This paper proposes a formal theoretical model that integrates classical strategic correlation, quantum game theory, and evolutionary dynamics in the context of institutional corruption. I introduce a classical parametric correlator K(λ), defined as a convex combination of strategic independence and the Fréchet–Hoeffding bounds (upper for positive λ and lower for negative λ), into a two-population Hawk–Dove game with enforcement. The correlator modifies both encounter probabilities and the effective transaction costs of legal compliance. The closed-form analysis and the baseline simulations are conducted on the symmetric invariant manifold x = y = z, while the general asymmetric system remains piecewise. A reduced-form alignment-calibration bridge maps the normalized analytical selection differential into material-payoff units and aligns its local stability root with the frequency-dependent replicator integrated numerically; the resulting baseline threshold is λ* = 1/3. For 0 ≤ λ < λ*, the baseline calibration exhibits a stable interior corruption branch, whereas above λ* the honest corner is locally asymptotically stable. Because convergence slows close to the bifurcation, the finite-horizon numerical honesty threshold is reached only when λ is sufficiently above λ*. I derive a restricted Eisert–Wilkens–Lewenstein I/X benchmark and use λ = sin2(γ), the squared concurrence (tangle), solely as a scalar calibration of entanglement strength. I show that, on the symmetric manifold, this restricted I/X distribution collapses to the independent product distribution for every γ, whereas the classical correlator remains concordant for λ > 0. The exact lower Fréchet–Hoeffding branch is used for λ < 0, preserving non-negativity and the stated marginals. The simulations identify a valid adversarial high-risk region under negative dependence and a welfare-improving cooperation corridor under sufficiently positive alignment. Policy implications focus on avoiding adversarial dependence and combining institutional alignment with conventional enforcement.
This paper investigates the stochastic adaptive linear-quadratic(LQ) two-person zero-sum differential game with asymmetric information. Compared with most existing literature, the study incorporates “parameter uncertainty” and “limited decision-making capability” into the dynamic game framework, thereby extending the applicable scope of traditional models. The information acquired by the two players is asymmetric, with an inclusion relation between their information sets. We analyze the adaptive problem where the system parameter matrix is unknown to both players. Based on the least squares estimator, the certainty equivalence principle, the stochastic filtering theory and the diminishing excitation technique, we construct a pair of adaptive strategies. It is proven that, when the system matrix is controllable and the algebraic Riccati equation associated with the game system admits a desired real symmetric solution, the Lyapunov equation, Cauchy–Schwarz inequality and Itô formula are adopted to prove that the designed adaptive strategies can guarantee global stability of the system and realize the Nash equilibrium. Moreover, the terminal objective payoff under the proposed adaptive strategies is derived by virtue of theories related to inner product spaces.
Mean field Game (MFG) Theory derived by combining stochastic optimal control with a statistical description of multiple populations of competing agents is increasingly used in many applications, especially those involving acquisition of a resource. Using an ergodic, 2-population MFG model it is demonstrated that nonlinear dynamics caused by the model’s underlying bifurcation structure can dominate predictions as the influence of the optimal control is enhanced by lowering the stochastic diffusivity σ . Multiple ergodic states are predicted, and rapid segregation of the distributions is observed over small ranges of σ close to bifurcation points. Introducing bias toward agents with more expertise breaks the bifurcation structure, however the effects of the bifurcations persist in the resulting multiple disconnected states. Including a common resource that is self-regenerating models the Tragedy of the Commons (TOTC) where the resource is depleted by over-acquisition by the competing populations. The dependence of the TOTC limit on σ and the regeneration rate (R_N ) exhibits the residual influence of the bifurcation structure and controls the sensitivity of the limiting value of R_N . Large differences in the maximum resource acquisition between the two populations are predicted.
Environmental sustainability is a shared responsibility in product and service markets: firms must make genuine efforts to reduce their environmental impact, and consumers must choose products that reflect their values. However, as the United Nations warns, greenwashing remains a major barrier to climate progress because it undermines public trust and distorts market incentives. To rigorously analyse this phenomenon, we model the strategic interactions between firms and consumers as a two-player asymmetric evolutionary game with continuous strategy sets and nonlinear payoffs. This setup captures realistic trade-offs and allows sustainability effort and consumer commitment to vary in degree rather than being treated as all-or-nothing decisions. We compute the strict Nash equilibria and derive sufficient conditions for the stability and instability of diverse outcomes. These include full disengagement, where neither firms nor consumers invest in sustainability; an interior outcome, where both choose moderate levels of effort; and boundary outcomes, where one side fully commits while the other remains only partially engaged. In particular, we apply the static concept of the Continuously Stable Strategy (CSS) to analyze the stability of partial cooperation, and Strong Uninvadability (SUP) to determine the conditions under which full sustainability, characterized by maximal genuine effort from firms and fully eco-conscious consumer behavior, becomes a robust outcome of the evolutionary dynamics. We also interpret these conditions economically, identifying the critical market thresholds required to break the greenwashing cycle.
While searching over large territories, solitary males find a potential mating location and encounter a rival there. This paper presents an evolutionary game with a sequence of moves that precede an ultimate hawk-dove subgame. First, the rival males face a choice between being aggressive (hawk) or passive (dove) when encountering each other before the female is present, and subsequently, each chooses whether or not to return for the ultimate hawk-dove subgame during mating season. Depending on the travel costs of returning to the location, there exist neutrally stable behavioral strategies (NSS) assigning different probability of aggression at the initial female-absent encounter than at the ultimate contest. Initial randomization over hawk-dove may facilitate partial coordination of the decisions to return to the mating spot. A male’s probability of returning to the potential mating spot is conditional on the outcome of the initial hawk-dove contest. The neutrally stable strategies of interest are not robust against indirect invasions.
Introduced 75 years ago, the Prisoner’s Dilemma (PD) game has been widely used to study the evolution of cooperation. In its classic form, which assumes uniform interaction rates and interaction times, cooperation is not evolutionarily viable because defection is always more profitable for individuals. This prediction contradicts empirical observations of cooperative behavior, particularly in mutualisms. In nature, mutualistic interactions do not occur randomly, and cooperating partners may evolve mechanisms or strategies to avoid interacting with low-quality partners or cheaters. In this article, we investigate the influence of non-uniform interaction rates on the evolutionary outcome of both the symmetric and asymmetric PD games, using explicit non-instantaneous pairing dynamics. In the symmetric version, cooperation is promoted by a lower cooperator–defector interaction rate, a higher cooperator–cooperator interaction rate, or a higher defector–defector interaction rate, highlighting the importance of cooperators interacting preferentially with one another. In the asymmetric version, the same pattern emerges: cooperation in both species is promoted by low interaction rates between cooperators and defectors. These results underscore the key role of non-uniform interaction rates in shaping evolutionary predictions in matrix games.
This paper concerns a class of deterministic infinite-horizon noncooperative difference (or discrete-time) games with discounted payoffs. Our main objective is to give conditions under which these are potential games. Hence, for each game in this class, there is an optimal control problem (OCP) whose optimal solutions are Nash equilibria for the given game. To this end, we propose an algorithm that provides first-order and separability conditions on the payoff’s stage reward functions, guaranteeing that a given function, which determines the corresponding OCP, is a potential function for the game. We show that for some difference games, these first-order conditions guarantee that the open-loop structure of the game’s Nash equilibria is contained in the open-loop structure of optimal solutions of the OCP determined by the potential. We also show that considering feedback controls instead of open-loop policies significantly reduces the classes of games that can be shown to be potential.
We consider a game in which individual players adopt strategies of moving left, moving right, or remain motionless, according to payoffs correlated to the overall population benefit and scaled by a state dependent modulation function. This results in an evolutionary game of motion under replicator dynamics; through a variety of simple forms, linear and piecewise linear, of the modulation function, we are able to capture how players local adaptation leads to the emergence of a group motion by analyzing both the stability and dynamic properties of the game. We uncover both swarm-style behaviors as well as group behaviors that counter-balance the motion of others. This includes the appearance of a continuum of equilibria akin to an evolutionary stable set. Numerical simulations are provided to validate the model analysis claims.
Spatial structure plays a crucial role in shaping evolutionary game dynamics in multi-strategy populations. However, the influence of spatial diffusion on players’ strategy evolution and interaction outcomes in non-zero-sum rock–paper–scissors (RPS) games with asymmetric payoffs has not been fully explored. This paper investigates an RPS evolutionary game in which the loser receives zero payoff. By incorporating spatial diffusion into replicator dynamics, we analyze the system under both homogeneous mixing and spatially structured environments. The main results can be summarized as follows: (i) in the homogeneous-mixing case, the system undergoes a Hopf bifurcation, leading from a stable equilibrium to persistent oscillations; (ii) spatial diffusion gives rise to Turing instability and Turing–Hopf bifurcation in certain parameter regimes, and numerical simulations reveal complex spatio-temporal patterns; (iii) in some parameter regions, chaotic dynamics are confirmed by positive maximum Lyapunov exponent, indicating increased uncertainty in strategic decision-making, and the system is in a disordered state; (iv) when the winner’s payoff exceeds twice the tie payoff, the homogeneous equilibrium corresponding to an equal distribution of the three strategies is stable in both the well-mixed system and the reaction-diffusion system. These results indicate that spatial diffusion can fundamentally alter evolutionary outcomes and induce complex and chaotic strategy dynamics. The proposed framework provides a reference for the analysis of multi-strategy evolutionary games with spatial diffusion.
This paper investigates the discounted criterion of stochastic Stackelberg games between an energy sharing provider (ESP) and prosumers in a peer-to-peer energy sharing market. The ESP acts as the leader in the Stackelberg game and possesses decision-making priority. As followers, prosumers choose their actions based on the leader’s strategy. The goal of the stochastic Stackelberg game is to maximize the ESP’s reward while minimizing the prosumers’ costs, formulated as a two-level optimization problem in which the objective functions are expressed in terms of an expected discounted criterion with certain interpretations. This model, which has not been previously considered, holds significant application value in smart grids. For any fixed leader’s strategy, the uniqueness of the followers’ best response is essential for determining the Stackelberg equilibrium. This uniqueness is equivalent to that of the optimal strategy in Markov decision processes (MDPs). We study a continuous model characterized by the uniqueness of the optimal strategy in MDPs, where uniqueness is established under convexity and monotonicity conditions. Based on these conditions, the existence of a stochastic Stackelberg equilibrium is proved. The results are illustrated through an analytical example involving stochastic dominance. Furthermore, for cases where uniqueness does not hold, we establish the existence of a stochastic Stackelberg ε -equilibrium, propose two convergent algorithms, and present a simulation case study to demonstrate their effectiveness.
In this paper, we study two-person zero-sum stochastic games with stopping and control for semi-Markov processes on a Borel state space under a risk-sensitive discounted cost criterion. In this framework, at each decision epoch, both players may either take an action or stop the game. Under suitable assumptions, we deduce that the game has a value obtained as the unique solution of certain dynamic programming inequalities with bilateral constraints. Moreover, we are able to show the existence of a saddle point equilibrium.
This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.
This paper explores stochastic maintenance for a large fleet of structures, focusing on the example of bridges. The approach involves replacing the states of degradation of a facility with a probability distribution over the states. Even though the degradation state of each individual structure may be available, it is often more convenient to work with the proportions of structures in each degradation state. This is particularly useful when incorporating constraints such as limited maintenance budgets or desired quality levels for the overall fleet. Probability distributions are commonly used when the initial state is unknown or when degradation is observed with error (i.e., under partial observation). However, the same methodology is applied here in a different context. Full information may be available, but using it directly can be too complex. Instead, partial information is used to reduce this complexity. The theory of Markov Decision Processes (MDPs) provides a framework for many applications in Operations Research and Management Science, and stochastic maintenance has become one such application. When working with probability distributions over states instead of individual states, the framework is referred to as a mean-field MDP. In this setting, the dynamic programming methodology for MDPs is extended to the mean-field case, tailored to fleets of structures. Both value iteration and policy iteration algorithms are considered to characterize the value function and determine the optimal (randomized) control policy.
We establish a modified notion of Nash equilibrium learning—convergence of the population state to the Nash equilibria set—in a generalization of the standard population games and evolutionary dynamics framework using system-theoretic passivity methods. In this setting, we allow each strategy to involve a sequence of sub-tasks that must be completed before strategy revision so long as the durations of the sub-tasks can be modeled with Erlang or exponential distributions. Furthermore, several canonical classes of natural learning rules are established and useful properties are derived.
Understanding the impact of population heterogeneity on the spread of vaccine-preventable diseases is crucial for containment and control. Here, we develop an experimental game model to examine how risks from disease and vaccination shape vaccination decisions in a population with heterogeneous vulnerability. Our results show that participants vaccinate strategically in line with Nash equilibrium. Specifically, vaccination rates were higher among more vulnerable individuals than among those less vulnerable. Additionally, we observed minimax behavior in a subset of individuals who consistently chose the secure option (vaccination) regardless of others’ actions. These findings underscore the epidemiological interdependence of vaccination decisions and the need for public health approaches that recognize the different risks and costs faced by vulnerable groups.
We develop a simple yet efficient Lagrangian method for computing equilibrium prices in a mean-field game price-formation model. We prove that equilibrium prices are optimal in terms of a suitable criterion and derive a primal-dual gradient-based algorithm for computing them. One of the highlights of our computational framework is the efficient, simple, and flexible implementation of the algorithm using modern automatic differentiation techniques. Our implementation is modular and admits a seamless extension to high-dimensional settings with more complex dynamics, costs, and equilibrium conditions. Additionally, automatic differentiation enables a versatile algorithm that requires only coding the cost functions of agents. It automatically handles the gradients of the costs, thereby eliminating the need to manually form the adjoint equations.
