Games doi: 10.3390/g17030029

Authors: Tyler C. Gardner Matthew W. Harris Logan Lancaster

Many engineering problems must account for the non-cooperative decisions and actions of multiple players. These problems can be modeled within a game-theoretic framework. The approach herein is to model such problems as mathematical games, convert them to semi-infinite programs, and utilize a semi-infinite program solver whose output is provably an ϵ-optimal Nash equilibrium. The approach is successfully benchmarked on two low-dimensional problems. Two types of higher-dimensional linear quadratic dynamic games are then investigated: ones where each player’s problem is convex and ones where at least one player’s problem is nonconvex. Within each type, variations based on information structure, control constraints, number of players, and semi-infinite objective are considered. The algorithm is tested with different internal solvers, and it successfully solves all test problems using MATLAB’s fmincon. The numerical solutions approximate analytical solutions (when they are known) within approximately one percent. For a three-player game with input saturation constraints, hundreds of variables, and no analytical solution, the computational time is approximately five minutes.

Games doi: 10.3390/g17030028

Authors: Feiran Xu Anna Tur

We study a continuous-time cooperative differential game of pollution control in which the pollution stock accumulates emissions and affects long-run welfare. The key feature is a one-time random increase in the public damage weight, interpreted as a regime shift in environmental policy, social damage assessment, or regulatory pressure. Using dynamic programming, we characterize the grand-coalition feedback solution from the Hamilton–Jacobi–Bellman equations and derive closed-form expressions for cooperative emissions, pollution dynamics, regime-specific steady states, and transition paths. Under emission caps, we construct the coalition characteristic function using a conservative worst-case benchmark for outsider behavior rather than an unlimited-pollution assumption. For payoff allocation, we derive a dynamic payment schedule that implements the Shapley allocation along the stochastic pollution path and keeps the remaining payoff consistent with the corresponding continuation game. Finally, we extend the framework to a threshold-triggered shifted-exponential switching mechanism. This extension gives a computable objective for the optimal threshold-hitting time and clarifies how the pollution threshold and switching hazard can be interpreted as policy-relevant indicators of regulatory or ecological regime change.

Games doi: 10.3390/g17030027

Authors: Chanalak Chaisrilak Thanee Chaiwat

Public goods provision is vulnerable to free riding, making sustained cooperation a central challenge in economics. Fehr and Schmidt’s inequity-aversion model explains how fairness concerns can support cooperation, but it treats preferences as fixed. Motivated by Kimbrough and Vostroknutov’s norm-sensitivity framework, this paper develops a reduced-form dynamic framework in which observed norm violations erode normative commitment over time. As normative commitment declines, the model maps this change into Fehr–Schmidt-style fairness parameters: guilt weakens and envy rises. These parameters provide an interpretive representation of norm erosion, while behavior is generated through a tractable contribution-scaling rule. The framework is calibrated illustratively to the public goods experiment of Fischbacher and Gächter. The calibration is not causal evidence of preference change and does not directly identify inequity-aversion parameters. It shows that a context-dependent preference channel can reproduce the observed aggregate decline in cooperation and generate testable implications. When no free-rider exposure is present, cooperation does not decline within the model. The model also predicts a nonlinear relationship between population-level free-rider prevalence and cooperation. Finally, because the model imposes a lower bound on normative commitment, this institutional floor determines long-run cooperation. The findings should be interpreted as model-based hypotheses for future experimental and field research.

Games doi: 10.3390/g17030026

Authors: Xiaoli Li Luo Wu

Frequent occurrences of inter-regional emergencies constitute critical impediments to global security and sustainable development, necessitating enhanced intergovernmental emergency collaboration. This study employs a network evolutionary game model (NEGM) to examine how vertical interventions shape diffusion mechanisms of cooperative strategies among local governments. The results show that (1) solely intensifying penalties or rewards yields diminishing marginal returns in incentivizing local governments to adopt a proactive cooperative strategy; (2) elevating the cost-sharing index significantly accelerates the diffusion rate of cooperative strategies, effectively mobilizing broader subnational engagement in public health emergency response; and (3) the tripartite integration of penalty-based enforcement, reward incentives, and cost-sharing mechanisms demonstrates synergistic superiority over alternative policy instruments—whether implemented individually or in pairwise combinations.

Games doi: 10.3390/g17030025

Authors: Michael R. Baye John Morgan

This paper examines drip pricing related to compulsory charges—a situation where firms intentionally make it costly for consumers to discover mandatory fees or surcharges that “drip” into the full (total) price, which is only revealed after incurring the hassle cost of completing a purchase. We show that drip pricing can arise as an equilibrium phenomenon with fully rational consumers and profit-maximizing firms. We also show that when consumers and firms are rational (a) situations where drip pricing raises prices and harms consumers are unlikely to arise from unilateral business decisions and (b) the most likely avenue by which drip pricing harms consumers is through the coordinated adoption of drip pricing.

Games doi: 10.3390/g17030024

Authors: Yang Chen Xiaofeng Wang

This paper investigates the evolution of public cooperation within a four-strategy public goods game that incorporates both consistently and inconsistently moralistic exclusion mechanisms. Using replicator dynamics in an infinite well-mixed population, we demonstrate that the presence of Inconsistent Moralists (IMs), i.e., non-contributors who hypocritically exclude other defectors, fundamentally reshapes the dynamical structure of the multi-player social dilemma game. While the system admits no interior fixed point and the IM strategy itself is evolutionarily unstable, IM acts as a critical catalyst by destabilizing pure defection and redirecting evolutionary trajectories toward exclusion-based cooperation. Ultimately, these findings reveal that diverse enforcement strategies can qualitatively alter evolutionary outcomes by providing a previously overlooked indirect pathway for cooperation to emerge and persist in social dilemmas.

Games doi: 10.3390/g17030023

Authors: Ying Zheng Jiayi Xu Xiao-Bing Zhang

This paper develops a dynamic game-theoretic model to evaluate market competitiveness in industries characterized by price competition and adjustment stickiness. We extend the dynamic oligopoly framework for estimating market competitiveness in the literature from a quantity-setting to a price-setting context with differentiated goods. By deriving the subgame perfect equilibrium in a linear-quadratic structure, we utilize an index analogous to the price conjectural variation to measure market competitiveness with differentiated goods. The model is applied to the Chinese retail oil market, and we find that the Chinese retail oil market, particularly dominated by two state firms, exhibits characteristics close to a collusive benchmark within the maintained model. The dynamic game model provides a tractable analytical tool for antitrust authorities to monitor strategic coordination in dynamic environments where price transparency or regulation may facilitate tacit coordination of pricing behavior to a high degree.

Games doi: 10.3390/g17030022

Authors: Noemi Navarro Jean-Christophe Pereau

This article offers a formal analysis of the paradoxes, dilemmas and strategic interactions explored in Liu Cixin’s trilogy The Three-Body Problem. Several games, such as the survival game, the deterrence game, the first contact game, and the big bang game, provide the foundations of cosmic sociology.

Games doi: 10.3390/g17030021

Authors: Chiu-Hui Li Jen-Yao Lee

This paper examines downstream firms’ incentives to accept equity participation by an upstream supplier in a vertically related market. We develop a multi-stage model in which the upstream firm offers an equity stake and sets input prices under price discrimination, while downstream firms subsequently compete à la Cournot. We show that upstream equity ownership induces the upstream firm to lower input prices by partially internalizing downstream profits. This mechanism generates a positive market-expansion effect for downstream firms through lower input costs, while equity ownership simultaneously creates a negative equity-dilution effect by reducing the share of profits retained by downstream firms. When products are homogeneous, the equity-dilution effect dominates the market-expansion effect, leading downstream firms into a Prisoner’s Dilemma. In contrast, under product differentiation, when the ownership share is sufficiently small, the market-expansion effect dominates the equity-dilution effect, resulting in higher downstream profits. In this case, accepting equity participation can be individually optimal for downstream firms, even though mutual acceptance may reduce their retained profits.

Games doi: 10.3390/g17020020

Authors: Mauro Passacantando Fabio Raciti

Eigenvector-based centrality captures recursive notions of importance in networks. While the direct problem computes centrality from given edge weights, the inverse eigenvector centrality problem seeks edge weights that reproduce a prescribed centrality profile; for directed multigraphs, this inverse task is typically non-unique and depends on the admissible arc structure. We study the direct and inverse problems on directed multigraphs and derive an explicit linear characterization of the set of admissible edge-weight vectors that are compatible with a given centrality target. On this feasible set, we formulate a generalized Nash equilibrium problem with shared centrality constraints, in which multiple agents select edge weights to maximize economically interpretable payoffs that incorporate arc-level competition effects. We provide conditions under which the induced game admits a concave potential function, yielding equilibrium existence and, under standard strict concavity assumptions, uniqueness. Finally, we illustrate the model on an airport network where nodes represent airports and parallel arcs represent airline-specific routes, showing that equilibrium selection produces a feasible and interpretable weight configuration that preserves the prescribed centrality.

Games doi: 10.3390/g17020019

Authors: Amit Gayer

This paper analyzes a monopolistic market with a continuum of consumers in the linear case. Consumers are vertically differentiated by a one-dimensional preference for quality, and the monopolist is allowed to offer a menu of quality-price pairs. The analysis shows that, in the linear case, the monopolist’s optimal offer endogenously collapses to a single quality-price pair, where the quality equals the highest feasible level. In addition, welfare maximization is achieved if and only if the market is fully served in equilibrium.

Games doi: 10.3390/g17020018

Authors: Timothy Flannery

Randomized response is a widely used survey technique for measuring stigmatized populations, but it may provide limited information in small samples. This paper introduces a method of elicitation through perfectly correlated questions, showing that correlation can substantially improve statistical performance and allow a dominant-strategy mechanism when either the interviewer or respondents hold symmetric beliefs. The framework also allows the interviewer to possess private information about the distribution of questions, further relaxing incentive constraints. Building on an existing survey design game framework, the paper introduces a novel efficiency-normalized comparison that holds statistical performance constant across mechanisms, enabling a direct comparison of incentives. The analysis characterizes incentive properties and estimation under correlated questions and randomized response, and identifies when it is optimal to ask respondents directly, use randomized response, or correlate questions.

Games doi: 10.3390/g17020017

Authors: Feng Luo Xinmiao Zhao Tiantong Xu

The rapid rise of live-streaming e-commerce has fostered a new “content clipping” model, in which secondary creators edit and republish anchors’ live-streaming content to promote product sales. While this model can expand market reach and enhance revenue, it also introduces copyright disputes, regulatory challenges, and profit-sharing conflicts among platforms, anchors, and secondary creators. This study develops a three-party evolutionary game model to examine strategic choices regarding platform regulation, anchor authorization, and secondary content creation. Results reveal that excessive regulation may undermine equilibrium and profitability, while appropriate authorization can balance risk and reward. Secondary creators’ participation is sensitive to commission rates and cost–benefit trade-offs. This research contributes to the literature by integrating copyright governance into live-streaming e-commerce game theory and offers actionable insights for designing regulatory mechanisms, optimizing authorization policies, and fostering sustainable multi-party collaboration.

Games doi: 10.3390/g17020016

Authors: Dingteng Wang Chengwei Liu Shuping Wang

The digital divide between large enterprises and SMEs (Small and Medium-sized Enterprises) within industrial clusters poses a significant challenge to achieving collective digital transformation, exacerbated by the quasi-public goods, attributes of digital inclusion ecosystems, and the prevalence of free-riding behavior. This paper investigates whether platform enterprises, as core actors occupying structural holes in cluster networks, can foster the co-construction of a digitally inclusive ecosystem. We developed a complex network public goods game model, incorporating performance feedback into a modified Fermi learning to capture firms’ adaptive decision-making based on historical and social aspirations. The model simulates strategic interactions on both small-world and scale-free networks, characteristic of industrial clusters. Numerical simulations reveal that: (1) The core driver of co-construction is the investment return coefficient; (2) Performance feedback amplifies individual rationality, accelerating the formation or collapse of cooperation depending on the investment return coefficient; (3) Platform empowerment—specifically, selectively connecting and incentivizing cooperative firms—effectively promotes ecosystem co-construction, with this strategy proving most impactful when investment returns are moderate. Furthermore, while this selective empowerment strategy benefits the cluster overall, its effect on the platform’s own revenue is network-dependent, showing a more pronounced decline in small-world structures. This study provides a novel analytical framework for understanding strategic interactions in digital inclusion and offers practical insights for policymakers and platform leaders in orchestrating collaborative digital transformation.

Games doi: 10.3390/g17020015

Authors: Michael McBride

This paper introduces fairer measures of individual pitcher performance in baseball using the Shapley Value from coalitional game theory. The paper’s key conceptual innovation is a novel two-stage procedure for constructing the coalitionary game value functions for runs allowed and outs recorded by a baseball team’s defense. This procedure enables the Shapley Value calculation to fairly divide credit for runs and out between different pitchers and between pitchers and fielders. It also results in two new statistics—Shapley Pitcher Runs (SPR) and Shapley Pitcher Outs (SPO)—that, unlike traditional pitching statistics, consistently satisfy several mathematical fairness axioms. A third statistic, called Shapley Run Average, provides a fairer measure of pitcher efficiency. I calculate these statistics for the 2022 Major League Baseball regular season and the 1955–2022 World Series championships. Using SPR and SPO as the standard for fairness, empirical analysis reveals that the traditional pitching statistics systematically and unfairly overcredit pitchers by 40–50%, with starting pitchers miscredited more severely than relievers. Analysis of SRA identifies efficient pitchers whose performance is obscured by conventional statistics and enables a reassessment of historic World Series performances. Overall, this work demonstrates another application of the Shapley Value to creating new performance measures in team sports.

Games doi: 10.3390/g17020014

Authors: Shade T. Shutters

Between 1959 and 1962, Alvin Scodel, J. Sayer Minas, and colleagues conducted some of the earliest laboratory studies of strategic interaction using non-zero-sum games. Working at the margins of economics in the Journal of Conflict Resolution, they documented a striking pattern: subjects frequently chose options that reduced an opponent’s payoff by more than their own, even when mutual cooperation was both individually and collectively optimal. These results—especially the behavior observed in their so-called Game H4, a Harmony Game in which cooperation strictly dominated defection—anticipate a central insight of evolutionary game theory: what matters for adaptation is relative payoff, not absolute gain. This essay reinterprets the Scodel–Minas experiments through a Darwinian lens, arguing that they provide an early empirical challenge to Nash-equilibrium reasoning and to models that evaluate strategies solely in terms of absolute utility. By reconstructing the H4 payoff structure and embedding it within a simple evolutionary framework, I show how small levels of “competitive” behavior can destabilize cooperative equilibria that appear self-evident under standard assumptions. I then revisit three later “puzzles” in the evolution of cooperation—altruistic punishment, the fragility of “win–win” treaties, and rejections in ultimatum bargaining—to ask how differently they might have been framed had the Scodel–Minas findings been part of the canonical experimental literature. Rather than treating these phenomena as surprising anomalies, a historically informed, relative-payoff perspective suggests that they could have been recognized much earlier as natural expressions of an already documented pattern.

Games doi: 10.3390/g17020013

Authors: Arup Daripa

Multi-unit common value auctions in which bidders submit demand functions are used for a variety of purposes, including selling government debt (Treasury auctions) and allocating liquidity (repo auctions). Typically, either a discriminatory or a uniform-price format is used. In this paper, we consider the incentive for participation by relatively uninformed bidders in the presence of more informed bidders under these formats. We characterize the equilibrium under a discriminatory auction and show that discriminatory pricing inhibits uninformed participation. In contrast, the equilibria we construct under a uniform pricing rule show that profitable uninformed participation can occur. The usefulness of widening participation in Treasury auctions makes the latter format a natural choice in these auctions, providing an explanation for the switch to the uniform-price format in US Treasury auctions. We also apply our results to repo auctions and show that a uniform-price format can reduce the ability of a central bank to steer interest rates. This sheds light on the reason for the switch away from the uniform-price format by several central banks in conducting repo auctions. We also consider the question of information aggregation and show that uniform-price auctions might fail to do so. The results also offer an explanation for the fact that the ECB, as well as several other central banks, prefer to allocate liquidity through a fixed-rate tender rather than either uniform-price or discriminatory auctions.

Games doi: 10.3390/g17010012

Authors: Katsuhiko Nishizaki

In a strategy-proof mechanism, implementation theory mostly assumes that each agent is rational in the sense that the agent reveals its true preference to induce its most preferred outcome. This assumption is sufficient to guarantee that the agent seeks such an outcome, but not necessary because the agent might be able to induce the outcome by revealing its other preference. On the basis of such an understanding, this paper considers an implementation problem with the bounded rationality of agents. The bounded rationality presented in this paper means that the agent might choose its best response which is different from its dominant strategy. To describe such behavior, this paper introduces a new notion of equilibrium, called (n−k)-dominant strategy Nash equilibrium at which at most k∈{0,1,…,n} boundedly rational agents might choose their best responses which are different from their dominant strategies, and at least (n−k) rational agents choose their dominant strategies. In addition, to show what a socially optimal outcome is collectively chosen under the existence of boundedly rational agents, this paper introduces a new notion of implementation, called k-secure implementation, which is a double implementation in dominant strategy equilibria and (n−k)-dominant strategy Nash equilibria. In specific environments with k≤(n+1)/2, this paper shows that majority rule satisfies k-secure implementability, but not secure implementability which is equivalent to n-secure implementability. In addition, this paper shows that majority rule realized the socially optimal outcome in the environments in laboratory experiments.

Games doi: 10.3390/g17010011

Authors: Robert Castro

Negotiation outcomes are commonly analyzed through equilibrium concepts, yet many agreements fail during implementation for reasons not captured by incentive structure alone This paper introduces a pre-equilibrium screening criterion for bargaining fragility based on a small set of agreement-level quantities characterizing dependency architecture: strain τ (the number of operative obligations requiring tracking), curvature κ (the density and strength of interdependencies among elements), compressibility σ (the extent to which complexity can be reduced through modularization without altering functional meaning), and the stability quotient Γ = κ/τ (average interdependence burden per element). We use the inequality Γ > σ as a classification rule; agreements with Γ > σ are classified as structurally fragile and, in the data, exhibit higher sensitivity to perturbations. Across 42 documented agreements, the diagnostic correctly classifies nearly all observed outcomes, with only a single false positive and no false negatives. The framework operates as a pre-equilibrium screen that complements (rather than replaces) Nash and bargaining equilibrium analyses by identifying agreement architectures that are structurally brittle under small shocks.

Games doi: 10.3390/g17010010

Authors: Alexander Matros Eren Bilen Leonid Matros

We examine the moral dilemma of how autonomous vehicles (AVs) should be programmed to act in unavoidable crash scenarios involving trade-offs between saving one life and saving many. We report results from three experimental studies that investigate individuals’ preferences over alternative AV decision rules in stylized crash scenarios. Across designs, we find robust support for a probabilistic decision rule that assigns passengers and pedestrians equal ex ante chances of survival (a 50:50 rule). This preference persists across different framings and remains salient even when additional probabilistic options are introduced.

Games doi: 10.3390/g17010009

Authors: Sam Ganzfried

There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer normal-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. We present an approach for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically-constrained program based on a nonlinear complementarity problem formulation from the sequence-form game representation. This approach capitalizes on recent advances for solving nonconvex quadratic programs. Our algorithm is able to quickly solve three-player Kuhn poker after removal of dominated actions. Of the available algorithms in the Gambit software suite, only the logit quantal response approach is successfully able to solve the game; however, the approach takes longer than our algorithm and also involves a degree of approximation. Our formulation also leads to a new approach for computing Nash equilibrium in multiplayer normal-form games which we demonstrate to outperform a previous quadratically-constrained program formulation.

Games doi: 10.3390/g17010008

Authors: Simen Aardal Ulsaker

This paper develops a game-theoretic model to study how exclusionary contracts affect firms’ incentives to invest in innovation. Several symmetric sellers compete to supply an identical product to a set of buyers, and each seller can invest in R&D to develop a higher-quality version of the product. Prior to choosing their R&D investments, sellers may offer exclusionary contracts to buyers. In equilibrium, all buyers sign an exclusionary contract with the same seller, which eliminates rival sellers’ incentives to invest in R&D and concentrates innovative effort in a single firm. Banning exclusionary contracts increases the aggregate probability of innovation and the joint surplus of buyers and sellers only when the R&D technology exhibits sufficiently strong diseconomies of scale.

Games doi: 10.3390/g17010007

Authors: Catherine Hafer

This paper examines the potential for pre-play communication to shorten the duration of two-player incomplete-information wars of attrition. If players’ types constitute costlessly verifiable information, then all types of players disclose their types, resulting in the war of attrition having duration zero. However, if type constitutes unverifiable information, the results are less sanguine. Pre-play cheap-talk communication has no effect on the play of the subsequent war of attrition. Mediated cheap-talk communication is no better: No institution that relies on players’ cheap-talk reports can systematically allocate the prize to the player who values it more highly at a lower resource cost than is entailed in equilibrium play of the war of attrition. Costly signaling in the form of burning money can effectively supplant the war of attrition as a means of allocating the prize, but it requires the same expected equilibrium resource expenditures, with the same expected distribution across types, as does the war of attrition. Thus, in spite of players’ unanimous preference for a system in which types are made known, and in spite of their disclosing type in equilibrium when type is verifiable, they nonetheless expend resources to credibly communicate their types when type is not verifiable, and the resources expended are, on average, equivalent to those expended in a war of attrition.

Games doi: 10.3390/g17010006

Authors: Esteban Colla-De-Robertis

We develop a model of monetary policy committee decision-making, building on the framework of games played through agents (GPTA). Interest groups seek to influence policy by offering action-contingent contracts to committee members. The resulting equilibrium admits a simple characterization and shows how institutional features—such as committee size—shape the extent of external influence. When political pressure pushes for expansive and inflationary policy, larger committees can enhance de facto independence by diluting this influence. We also show that when anti-inflationary pressures dominate, an appropriate choice of committee size can replicate the preference shift towards more conservativeness familiar from delegation frameworks, even when it is not feasible to appoint a conservative central banker in a systematic way.

Games doi: 10.3390/g17010005

Authors: Marc Escrihuela-Villar

We analyze the relationship between collusion sustainability in an infinitely repeated game using trigger strategies and the elasticity of substitution. To this end, we adopt a demand function with constant elasticity of substitution between the differentiated goods. Since our model exhibits a one-to-one relationship between the elasticity of substitution and demand price elasticity, we demonstrate that a larger elasticity decreases the sustainability of collusion. Intuitively, a more elastic demand function causes the increase in deviation profits to compensate for the increase in collusive profits, making collusion less easily sustained. This result holds regardless of whether firms compete in quantities or prices.

Games doi: 10.3390/g17010004

Authors: Steven J. Brams D. Marc Kilgour Christian Klamler

We study conflicts between envy-based fairness and efficiency for allocating indivisible items under additive utilities. We formalize several small, transparent instances showing that standard envy-freeness (EF) or its relaxations EFX and EFX0—i.e., envy-freeness up to any item, where EFX restricts attention to positively valued items and EFX0 allows removing zero-valued items as well—can conflict with Pareto-optimality (PO), maximin (MM), or maximum Nash welfare (MNW). Normatively, we argue that envy-freeness (even as EFX or EFX0) is not a panacea for allocating indivisible items and should be weighed against efficiency and welfare criteria.

Games doi: 10.3390/g17010003

Authors: Guanchu He Colton Hill Joshua H. Seaton Philip N. Brown

This paper considers how a system designer can program a team of autonomous agents to coordinate with one another such that each agent selects (or covers) an individual resource with the goal that all agents collectively cover the maximum number of resources. Specifically, we study how agents can formulate strategies without information about other agents’ actions so that system-level performance remains robust in the presence of communication failures. First, we use an algorithmic approach to study the scenario in which all agents lose the ability to communicate with one another, have a symmetric set of resources to choose from, and select actions independently according to a probability distribution over the resources. We show that the distribution that maximizes the expected system-level objective under this approach can be computed by solving a convex optimization problem, and we introduce a novel polynomial-time heuristic based on subset selection. Further, both of the methods are guaranteed to be within 1−1/e of the system’s optimal in expectation. Second, we use a learning-based approach to study how a system designer can employ neural networks to approximate optimal agent strategies in the presence of communication failures. The neural network, trained on system-level optimal outcomes obtained through brute-force enumeration, generates utility functions that enable agents to make decisions in a distributed manner. Empirical results indicate the neural network often outperforms greedy and randomized baseline algorithms. Collectively, these findings provide a broad study of optimal agent behavior and its impact on system-level performance when the information available to agents is extremely limited.

Games doi: 10.3390/g17010002

Authors: Benjamin Kang James Unwin

In an auction, each party bids a certain amount, and the one who bids the highest is the winner. Interestingly, auctions can also be used as models for other real-world systems. In an all-pay auction all parties must pay a forfeit for bidding. In the most commonly studied all-pay auction, parties forfeit their entire bid, and this has been considered as a model for expenditure on political campaigns. Here, we consider a number of alternative forfeits that might be used as models for different real-world competitions, such as preparing bids for defense or infrastructure contracts.

Games doi: 10.3390/g17010001

Authors: Garrett Milam Andrew Monaco

This work addresses whether heterogeneity in player endowments influences investment decisions in common pool resource (CPR) games, shedding light on the relationship between inequality and economic decision making. We explore two theoretical avenues from behavioral economics—linear other-regarding preferences and inequity aversion—and examine the predictions of each with a laboratory experiment. Our experimental results roundly reject the majority of these explanations: in treatments with endowment inequality, high endowment individuals invest more in the common pool resource than low endowment individuals. We discuss these results in the context of the literature on psychological entitlement and positional preferences.

Games doi: 10.3390/g16060066

Authors: Xingwei Hu

This paper introduces a novel framework for designing fair and sustainable unemployment benefits, grounded in cooperative game theory and real-time fiscal policy. The labor market is modeled as a coalitional game, where a random subset of participants is employed, generating stochastic economic output. To ensure fairness, we adopt equal employment opportunity as a normative benchmark and propose a dichotomous valuation rule that assigns value to both employed and unemployed participants. Within a continuous-time, balanced budget framework, we derive a closed-form payroll tax rate that is fair, debt-free, and asymptotically risk-free. This tax rule is robust across alternative objectives and promotes employment, productivity, and equality of outcome. The framework naturally extends to other domains involving random bipartitions and shared payoffs, such as voting rights, health insurance, road tolling, and feature selection in machine learning. Our approach offers a transparent, theoretically grounded policy tool for reducing poverty and economic inequality while maintaining fiscal discipline.

Games doi: 10.3390/g16060065

Authors: Esther Hauk

This paper develops a formal model to analyze how foreign interventions—via resource transfers towards mobilization, technological upgrades of the mobilization technology, and various forms of conditional aid—reshape identity choices and conflict dynamics in divided societies. After a foreign intervention occurred, individuals simultaneously decided how many resources to allocate to conflict and whether to identify as ethnic or national. The utility derived from identity decreases with the perceived social distance from the chosen group and increases with the group’s status. Foreign interventions can modify identity choices by affecting perceived social distance or group status. Our results reveal that inclusive aid and material support for mobilization are likely to induce national identification. Conversely, exclusive or ethnically targeted aid and technological upgrades of mobilization technology are likely to result in ethnic identification. We show that for all types of interventions analyzed, conflict mobilization is lower and the intervened nation’s material payoff is higher when individuals identify nationally than ethnically.

Games doi: 10.3390/g16060064

Authors: Luka Secilmis Teo Secilmis Simon Jantschgi Heinrich H. Nax

Von Neumann’s minimax theorem defines optimal strategic unpredictability in zero-sum games. Empirical evidence from professional sports has been interpreted as positive behavioral evidence for minimax. In this article, we analyze the strategic optimality of offensive plays in the basketball endgame when a team has a final possession and trails by no more than a single basket. This final moment of the game most closely approximates the simultaneous-move conditions of a game where minimax theory applies. Using comprehensive NBA data from 2010 to 2025, we test for equality of success rates across shooter types (star vs. non-stars) and shot selection (two-point vs. three-point). Our analysis reveals systematic violations of minimax play that have intensified with basketball’s shift to three-pointers and higher expected points. In the final decisive moment of the game, we find that teams systematically overuse three-point shots even though the two-point attempt yields higher field goal percentages. In addition, teams over-rely on star players for the final shot; non-star two-point shots have been the top-performing endgame option in 2022–2025.

Games doi: 10.3390/g16060063

Authors: Garret Ridinger Anne Carpenter

This paper studies the influence of moral emotions and beliefs on understanding charitable giving. While specific moral emotions such as empathy, guilt, and shame have been associated with prosocial behavior, how they impact giving behavior may depend on beliefs about the giving of others. Using a laboratory experiment, individuals participated in a dictator game with charity and completed measures of beliefs, empathy, guilt, and shame. Results show that while individual variation in empathy, guilt, and shame is important in explaining charitable giving, these effects depend crucially on individual beliefs.

Games doi: 10.3390/g16060062

Authors: Prarthana Saikia Ankita Sharma

Wisdom in negotiation is increasingly vital in managing conflicts, yet its behavioural expression remains underexplored. This study explores the behavioural signatures of individuals nominated as wise negotiators within an organisational context. There were 48 participants recruited as wise negotiators from a larger pool of 313 participants. There were three manipulations used: archetypes (personality), reciprocity style, and emotionality, resulting in a 4X3X2 design (24 conditions). Participants were also asked to fill out various wisdom related questionnaires. Each participant had to go through 24 conditions separately before playing an investment game each time. For the analysis purpose, three-way repeated ANOVA and three-way repeated ANCOVA were used. The results revealed that there was a difference in how wise negotiators negotiate differently with different archetypes (p < 0.01), reciprocity (p < 0.01) and emotional situations (p < 0.01). Additionally, there were also interaction effects of archetypes, reciprocity and emotional situations (p < 0.05). Notably, when wisdom variables were statistically controlled, these differences became nonsignificant. A supplementary 2 × 2 design explored gender interactions, showing that outcomes differed by opponents’ gender but not by the gender of the wise negotiator. This finding highlights the role of wisdom traits in strategic negotiation and has implications for training and selection in a high-stakes negotiation context.

Games doi: 10.3390/g16060061

Authors: Karam Allali

We investigate a susceptible–exposed–infected–recovered (SEIR) epidemic model that distinguishes between two subpopulations: individuals favoring vaccination (pro-vaccination), represented by the compartments (SP,EP,IP,RP), and those opposing vaccination (anti-vaccination), described by (SA,EA,IA,RA). The two systems are interconnected through flows between the susceptible classes, capturing the possibility of individuals switching their vaccination strategy, as well as through transitions involving recovered individuals. This framework captures the behavioral interplay during an epidemic, where individuals may reconsider their strategies depending on infection prevalence and the perceived costs and benefits of vaccination versus infection. In the model, immunity may be acquired either through vaccination or after being infected, while waning immunity adds further complexity to individual decision-making. To study these dynamics, we embed the epidemiological system into an evolutionary game framework, where strategy adoption depends on infection levels and associated payoffs. Our analysis shows that, at Nash equilibrium, both pro- and anti-vaccination groups exhibit similar behavior. Numerical simulations further indicate that greater vaccination coverage mitigates the social dilemma, whereas higher rates of waning immunity intensify it.

Games doi: 10.3390/g16060060

Authors: Ben Depoorter Simon Jantschgi Ivan Lendl Miha Mlakar Heinrich Nax

A series of articles has tested von Neumann’s minimax theory against behavioral evidence based on field data from professional sports. The evidence has been viewed and collectively cited as positive evidence that elite athletes in their familiar sports contexts mix well and behave in line with minimax. In this paper, based on open state-of-the-art tennis data and analytics, we shall uncover new and significant evidence against minimax at the very top of the game, where previously, such results had not been obtained. The kinds of behavioral deviations from minimax that we find become apparent, because we enrich the test strategy to take into account whether or not players face ‘pressure’ situations like break points and other decisive points. Our paper highlights that the prior literature’s failure to reject minimax does not constitute positive behavioral evidence, as some of that literature argued, because it is not robust to data aggregations and separations that are psychologically natural given the relevant real-world context. In this case, this means separating serves into the serve types that players actually consider and separating situations by pressure levels, which leads to clear and sound rejection of minimax.

Games doi: 10.3390/g16060059

Authors: Zhuoer Zhang Bryce Morsky

Linear programming and polyhedral representation conversion methods have been widely applied to game theory to compute equilibria. Here, we introduce new applications of these methods to two game-theoretic scenarios in which players aim to secure sufficiently large payoffs rather than maximum payoffs. The first scenario concerns truncation selection, a variant of the replicator equation in evolutionary game theory where players with fitnesses above a threshold survive and reproduce while the remainder are culled. We use linear programming to find the sets of equilibria of this dynamical system and show how they change as the threshold varies. The second scenario considers opponents who are not fully rational but display partial malice: they require a minimum guaranteed payoff before acting to minimize their opponent’s payoff. For such cases, we show how generalized maximin procedures can be computed with linear programming to yield improved defensive strategies against such players beyond the classical maximin approach. For both scenarios, we provide detailed computational procedures and illustrate the results with numerical examples.

Games doi: 10.3390/g16060058

Authors: Jobst Heitzig

We propose a one-shot, non-cooperative mechanism that implements the core in a large class of public goods games. Players simultaneously choose conditional commitment functions, which are binding unilateral commitments that condition a player’s contribution on the contributions of others. We prove that the set of strong Nash equilibrium outcomes of this mechanism coincides exactly with the core of the underlying cooperative game. We further show that these core outcomes can be found via simple individual learning dynamics.

Games doi: 10.3390/g16060057

Authors: Kevin Fathi

This paper introduces the Bateson Game, a signaling game in which ambiguity over the governing rules of interaction (interpretive frames), rather than asymmetry of information about player types, drives strategic outcomes. We formalize the communication paradox of the “double bind” by defining a class of games where a Receiver acts under uncertainty about the operative frame, while the Sender possesses private information about the true frame, benefits from manipulation, and penalizes attempts at meta-communication (clarification). We prove that the game’s core axioms preclude the existence of a separating Perfect Bayesian Equilibrium. More significantly, we show that under boundedly rational learning dynamics, the Receiver’s beliefs can become locked into one of two pathological states, depending on the structure of the Sender’s incentives. If the Sender’s incentives are cyclical, the system enters a persistent oscillatory state (an “ambiguity trap”). If the Sender’s incentives align with reinforcing a specific belief or if the Sender has a dominant strategy, the system settles into a stable equilibrium (a “certainty trap”), characterized by stable beliefs dictated by the Sender. We present a computational analysis contrasting these outcomes, demonstrating empirically how different parametrizations lead to either trap. The Bateson Game provides a novel game-theoretic foundation for analyzing phenomena such as deceptive AI alignment and institutional gaslighting, demonstrating how ambiguity can be weaponized to create durable, exploitative strategic environments.

Games doi: 10.3390/g16060056

Authors: Shacked Avrashi Lior Givon Ilan Fischer

Classifying games according to their strategic properties provides meaningful insights into the motivations driving the interacting parties, suggests possible future trajectories, and in some cases also points to potential interventions aiming to influence the interactions’ outcomes. Here, we present a new classification that merges two perspectives: (i) a revised version of Rapoport and Guyer’s taxonomy, which extends beyond the original 78 games they describe by classifying all two-by-two games according to fundamental strategic properties, and (ii) a novel classification grounded in the theory of subjective expected relative similarity, which addresses not only the games’ payoffs but also the players’ strategic perceptions of their opponents. While Rapoport and Guyer’s original taxonomy classifies only strictly-ordinal games, the revised classification addresses all two-by-two games. It comprises eleven categories that are further grouped into five super-categories that focus on the game’s expected outcome and its strategic stability. The second, similarity-based, classification comprises four main categories, specifying whether players’ perceptions of their opponents have the potential to influence strategic decision-making. The merged classification comprises 14 game types, offering a holistic account of the strategic interaction, the players’ underlying motivations, and the expected outcome. It combines the fixed strategic properties with the variable social aspects of the interaction. Moreover, the novel classification points to the potential of social interventions that may influence the game’s outcome by altering strategic similarity perceptions. Therefore, the present work is relevant for both theoretical and experimental research, providing insights into actual choices expected inside and outside of the laboratory.

Games doi: 10.3390/g16050055

Authors: Fei Jing Junjie Dong

When a technology provider (entrant) enters an emerging end market, he may outsource critical components from a competing conventional manufacturer (incumbent) or insource critical components. Under the outsourcing strategy, brand reputation spills over from the incumbent to the entrant—a phenomenon termed brand spillover. This paper investigates the sourcing strategy (insourcing or outsourcing) and contract choice (wholesale price contract or revenue share contract) in markets subject to brand spillover. We develop a game theoretic model consisting of one entrant with a new technology and one incumbent who sells the traditional product in the end market and the critical component to the entrant. We find that the entrant adopts the insourcing strategy only if his optimal quantity, including original market power and brand spillover, is intermediate. Otherwise, the outsourcing strategy with wholesale price contract is selected when his optimal quantity is low, while revenue-sharing contracts dominate at high quantity. Interestingly, when brand spillover intensity exceeds a threshold, both parties benefit from a higher level of brand spillover under the wholesale price contract.

Games doi: 10.3390/g16050054

Authors: Leandro Chaves Rêgo France Evellyn Gomes de Oliveira Giannini Italino Alves Vieira D. Marc Kilgour

The 2×2 game is the simplest non-trivial model of strategic interaction: there are two players, each has two strategies, and each has a strict preference ranking over the four possible outcomes. For models of play that depend only on the ranking of the outcomes, the catalog of 2×2 games permits many useful comparisons and contrasts. By interpreting a 2×2 game as a graph model, we obtain new data on the properties of limited-move (Lh) stability. Specifically, for each 2×2 strict ordinal game, we determine the Lh-stable outcomes; show how stability depends on the horizon, h; and find the lengths of cycles and the numbers of moves until cycling begins. We then compare our observations with other classifications of these games and with the values of the conflict and harmony indices.

Games doi: 10.3390/g16050053

Authors: Chao Zhao Yongmei Chi Nini Gao Jixiang Song

Fresh agricultural products have significant seasonality and perishability, and their cross-regional sales often face differences in market demand, price, and sales volume. In the context of government quality regulation, competition among retailers in different regions drives supply chain members to improve product quality, expand sales, and reduce losses. However, conflicts of interest under decentralized decision-making may lead to overall inefficiency. This article constructs a supply chain model consisting of a single Manufacturer and two regional Retailers to study the quality competition and coordination mechanism of cross-regional fresh food supply chains under government supervision. By comparing centralized and decentralized decision-making, it is found that although quality improvement in decentralized mode helps enhance competitiveness and sales performance, it is difficult to effectively increase profits and may even lead to a decline in profits. Therefore, this article proposes a cost-sharing contract to achieve supply chain coordination. Research has shown that this contract can effectively improve the overall profit of the supply chain and achieve Pareto improvement; under high market demand and strict regulatory penalties, the total profit of the supply chain increases, but the dominant Retailer benefits more, which can easily trigger the “Matthew effect”. The research results reveal the comprehensive impact of quality investment, contract coordination, market demand fluctuations, and regulatory intensity on supply chain performance, providing theoretical basis and management insights for improving the collaborative efficiency and policy design of cross-regional fresh food supply chains.

Games doi: 10.3390/g16050052

Authors: Peichen Ye Anna Tur Yilun Wu

This paper delves into a multi-player non-renewable resource extraction differential game model, where the duration of the game is a random variable with a composite distribution function. We first explore the conditions under which the cooperative solution also constitutes a Nash equilibrium, thereby extending the theoretical framework from a fixed duration to the more complex and realistic setting of random duration. Assuming that players are unaware of the switching moment of the distribution function, we derive optimal estimates in both time-dependent and state-dependent cases. The findings contribute to a deeper understanding of strategic decision-making in resource extraction under uncertainty and have implications for various fields where random durations and cooperative strategies are relevant.

Games doi: 10.3390/g16050051

Authors: Sijia Xu Peng Liu

In this paper, we study the multidimensional claims problem and introduce the eating algorithm to this problem. It is shown that a solution is efficient if and only if it is the outcome of the eating algorithm with a profile of specific eating functions. Moreover, we adapt three classical solutions, i.e., the constrained equal awards rule, constrained equal losses rule, and proportional rule, to the current setting and show that they are special cases of the eating algorithm with specific eating functions.

Games doi: 10.3390/g16050050

Authors: Patrick Legros Andrew F. Newman

The value principle in organizational economics states that the net market value of the goods that a firm sells is a key determinant of its organizational design. We survey and extend some recent developments in the theoretical literature at the nexus of organizational and industrial economics, focusing on this precept as the unifying theme. Under perfect competition, we study how market price influences the use of scarce professional management and the degree of organizational heterogeneity in an industry. In a more general setting, we show how changes in demand influence not only the use of professional management, but also the size and the market power of firms. And we show how prices can affect the internal control structure of firms, sometimes in highly distorted ways. We discuss applications to comparative industrial organization and to technological diffusion.

Games doi: 10.3390/g16050049

Authors: Amit Gayer

This paper introduces a novel ratio-based framework for analyzing how consumer heterogeneity translates into product differentiation in vertically structured monopoly markets. We consider a monopolist facing a continuum of consumers and a strictly convex production cost function and identify conditions under which the heterogeneity of preferences, measured by the length of the consumer type interval, maps into a corresponding range of offered qualities. The analysis shows that this mapping depends on the curvature of the marginal cost function: under linear costs, the relationship is proportional; under convex costs, heterogeneity expands faster than segmentation; and under concave costs, the reverse occurs. These findings offer a new lens for understanding endogenous market granularity in monopoly settings and have potential applicability in markets with vertically differentiated goods. We also show that under partial market coverage, this proportionality breaks down - even in the linear case - revealing a critical asymmetry in equilibrium structure.

Games doi: 10.3390/g16050048

Authors: Arturo Macías

Mainstream experimental economics is characterized by its focus on theory testing and “treatment effects” on aggregate outcomes. The “agentic” alternative is concerned with the econometric specification of individual behavior. In this essay, first, a literature review of agentic experimental economics is provided, and a stylized workflow is proposed to produce and validate econometric models of individual behavior based on experimental data: (i) create a baseline (“optimal”) behavioral benchmark (by analytical means or reinforcement learning) for the considered multi-agent game, (ii) conduct experiments with human subjects, (iii) use the experimental results to characterize the structure of the deviations from the baseline behavior, and (iv) re-run the experiment with artificial agents calibrated in the previous step, and compare the outcomes with those of the human experiment. Two papers have been selected to illustrate the successful use of the proposed workflow. Finally, the relations between agent-based and experimental economics are discussed after deep learning has “tamed” the curse of dimensionality.

Games doi: 10.3390/g16050047

Authors: Adnane Fouadi Mourad El Ouali Anand Srivastav

We investigate the (a:b) Maker–Breaker subgraph game played on the edge set of the complete graph Kn, where n,a,b∈N, and Maker’s objective is to construct a member of a prescribed family of graphs H, while Breaker aims to prevent this. In our study, Breaker moves first, and H is taken to be either the family of connected spanning subgraphs or the family of spanning subgraphs with minimum-degree at least k=k(n). For the (a:b) minimum-degree-k game, we determine the asymptotically optimal threshold bias across a wide range of values for a. For the (a:b) connectivity game, we resolve an open problem posed by Hefetz et al. (2012) by identifying the exact leading term in the asymptotic behavior of the threshold bias when a=clnn.

Games doi: 10.3390/g16050046

Authors: Yu-Hsien Liao

This paper establishes dividend-based representations for two influence assessments. First, we define a system of min-dividends derived from the minimal-influence evaluation via a unique linear decomposition using unanimity-type spanning models. Building on this, we further construct a pair of internal and external min-dividends satisfying Completeness and Balancedness conditions, through which we express the stable min-value as the net difference of internal gains and external losses. We then demonstrate that the minimal self-stable value can be represented as accumulated average min-dividends across all coalitions they have participated in. Furthermore, the proposed expression also is adopted to analyze the stability of the minimal self-stable value. These results extend the classical notion of dividends into a minimal-influence-based framework with potential applications in fair resource allocation and responsibility apportionment.

Games doi: 10.3390/g16050045

Authors: Alfredo Martinez Jonathan Machado Eric Sanchez Igor V. Erovenko

Meningococcal meningitis is a deadly acute bacterial infection caused by the Neisseria meningitidis bacterium that affects the membrane covering the brain and spinal cord. The World Health Organization launched the “Defeating bacterial meningitis by 2030” initiative in 2018, which relies on recent discoveries of cheap and effective vaccines. Here, we consider one important factor—human behavior—which is often neglected by immunization campaigns. We constructed a game-theoretic model of meningitis in the meningitis belt, where individuals make selfish rational decisions whether to vaccinate based on the assumed costs and the vaccination decisions of the entire population. We identified conditions when individuals should vaccinate, and we found the optimal (equilibrium) population vaccination rate. We conclude that voluntary compliance significantly reduces the endemic levels of meningitis if the cost of vaccination relative to the cost of the disease is sufficiently low, but it does not eliminate the disease. We also performed uncertainty and sensitivity analysis on our model.

Games doi: 10.3390/g16050044

Authors: Ulrike Vollstädt Patrick Imcke Franziska Brendel Christiane Ehses-Friedrich

Certifiers such as Stiftung Warentest (Germany), Which? (UK), and Consumer Reports (US) reduce asymmetric information between buyers and sellers by providing credible information about product quality. However, due to their limited testing capacities, they face a set selection problem and test only a subset of all available product models. We show theoretically that, under any current mechanism to select product models for testing, buyers always end up buying suboptimal product models, unless all product models which would be sold under complete information (or all but the overall cheapest one) happen to be tested. Instead, we propose a novel mechanism based on voluntary disclosure, but with the same testing capacity, which always yields the maximum possible consumer surplus and thus weakly dominates any current mechanism. Furthermore, we confirm in a controlled laboratory experiment that our mechanism significantly increases consumer surplus.

Games doi: 10.3390/g16050043

Authors: Guang-Zhen Sun

Hirshleifer’s difference-form contest technology is a useful tool in the study of a class of conflict, especially military combats. We aim to highlight an important feature that the Hirshleifer contest model distinctively has, namely passivity (bidding zero effort) may stand as an effective choice in conflict even when the contest is highly deterministic (i.e., with small noise). For that purpose, we establish two propositions on the contest with n≥2 risk-neutral contestants under small noise. The first proposition states that every contestant bids arbitrarily close to zero (if not bidding zero with positive probability at all) under sufficiently small noise. The second proposition, more strikingly, states that every contestant either bids arbitrarily close to the second-highest valuation (among all the contestants’ valuations), or simply remains passive with certainty under any sufficiently small noise. We further show that the first proposition holds for the contest between risk-averse individuals endowed with constant absolute risk aversion as well, and illustrate by an example how quickly polarization in bidding among contestants, as is predicted by the propositions, may emerge as the noise of the contest abates. These results help pave the way toward a complete characterization of the difference-form contest.

Games doi: 10.3390/g16050042

Authors: Lorenzo Ferrari Werner Güth Vittorio Larocca Luca Panaccione

This paper examines stochastic cooperation in markets with two sellers who exhibit one-sided dependency. The independent seller’s pricing influences the dependent seller’s demand, but not vice versa. We study the one-dimensional hybrid game class whose parameter is the exogenously given probability of cooperation. In each game of this class, both sellers simultaneously choose prices that determine their endogenous threats, i.e., conflict profits. The sellers are aware of the cooperation probability but cannot condition prices on whether or not there is cooperation. We characterize the equilibrium prices and the sellers’ expected profits. Our main result shows that the independent seller earns higher expected profits when cooperation is more likely. In contrast, the dependent seller earns lower expected profits when the likelihood of cooperation is below a threshold that we characterize explicitly, and higher profits are earned thereafter. These findings suggest that, within our framework, antitrust concerns may be mitigated. Since dependent sellers can incur losses from cooperation, collusion attempts become less viable in markets with one-sided dependency.

Games doi: 10.3390/g16050041

Authors: Jochen Staudacher

We investigate power indices for simple games with precoalitions which distribute power among players in an external and an internal step. We extend an existing approach which uses the Public Good index both on the external level in the quotient game as well as on the internal level for measuring the leverage of players to threaten their peers through departing the precoalition. We replace the Public Good index in that model by five other efficient power indices, i.e., the Shapley–Shubik index, the Deegan–Packel index, the Johnston index and two indices based on null player free winning coalitions. Axiomatizations of the novel power indices with threat partitions are presented. We also propose a slight modification to the existing framework for threat power indices which guarantees that null players are always assigned zero power. Numerical results for all power indices combined with different threat partitions are presented and discussed.

Games doi: 10.3390/g16040040

Authors: Henrique Santos Thyago Nepomuceno

This paper analyzes the pricing dynamics of product warranties by developing a three-player hierarchical game model involving a manufacturer, an independent service agent, and a consumer. The model provides a scenario where the manufacturer and the agent form a coalition to coordinate pricing strategies, while interacting non-cooperatively with the consumer. In this framework, the manufacturer sets the product’s sale price, including the base warranty, while the agent determines the price of extended maintenance services. The key contribution is the application of the Shapley value to equitably distribute the coalition’s profits based on each member’s contribution—a novel approach in the warranty pricing literature. We detail the characteristic functions that define the coalition’s structure and present computer simulations to estimate the expected costs associated with maintenance services. A comprehensive sensitivity analysis is applied to report how changes in parameters influence equilibrium strategies and players’ payoffs. The results provide strategic insights into how manufacturers and agents can coordinate to optimize pricing, capture consumer surplus, and improve decision-making in warranty service markets.

Games doi: 10.3390/g16040039

Authors: Justin Chan

Non-cooperative games in normal form are specified by a player set, sets of player strategies, and payoff functions. Cooperative games, meanwhile, are specified by a player set and a worth function that maps coalitions of players to payoffs they can feasibly achieve. Although these games study distinct aspects of social behavior, this paper proposes a novel attempt at relating the two models. In particular, cooperative games may be represented by a non-cooperative game in which players can freely sign binding agreements to form coalitions. These coalitions inherit a joint strategy set and seek to maximize collective payoffs. When these coalitions play against one another, the equilibrium payoffs for each coalition coincide with what is predicted by the worth function. This paper proves sufficient conditions under which cooperative games can be represented by non-cooperative games. This paper finds that all strictly superadditive partition function form (PFF) games can be represented under Nash equilibrium (NE) and rationalizability; that all weakly superadditive characteristic function form (CFF) games can be represented under NE; and that all weakly superadditive PFF games can be represented under trembling hand perfect equilibrium (THPE).

Games doi: 10.3390/g16040038

Authors: Petra Nieken Abdolkarim Sadrieh Nannan Zhou

Overconfidence is a well-established behavioral bias that involves the overestimation of one’s own capabilities. We introduce a model in which managers and agents exert effort in a joint production, after the manager decides on the allocation of the tasks. A rational manager tends to reduce their own effort by delegating the critical task to the agent more often than in an efficient task allocation. In contrast, an overconfident manager engages in responsibility hoarding, i.e., is likely to delegate a critical task less often to the agent than a rational manager. In fact, a manager with a sufficiently high ability and a moderate degree of overconfidence increases the total welfare by refusing to delegate critical tasks and by exerting more effort than a rational manager. Finally, we derive the conditions under which the responsibility hoarding can persist in an organization, showing that the bias survives as long as the overconfident manager can rationalize the observed output by underestimating the ability of the agent.

Games doi: 10.3390/g16040037

Authors: Sébastien Huot Abbas Edalat

In this work, we study n-agent Bayesian games with m-dimensional vector types and linear payoffs, also called linear multidimensional Bayesian games. This class of games is equivalent with n-agent, m-game uniform multigames. We distinguish between games that have a discrete type space and those with a continuous type space. More specifically, we are interested in the existence of pure Bayesian Nash equilibriums for such games and efficient algorithms to find them. For continuous priors, we suggest a methodology to perform Nash equilibrium searches in simple cases. For discrete priors, we present algorithms that can handle two-action and two-player games efficiently. We introduce the core concept of threshold strategy and, under some mild conditions, we show that these games have at least one pure Bayesian Nash equilibrium. We illustrate our results with several examples like the double-game prisoner’s dilemma (DGPD), the game of chicken, and the sustainable adoption decision problem (SADP).

Games doi: 10.3390/g16040036

Authors: Peter Paul Vanderschraaf

I consider an underexplored possible explainer of the “surprising” results of Ultimatum Game experiments, namely, that Proposers and Recipients consider following only some of all the logically possible strategies of their Ultimatum Game. I present an evolutionary analysis of different games having the same set of allowable Proposer offers and functions that determine Proposer and Recipient payoffs. For Unrestricted Ultimatum Games, where Recipients may choose from among any of the logically possible pure strategies, populations tend to evolve most often to Nash equilibria where Proposers make the lowest allowable offer. However, for Threshold Reduced Ultimatum Games, where Recipients must choose from among minimum acceptable offer strategies, and for Range Reduced Ultimatum Games, where Recipients must choose from among pure strategies that spurn offers that are “too high” as well as “too low”, populations tend to evolve most often to Nash equilibria where Proposers offer substantially more than the lowest possible offer, a result that is consistent with existing Ultimatum Game experimental results. Finally, I argue that, practically speaking, actual Proposers and Recipients will likely regard some reduction of the Unrestricted Ultimatum Game as their game because, for them, the strategies of this reduction are salient.

Games doi: 10.3390/g16040035

Authors: Hao Liu Xuewen Qian Chen Qu Jingyi Shen

We analyze the endogenous formation of patent pools among three patent owners and the associated welfare effects. Under a condition of synergistic three-patent combination, either a unique symmetric equilibrium or infinitely many asymmetric equilibria can arise when patents are fragmented. By using the notion of equilibrium binding agreements, we show that (1) when there is a unique symmetric equilibrium under fragmented patents, the complete pool is both stable and welfare-maximizing; (2) fragmented patents are stable in the presence of infinitely many asymmetric equilibria; and (3) when considering only a single specific asymmetric equilibrium under fragmented patents, the complete pool is welfare-maximizing if it is stable, while fragmented patents can be both stable and welfare-maximizing under certain conditions. We also discuss an alternative version of synergism and an alternative bargaining protocol for patent pool formation.

Games doi: 10.3390/g16040034

Authors: B. Pablo Montagnes Stephane Wolton Junyan Jiang

Hierarchies are common in political settings. From judges to elected politicians, as well as from activists to bureaucrats, political agents compete for promotion to higher positions. This paper studies political tournaments and their impact on two aspects of political performance: accountability and selection. While larger tournaments discourage effort, they improve selection. We also discuss the optimal design of tournaments as a function of the principal’s objectives and the features of the environment. We find that tournaments of size two (such as two-candidate elections) are generally suboptimal. Our analysis also highlights that increased desirability of promotion always increases effort but can reduce the optimal tournament size under certain conditions. We also present a range of other comparative statics.

Games doi: 10.3390/g16040033

Authors: Sung-Hoon Park Jason F. Shogren

Herein, we further examine how we can value nonmarket goods and services by considering the costs associated with environmental conflicts. Focusing on two American rules—the asymmetric reimbursement system and the contingent fee contract—we develop a strategic game-theoretic model in which a citizens group engages a delegate through a contingent fee compensation contract, while a polluter engages a delegate through an hourly fee compensation contract. If the citizens group prevails, the polluter is obligated to contribute a portion of the contingent fee. Solving for the subgame perfect equilibrium, two results emerge. First, the 4x-rule can be maintained through the adjustment of the asymmetric reimbursement system. Second, the asymmetric reimbursement system can serve both as a supplementary method to measure nonmarket valuation and to reduce the rent dissipation resulting from environmental conflicts under general circumstances.

Games doi: 10.3390/g16040032

Authors: Ruohan Wang Jing Wang Zhi Yang

According to the research, the proportions of various types of investors may or may not change in response to different trading strategies or complex trading environments. This paper investigates a single-period trading model for multiple risk-averse and risk-neutral insider traders. The distinction is that we consider the impact of the number of risk-averse traders and different types of insider traders on market liquidity. The model contains four types of trading entities: risk-neutral and risk-adverse insider traders, risk-neutral market makers, and noise traders. Firstly, we prove the existence and uniqueness of the model’s linear Nash equilibrium; secondly, we compare the model with multiple risk-neutral and risk-adverse insider traders in the market to the model with only risk-neutral insider traders and risk-adverse insider traders. It is shown that the market liquidity parameter λ decreases with the increase in the number of risk-averse persons N2 in a particular range and increases with the increase in the number of risk-averse persons N2 in another range. Markets with risk-neutral and risk-averse insider traders have consistently lower liquidity than markets with only risk-neutral insider traders. Comparing this to markets with only risk-adverse insider traders reveals that the number of risk-adverse traders heavily influences market liquidity.

Games doi: 10.3390/g16030031

Authors: Laura Levaggi Rosella Levaggi

Health care systems rely on geographical boundaries that secure financial stability and adequate planning. Quality differences across regions often arise for efficiency reasons, causing patient flows if mobility is free. In this paper, a theoretical spatial competition model is developed to study the role of patients’ mobility on quality setting and to draw policy implications on its use as an instrument to reduce disparities, in a setting where regions differ in efficiency, costs, and market structure. From the analysis, it emerges that the institutional setting matters and a trade-off may appear between equity (in terms of quality difference across patients) and welfare (finding an allocation that maximizes social benefits). In a centralized setting, it is optimal to regulate mobility and increase the quality gap, while allowing free mobility calls for a refined quality setting, in which, depending on a balance between costs and benefits, the quality gap may be either increased or decreased. In decentralization the gap is generally lower, compared to centralization: the different consideration of benefits from local quality provision results in higher quality levels where the market structure is vertically integrated.

Games doi: 10.3390/g16030030

Authors: Jingyi Shen Chen Qu

In symmetric games with externalities across coalitions, we investigate how three players form coalitions using two solutions: n∗, which is a focal prediction of coalition structure in a class of noncooperative coalitional bargaining games, and equilibrium binding agreements, which represent the cooperative blocking approach. We find that the coarsest equilibrium coalition structure (based on the latter notion) is never finer than n∗, and we provide a sufficient and necessary condition for these two solutions to generate the same coalition structure (i.e., the two solutions coincide if and only if the first coalition to form in n∗ is not a two-player coalition or a particular condition about average coalitional worths is satisfied). In symmetric games with more than three players, we demonstrate through a series of examples that any relationship between these two solutions is possible. We also discuss symmetric games with positive externalities or equal division in which these two solutions coincide.

Games doi: 10.3390/g16030029

Authors: Yigal Gerchak Eugene Khmelnitsky

Establishing a partnership necessitates agreeing on how to divide future profits or losses. We consider parties who wish to contract on the division of uncertain future profits. We propose to divide profits according to the Raiffa–Kalai–Smorodinsky (K-S) solution, which is the intersection point of the feasible region’s boundary and the line connecting the disagreement and ideal points. It is the only function which satisfies invariance to linear transformations, symmetry, strong Pareto optimality, and monotonicity. We formulate the general problem of designing a contract which divides uncertain future profit between the partners and determines shares of each partner. We first focus on linear and, later, non-linear contracts between two partners, providing analytical and numerical solutions for various special cases in terms of the utility functions of the partners, their beliefs, and the disagreement point. We then generalize the analysis to any number of partners. We also consider a contract which is partially based on the parties’ financial contribution to the partnership, which have a positive impact on profit. Finally, we address asymmetric K-S solutions. K-S solutions are seen to be a useful predictor of the outcome of negotiations, similar to Nash’s bargaining solution.

Games doi: 10.3390/g16030028

Authors: Maximilian Olaf Hoyer Frans van Winden

This paper experimentally investigates the relationship between an investor and a project manager. Project managers choose from a pool of projects, the success probabilities of which are uncertain. Investors can change projects, but also have to change project managers if they want to do so. An additional joint project or a voluntary money transfer precedes their interaction. We hypothesize that investors favor projects of managers with whom they share positive experiences at that stage, even though these experiences do not provide any information about the subsequent project’s success probability. Interaction through a voluntary transfer plays a clear and significant role in the investors’ decision-making via bonding, whereas the influence of merely sharing a positive or negative experience proves more complex.

Games doi: 10.3390/g16030027

Authors: James Gilmore David Porter

One of the main goals of two-sided matching mechanisms is to pair two groups of agents in a stable manner. Stability means that no pair of agents has an incentive to deviate from their assigned match. The outcome of such a match can have significant consequences for the participants involved. Most existing research in this field assumes that the quotas of organizations are fixed and externally determined, which may not always be realistic. We introduce the concept of slot stability, which considers the possibility that organizations may want to adjust their quotas after the match process. To address this issue, we propose an algorithm that generates both stable and slot-stable matches by using flexible, endogenous quotas.

Games doi: 10.3390/g16030026

Authors: Blake A. Allison Jason J. Lepore

This paper studies a class of two-player all-pay contests with externalities that encompass a general version of duopoly price competition. This all-pay contest formulation puts little restriction on production technologies, demand, and demand rationing. There are two types of possible equilibria: In the first type of equilibrium, the lower bound to pricing is the same for each firm, and the probability of any pricing tie above this price is zero. Each firm’s equilibrium expected profit is their monopoly profit at the lower bound price. In the second type of equilibrium, one firm prices at the lower bound of the other firm’s average cost and other firm prices according to a non-degenerate mixed strategy. This type of equilibrium can only occur if production technologies are sufficiently different across firms. We derive necessary and sufficient conditions for the existence of pure strategy equilibrium and use these conditions to demonstrate the fragility of deterministic outcomes in pricing games.

Games doi: 10.3390/g16030025

Authors: Yoan Hermstrüwer

This article explores the impact of procedural information on the behavior of students under two school admission procedures commonly used in the US, the EU, and other jurisdictions: the Gale–Shapley mechanism and the Boston mechanism. In a lab experiment, I compare the impact of information about the mechanism, information about individually optimal application strategies, and information about both. I find that strategic and full information increases truth-telling and stability under the Gale–Shapley mechanism. Under the Boston mechanism, however, the adoption of equilibrium strategies remains unaffected. Contrary to the prevailing assumptions in matching theory, the Boston mechanism improves perceived fairness. These results underscore the importance of procedural transparency and suggest that eliminating justified envy may not be sufficient to foster fairness and mitigate litigation risks.

Games doi: 10.3390/g16030024

Authors: Michele Bernasconi Giulio Cinquanta Valentino Dardanoni Vincenzo Prete

Comparing mobility is an important but controversial issue. In this paper, we argue that in a specific and relevant case, there exists a univocal and non-controversial definition of greater (exchange) mobility that allows for unambiguous comparisons. We conducted a questionnaire experiment to investigate whether people’s perceptions of social mobility align with this definition, and we found that people’s choices are broadly in line with the theoretical predictions.

Games doi: 10.3390/g16030023

Authors: Khac Minh Duc Do Dmitriy Knyazev

This paper develops a simple model of sequel production for experience goods, showing how reputation shapes a producer’s incentives. Producers differ in productivity, which determines how much effort they invest. A sequel is made only if the previous installment exceeds a quality threshold, capturing the idea that consumers base future consumption on past success. Although high-productivity producers create higher-quality originals and sequels, the conditioning on successful originals still makes sequels, on average, worse than their predecessors. This aligns with evidence of sequel underperformance in media markets.

Games doi: 10.3390/g16030022

Authors: Klaudia Schäffer Adrienn Král Ádám Kun

Human cooperation is ubiquitous and instinctive. We are among the most cooperative species on Earth. Still, research mostly focuses on why we cooperate, instead of understanding why some of us do not do so. The public goods game can be used to map human cooperation as well as to study free riding. We acquired data through an online, unincentivized questionnaire which prompted respondents to choose how much of an initial endowment to contribute to a common pool. The respondents contributed, on average, 54% of their initial endowment to the common pool. The usual categorization scheme of the elicited conditional contribution pattern discerns unconditional free riders who do not contribute irrespective of the contributions of others and calls everyone a conditional cooperator who correlates their contribution with that of the others. However, someone consistently offering less than the others should not be called a cooperator. Consequently, based on the conditional contribution patterns among our respondents, we suggest a recategorization of contribution patterns into the following categories: unconditional cooperator (1.5%), unconditional free rider (10.6%), perfect conditional cooperator (42.6%), hump-shaped contributor (0.7%), V-shaped contributor (0.4%), conditional cooperator (16.6%), conditional free rider (13.6%), conditional contributor (6.4%), negative conditional contributor (0%), and others (7.6%). We only call someone a cooperator if the respondent at least matches others’ contribution, and call everyone consistently offering less a free rider. Furthermore, we found no difference between the contributions of women and men. No correlation of contribution with age, educational attainment, and size of the residential settlement was found. Students’ contributions were not different from non-students’ contributions. We found a significant correlation of the contribution to the common pool with hypercompetitive orientation (negative correlation) and the self-assessed willingness to take risks in general (positive correlation).

Games doi: 10.3390/g16030021

Authors: Anna Kerkhof Johannes Münster

Media content is an important privately supplied public good. While it has been shown that contributions to a public good crowd out other contributions in many cases, the issue has not been thoroughly studied for media markets yet. We show that in a standard model of commercial media bias, qualities of media content are strategic complements, whereby investments into quality can crowd in further investments and engage competitors in a race to the top. Therefore, financially strong public service media can mitigate commercial media bias: the content of commercial media can be more in line with the preferences of the audience and less advertiser-friendly in a dual (mixed public and commercial) media system than in a purely commercial media market.

Games doi: 10.3390/g16020020

Authors: Samira Habli Rachid El-Azouzi Essaid Sabir Mandar Datar Halima Elbiaze Mohammed Sadik

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.

Games doi: 10.3390/g16020019

Authors: Werner Güth Ludivine Martin Tibor Neugebauer Sotiria Xanalatou

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.

Games doi: 10.3390/g16020018

Authors: Yuri Perez Fabio Henrique Pereira Pedro Henrique Triguis Schimit

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.

Games doi: 10.3390/g16020017

Authors: Stefano Vannucci

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.

Games doi: 10.3390/g16020016

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

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.

Games doi: 10.3390/g16020015

Authors: Shih-Hao Huang Chien-Shu Tsai Jen-Yao Lee Su-Ching Tsai

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.

Games doi: 10.3390/g16020014

Authors: Arka Mukherjee Subhadip Ghosh

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.

Games doi: 10.3390/g16020013

Authors: Maarten Janssen Vladimir Karamychev

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.

Games doi: 10.3390/g16020012

Authors: Nasimeh Heydaribeni Achilleas Anastasopoulos

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.

Games doi: 10.3390/g16020011

Authors: Yahel Giat Amichai Mitelman

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.

Games doi: 10.3390/g16010010

Authors: Haitao Li Zhenping Geng Mengyuan Qin

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.

Games doi: 10.3390/g16010009

Authors: Guennady Ougolnitsky Alexey Korolev

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.

Games doi: 10.3390/g16010008

Authors: Angelos Gkekas Athina Apostolidou Artemis Vernadou Athanasios Kehagias

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.

Games doi: 10.3390/g16010007

Authors: Congcong Wang Yuhan Wang Shanshan Chen Shravan Luckraz Bruno Antonio Pansera

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.

Games doi: 10.3390/g16010006

Authors: Diego Armando Canales

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.

Games doi: 10.3390/g16010005

Authors: Hiroto Kuramitsu Kaiyu Suzuki Tomofumi Matsuzawa

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.

Games doi: 10.3390/g16010004

Authors: Oleg Smirnov

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.

Games doi: 10.3390/g16010003

Authors: Ilan Fischer Shacked Avrashi Lior Givon

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.

Games doi: 10.3390/g16010002

Authors: Marco Biella Max Hennig Laura Oswald

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.

Games doi: 10.3390/g16010001

Authors: Cemil Selcuk

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.

Games doi: 10.3390/g15060045

Authors: Sam Ganzfried

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.

Games doi: 10.3390/g15060044

Authors: Anna De Simone K. P. S. Bhaskara Rao

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.

Games doi: 10.3390/g15060043

Authors: Achille Basile K. P. S. Bhaskara Rao Anna De Simone Ciro Tarantino

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.

Games doi: 10.3390/g15060042

Authors: Alexandra Fedrigo

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.

Games doi: 10.3390/g15060041

Authors: Alexandra Fedrigo

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.