Games

26 pages, 1360 KB

Open AccessArticle

Efficient Favouritism with Status Incentives: A Moral-Hazard Game of Delegation and Recognition

by Swapnendu Banerjee, Oindrila Dey and Subhadip Ghosh

Games 2026, 17(4), 43; https://doi.org/10.3390/g17040043 - 7 Aug 2026

This paper studies the interaction between status incentives and organizational design in a two-agent moral hazard framework with limited liability. A risk-neutral principal chooses between two regimes: favouritism, under which one agent receives exclusive decision rights and status recognition for successful project [...] Read more.

This paper studies the interaction between status incentives and organizational design in a two-agent moral hazard framework with limited liability. A risk-neutral principal chooses between two regimes: favouritism, under which one agent receives exclusive decision rights and status recognition for successful project implementation, and fairness, under which both agents share equal decision rights and status is distributed across agents. We show that status incentives can make ex-post favouritism optimal even when the principal does not exhibit any ex-ante preferential bias toward any particular agent. Introduction of status incentives shrink the parameter region supporting interior inefficient favouritism, and eliminate it entirely whenever project returns are large enough to sustain interior contracts for both agents. We also show that, for a non-empty set of primitive parameter values, the principal’s optimal regime can be non-monotonic in status: favouritism is optimal when status valuation is sufficiently low or sufficiently high, while fairness may dominate for intermediate values. The results shed light on why selective, hierarchical recognition systems coexist with flat team-credit structures across organizations. Full article

(This article belongs to the Section Applied Game Theory)

21 pages, 1022 KB

Open AccessArticle

An Extreme Learning Machine-Based Method for Solving Linear–Quadratic Nonzero-Sum Differential Games

by Changdong Duan and Yuefei Yuan

Games 2026, 17(4), 42; https://doi.org/10.3390/g17040042 - 6 Aug 2026

Abstract

Multi-agent interaction in linear–quadratic (LQ) differential games gives rise to open-loop Nash equilibria that rarely admit closed-form expressions, motivating the development of reliable numerical solvers. Classical approaches such as shooting and spectral collocation are sensitive to the initial guess on the unknown boundary [...] Read more.

Multi-agent interaction in linear–quadratic (LQ) differential games gives rise to open-loop Nash equilibria that rarely admit closed-form expressions, motivating the development of reliable numerical solvers. Classical approaches such as shooting and spectral collocation are sensitive to the initial guess on the unknown boundary values and accumulate discretisation error over long horizons, while deep-learning alternatives require iterative gradient-based training with architecture- and convergence-specific overhead. To overcome these limitations, we recast the LQ nonzero-sum game as a linear two-point boundary value problem (TPBVP) via the Pontryagin maximum principle (PMP) and solve it with a single-layer feedforward neural network (SLFN) in which hidden-layer parameters are sampled once and fixed. The state and all player-specific costates are parameterised by random hidden features on a uniform time grid, the boundary conditions are appended as dedicated rows of the linear collocation system, and the output weights follow from a single Moore–Penrose pseudoinverse, entirely bypassing gradient-based iteration. For the scalar LQ optimal-control TPBVP, a residual-to-solution stability theorem converts the continuous equation and boundary residuals into uniform state, costate, control, and cost error bounds. Validation across two-player low- and high-dimensional benchmarks, a heterogeneous three-player game, and paired seed sweeps confirms high accuracy against analytical and matrix-exponential references, while revealing that no single activation function dominates across all problem types: tanh is most accurate in one-dimensional settings, and Gaussian RBF leads in multidimensional cases. Full article

(This article belongs to the Special Issue New Advances in Computational Game Theory and Its Applications)

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27 pages, 2148 KB

Open AccessArticle

Endogenous Agreement Geometry in Nash Bargaining over a Continuum of Issues

by Alessio Staffini

Games 2026, 17(4), 41; https://doi.org/10.3390/g17040041 - 6 Aug 2026

Abstract

Many bargaining problems allocate control over heterogeneous issues rather than a single scalar surplus. This paper studies a two-player game-theoretic Nash bargaining problem over a continuum of issues with stochastic valuation fields. Taking the classical maximum Nash welfare cutoff allocation as a benchmark, [...] Read more.

Many bargaining problems allocate control over heterogeneous issues rather than a single scalar surplus. This paper studies a two-player game-theoretic Nash bargaining problem over a continuum of issues with stochastic valuation fields. Taking the classical maximum Nash welfare cutoff allocation as a benchmark, we characterize the selected agreement as an endogenous excursion set of the log-relative valuation field. We then study its economic geometry: stationarity and ergodicity yield deterministic many-issue payoff limits, a finite-domain perturbation formula separates local shocks from global cutoff feedback, Kac–Rice methods describe boundary intensity, and a perimeter penalty for fragmented contracts turns the bargain into a finite perimeter variational problem. At regular boundary points, the complexity-penalized bargain satisfies a curvature-adjusted bargaining condition. The analysis connects cooperative game theory, Nash bargaining, fair division, random field geometry, and contract complexity. Full article

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

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21 pages, 384 KB

Open AccessFeature PaperArticle

Credence Goods in Markets with Collective Reputation: Nature of Equilibria and the Value of Certification

by Erin Krupka, Thomas P. Lyon and Arnab Mitra

Games 2026, 17(4), 40; https://doi.org/10.3390/g17040040 - 24 Jul 2026

Viewed by 204

Abstract

Fraud can cause markets for credence goods to collapse, but collective reputation has been suggested as a potential solution, especially when buyer valuations for quality are high. We develop a discrete model of a market for label credence goods and provide conditions for [...] Read more.

Fraud can cause markets for credence goods to collapse, but collective reputation has been suggested as a potential solution, especially when buyer valuations for quality are high. We develop a discrete model of a market for label credence goods and provide conditions for no-fraud equilibria based on collective reputation. We identify conditions under which all equilibria may be asymmetric or involve mixed strategies and hence sellers may face coordination problems. Our analysis suggests such coordination problems are the greatest when the size of the high-quality market is approximately half of the overall market (that incorporates both high- and low-quality goods), in the absence of any focal point or coordination device. Concerning certification, we find that it may not be particularly useful when buyer valuations for credence attributes are high under a well-performing collective reputation regime and a relatively small-sized low-quality market. However, we also determine that certification may serve its purpose the most when buyer valuations for credence attributes are low (contrary to standard intuition) and/or the size of the high-quality market is around half of the overall market (i.e., the high- and low-quality markets are of approximately equal size). Finally, our results indirectly indicate that agencies/NGOs concerned about fraudulent behavior may focus their attention on markets where heterogenous firms enjoy collective reputation. Full article

(This article belongs to the Section Applied Game Theory)

18 pages, 5336 KB

Open AccessFeature PaperArticle

Inefficient Learning Leads to Efficient Coordination: A Repeated Stag-Hunt Game with Learning Agents

by Ren Manfredi, Daniele Vilone, Tijan J. Cvetkovic, Franco Bagnoli and Andrea Guazzini

Games 2026, 17(4), 39; https://doi.org/10.3390/g17040039 - 23 Jul 2026

Viewed by 269

Abstract

In coordination games, the distinctive presence of multiple equilibria poses a challenge to coordination, and this has led over time to the proposal of numerous determinants of the latter. In this paper, we study coordination in a repeated Stag-hunt game played by two [...] Read more.

In coordination games, the distinctive presence of multiple equilibria poses a challenge to coordination, and this has led over time to the proposal of numerous determinants of the latter. In this paper, we study coordination in a repeated Stag-hunt game played by two agents who are able to learn from the outcomes of each game, but do not know the payoff matrix. By modulating their learning capabilities, we show that agents are able to coordinate on the social optimum when they learn slowly from experience and are prone to making mistakes. A comparison with the results from a recent laboratory experiment is also provided. Full article

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

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43 pages, 961 KB

Open AccessFeature PaperArticle

Promotion Thresholds, Revenue Sharing, and Delivery Risk in Reward-Based Crowdfunding

by Joyaditya Laik, Esther Gal-Or and Prakash Mirchandani

Games 2026, 17(4), 38; https://doi.org/10.3390/g17040038 - 21 Jul 2026

Viewed by 296

Abstract

This paper investigates two marketing strategies a reward-based crowdfunding platform employs to align its preferences with an entrepreneur’s choice of pledge and target levels. These are (a) how to promote campaigns to potential backers, and (b) how to share campaign revenues with the [...] Read more.

This paper investigates two marketing strategies a reward-based crowdfunding platform employs to align its preferences with an entrepreneur’s choice of pledge and target levels. These are (a) how to promote campaigns to potential backers, and (b) how to share campaign revenues with the entrepreneur. Kickstarter, for instance, promotes a set of campaigns by compiling a list of “recommended” projects. This research shows that the platform’s choice of the promotion rule may expose entrepreneurs to the risk of not generating sufficient funds to start production, which can damage their and the platform’s reputation. When the platform’s reputational risk is not very high, it reduces the risk of non-delivery by increasing the revenue share of the entrepreneur. The platform’s strategies are likely to ensure production when backers derive warm glow from pledging, when the entrepreneur’s development cost is low, or when the entrepreneur has minimal reputational cost if production fails. However, low reputational costs motivate the entrepreneur to lower the target, thus increasing the likelihood of insufficient funds to start production. We propose strategies the platform can use, including customizing the revenue share based on the campaign characteristics, to rectify such misalignments. Full article

(This article belongs to the Section Applied Game Theory)

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39 pages, 1516 KB

Open AccessFeature PaperArticle

Decentralized, Efficient, and Fair: Mean-Field Predictive Control for Bidirectional EV Coordination Under Uncertainty

by Samuel M. Muhindo

Games 2026, 17(4), 37; https://doi.org/10.3390/g17040037 - 9 Jul 2026

Viewed by 358

Abstract

We propose a decentralized strategy for coordinating the bidirectional charging and discharging of battery electric vehicles (BEVs) in renewable-powered parking lots. The framework combines mean-field games (MFGs) and model predictive control (MPC) to address the coupled stochastic dynamics induced by uncertain renewable generation [...] Read more.

We propose a decentralized strategy for coordinating the bidirectional charging and discharging of battery electric vehicles (BEVs) in renewable-powered parking lots. The framework combines mean-field games (MFGs) and model predictive control (MPC) to address the coupled stochastic dynamics induced by uncertain renewable generation and random vehicle arrivals and departures. Solar and wind power fluctuations are modeled using autoregressive moving-average (ARMA) processes, while the time-varying vehicle population is represented through finite Poisson processes. The coordination problem is formulated as a large-scale game, where an aggregator designs individual cost functions to maximize available energy utilization while promoting fairness through near-equal states of charge (SOCs) at departure. Scalability is achieved through MFG theory, ensuring convergence and stability even under highly volatile generation and fluctuating agent populations. Numerical simulations validate the proposed strategy against two straightforward algorithms: capacity-ordered saturation allocation (COSA) and capacity-ordered fair allocation (COFA). These centralized approaches achieve high target fulfillment in static, low-intensity environments, where available energy accommodates a stable fleet without exceeding power limits. However, their efficacy degrades significantly in dynamic, high-intensity environments, where the interplay of volatile generation, continuous fleet turnover, and strict power constraints strains the system. In contrast, the proposed MFG-MPC framework provides a decentralized response that elegantly navigates the trade-offs between energy availability, demand stochasticity, and power limits. Ultimately, this approach ensures robust energy utilization while safeguarding vehicle equity, confirming its strong suitability for real-time deployment. Full article

(This article belongs to the Special Issue Dynamic Game Theory in Sustainability)

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34 pages, 6647 KB

Open AccessArticle

Engineered Misunderstanding Under Psychological Warfare: A Bayesian Signaling Game of Felt-Understanding Collapse in the German Atomausstieg

by Ryanne R. L. Fairchild

Games 2026, 17(4), 36; https://doi.org/10.3390/g17040036 - 2 Jul 2026

Viewed by 487

Abstract

Russian state-sponsored disinformation has been described in policy and the operational literature, but it is less often formalized in game-theoretic terms. Here, a two-layered formal model is developed showing how adversarial perturbation of a communication channel can collapse cross-group felt understanding—the third-order intentional [...] Read more.

Russian state-sponsored disinformation has been described in policy and the operational literature, but it is less often formalized in game-theoretic terms. Here, a two-layered formal model is developed showing how adversarial perturbation of a communication channel can collapse cross-group felt understanding—the third-order intentional state/belief structure, established empirically by Livingstone, in which one group believes its perspectives are recognized and accepted as valid by another. The Analytical Model is a static Bayesian signaling game with binary types and a noisy channel parameterized by perturbation rate π. The Analytical Model shows that when recognition benefits exceed signaling costs, there exists a perturbation threshold π* = 1 − c_R/(u_R · p) above which mutual misrecognition becomes the unique Perfect Bayesian Equilibrium outcome. The Computational Model embeds this logic in an agent-based simulation on a homophilic stochastic block model and scale-free networks with continuous recognition capacity. Four substantive findings emerge: the closed-form analytical threshold from the Analytical Model predicts the boundary of collapse in the dynamic networked simulation; high network homophily protects cooperative behavior below π* but provides no rescue above it; bridge seeding—the placement of recognition-capable agents at structurally central cross-group positions—is the most effective of three policy interventions tested, rescuing cooperation even above π*; and uniform adversarial volume is approximately as damaging as strategically targeted adversarial precision across both small dense and large scale-free topologies, qualifying the operational claim that targeted disinformation should strictly outperform volume-based approaches. The model is illustrated with the German Atomausstieg (nuclear phase-out) case, and implications for clinical psychology, public policy, and intergroup recognition under psychological warfare are discussed. Full article

(This article belongs to the Special Issue Games with Incomplete Information)

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15 pages, 300 KB

Open AccessArticle

Fixed-Time Pursuit–Evasion Differential Game with Grönwall Constraints in Hilbert Space l₂

by Gafurjan Ibragimov, Saidakbar Gulomov and Bruno Antonio Pansera

Games 2026, 17(4), 35; https://doi.org/10.3390/g17040035 - 2 Jul 2026

Viewed by 296

Abstract

This paper studies a pursuit–evasion differential game in the Hilbert space

l_{2}

involving countably many pursuers and one evader. The players follow simple motion dynamics, while their control functions are subject to Grönwall-type constraints. The value of the game

λ

is defined [...] Read more.

This paper studies a pursuit–evasion differential game in the Hilbert space

l_{2}

involving countably many pursuers and one evader. The players follow simple motion dynamics, while their control functions are subject to Grönwall-type constraints. The value of the game

λ

is defined as the infimum of the distances between the evader and the pursuers at a given terminal time

θ

. The pursuers aim to minimize this distance

λ

by the end of the game, while the evader aims to keep it as large as possible. Optimal strategies for the pursuers are constructed via the method of fictitious pursuers, and an explicit evasion strategy is derived. As a result, the optimal strategies of both sides and the value of the game are obtained. Full article

22 pages, 1776 KB

Open AccessFeature PaperArticle

Markovian Analysis of a Small Population Evolutionary Prisoner’s Dilemma Game

by Athanasios Kehagias

Games 2026, 17(4), 34; https://doi.org/10.3390/g17040034 - 25 Jun 2026

Viewed by 292

Abstract

In this work we consider an evolutionary game in which a small population repeatedly plays the Iterated Prisoner’s Dilemma (IPD) game using the pairwise proportional imitation (PPI) revision protocol. Since we are dealing with a small population, we can explicitly formulate a Markov [...] Read more.

In this work we consider an evolutionary game in which a small population repeatedly plays the Iterated Prisoner’s Dilemma (IPD) game using the pairwise proportional imitation (PPI) revision protocol. Since we are dealing with a small population, we can explicitly formulate a Markov chain (MC) model of the evolutionary game and study its properties. In particular, we identify the absorbing states and their basins of attraction. In addition, we briefly study the replicator dynamics of the same game for the case of a large population. We find that, as the size of the population increases, the Markovian dynamics approximate the replicator dynamics. Full article

(This article belongs to the Special Issue Evolutionary Games on Networks and Biological Systems)

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Games, Volume 17, Issue 4 (August 2026) – 10 articles

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