International Game Theory Review

Ecological and environmental economics are inherently dynamic systems requiring dynamic optimization tools. Understanding the interplay between environmental economics, ecological economics, and dynamic games is crucial. This paper presents a systematic literature review on ecological and environmental economics modeled with dynamic games. To be more specific, this systematic literature review analyzes a dataset of 88 peer-reviewed articles from international journals. This study identifies clusters related to resource management, taxation, and policy. It also reveals niche and motor research themes, such as policy, biological invasions, pollution, taxation, abatement, and efficiency, paving the way for future research avenues. By comprehensively examining the literature, this review provides insights into current and future challenges faced by companies, consumers, regulators, and society. It contributes to a deeper understanding of the complex relationship between ecological and environmental dynamics and the field of dynamic games in economics.

This paper develops a model to assess the number of nurses needed to ensure both healthy patients and caregivers. We propose a model with random arrivals and exits of patients who may be of a single type (or several), and calculate the average care time they can receive. We show that the mean care time does not depend only on the mean number of patients in the unit. Actually, the probability distribution of the new patients per time step plays a central role. In the Poissonian case, we obtain totally explicit results, prove that the mean care time converges to a constant and give numerical examples. We also propose an analysis of the impact of working conditions on the average time that can be devoted to a patient. Four scenarios are proposed with numerical applications. Our analysis provides insights into current discussions on the introduction of caregiver ratios in hospitals to improve both the quality of care and caregivers’ working conditions.

Nowadays, to cater the increasing green customers, firms have switched to green manufacturing. In a dual-channel green supply chain (DCGSC), customers experience products at offline stores and buy them online (free-riding). Often imprecise cognitive biases (“fairness concern” and “overconfidence”) are observed among the supply chain (SC) members. With these facts, this study introduces the free-riding and above cognitive biases in a DCGSC with a manufacturer selling a green product through own online and offline retail channels and examines their effects. A centralized and four decentralized models (for green and nongreen products) are formulated depending upon channel members’ cognitive biases individually and jointly with and without free-riding. The fuzzy objectives and constraints are made deterministic using expectation and possibility measures, respectively. Models are solved and illustrated numerically. The results indicate that free-riding is harmful and beneficial to retailer and manufacturer, respectively. Manufacturer’s overconfidence enhances the retailer’s profit but decreases or increases own profit depending upon the salvage value. Retailer’s fairness concern is catastrophic for manufacturer but beneficial for her. Product greening increases manufacturer’s profit than the carbon tax regulation for lower emissions. In addition to above observations, for maximum profit, management should not go for greening beyond an optimum level.

The concept of swarm comes from the biological world. Drones gather in groups of 100 or even 1000 to fly like a flock of birds, called swarms. Swarm systems satisfy several assumptions such as decentralized controls, local information and simple platforms. Swarm systems have attractive properties such as resilience, scalability, and ease of development and implementation. Swarm techniques can perform simple tasks such as moving in a coordinated direction. The flocking behavior of a group of animals that converge by local interactions toward the same heading is an example of simple consensus for decentralized dynamic systems. However, the notion of decentralized control based on local information suffers from taking into account the overall behavior of the group. For example, in a complex environment, a swarm will adapt to the presence of obstacles and congestion reactively, whereas we would like more anticipatory control. The objective of this paper is to propose a solution based on Mean Field Game (MFG) concepts to integrate macro-level knowledge at the micro-level in decentralized flocking.

Security Games traditionally assume sequential interactions, with the defender acting as the leader and the attacker as the follower. In this paper, we challenge such an assumption by showing that security interactions can also be simultaneous or that even the attacker might be the leader. Specifically, we consider a simple security game framework consisting of two possible targets, one attacker and one defender. Then, we extend that basic interaction using a well-known endogenous timing framework (the Game with Observable Delay) to endogenously obtain the timing of the game as part of the equilibrium outcome. Our results show that, for the same set of general assumptions, the interaction can be simultaneous or sequential (with either the defender or attacker being the leader). Although the simultaneous case could be discarded using payoff dominance, it is impossible to compare the two sequential configurations using the same criterion. The risk dominance criterion also turns out to be helpless in this case. Thus, we argue that considering attackers as leaders in Security Games is a plausible scenario that merits further consideration, not only from a theoretical point of view but also due to its alignment with certain types of real-world crime.

Game-theoretic behavioral and epidemiological models suggest that it is impossible to eradicate a disease through voluntary control actions when individuals behave according to their own interests. The simple reason is that free-riding behavior, which is best for self-interest, leads to a control coverage on the group level that is insufficient to eradicate the disease. Here we show that, for diseases that are caused by coinfection, self-interest can actually result in the socially optimal outcome of disease eradication. Our result challenges the conventional wisdom that selfish behavior undermines the group interest; it resolves a social dilemma in the absence of any cooperation, institutional arrangements, or social norms. Our model is motivated by coinfecting plant viruses, common among agricultural crops and natural plants, and the behavioral dynamics of growers to adopt protective action (biological or chemical control). The epidemiological scenario, in which the disease is eradicated by self-interest, is characterized by a positive feedback process in which coinfection enhances infectivity. Similar feedback structures exist for a range of typical epidemiological processes that facilitate disease persistence if prevalence is sufficiently large. The underlying mechanism may therefore be applicable to other diseases.

Why use a viabilistic approach to model economy? Because it is urgent to re-assess the modeling of the business cycle that integrates dynamic money creation and that has at its core the notion of constraints. The purpose of this paper is thus to propose an evolutionary economic model using a mathematical framework derived from set-valued analysis, differential inclusions and viability theory. As this approach is dynamical, the model will not use two concepts on which neo-classical economics are built i.e., (1) the existence of static repeated equilibria, (2) the optimization by an economic agent of a utility function. Furthermore, we will not use any self-organization process or “invisible hand”, as in our viabilist framework the economic viability will be maintained by explicit choices of the economic agents and of the lender of last resort over the budgetary and monetary rules to apply. Indeed, even simple constraints on the second-derivative of the means of payments already induce four-phased business cycles. Coupling these constraints over the creation of means of payment with economic evolutions of prices and commodities through a budgetary constraint allows for a single formalism, tackling jointly the question of the viability of economic and monetary evolutions.

Differential games with additive payoffs are considered. A new approach to constructing the characteristic function for such games is defined. Properties of optimality principles such as the core and the Shapley value constructed by applying the new characteristic function are studied. The solutions obtained are demonstrated by means of an example.

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Identifying a Subgame Perfect Nash Equilibrium (SPNE) of a two-player Stackelberg game could be not a manageable task, especially when the players have a continuum of actions and the follower’s best reply correspondence is not single-valued. Aim of the paper is to investigate the issue of construction of SPNEs in Stackelberg games by exploiting perturbations of both the action sets and the payoff functions of the leader and the follower. To achieve the goal, since the limit of SPNEs of perturbed games is not necessarily an SPNE of the original game even for classical perturbations, we prove under nonrestrictive convergence conditions how to produce an SPNE starting from a sequence of SPNEs of general perturbed games. This result allows to describe a procedure to find SPNEs that can accommodate various types of perturbations. More precisely, under mild assumptions on the data of the original game, we show that a large class of perturbed games (including, for example, perturbation approaches relying on the Tikhonov and entropic regularizations or motivated by altruistic and antagonistic behaviors) satisfies the convergence conditions for constructing an SPNE. The specific SPNE selections associated to such a class, together with their possible behavioral interpretations, are discussed and an illustrative example is provided.

Zero-sum two-person finite undiscounted (limiting ratio average) semi-Markov games are considered where both the transition probabilities and the transition times are controlled by a fixed player in all states. We prove that, if such a game is unichain, the value and optimal stationary strategies of the players can be obtained from an optimal solution of the linear programming algorithm for the associated one player control stochastic game using a data transformation method. An example is provided for such class of games.

Terms like inducement, incentivization, persuasion, and to some extent enticement, are used in our daily lives to describe situations where one individual (decision maker, or entity) acts in a way to influence the decision-making process of another individual or individuals, where the outcome could benefit all involved or only the one who has initiated the process. Such influence could be exerted in two different ways (though variations do exist): via a direct input by the influencer into the utility or reward (or loss) of the receiving party, or by controlling (and possibly crafting) the information flow to the latter, in an attempt to shape beliefs at the receiving end (as in spread of disinformation). Both scenarios (and those that fall in between) could be analyzed within a dynamic Stackelberg game-theoretic framework, with a precise notion of equilibrium, which this paper addresses. The focus will naturally be on soft inducement (incentivization, persuasion) policies, rather than hard enforcement (such as threat) ones which are not that interesting or practical. This overview paper introduces some explicit models that lead to appealing such policies. It also includes a discussion on the impact of various factors, such as population size and uncertainty in modeling, on the resulting equilibria, and identify several challenges that lie ahead.

In this paper, we consider a singular control optimization problem which generalizes the Gordon–Schaefer model. A population of individuals with endogenous growth is harvested by two independent agents, and their joint profit is maximized. We show that adding a second control to the standard model greatly complicates the standard solution, which is a Most Rapid Approach. We prove that no solution where both controls are singular can exist. Using numerical experiments, we exhibit situations where the optimal steady state corresponds to one of the controls singular, and the other one is equal to one of its specific bounds. In some cases, the optimal path resembles the one-control solution, but with an additional threshold on the population beyond which both controls must be used. However, in other cases, the optimal control has more switching points and is not monotonous. We even exhibit cases where the optimal policy admits two steady states.

We consider a finite-horizon discrete-time dynamic model of oligopolistic competition, with uncertain supply. The firms (retailers) make their decisions on quantities (sales) and orders of goods from a nonstrategic manufacturer taking into account uncertainty in demand parameters and available supply capacity. The retailers need to satisfy the joint available supply capacity constraint. The model is described as a game played over an event tree with given transition probabilities. We find a normalized equilibrium for the game with coupling constraints. The two-step procedure to find this equilibrium is described. We provide the necessary conditions of the equilibrium and examine different scenarios of uncertainties in the model by numerical simulations.

It is well known that laboratory subjects often do not play mixed strategy equilibria games according to the theoretical predictions. However, little is known about the role of cognition in these strategic settings. We therefore conduct an experiment where subjects play a repeated hide and seek game against a computer opponent. Subjects play with either fewer available cognitive resources (high cognitive load treatment) or with more available cognitive resources (low cognitive load treatment). Surprisingly, we find some evidence that subjects in the high load treatment earn more than subjects in the low treatment. However, we also find that subjects in the low treatment exhibit a greater rate of increase in earnings across rounds, thus suggesting more learning. Our evidence is consistent with subjects in the low load treatment over-experimenting. Further, while we observe that subjects do not mix in the predicted proportions and that their actions exhibit serial correlation, we do not find strong evidence these are related to their available cognitive resources. This suggests that the standard laboratory deviations from the theoretical predictions are not associated with the availability of cognitive resources. Our results shed light on the extent to which cognitive resources affect (and do not affect) behavior in games with mixed strategy equilibria.

Game theory is of substantial significance in diverse domains, acting as a potent instrument to comprehend and assess strategic engagements among rational decision-makers. It formulates mathematical models to represent strategic interactions among rational decision-makers in the competitive world. Due to ambiguity in the real-world problems, acquiring the precise payoff values of a matrix game proves challenging. However, in numerous scenarios, these payoffs fluctuate within specific ranges, making them suitable for consideration as interval numbers. This leads to the formulation of a special form of game problem known as the interval valued game problem (IVGP). Some methodologies exist in the literature to find the optimal strategies as well as the value of game for IVGP, but most of them possess some limitations, resulting in the need for proposing a new methodology to find the optimal strategies and value of game. Thus, in this paper, a new solution method for game problems with payoffs represented as interval numbers is presented, utilizing the fuzzy concept. The process begins by transforming the interval payoffs into fuzzy numbers using a ranking function. Subsequently, these fuzzy payoffs are converted into crisp values, leading to the formulation of the crisp matrix game. The resulting crisp matrix game is then solved using linear programming approach. Additionally, MATLAB code for the proposed method is developed and proposed to streamline the computation process, enhancing comparison and decision-making efficiency, particularly when dealing with large payoff matrices. Furthermore, three numerical examples are provided to illustrate the validity of the proposed approach as well as its MATLAB code. A real-life example of IVGP in the realm of tourism industry is also provided. Finally, a comparative analysis is conducted, comparing the proposed method with some existing methods.

In this paper, we propose a novel approach to study the production of consumption goods within a price floor carbon emission trading scheme. A stochastic Stackelberg game is set up between the environment authority (Player 1), and the manufacturer/producer (Player 2). In such a hybrid emission trading scheme, the authority sets an emissions target and producers will choose to participate in carbon trading on this basis, subject to a stochastic demand with information delayed. When the stochastic demand is determined by the market sale price and the scale of carbon emission reduction, we apply the maximum principle to obtain both the optimal production and market price. We show when the demand is determined by both the market sales price and carbon emission reduction, manufacturers are persuaded to set the market sales price first (ex ante incentive) and then proceed to a carbon trading scheme, thus providing a backward-looking advantage to manufacturers. Consequently, the manufacturer and the environmental authority can find a subgame-perfect Nash equilibrium via backward induction.

The principle of minimum differentiation stated by Hotelling in his pioneer paper was proved to be invalid 50 years later, since there is a region where the players are so close that Nash equilibrium does not exist. Some authors have contributed with amendments to the original game in the aim of ensuring the existence of Nash equilibrium independently of the location of the players. However, this work analyzes the less frequently studied case of the Hotelling game with close players under different implementations of the game, considering elastic and inelastic demand (conventional Hotelling game, the Hotelling game with reservation cost and Hotelling-Smithies game). The study is supported by means of numerical simulation of the game.

In the Marriage Game, women decide whether to get married and afterward men decide whether to cooperate in the domestic tasks. Then, if a husband is not cooperative, a wife can divorce or remain married. The sequence of decisions can be formalized as a game in extensive form, in which the strategic interaction between partners turns out evident. A husband may deceive, hoping that the divorce threat is not realistic, and a woman may decide not to get married from the beginning. If players had perfect knowledge about partners’ attitude, then divorce would not exist. Unfortunately, partners’ type is revealed only after the wedding. So, it is assumed that the game is played by different type of players, both men and women, and they interact with incomplete information. The model is solved by calculating all the perfect Bayes–Nash equilibria and it is shown that they depend on a complex interaction between model parameters, as multiple or mixed strategy equilibria can be found same parameters configuration. Comparing equilibria, we argue that the model can describe the marriage evolution that has been experienced by contemporary society. Following the increasing women’s participation in the job market, the number of marriages started to decline, but surprisingly, we are observing its resurgence in the most advanced societies. Our solution to the Marriage Game points to both the active role of autonomous women and the men’s expectations about marriage. As the number of autonomous women increases, men realize that divorce is credible and they change their expectations on women.