International Game Theory Review

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.

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 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.

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.

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.

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.

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.

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.

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.

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.