Games

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 (registering DOI) - 1 Feb 2025

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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14 pages, 591 KiB

Open AccessArticle

Asymmetric Information and Credit Rationing in a Model of Search

by Cemil Selcuk

Games 2025, 16(1), 1; https://doi.org/10.3390/g16010001 - 2 Jan 2025

Abstract

This paper presents a competitive search model focusing on the impact of asymmetric information on credit markets. We show that limited entry by lenders results in endogenous credit rationing, which, in turn, plays a key role in managing adverse selection and prevents the [...] Read more.

This paper presents a competitive search model focusing on the impact of asymmetric information on credit markets. We show that limited entry by lenders results in endogenous credit rationing, which, in turn, plays a key role in managing adverse selection and prevents the credit market from collapsing. Full article

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7 pages, 183 KiB

Open AccessArticle

Computing Stackelberg Equilibrium for Cancer Treatment

by Sam Ganzfried

Games 2024, 15(6), 45; https://doi.org/10.3390/g15060045 - 23 Dec 2024

Abstract

Recent work by Kleshnina et al. has presented a Stackelberg evolutionary game model in which the Stackelberg equilibrium strategy for the leading player corresponds to the optimal cancer treatment. We present an approach that is able to quickly and accurately solve the model [...] Read more.

Recent work by Kleshnina et al. has presented a Stackelberg evolutionary game model in which the Stackelberg equilibrium strategy for the leading player corresponds to the optimal cancer treatment. We present an approach that is able to quickly and accurately solve the model presented in that work. Full article

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

9 pages, 252 KiB

Open AccessArticle

Two-Valued Strongly Group Strategy-Proof Social Choice Functions

by Anna De Simone and K. P. S. Bhaskara Rao

Games 2024, 15(6), 44; https://doi.org/10.3390/g15060044 - 10 Dec 2024

Abstract

We present simple and direct arguments to characterize strongly group strategy-proof social choice functions whose range is of cardinality two. The underlying society is of arbitrary cardinality, and agents can be indifferent among alternatives. Full article

(This article belongs to the Section Cooperative Game Theory and Bargaining)

18 pages, 333 KiB

Open AccessArticle

Isotone Classes of Social Choice Functions with Binary Range

by Achille Basile, K. P. S. Bhaskara Rao, Anna De Simone and Ciro Tarantino

Games 2024, 15(6), 43; https://doi.org/10.3390/g15060043 - 9 Dec 2024

Abstract

Recently, it has been shown that the characterizations of different classes of non-manipulable social choice functions with binary range can be reduced to a common functional form. In the present paper, we investigate the reasons why this happens. We show that all the [...] Read more.

Recently, it has been shown that the characterizations of different classes of non-manipulable social choice functions with binary range can be reduced to a common functional form. In the present paper, we investigate the reasons why this happens. We show that all the classes considered share a common mathematical property. We name this property, which is lattice theoretical in nature, isotonicity. Full article

27 pages, 587 KiB

Open AccessArticle

Threshold Protocol Game on Graphs with Magic Square-Generalization Labelings

by Alexandra Fedrigo

Games 2024, 15(6), 42; https://doi.org/10.3390/g15060042 - 3 Dec 2024

Abstract

Graphical games describe strategic interactions among a specified network of players. The threshold protocol game is a graphical game that models the adoption of a lesser-used product in a population when individuals benefit by using the same product. The threshold protocol game has [...] Read more.

Graphical games describe strategic interactions among a specified network of players. The threshold protocol game is a graphical game that models the adoption of a lesser-used product in a population when individuals benefit by using the same product. The threshold protocol game has historically been considered using infinite, simple graphs. In general, however, players might value some relationships more than others or may have different levels of influence in the graph. These traits are described by weights on graph edges or vertices, respectively. Relative comparisons on arbitrarily weighted graphs have been studied for a variety of graphical games. Alternatively, graph labelings are functions that assign values to the edges and vertices of graphs based on a particular set of rules. This work demonstrates that the outcome of the threshold protocol game can be characterized on a magic square-generalization labeled graph. There are a variety of graph labelings that generalize the concept of magic squares. In each, the labels on similar sets of graph elements sum to a constant. The constant sums of magic square-generalization labelings mean that each player experiences a constant level of influence without needing to specify the value of players relative to one another. The game outcome is compared across different types and features of labelings. Full article

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16 pages, 591 KiB

Open AccessArticle

End Behavior of the Threshold Protocol Game on Complete and Bipartite Graphs

by Alexandra Fedrigo

Games 2024, 15(6), 41; https://doi.org/10.3390/g15060041 - 2 Dec 2024

Abstract

The threshold protocol game is a graphical game that models the adoption of an idea or product through a population. There are two states players may take in the game, and the goal of the game is to motivate the state that begins [...] Read more.

The threshold protocol game is a graphical game that models the adoption of an idea or product through a population. There are two states players may take in the game, and the goal of the game is to motivate the state that begins in the minority to spread to every player. Here, the threshold protocol game is defined, and existence results are studied on infinite graphs. Many generalizations are proposed and applied. This work explores the impact of graph topology on the outcome of the threshold protocol game and consequently considers finite graphs. By exploiting the well-known topologies of complete and complete bipartite graphs, the outcome of the threshold protocol game can be fully characterized on these graphs. These characterizations are ideal, as they are given in terms of the game parameters. More generally, initial conditions in terms of game parameters that cause the preferred game outcome to occur are identified. It is shown that the necessary conditions differ between non-bipartite and bipartite graphs because non-bipartite graphs contain odd cycles while bipartite graphs do not. These results motivate the primary result of this work, which is an exhaustive list of achievable game outcomes on bipartite graphs. While possible outcomes are identified, it is noted that a complete characterization of when game outcomes occur is not possible on general bipartite graphs. Full article

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15 pages, 294 KiB

Open AccessArticle

Bayesian Fictitious Play in Oligopoly: The Case of Risk-Averse Agents

by Julide Yazar

Games 2024, 15(6), 40; https://doi.org/10.3390/g15060040 - 27 Nov 2024

Abstract

A number of learning models have been suggested to analyze the repeated interaction of boundedly rational agents competing in oligopolistic markets. The agents form a model of the environment that they are competing in, which includes the market demand and price formation process, [...] Read more.

A number of learning models have been suggested to analyze the repeated interaction of boundedly rational agents competing in oligopolistic markets. The agents form a model of the environment that they are competing in, which includes the market demand and price formation process, as well as their expectations of their rivals’ actions. The agents update their model based on the observed output and price realizations and then choose their next period output levels according to an optimization criterion. In previous works, the global dynamics of price movement have been analyzed when risk-neutral agents maximize their expected rewards at each round. However, in many practical settings, agents may be concerned with the risk or uncertainty in their reward stream, in addition to the expected value of the future rewards. Learning in oligopoly models for the case of risk-averse agents has received much less attention. In this paper, we present a novel learning model that extends fictitious play learning to continuous strategy spaces where agents combine their prior beliefs with market price realizations in previous periods to learn the mean and the variance of the aggregate supply function of the rival firms in a Bayesian framework. Next, each firm maximizes a linear combination of the expected value of the profit and a penalty term for the variance of the returns. Specifically, each agent assumes that the aggregate supply of the remaining agents is sampled from a parametric distribution employing a normal-inverse gamma prior. We prove the convergence of the proposed dynamics and present simulation results to compare the proposed learning rule to the traditional best response dynamics. Full article

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

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16 pages, 284 KiB

Open AccessArticle

A Stepwise Conflict Analysis Using the Graph Model

by Raí dos Santos Mota, Maisa Mendonça Silva and Leandro Chaves Rêgo

Games 2024, 15(6), 39; https://doi.org/10.3390/g15060039 - 27 Nov 2024

Abstract

Information about decision-makers’ preferences is essential for the efficient modeling of a conflict. However, obtaining this information becomes more challenging as the size of the conflict increases. To address this issue, this study proposes a new approach to the option prioritizing method within [...] Read more.

Information about decision-makers’ preferences is essential for the efficient modeling of a conflict. However, obtaining this information becomes more challenging as the size of the conflict increases. To address this issue, this study proposes a new approach to the option prioritizing method within the graph model for conflict resolution. The approach aims to gather preferences from decision-makers in a more consistent and practical manner. The proposed method involves partitioning the set of conflict options based on their importance, then applying the option prioritizing method and conflict stability analysis to subconflicts, where only the options in each partition set are considered. Additionally, only states that are equilibria in a given step are deemed feasible in subsequent steps. The main findings highlight a reduction in the cognitive effort required from decision-makers and the generation of more effective and consistent solutions that address the core needs of the conflict. By working with subsets of options incrementally, the method offers a more simplified and robust understanding of the problem. To demonstrate the proposed method, a real hydrological conflict was used as a case study. Full article

(This article belongs to the Section Cooperative Game Theory and Bargaining)

6 pages, 243 KiB

Open AccessArticle

Evasion Differential Games in the Space of Square Summable Sequences

by Bekhzod Aminov and Marks Ruziboev

Games 2024, 15(6), 38; https://doi.org/10.3390/g15060038 - 19 Nov 2024

Abstract

In this article, we consider simple-motion pursuit–evasion differential games in the Hilbert space of square summable sequences. We show that when the players have the same dynamic capabilities, evasion is possible under some assumptions about the initial positions of the players. Full article

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

9 pages, 221 KiB

Open AccessArticle

On Remoteness Functions of k-NIM with k + 1 Piles in Normal and in Misère Versions

by Vladimir Gurvich, Vladislav Maximchuk, Georgy Miheenkov and Mariya Naumova

Games 2024, 15(6), 37; https://doi.org/10.3390/g15060037 - 13 Nov 2024

Abstract

Given integer n and k such that

0 < k \leq n

and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who [...] Read more.

Given integer n and k such that

0 < k \leq n

and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who has to move but cannot is the loser in the normal version of the game and (s)he is the winner in the misère version. Cases

k = 1

and

k = n

are trivial. For

k = 2

, the game was solved for

n \leq 6

. For

n \leq 4

, the Sprague–Grundy function was efficiently computed (for both versions). For

n = 5, 6

, a polynomial algorithm computing P-positions was obtained for the normal version. Then, for the case

k = n - 1

, a very simple explicit rule that determines the Smith remoteness function was found for the normal version of the game: the player who has to move keeps a pile with the minimum even number of stones; if all piles have an odd number of stones, then (s)he keeps a maximum one, while the

n - 1

remaining piles are reduced by one stone each in accordance with the rules of the game. Computations show that the same rule works efficiently for the misère version too. The exceptions are sparse. We list some. Denote a position by

x = (x_{1}, \dots, x_{n})

. Due to symmetry, we can assume wlog that

x_{1} \leq \dots \leq x_{n}

. Our computations partition all exceptions into the following three families:

x_{1}

is even,

x_{1} = 1

, and odd

x_{1} \geq 3

. In all three cases, we suggest formulas covering all found exceptions, but it is not proven that there are no others. Full article

12 pages, 266 KiB

Open AccessArticle

Monopoly and Quality Omission

by Amit Gayer

Games 2024, 15(6), 36; https://doi.org/10.3390/g15060036 - 29 Oct 2024

Abstract

This study delves into a market characterized by vertical product differentiation. Product qualities are represented on a one-dimensional interval scale. The research investigates the equilibrium within a monopoly scenario, considering a production cost that is strictly convex. The monopoly offers a strategy comprising [...] Read more.

This study delves into a market characterized by vertical product differentiation. Product qualities are represented on a one-dimensional interval scale. The research investigates the equilibrium within a monopoly scenario, considering a production cost that is strictly convex. The monopoly offers a strategy comprising various quality–price combinations, with consumer choices determining profits. The analysis involves a comparison between two analogous models: one with a continuous range of consumers and the other with a finite number of consumers. The study explores disparities in the potential for market failure between these two settings. Notably, numerical illustrations underscore these divergences in both market contexts. Full article

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

18 pages, 740 KiB

Open AccessArticle

On Isaac’s War Game of Attrition and Attack Using Dynamic Programming Approach

by Benghebrid Safa, Bouremani Touffik and Benterki Djamel

Games 2024, 15(6), 35; https://doi.org/10.3390/g15060035 - 24 Oct 2024

Abstract

In this study, we use the dynamic programming method introduced by Mirică (2004) to solve the well-known war game of attrition and attack as formulated by Isaacs (1965). By using this modern approach, we extend the classical framework to explore optimal strategies within [...] Read more.

In this study, we use the dynamic programming method introduced by Mirică (2004) to solve the well-known war game of attrition and attack as formulated by Isaacs (1965). By using this modern approach, we extend the classical framework to explore optimal strategies within the differential game setting, offering a complete, comprehensive and theoretically robust solution. Additionally, the study identifies and analyzes feedback strategies, which represent a significant advancement over other strategy types in game theory. These strategies dynamically adapt to the evolving state of the system, providing more robust solutions for real-time decision-making in conflict scenarios. This novel contribution enhances the application of game theory, particularly in the context of warfare models, and illustrates the practical advantages of incorporating feedback mechanisms into strategic decision-making. The admissible feedback strategies and the corresponding value function are constructed through a refined application of Cauchy’s Method of characteristics for stratified Hamilton–Jacobi equations. Their optimality is proved using a suitable Elementary Verification Theorem for the associated value function as an argument for sufficient optimality conditions. Full article

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25 pages, 22933 KiB

Open AccessArticle

Simulation of the Stackelberg–Hotelling Game

by Luis Garcia-Perez, Juan Grau-Climent, Juan C. Losada and Ramon Alonso-Sanz

Games 2024, 15(5), 34; https://doi.org/10.3390/g15050034 - 11 Oct 2024

Abstract

This work studies the Hotelling game with sequential choice of prices, that is, the Stackelberg–Hotelling (SHOT) game. The game is studied through numerical simulation, which provides the subgame perfect equilibrium solution not only in the unrestricted game but also in the game with [...] Read more.

This work studies the Hotelling game with sequential choice of prices, that is, the Stackelberg–Hotelling (SHOT) game. The game is studied through numerical simulation, which provides the subgame perfect equilibrium solution not only in the unrestricted game but also in the game with reservation cost and with elastic demand. The simulation technique is tested first in the unconstrained game, where the analytical subgame perfect equilibrium solution was already known. Then, the numerical procedure is generalized to cope with the SHOT game with reservation cost and with elastic demand. These enriched formulations of the SHOT game have not been studied so far, so this article provides an exploratory study of them. Full article

(This article belongs to the Section Algorithmic and Computational Game Theory)

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16 pages, 2265 KiB

Open AccessArticle

A Solution to Dynamic Weapon Assignment Problem Based on Game Theory for Naval Platforms

by Oğuzkan Akbel and Aykut Kalaycıoğlu

Games 2024, 15(5), 33; https://doi.org/10.3390/g15050033 - 6 Oct 2024

Abstract

Weapon target assignment is a critical challenge in military contexts. Traditionally, commanding officers manually decide weapon assignments, but the problem’s complexity has grown over time. To address this, automated systems have been introduced. These systems fall into two categories, which are static (time-independent) [...] Read more.

Weapon target assignment is a critical challenge in military contexts. Traditionally, commanding officers manually decide weapon assignments, but the problem’s complexity has grown over time. To address this, automated systems have been introduced. These systems fall into two categories, which are static (time-independent) and dynamic (considering changes over time). Static systems solve the problem for a single time step without considering temporal changes. Dynamic systems incorporate time as a variable, adapting to evolving scenarios. Two main approaches exist, which are asset-based and target-based. Asset-based approach maximizes the survival probability of assets, which is our focus in this study. We propose a solution using game theory that spans the entire area and all time frames. We employ game theory, treating continuous functions of time as utility functions for vessels. Continuous probability-to-kill values for weapons are defined across the area. Threat trajectories yield continuous kill probabilities for the weapons, translating to vessel utility. To avoid inefficiencies, we align individual vessel utility with global utility. The Nash Equilibrium provides the optimal weapon assignment strategy. Our study uses a naval environment for analysis. In summary, our research leverages game theory to dynamically assign weapons to naval vessels, aiming to maximize asset survival. Full article

(This article belongs to the Section Applied Game Theory)

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