Games

28 pages, 832 KiB

Open AccessFeature PaperArticle

Two-Tier Marketplace with Multi-Resource Bidding and Strategic Pricing for Multi-QoS Services

by Samira Habli, Rachid El-Azouzi, Essaid Sabir, Mandar Datar, Halima Elbiaze and Mohammed Sadik

Games 2025, 16(2), 20; https://doi.org/10.3390/g16020020 - 21 Apr 2025

Abstract

Fog computing introduces a new dimension to the network edge by pooling diverse resources (e.g., processing power, memory, and bandwidth). However, allocating resources from heterogeneous fog nodes often faces limited capacity. To overcome these limitations, integrating fog nodes with cloud resources is crucial, [...] Read more.

Fog computing introduces a new dimension to the network edge by pooling diverse resources (e.g., processing power, memory, and bandwidth). However, allocating resources from heterogeneous fog nodes often faces limited capacity. To overcome these limitations, integrating fog nodes with cloud resources is crucial, ensuring that Service Providers (SPs) have adequate resources to deliver their services efficiently. In this paper, we propose a game-theoretic model to describe the competition among non-cooperative SPs as they bid for resources from both fog and cloud environments, managed by an Infrastructure Provider (InP), to offer paid services to their end-users. In our game model, each SP bids for the resources it requires, determining its willingness to pay based on its specific service demands and quality requirements. Resource allocation prioritizes the fog environment, which offers local access with lower latency but limited capacity. When fog resources are insufficient, the remaining demand is fulfilled by cloud resources, which provide virtually unlimited capacity. However, this approach has a weakness in that some SPs may struggle to fully utilize the resources allocated in the Nash equilibrium-balanced cloud solution. Specifically, under a nondiscriminatory pricing scheme, the Nash equilibrium may enable certain SPs to acquire more resources, granting them a significant advantage in utilizing fog resources. This leads to unfairness among SPs competing for fog resources. To address this issue, we propose a price differentiation mechanism among SPs to ensure a fair allocation of resources at the Nash equilibrium in the fog environment. We establish the existence and uniqueness of the Nash equilibrium and analyze its key properties. The effectiveness of the proposed model is validated through simulations using Amazon EC2 instances, where we investigate the impact of various parameters on market equilibrium. The results show that SPs may experience profit reductions as they invest to attract end-users and enhance their quality of service QoS. Furthermore, unequal access to resources can lead to an imbalance in competition, negatively affecting the fairness of resource distribution. The results demonstrate that the proposed model is coherent and that it offers valuable information on the allocation of resources, pricing strategies, and QoS management in cloud- and fog-based environments. Full article

(This article belongs to the Section Non-Cooperative Game Theory)

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28 pages, 768 KiB

Open AccessArticle

Procedurally Fair Co-Determination with Endogeneous Value Uncertainty: An Experiment

by Werner Güth, Ludivine Martin, Tibor Neugebauer and Sotiria Xanalatou

Games 2025, 16(2), 19; https://doi.org/10.3390/g16020019 - 21 Apr 2025

Abstract

We present an experimental test of a procedurally fair co-determination mechanism where group members reduce their value uncertainty before submitting bids for a joint project. The results suggest a relatively efficient mechanism, with unprofitable projects being largely rejected and profitable ones accepted. Repeated [...] Read more.

We present an experimental test of a procedurally fair co-determination mechanism where group members reduce their value uncertainty before submitting bids for a joint project. The results suggest a relatively efficient mechanism, with unprofitable projects being largely rejected and profitable ones accepted. Repeated interactions tended to enhance the efficiency, while uncertain information reduced it. The subjects invested surprisingly little search effort to reduce the uncertainty about the costs and benefits, and appeared to trade off search costs against higher bids. Full article

(This article belongs to the Section Behavioral and Experimental Game Theory)

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16 pages, 5600 KiB

Open AccessArticle

Cultural Dissemination on Evolving Networks: A Modified Axelrod Model Based on a Rewiring Process

by Yuri Perez, Fabio Henrique Pereira and Pedro Henrique Triguis Schimit

Games 2025, 16(2), 18; https://doi.org/10.3390/g16020018 - 17 Apr 2025

Abstract

In this paper, we investigate the classical Axelrod model of cultural dissemination under an adaptive network framework. Unlike the original model, we place agents on a complex network, where they cut connections with any agent that does not share at least one cultural [...] Read more.

In this paper, we investigate the classical Axelrod model of cultural dissemination under an adaptive network framework. Unlike the original model, we place agents on a complex network, where they cut connections with any agent that does not share at least one cultural trait. This rewiring process alters the network topology, and key parameters—such as the number of traits, the neighborhood search range, and the degree-based preferential attachment exponent—also influence the distribution of cultural traits. Unlike conventional Axelrod models, our approach introduces a dynamic network structure where the rewiring mechanism allows agents to actively modify their social connections based on cultural similarity. This adaptation leads to network fragmentation or consolidation depending on the interaction among model parameters, offering a framework to study cultural homogeneity and diversity. The results show that, while long-range reconnections can promote more homogeneous clusters in certain conditions, variations in the local search radius and preferential attachment can lead to rich and sometimes counterintuitive dynamics. Extensive simulations demonstrate that this adaptive mechanism can either increase or decrease cultural diversity, depending on the interplay of network structure and cultural dissemination parameters. These findings have practical implications for understanding opinion dynamics and cultural polarization in social networks, particularly in digital environments where rewiring mechanisms are analogous to recommendation systems or user-driven connection adjustments. Full article

(This article belongs to the Section Learning and Evolution in Games)

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6 pages, 190 KiB

Open AccessArticle

Von Neumann–Morgenstern Hypergraphs

by Stefano Vannucci

Games 2025, 16(2), 17; https://doi.org/10.3390/g16020017 - 15 Apr 2025

Abstract

A simple hypergraph H with vertex set X and edge set

E

is representable by Von Neumann–Morgenstern (VNM)-stable sets—or VNM—if there exists an irreflexive simple digraph D with vertex set X such that each edge of H is a VNM-stable set of D [...] Read more.

A simple hypergraph H with vertex set X and edge set

E

is representable by Von Neumann–Morgenstern (VNM)-stable sets—or VNM—if there exists an irreflexive simple digraph D with vertex set X such that each edge of H is a VNM-stable set of D. It is shown that a simple hypergraph H is VNM if and only if each edge of H is a maximal clique of the conjugation graph of H. A related algorithm that identifies finite VNM hypergraphs is also provided. Full article

19 pages, 392 KiB

Open AccessArticle

Analysis of the Competition of the South-Eastern Railway of Peru Through a Timetable Auction

by Augusto Aliaga-Miranda, Luis Ricardo Flores-Vilcapoma, Christian Efrain Raqui-Ramirez, José Luis Claudio-Pérez, Yadira Yanase-Rojas and Jovany Pompilio Espinoza-Yangali

Games 2025, 16(2), 16; https://doi.org/10.3390/g16020016 - 7 Apr 2025

Abstract

Our research analyzes the design of an auction model for railway transportation on the South-East Railway of Peru, managed by Ferrocarril Transandino S.A. (Fetransa) and operated by PeruRail. Initially, the regulatory framework aimed to promote competition in railway transportation through timetable auctions and [...] Read more.

Our research analyzes the design of an auction model for railway transportation on the South-East Railway of Peru, managed by Ferrocarril Transandino S.A. (Fetransa) and operated by PeruRail. Initially, the regulatory framework aimed to promote competition in railway transportation through timetable auctions and infrastructure access. However, the concession has resulted in a vertically integrated structure that favors PeruRail, which faces minimal direct competition, controls high-demand time slots, and hinders the entry of other operators due to strategic and structural access barriers. To address these distortions, we propose reforming the auction mechanism to neutralize these advantages and enhance competition. In this revised framework, the track usage fee will serve as the competitive factor, with the highest bid above a minimum base rate securing the allocation. Additionally, we propose the implementation of asymmetric tariffs to compensate for the higher costs faced by operators with fewer economies of scale, technological optimizations to facilitate equitable access to time slots, and stricter oversight mechanisms to ensure transparency in timetable allocation. These measures aim to balance the market and safeguard competition through a more equitable and efficient auction design. Full article

(This article belongs to the Special Issue Applications of Game Theory to Industrial Organization)

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8 pages, 215 KiB

Open AccessArticle

Biased-Manager Hiring in a Market with Network Externalities and Product Compatibility

by Shih-Hao Huang, Chien-Shu Tsai, Jen-Yao Lee and Su-Ching Tsai

Games 2025, 16(2), 15; https://doi.org/10.3390/g16020015 - 21 Mar 2025

Abstract

This paper studies biased-manager hiring in a market with network externalities and product compatibility. We show that the aggressivity of a biased manager has a non-linear relationship with product compatibility; however, since both owners want to hire aggressive managers, product compatibility is irrelevant [...] Read more.

This paper studies biased-manager hiring in a market with network externalities and product compatibility. We show that the aggressivity of a biased manager has a non-linear relationship with product compatibility; however, since both owners want to hire aggressive managers, product compatibility is irrelevant to the type of manager the owner hires. In Cournot competition, product compatibility is crucial in alleviating the “prisoner’s dilemma” due to the net network effect of network externalities with product compatibility. In Bertrand competition, the “prisoner’s dilemma” is resolved when the augmented net network effect of product compatibility is large. Full article

(This article belongs to the Section Applied Game Theory)

17 pages, 530 KiB

Open AccessArticle

Greenwashing Risks in Environmental Quality Competition: Detection and Deterrence

by Arka Mukherjee and Subhadip Ghosh

Games 2025, 16(2), 14; https://doi.org/10.3390/g16020014 - 11 Mar 2025

Abstract

The rising prevalence of greenwashing by firms has emerged as a major concern for regulatory authorities over the past decade. This paper examines the impact of regulation on firms’ strategic decisions regarding greenwashing and environmental quality in an oligopolistic market. We model two [...] Read more.

The rising prevalence of greenwashing by firms has emerged as a major concern for regulatory authorities over the past decade. This paper examines the impact of regulation on firms’ strategic decisions regarding greenwashing and environmental quality in an oligopolistic market. We model two firms that compete on environmental quality and greenwashing levels, operating under the oversight of a regulatory authority. The authority’s policy instruments include a detection mechanism and fines imposed on firms engaging in greenwashing. Using a differential game-theoretical framework, we examine the effectiveness of regulatory interventions like detection and penalties in reducing greenwashing behavior and enhancing environmental quality. Additionally, we discuss the post-detection trajectories of both firms, providing insights into the effects on consumer perceptions and market competition. We find that while regulation can reduce greenwashing as expected, it may also reduce firms’ environmental quality efforts. Indeed, when penalties are sufficiently high, the marginal returns on investment in greenwashing exceed those from actual green quality improvements. Full article

(This article belongs to the Special Issue Applications of Differential Games and Related Models in Mathematics and Economics)

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25 pages, 808 KiB

Open AccessArticle

Sharing Price Announcements

by Maarten Janssen and Vladimir Karamychev

Games 2025, 16(2), 13; https://doi.org/10.3390/g16020013 - 10 Mar 2025

Abstract

We present a simple model where, before competing in prices, firms announce which prices they intend to choose. Deviating from these announcements involves a cost. We show that sharing pricing intentions results in prices being set above their competitive levels. All equilibria result [...] Read more.

We present a simple model where, before competing in prices, firms announce which prices they intend to choose. Deviating from these announcements involves a cost. We show that sharing pricing intentions results in prices being set above their competitive levels. All equilibria result in prices that are higher than in the absence of announcements. When the deviation cost of not sticking to the price announcement is high, the unique equilibrium market outcome is asymmetric, as with price leadership. When this cost is low, a symmetric equilibrium exists with even higher prices. Product differentiation is a key ingredient to these results. Full article

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37 pages, 427 KiB

Open AccessArticle

Structured Equilibria for Dynamic Games with Asymmetric Information and Dependent Types

by Nasimeh Heydaribeni and Achilleas Anastasopoulos

Games 2025, 16(2), 12; https://doi.org/10.3390/g16020012 - 3 Mar 2025

Abstract

We consider a dynamic game with asymmetric information where each player privately observes a noisy version of a (hidden) state of the world V, resulting in dependent private observations. We study the structured perfect Bayesian equilibria (PBEs) that use private beliefs in [...] Read more.

We consider a dynamic game with asymmetric information where each player privately observes a noisy version of a (hidden) state of the world V, resulting in dependent private observations. We study the structured perfect Bayesian equilibria (PBEs) that use private beliefs in their strategies as sufficient statistics for summarizing their observation history. The main difficulty in finding the appropriate sufficient statistic (state) for the structured strategies arises from the fact that players need to construct (private) beliefs on other players’ private beliefs on V, which, in turn, would imply that one needs to construct an infinite hierarchy of beliefs, thus rendering the problem unsolvable. We show that this is not the case: each player’s belief on other players’ beliefs on V can be characterized by her own belief on V and some appropriately defined public belief. We then specialize this setting to the case of a Linear Quadratic Gaussian (LQG) non-zero-sum game, and we characterize structured PBEs with linear strategies that can be found through a backward/forward algorithm akin to dynamic programming for the standard LQG control problem. Unlike the standard LQG problem, however, some of the required quantities for the Kalman filter are observation-dependent and, thus, cannot be evaluated offline through a forward recursion. Full article

(This article belongs to the Section Learning and Evolution in Games)

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14 pages, 352 KiB

Open AccessArticle

Game Theory Framework for Mitigating the Cost Pendulum in Public Construction Projects

by Yahel Giat and Amichai Mitelman

Games 2025, 16(2), 11; https://doi.org/10.3390/g16020011 - 3 Mar 2025

Abstract

The coexistence of the winner’s curse and cost overruns in the construction industry implies a cost pendulum in which the winning bid is undervalued, whereas the final payment to the contractor is overvalued. We posit that this results from a strategic interaction between [...] Read more.

The coexistence of the winner’s curse and cost overruns in the construction industry implies a cost pendulum in which the winning bid is undervalued, whereas the final payment to the contractor is overvalued. We posit that this results from a strategic interaction between three stakeholders: the public agency (PA), the project manager (PM), and the winning contractor, and we propose a game-theoretic framework to model this dynamic. In the current state of practice, the subgame between the contractor and the PM leads to opportunistic contractor behavior and lenient supervision, resulting in increased costs for the PA. We analyze how procedural and cultural interventions by the PA, specifically shifting from a low-bid to an average-bid auction and incentivizing stricter PM oversight, alter the strategic equilibrium. Our findings indicate that while each change alone provides limited improvement, implementing both significantly reduces cost overruns by aligning stakeholder incentives. The findings of this analysis provide insight into how public agencies can mitigate the widespread problem of cost overruns. Full article

(This article belongs to the Section Applied Game Theory)

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14 pages, 814 KiB

Open AccessArticle

Strategy Consensus of Networked Evolutionary Games Based on Network Aggregation and Pinning Control

by Haitao Li, Zhenping Geng and Mengyuan Qin

Games 2025, 16(1), 10; https://doi.org/10.3390/g16010010 - 11 Feb 2025

Abstract

The computational complexity of large-scale networked evolutionary games has become a challenging problem. Based on network aggregation and pinning control methods, this paper investigates the problem of control design for strategy consensus of large-scale networked evolutionary games. The large-size network is divided into [...] Read more.

The computational complexity of large-scale networked evolutionary games has become a challenging problem. Based on network aggregation and pinning control methods, this paper investigates the problem of control design for strategy consensus of large-scale networked evolutionary games. The large-size network is divided into several small subnetworks by the aggregation method, and a pinning control algorithm is proposed to achieve the strategy consensus of small subnetworks. Then, the matchable condition between the small subnetworks is realized by the input–output control. Finally, some sufficient conditions as well as an algorithm are proposed for the strategy consensus of large-scale networked evolutionary games. Full article

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22 pages, 578 KiB

Open AccessArticle

Differential Games of Cournot Oligopoly with Consideration of Pollution, Network Structure, and Continuous Updating

by Guennady Ougolnitsky and Alexey Korolev

Games 2025, 16(1), 9; https://doi.org/10.3390/g16010009 - 9 Feb 2025

Abstract

We have built and investigated analytically and numerically a differential game model of Cournot oligopoly with consideration of pollution, network structure, and continuous updating. Up to this time, games with network structure and continuous updating were considered separately. We analyzed time consistency for [...] Read more.

We have built and investigated analytically and numerically a differential game model of Cournot oligopoly with consideration of pollution, network structure, and continuous updating. Up to this time, games with network structure and continuous updating were considered separately. We analyzed time consistency for a cooperative solution of the game. For a specific example, we built a non-empty subgame perfect subcore. We considered stochastic versions of the proposed model and received results similar to the deterministic case. Full article

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16 pages, 461 KiB

Open AccessArticle

Learning Optimal Strategies in a Duel Game

by Angelos Gkekas, Athina Apostolidou, Artemis Vernadou and Athanasios Kehagias

Games 2025, 16(1), 8; https://doi.org/10.3390/g16010008 - 5 Feb 2025

Abstract

We study a duel game in which each player has incomplete knowledge of the game parameters. We present a simple, heuristically motivated and easily implemented algorithm by which, in the course of repeated plays, each player estimates the missing parameters and consequently learns [...] Read more.

We study a duel game in which each player has incomplete knowledge of the game parameters. We present a simple, heuristically motivated and easily implemented algorithm by which, in the course of repeated plays, each player estimates the missing parameters and consequently learns his optimal strategy. Full article

(This article belongs to the Section Learning and Evolution in Games)

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7 pages, 862 KiB

Open AccessArticle

A Note on the Welfare and Policy Implications of a Two-Period Real Option Game Under Imperfect Information

by Congcong Wang, Yuhan Wang, Shanshan Chen, Shravan Luckraz and Bruno Antonio Pansera

Games 2025, 16(1), 7; https://doi.org/10.3390/g16010007 - 3 Feb 2025

Abstract

We show that the discrete real option game model proposed in the recent literature can be extended to the case of imperfect information. As a result, the model can cover a wider range of applications. However, we also observe that the effectiveness of [...] Read more.

We show that the discrete real option game model proposed in the recent literature can be extended to the case of imperfect information. As a result, the model can cover a wider range of applications. However, we also observe that the effectiveness of implementing the subsidy is affected by the imperfect informational structure. Full article

(This article belongs to the Section Applied Game Theory)

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15 pages, 269 KiB

Open AccessArticle

A Model of k-Winners

by Diego Armando Canales

Games 2025, 16(1), 6; https://doi.org/10.3390/g16010006 - 1 Feb 2025

Abstract

The concept of the Condorcet winner has become central to most electoral models in the political economy literature. A Condorcet winner is the alternative preferred by a plurality in every pairwise competition; the notion of a k-winner generalizes that of a Condorcet [...] Read more.

The concept of the Condorcet winner has become central to most electoral models in the political economy literature. A Condorcet winner is the alternative preferred by a plurality in every pairwise competition; the notion of a k-winner generalizes that of a Condorcet winner. The k-winner is the unique alternative top-ranked by the plurality in every competition comprising exactly k alternatives (including itself). This study uses a spatial voting setting to characterize this theoretical concept, showing that if a k-winner exists for some

k > 2

, then the same alternative must be the

k^{'}

-winner for every

k^{'} > k

. We derive additional results, including sufficient and necessary conditions for the existence of a k-winner for some

k > 2

. Full article

(This article belongs to the Special Issue Conflict and Communication: Applications of Game Theory to Political Economy)

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11 pages, 890 KiB

Open AccessArticle

N-Tuple Network Search in Othello Using Genetic Algorithms

by Hiroto Kuramitsu, Kaiyu Suzuki and Tomofumi Matsuzawa

Games 2025, 16(1), 5; https://doi.org/10.3390/g16010005 - 9 Jan 2025

Abstract

As one of the strongest Othello agents, Edax employs an n-tuple network to evaluate the board, with points of interest represented as tuples. However, this network maintains a constant shape throughout the game, whereas the points of interest in Othello vary with respect [...] Read more.

As one of the strongest Othello agents, Edax employs an n-tuple network to evaluate the board, with points of interest represented as tuples. However, this network maintains a constant shape throughout the game, whereas the points of interest in Othello vary with respect to game’s progress. The present study was conducted to optimize the shape of the n-tuple network using a genetic algorithm to maximize final score prediction accuracy for a certain number of moves. We selected shapes for 18-, 22-, 26-, 30-, 34-, 38-, 42-, and 46-move configurations, and constructed an agent that appropriately shapes an n-tuple network depending on the progress of the game. Consequently, agents using the n-tuple network developed in this study exhibited a winning rate of 75%. This method is independent of game characteristics and can optimize the shape of larger (or smaller) N-tuple networks. Full article

(This article belongs to the Section Learning and Evolution in Games)

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12 pages, 759 KiB

Open AccessArticle

High Cost of Survival Promotes the Evolution of Cooperation

by Oleg Smirnov

Games 2025, 16(1), 4; https://doi.org/10.3390/g16010004 - 9 Jan 2025

Abstract

Living organisms expend energy to sustain survival, a process which is reliant on consuming resources—termed here as the “cost of survival”. In the Prisoner’s Dilemma (PD), a classic model of social interaction, individual payoffs depend on choices to either provide benefits to others [...] Read more.

Living organisms expend energy to sustain survival, a process which is reliant on consuming resources—termed here as the “cost of survival”. In the Prisoner’s Dilemma (PD), a classic model of social interaction, individual payoffs depend on choices to either provide benefits to others at a personal cost (cooperate) or exploit others to maximize personal gain (defect). We demonstrate that in an iterated Prisoner’s Dilemma (IPD), a simple “Always Cooperate” (ALLC) strategy evolves and remains evolutionarily stable when the cost of survival is sufficiently high, meaning exploited cooperators have a low probability of survival. We derive a rule for the evolutionary stability of cooperation, x/z >T/R, where x represents the duration of mutual cooperation, z the duration of exploitation, T the defector’s free-riding payoff, and R the payoff for mutual cooperation. This finding suggests that higher survival costs can enhance social welfare by selecting for cooperative strategies. Full article

(This article belongs to the Special Issue Evolution of Cooperation and Evolutionary Game Theory)

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23 pages, 3607 KiB

Open AccessArticle

Subjective Game Structures: A Behavioral Game Theoretic Analysis of Hidden Perceptions and Strategic Properties Underlying the Israeli–Palestinian Conflict

by Ilan Fischer, Shacked Avrashi and Lior Givon

Games 2025, 16(1), 3; https://doi.org/10.3390/g16010003 - 7 Jan 2025

Abstract

Here, we apply a novel framework, termed Subjective Game Structures (SGSs), for uncovering and analyzing hidden motivations in ecological conflicts. SGSs enable the examination of implicit attitudes and motivations within individuals and groups. We elicited SGSs from Israeli and Palestinian participants between March [...] Read more.

Here, we apply a novel framework, termed Subjective Game Structures (SGSs), for uncovering and analyzing hidden motivations in ecological conflicts. SGSs enable the examination of implicit attitudes and motivations within individuals and groups. We elicited SGSs from Israeli and Palestinian participants between March 2019 and February 2020 (approximately three years before 7 October 2023), trying to answer the questions of whether Israelis and Palestinians perceived the conflict in a similar manner, whether they have identical assessments of the associated payoffs, and what can be done to reduce future hostilities and attain peaceful solutions. The results reveal meaningful differences between the parties. Israeli SGSs largely reflected expectations of mutually cooperative outcomes, while Palestinian SGSs exhibited ambivalence and a higher occurrence of confrontational expectations from both parties. Approximately 70% of Israeli SGSs and 40% of Palestinian SGSs were categorized as absolutely stable games, indicating that a meaningful portion of participants implicitly anticipated cooperative and mutually beneficial resolutions. Additionally, Palestinian participants’ perceptions of strategic similarity with Israelis were considerably lower than the perceptions of Israeli participants, pointing to meaningful gaps in the alternatives each side was expecting the other side to choose. The discussion highlights the importance of enhancing subjective perceptions of similarity and shaping parties’ perceived payoff structures as two key pathways to fostering peaceful interactions in diverse social and political conflicts. Full article

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21 pages, 890 KiB

Open AccessArticle

Investigating the Social Boundaries of Fairness by Modeling Ultimatum Game Responders’ Decisions with Multinomial Processing Tree Models

by Marco Biella, Max Hennig and Laura Oswald

Games 2025, 16(1), 2; https://doi.org/10.3390/g16010002 - 3 Jan 2025

Abstract

Fairness in competitive games such as the Ultimatum Game is often defined theoretically. According to some of the literature, in which fairness is determined only based on resource allocation, a proposal splitting resources evenly (i.e., 5:5) is generally assumed as fair, and minimal [...] Read more.

Fairness in competitive games such as the Ultimatum Game is often defined theoretically. According to some of the literature, in which fairness is determined only based on resource allocation, a proposal splitting resources evenly (i.e., 5:5) is generally assumed as fair, and minimal deviation (i.e., 4:6) is considered enough to classify the proposal as unfair. Relying on multinomial processing tree models (MPTs), we investigated where the boundaries of fairness are located in the eye of responders, and pit fairness against relative and absolute gain maximization principles. The MPT models we developed and validated allowed us to separate three individual processes driving responses in the standard and Third-Party Ultimatum Game. The results show that, from the responder’s perspective, the boundaries of fairness encompass proposals splitting resources in a perfectly even way and include uneven proposals with minimal deviance (4:6 and 6:4). Moreover, the results show that, in the context of Third-Party Ultimatum Games, the responder must not be indifferent between favoring the proposer and the receiver, demonstrating a boundary condition of the developed model. If the responder is perfectly indifferent, absolute and relative gain maximization are theoretically unidentifiable. This theoretical and practical constraint limits the scope of our theory, which does not apply in the case of a perfectly indifferent decision-maker. Full article

(This article belongs to the Special Issue Fairness in Non-cooperative Strategic Interactions)

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