key proliferative pathways, creating potentially targetable vulnerabilities to be exploited. While recent studies have applied theoretical ecological and evolutionary principles in understanding cancer progression, it remains a new and exciting endeavor to quantify effective evolutionary games in refractory cancer to design alternative treatment strategies that induce an evolutionary double-bind. Here, we developed ER+ breast cancer cell lines sensitive and resistant to paclitaxel or doxorubicin to test the cross-sensitivity and efficacy of chemotherapy and disulfiram as an alternative treatment strategy. To quantify this cross-sensitivity of chemotherapy and disulfiram and use the double-bind to mitigate resistance, we also developed a game-theoretic mathematical modeling framework with parametrization performed using the “evolutionary game assay” technique, a direct measurement of payoff matrices that specify evolutionary games measured in cancer. Spheroids of varied initial frequencies of sensitive and resistant cells (100% monoculture; 75/25, 50/50, and 25/75 sensitive and resistant coculture) were treated with chemotherapy and disulfiram. The mathematical model was parameterized from the in vitro experimental data and as expected, treating with chemotherapy selects for resistant cells. However, disulfiram alone shifts the selection dynamics to favor sensitive cells. These modeling results predict the existence of a double-bind between chemotherapy and disulfiram where disulfiram selects for the sensitive subpopulation, which then re-sensitizes the spheroids to subsequent administration of chemotherapy. Importantly, the model predicts that the double-bind is non-symmetric and does not exist when treatments are given in combination. These predictions suggest the importance of timing/dosing strategies in which several schemes were further validated in vitro, confirming the superior efficacy of chemotherapy followed by disulfiram in blocking resistance emergence. This project integrates experimental data with mathematical modeling to identify a treatment double-bind in ER+ breast cancer. The integrative experimental-mathematical framework developed here has broad applicability in designing evolution-based or adaptive therapy treatment strategies across a wide range of cancer types. Citation Format: Rena Emond, Jeffrey West, Alexander Anderson, Andrea H. Bild. Disulfiram re-sensitizes ER+ breast cancer cells resistant to chemotherapy through an evolutionary double-bind [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 3164.

Performance Analysis of MIMO Information Time-Delay System Under Bandwidth, Cyber-Attack, and Gaussian White Noise

The performance limitations of multi-input–multi-output (MIMO) information time-delay system (ITDS) due to bandwidth, additive white Gaussian noise (AWGN), and cyber-attacks are investigated in this work. The game theory methods are used to analyze the impact of intrusion detection systems on systems performance in order to deal with network attacks under incomplete information. The expression of performance limitations is obtained by using the spectrum decomposition technique, inner–outer factorization, and partial factorization techniques under the two-degree-of-freedom (2DOF) compensator. The results demonstrate that the performance limitations of the system are not only influenced by the inherent properties, including time delay, nonminimum phase (NMP) zeros, and unstable zeros and their directions of the given object but are also influenced by the communication parameters, such as AWGN and bandwidth. In addition, the results show that under the cyber-attacks with incomplete information, the performance is only influenced by the payoff matrix of the defense, and not by the payoff matrix of the offense. Finally, the theoretical results are verified by performing simulations.

Evolution of parental roles in phase portraits of bimatrix asymmetric games

In this paper, we address the evolutionary dynamic of parental roles using game theory. The main purpose of the article was to expand a classical list of evolutionary dynamic parental conflicts by adding some important cases which hitherto have not been intensively studied. Our models are apt to deliver some novel insights into the evolution of parental care. We also introduced several hypothetical events that served as illustrations of an arising alteration in cost-benefits for both parents and simulated a subsequent evolutionary endpoint. Our models revealed that evolutionary outcomes for reproductive decisions of both parents could be completely predicted by certain payoff matrices, which serve as proxies for a Darwinian fitness gain. In this sense, the result of a frequency-dependent selection on reproductive traits would inevitably depend on fitness costs and benefits arising for both parents in various circumstances. We demonstrated that population division could be a plausible evolutionary consequence for any human mating game where ‘reproductive defection’ represents the best response to any action by the reproductive opponent. We conclude that future evolutionary studies of human reproductive behavior should be more oriented on estimating a sex-biased asymmetry in potential fitness gains obtained by cooperative and deceptive parents in diverse environments and cultures.

Research on Trajectory Tracking Control of Driverless Electric Formula Racing Car Based on Game Theory

A game theory based trajectory tracking control method is studied for the dual-objective optimization problem of trajectory tracking the accuracy and driving stability of driverless electric formula racing cars in high-speed trajectory conditions. The general control strategy and the model predictive controller based on the evolutionary game between the two players are designed to optimize their own decisions to achieve the optimal payoff for themselves, and to obtain the optimal solution to the dual-objective optimization problem, by taking the dual objectives of trajectory tracking accuracy and driving stability as the two players in the game. Considering the influence of the dynamic environment, the time-varying interactive game mechanism between two plays is introduced, the game payoff matrix is established, the weights of each subject are determined, and a dynamic replication system is constructed by weight evolution to find the optimal equilibrium strategy for the model prediction controller. The simulated results show that the designed controller can play a significant role in optimizing the trajectory tracking accuracy and driving stability compared to a single model predictive controller under different speed tracking conditions.

Half a century of evolutionary games: a synthesis of theory, application and future directions.

Half a century of evolutionary games: a synthesis of theory, application and future directions.

Research on the Game Mechanism of Rational Outsourcing Database

While outsourcing databases are widely used, their security is also facing great challenges. In order to reduce the validation and communication overheads of outsourcing solutions, this paper designs a game mechanism for rational outsourcing databases based on game theory, setting incentives and defining a payoff matrix to ensure that each rational participant will only maintain an honest strategy so as not to contradict the self-interest nature and achieve a Nash equilibrium. Finally, through simulation experiments, this paper proves that the use of utility functions to secure data can reduce the user's verification overhead and communication overhead, which has important research value.

Cooperators Pay More Attention to the Outcome of Mutual Cooperation in the One-Shot Prisoner’s Dilemma Game: Empirical Evidence From an Eye-Tracking Study

Based on the dual-process theory, we hypothesized that cooperators in the one-shot prisoner’s dilemma game would exhibit intuitive decision-making. We also hypothesized that an experimental manipulation that may prevent intuitive cooperation (in this study, inverting the payoff matrix) would result in more protracted decision-making and thus lower cooperation rates. To examine these hypotheses, we designed an experiment using an eye-tracking device. In one condition, the outcome of mutual cooperation in the one-shot prisoner’s dilemma game was presented in the upper left-hand corner explaining the result of mutual cooperation first, as in many previous studies (control condition). To suppress the function of intuitive cooperation, we also set the inverted payoff matrix condition in which the positions of the outcomes of mutual cooperation and mutual non-cooperation in the payoff matrix were inverted, explaining the result of mutual non-cooperation first. Although our results did not show the expected differences between the two conditions, it was suggested that cooperators’ decision-making time was shorter in the control condition and that cooperators consistently paid more attention to the outcome of mutual cooperation regardless of the conditions. These results are discussed in relation to the intuitive cooperation model and the social exchange heuristic.

Analysis of the n-person noncooperative supermodular multiobjective games

This paper investigates a class of noncooperative multiobjective games in the supermodular form played by a finite number of players. We prove the existence of two different equilibria of those games, namely, the Pareto equilibrium and the weighted Nash equilibrium. Using Python programming, we apply a genetic algorithm to obtain the Pareto equilibrium and a modified payoff matrix to obtain the weighted Nash equilibrium. We demonstrate the application of our result in an inventory game problem.

Eco-evolutionary cyclic dominance among predators, prey, and parasites

Predator–prey interactions are one of ecology’s central research themes, but with many interdisciplinary implications across the social and natural sciences. Here we consider an often-overlooked species in these interactions, namely parasites. We first show that a simple predator–prey–parasite model, inspired by the classical Lotka–Volterra equations, fails to produce a stable coexistence of all three species, thus failing to provide a biologically realistic outcome. To improve this, we introduce free space as a relevant eco-evolutionary component in a new mathematical model that uses a game-theoretical payoff matrix to describe a more realistic setup. We then show that the consideration of free space stabilizes the dynamics by means of cyclic dominance that emerges between the three species. We determine the parameter regions of coexistence as well as the types of bifurcations leading to it by means of analytical derivations as well as by means of numerical simulations. We conclude that the consideration of free space as a finite resource reveals the limits of biodiversity in predator–prey–parasite interactions, and it may also help us in the determination of factors that promote a healthy biota.

Momentum Ranking Function of Z-Numbers and its Application to Game Theory

After Zadeh introduced the concept of z-number scientists in various fields have shown keen interest in applying this concept in various applications. In applications of z-numbers, to compare two z-numbers, a ranking procedure is essential. While a few ranking functions have been already proposed in the literature there is a need to evolve some more good ranking functions. In this paper, a novel ranking function for z-numbers is proposed- "the Momentum Ranking Function"(MRF). Also, game theoretic problems where the payoff matrix elements are z-numbers are considered and the application of the momentum ranking function in such problems is demonstrated.

STSIR: An individual-group game-based model for disclosing virus spread in Social Internet of Things

Social Internet of Things (SIoT) with deep integration of Internet of Things and social networks has become a target of a large number of hackers who attempt to spread viruses for breaching data confidentiality and service reliability. Therefore, exposing the law of virus spread with social characteristics and addressing historical dependence of infection and recovery rates in an SIoT are urgent problems to be solved at present. To this end, we propose a novel virus spread model (STSIR) based on an epidemic theory's analysis framework and individual-group game theory, which more reasonably describes viruses spread among devices considering people behavior. Aiming at the characteristics of SIoTs including limited social distance and dynamic number variation of people and devices, we adopt and improve the traditional epidemic model SIR to reveal the form of viruses continuously spreading to neighbor nodes. We then introduce an individual-group game to establish the attack and defense model between infected SIoT nodes and susceptible SIoT nodes, in order to not only obtain the mixed Nash equilibrium solution by using a payoff matrix but also solve the dependence of the infection and recovery rates on historical experience. Further, we establish differential equations to represent the model STSIR, which are the basis of proving the existence of model equilibrium points and analyzing stability mathematically. Finally, the effectiveness of the model STSIR in curbing virus spread is verified by simulating two equilibrium points. Under the same conditions in an SIoT, the model STSIR reduce viruses by ∼45% more than the model SIS, and saves stabilization time cost by ∼66.7% compared with the model SIR, which proves that the model STSIR is obviously more effective.