Three-player Game Research Articles

The Mercurial Commitment: Revisiting the Unintended Consequences of Military Humanitarian Intervention and Anti-Atrocity Norms

Abstract The debate on whether military humanitarian intervention and anti-atrocity norms, such as the responsibility to protect, cause perverse incentives, and provocative violence by a rebel group, has yet to reach a consensus. Specifically, existing theories are unable to fully explain why “emboldened” rebel groups provoke the government in some cases but not others. This paper reconciles this unresolved debate by arguing that it is not anti-atrocity norms but the temporary feasibility of humanitarian intervention that induces the potential rebel group to actually provoke the government. The inherent temporariness of the feasibility of large-scale intervention causes a commitment problem: An expectation of a temporary intervention shifts the balance of power toward a potential rebel and opens a quickly closing window of opportunity. A three-player game shows that, among other findings, a higher probability of humanitarian intervention alleviates the potential rebel’s incentive to fight, implying that the criticism of anti-atrocity norms is a “false charge.” The model also (i) explains why weaker groups can rebel in the shadow of external intervention and (ii) presents a novel implication that lowering the intervention cost can exacerbate the adverse incentive when a domestic commitment problem caused by the consolidation of government power is not severe.

A three-player game model for promoting enterprise green technology innovation from the perspective of media coverage.

The objective of this study was to explore the game relationship among enterprise, the government, and the public under the new media environment, so as to provide decision-making reference for improving enterprise green technology innovation and promoting economy high-quality development with new media participation. This study constructs a three-subject evolutionary game model of enterprise, government, and public based on multi-agent relationship analysis and evolutionary game theory. In addition, the derivation of an evolutionary equilibrium strategy and numerical simulation analysis is carried out to comprehensively explore the evolution trajectory of green technology innovation system under the new media environment. (1) The system may have four stable evolutionary strategies: (1,0,0), (0,0,1), (1,0,1), and (1,1,1). (2) The initial strategy probability of various actors would affect the system evolution speed but not the evolution result, and the authenticity of new media reports is an important factor determining the system evolution of green technology innovation. (3) Numerical simulation results show that a fair and just new media environment can effectively constrain the traditional production behavior of enterprise, actively guide the public to participate in supervision, and play an alternative role to government regulation to a certain extent. This study explores the evolutionary balance strategy of green technology innovation system under the new media environment, which not only enriches relevant theories of media environment governance but also has important reference value for promoting enterprises' green technology innovation and establishing an environmental governance system jointly governed by government, enterprise, public, and media.

Plea bargaining and investigation effort: inquisitorial criminal procedure as a three-player game

Plea bargaining and investigation effort: inquisitorial criminal procedure as a three-player game

Exploring the Benefits of Representing Multiplayer Game Data in a Coordinate System

In the realm of game theory, a range of mathematical approaches exists for the representation of game data, with the extensive form (depicted as a game tree) and the normal form (illustrated as a payoff matrix) standing out as the most prevalent. However, a significant drawback associated with these approaches is their limited scalability. As the number of players or their strategic options increases, these techniques progressively lose their feasibility and become less practical for meaningful analysis. The present work proposes an alternative approach that significantly enhances the representation of data in two- or three-player games. Within this framework, the conventional payoff matrix is substituted with a payoff coordinate system, employing a coordinate plane for two-player games and a coordinate space for three-player games. This approach offers numerous advantages when compared to other methods. For instance, the Nash equilibrium can be readily identified within a game without requiring an extensive duration to exhaustively examine all strategies for its determination. By employing this approach, the representation of game data becomes more convenient and efficient, making it easier to analyze and comprehend the underlying strategies employed by players.

A cooperative three-player game theory approach for designing an ideal video steganography framework

ABSTRACT This paper presents a cooperative game theory approach to improve the video steganography framework. Wherein, the video steganography framework comprises the following steps: (1) Cover video devising, (2) Secret image pre-processing, and (3) Embedding process. In the first step, the cover video is segmented using scene change detection method. Once the scenes are segmented the motion vectors are identified by Block Matching Motion Estimation Algorithm (BMMEA). Based on these motion vectors, the Region of Interest (ROI) is selected. The selected ROI is grouped using their momentum. In order to add an additional layer of security, the secret image is scrambled using pixel-wise Arnold Transform in the next step. Finally, based on the ROI groups, the scrambled secret data is embedded into the ROI's locations. An ideal and perfect video steganography system should be able to maintain the quality of the video after embedding the secret data. The quality factors of the video steganography approach include perceptible invisibility, payload capacity, and robustness. However, these quality factors are conflicted with each other. In order to address this issue, having these three quality factors as players a cooperative 3-players game theory approach is proposed. This technique provides an optimal solution for the video steganography framework using strategy adaption. The optimal solution is acquired by applying the Iterative Elimination of Strictly Dominant Strategies (IESDS) method. The achieved optimal solution comes up with the best trade-off between the quality factors. Experimental results have deduced the best optimal solution from the proposed three-player game theoretic approach that helps the video steganography approach to resolve the quality and security issues. It is also observed that the optimized proposed method outperforms the contemporary methods by attaining significant outcomes.

Cooperative Game-Theoretic Models of the Cournot Oligopoly

In this paper, we build and investigate cooperative games with different characteristic functions (von Neumann–Morgenstern, Petrosyan–Zaccour, Gromova–Petrosyan) on the base of symmetrical Cournot oligopoly game-theoretic models in normal form. We find Nash and Stackelberg equilibria and cooperative solutions for nonsymmetrical Cournot oligopoly game-theoretic models in normal form. Also, we build and investigate cooperative three-player games with the same characteristic functions on the base of these models, including the calculation of Shapley value. We compare payoffs received by the players according to different optimality principles of cooperative games and games in normal form.

Asymmetric guessing games

This paper theoretically and experimentally investigates the behavior of asymmetric players in guessing games. The asymmetry is created by introducing $$r>1$$ replicas of one of the players. Two-player and restricted N-player cases are examined in detail. Based on the model parameters, the equilibrium is either unique in which all players choose zero or mixed in which the weak player ( $$r=1$$ ) imitates the strong player ( $$r>1$$ ). A series of experiments involving two and three-player repeated guessing games with unique equilibrium is conducted. We find that equilibrium behavior is observed less frequently and overall choices are farther from the equilibrium in two-player asymmetric games in contrast to symmetric games, but this is not the case in three-player games. Convergence towards equilibrium exists in all cases but asymmetry slows down the speed of convergence to the equilibrium in two, but not in three-player games. Furthermore, the strong players have a slight earning advantage over the weak players, and asymmetry increases the discrepancy in choices (defined as the squared distance of choices from the winning number) in both games.

Evolutionary game analysis of three parties in logistics platforms and freight transportation companies’ behavioral strategies for horizontal collaboration considering vehicle capacity utilization

In China, logistics platforms are an effective way to solve vehicle capacity utilization using information sharing. However, most logistics platforms do not possess operational sustainability due to excessive profit-seeking. To address this problem, conflicts of interest among freight transportation participants are discussed using a stakeholder approach. A three-player evolutionary game model (TEGM) is developed to analyze the interactions among freight carriers, freight shippers, and logistics platforms. Then, the asymptotic equilibrium and evolutionary stability strategies of the three-player game are analyzed. The results indicate that a high-level positive network externality is the driving force behind the logistics platform’s “high-level service”. A fairness payment incentive guarantees a “sharing” strategy for freight carriers and shippers. When the high-level positive network externality is limited and lower than a threshold value, there is no stable equilibrium point in the TEGM. A government tax incentive cannot change the freight carriers’ and shippers’ strategy to participate in this horizontal collaboration system, except for the logistics platform’s probability of providing “high-level service”. However, the behavioral strategies of the freight transportation participants can be changed to achieve the sustainability of freight transportation by reducing the value-added tax rate through the logistics platform and increasing the high-level positive network externality of the logistics platform and other participants’ perceived fairness through a payment incentive. Finally, suggestions for regulating the behaviors of freight transportation participants and promoting the sustainability of freight transportation are discussed.

Regret-Based Nash Equilibrium Sorting Genetic Algorithm for Combinatorial Game Theory Problems with Multiple Players.

We introduce a regret-based fitness assignment strategy for evolutionary algorithms to find Nash equilibria in noncooperative simultaneous combinatorial game theory problems where it is computationally intractable to enumerate all decision options of the players involved in the game. Applications of evolutionary algorithms to non-cooperative simultaneous games have been limited due to challenges in guiding the evolutionary search toward equilibria, which are usually inferior points in the objective space. We propose a regret-based approach to select candidate decision options of the players for the next generation in a multipopulation genetic algorithm called Regret-Based Nash Equilibrium Sorting Genetic Algorithm (RNESGA). We show that RNESGA can converge to multiple Nash equilibria in a single run using two- and three- player competitive knapsack games and other games from the literature. We also show that pure payoff-based fitness assignment strategies perform poorly in three-player games.

A three-player game theory model for carbon cap-and-trade mechanism with stochastic parameters

The cap-and-trade (C&T) mechanism is a widely used tool by governments to reduce the emission of greenhouse gases. Naturally, businesses operating under a C&T scheme adopt strategies to ensure profits maximization and emission minimization. Effectiveness of such strategies depends on mutual interaction of external and internal factors. This article develops a stochastic game theoretical model consisting of a manufacturer, a third-party carbon emission verifier, and the government to study the necessary trade-offs to maximize the stated objectives. The proposed model is validated using a numerical example. Furthermore, it is demonstrated that the proposed model maximizes social welfare by finding the best penalty for bribery and violating the assigned carbon emission quota through advising a re-verification strategy to detect possible collusion between the manufacturer and verifier.

A novel version for three-player gambler's ruin problem

In the classic three-player ruin problem, the play continues until at least one of the players is completely ruined. In this research, we present a novel version of the classic three-player game with interest lies in a specific player. We determine the ruin probabilities and expected durations of the game given that our player (i.e. the gambler or the casino's player) wins or loses. The desired game plan is executed for both the cases, i.e. symmetric and asymmetric with all of the players having equal initial stakes. We also obtained the asymptotic results of the ruin probabilities and expected durations of the proposed game plan. Further, the validity of the desired game structure is also verified through a Monte Carlo simulation study.

Existence of Stable Coalition Structures in Three-player Games with Graph-constrained Solution

The stability of coalition structures is investigated in the sense that no player prefers to individually deviate from the current coalition considering his utility. This principle is close to the concept of the Nash equilibrium. The evaluation of each player's utility is determined with respect to the solution concept- the Shapley value with exogenous directed graph constraint. The existence of a stable coalition structure with respect to such a solution is examined for two-player as well as three-player games.

Strange attractors and nontrivial solutions in games with three players

Two-player games have been sufficiently researched, but three-player games have a much greater wealth of solutions, in particular, nonstationary solutions. However, unsteady solutions of three-player games have not been studied in the literature. On the basis of numerical modeling, the dynamic behavior of a 2 × 2 × 2 game with three players is investigated in the range of payoff matrices from -1 to 1. The system of dynamic equations includes three variables (probabilities of pure strategies) and twelve parameters associated with cubic payoff matrices of players. The phase space of a three-player game is a cube (probabilities vary from 0 to 1). It was found that for some payoff matrices, there are no stable solutions in the system. In addition, the structure of the phase space is complex: the volume of the cube is a fractal repulsive region from which the phase trajectories leave, a strange attractor is located near the edge corresponding to the solution of the "node-center" type, and another strange attractor is a chaotic motion between adjacent vertices of the cube. The obtained dynamic properties of the game can be used for making decisions in economics, negotiations between three persons, and modeling relationships between biological species.

Multi-Player Evolutionary Game of Network Attack and Defense Based on System Dynamics

We formalize the adversarial process between defender and attackers as a game and study the non-cooperative evolutionary game mechanism under bounded rationality. We analyze the long-term dynamic process between the attacking and defending parties using the evolutionary stable strategies derived from the evolutionary game model. First, we construct a multi-player evolutionary game model consisting of a defender and multiple attackers, formally describe the strategies, and construct a three-player game payoff matrix. Then, we propose two punishment schemes, i.e., static and dynamic ones. Moreover, through the combination of mathematical derivation with simulation, we obtain the evolutionary stable strategies of each player. Different from previous work, in this paper, we consider the influence of strategies among different attackers. The simulation shows that (1) in the static punishment scheme, increasing the penalty can quickly control the occurrence of network attacks in the short term; (2) in the dynamic punishment scheme, the game can be stabilized effectively, and the stable state and equilibrium values are not affected by the change of the initial values.

Three-Player Gambler’s Ruin Problem: Some Extensions

For calculating the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. However, the martingale approach is not appropriate to determine the ruin time of asymmetric game. In general, ruin probabilities in both cases, that is, symmetric and asymmetric games as well as the expected ruin time for the asymmetric games are still need to be calculated. The current work is also about three-player gambler’s ruin problem with some extensions. We provide expressions for the ruin time with or without ties when all the players have equal or unequal initial fortunes. Finally, the validity of the asymmetric game is also tested through a Monte Carlo simulation study.
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A three-player gambler's ruin problem: some extensions

For the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. For the case of asymmetric game, the martingale approach is not valid to determine the ruin time. In general, the ruin probabilities for both cases, i.e. symmetric and asymmetric game and expected ruin time for asymmetric game are still awaiting to be solved for this game. The current work is also about three-player gambler’s ruin problem with some extensions as well. We provide expressions for the ruin time with (without) ties when all the players have equal (unequal) initial fortunes. Finally, the validity of asymmetric game is also tested through a Monte Carlo simulation study.

STRATEGIC INTERACTIONS AND NEGATIVE OIL PRICES

This paper argues on theoretical grounds that the negative oil prices event on April 20, 2020, was mainly due to the strategic interactions among some active traders on both sides of the futures contract. We present a three-player game of futures trading in which a continuum range of negative price can be supported as (strong) Nash equilibrium, yet none of those constitutes an [Formula: see text]-equilibrium originally developed by Ma (2009). We further propose the notion of coalition-with-side-payment as a solution concept for the environment where strategic interactions and transfer payments among players are allowed. Our model captures the mechanism underlying futures price manipulation, and its predictions largely agree with the observations on that day, which are beyond the scope of demand–supply and physical delivery narratives.

Government subsidies and online branding for agricultural products

Online branding plays an increasingly vital role for agricultural products in agriculture and e-commerce industries alike. To improve the cooperation between farmers and e-retailers, and to increase awareness of agricultural products through online branding, many governments implement series of subsidy policies. We investigate the impact of government subsidies by proposing a three-player game model (consisting of a government, a farmer, and an e-retailer) in supply chains. More specifically, we develop four Stackelberg game scenarios, including a benchmark scenario in which government provides no subsidies, two scenarios in which government subsidizes either farmer or e-retailer without subsidizing the other, and one in which both are subsidized. We find awareness of online branding for agricultural products is positively related to farmers? market sensitivity while e-retailers? cost factor shows a negative relation; optimal awareness emerges when both farmers and retailers are subsidized. In turn, farmers and retailers achieve their best profits when both are subsidized. Furthermore, farmers? profits are higher than e-retailers? in every subsidy scenario. We obtain the most effective parameters for government subsidies using ex ante and ex post strategies, extending our model by incorporating spillover effects. Most of our conclusions are consistent with intuition and propositions we began with, but it is interesting to note that farmers? best profits appear when farmers receive subsidies and e-retailers do not.

The impact of government subsidies on contract design of green technology R&D cooperation

ABSTRACT Different types of government subsidies are used to promote green technology innovation in many countries. Driven by government subsidies and consumer green preference, it’s common for manufacturers (the marketer) to cooperate with research institutes (the innovator) in green technology research and development (R&D). Managing R&D collaboration under different government subsidies is a challenge for the marketer. Meanwhile, how can the government design the subsidy policies to achieve a better effect is also vital. Thus, a two-level three-player game is proposed. The upper layer is the government's decision on R&D subsidy or per-unit production subsidy, and the lower layer is a principal-agent model composed of a marketer and an innovator, who cooperate simply or deeply with a contract with or without milestone payment. We find that: (a) It is always advantageous for the marketer to offer a milestone payment after successful R&D; (b) Deep R&D cooperation is more profitable for the marketer; (c) R&D subsidy is better at stimulating the environmental performance of green technology cooperation.

Defender–attacker–operator: Tri-level game-theoretic interdiction analysis of urban water distribution networks

The aim of this paper is to advance the field of network interdiction analysis by introducing an application to the urban water distribution networks (WDNs), deploying protective resources against intentional attacks. The resource allocation problem for urban water supply systems is considered as a three-player (i.e., defender–attacker–operator) game, in which the attacker aims to maximize disruption impacts via interdicting water plants in the network, the defender aims to minimize the worst-case disruption impacts achieved by the attacker while the system operators fulfill the water demand in the residual urban water supply network. Considering the operating characteristics of the water supply network, we adopted the method of hydraulic analysis in the third level to obtain its reliability, and use this as the game equilibrium index of the first two levels. An effective modified variable neighborhood search method is devised to obtain the solution to the game. Finally, a case study was conducted based on the data of water supply network of a certain city in China to evaluate the effectiveness of protection resources against intentional attacks.