Remarks on capture-avoidance games

We formulate the problems of capture and avoidance as differential games with terminal cost, thus avoiding some difficulties arising from choosing the terminal time or the closest distance as the cost. This puts capture and avoidance in the context of differential games, in the sense of Isaacs. The results are illustrated via linear differential games with hard control constraints.     

A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward–backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzero-sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.     

Existence of value in pursuit-evasion games with restricted phase coordinates

Almost all results referring to the problem of the existence of a value in differential games concern games without restricted phase coordinates. In this paper, we introduce a concept of value for differential games of pursuit and evasion and prove, under some general assumption, the existence of it. The players are required to satisfy some general phase constraints. The arguments employed in this paper are based on some extent on Krasovskii's method of extremal construction. We also show that the lower value in the Friedman sense is a generalization of our value. In a special linear case, the equivalence between pursuit differential games and time-optimal control problems is established.     

Cheap play with no regret

This paper studies a special class of differential information games with pre-play communication —games with “cheap play”. We consider problems in which there are several rounds of payoff-irrelevant publicly observable choice (or discussion) of actions, followed by a final round in which actions are binding and payoff relevant. A natural focal subset of equilibria of such games in one that consists of equilibria involvingno regret. Such games were first studied by Green and Laffont (1987), where a criterion calledposterior implementability is introduced with the intention of identifying regret-free equilibria in games with cheap play. This is simply a restriction on the Bayesian equilibrium of the underlying one-shot game. If indeed such a restriction does characterize regret-freeness, then the analytics of such situations would be enormously simplified since one can ignore the messy extended-form of the cheap play game; merely examining the one-shot game is sufficient. We argue that regret-freeness of an equilibrium has a subtle distinction: regret-freeness in moves and regret-freeness in assessments. We show that the former causes the extended-form to be irrelevant; posterior implementability completely characterizes equilibria with regret-freeness in moves. The latter, on the other hand, does not yield a similar principle: the extended-form cannot be ignored.     

Nash Equilibria of Risk-Sensitive Nonlinear Stochastic Differential Games

This paper considers a class of risk-sensitive stochastic nonzero-sum differential games with parametrized nonlinear dynamics and parametrized cost functions. The parametrization is such that, if all or some of the parameters are set equal to some nominal values, then the differential game either becomes equivalent to a risk-sensitive stochastic control (RSSC) problem or decouples into several independent RSSC problems, which in turn are equivalent to a class of stochastic zero-sum differential games. This framework allows us to study the sensitivity of the Nash equilibrium (NE) of the original stochastic game to changes in the values of these parameters, and to relate the NE (generally difficult to compute and to establish existence and uniqueness, at least directly) to solutions of RSSC problems, which are relatively easier to obtain. It also allows us to quantify the sensitivity of solutions to RSSC problems (and thereby nonlinear H-control problems) to unmodeled subsystem dynamics controlled by multiple players.     

Linear differential pursuit games with integral constraints in the presence of delay

Problems involving linear differential pursuit games were studied by many authors; their work served as a basis for studying pursuit problems in linear differential games with integral constraints. In the present paper, we obtain sufficient conditions for the solvability of linear pursuit problems with integral constraints on the control of the players in the presence of delay.     

A double obstacle problem arising in differential game theory

A Hamilton–Jacobi equation involving a double obstacle problem is investigated. The link between this equation and the notion of dual solutions—introduced in [S. As Soulaimani, Infinite horizon differential games with asymmetric information, PhD thesis; P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim. 46 (3) (2007) 816–838; P. Cardaliaguet, C. Rainer, Stochastic differential games with asymmetric information, Appl. Math. Optim. 59 (1) (2009) 1–36] in the framework of differential games with lack of information—is established. As an application we characterize the convex hull of a function in the simplex as the unique solution of some nonlinear obstacle problem.     

Credibility of Incentive Equilibrium Strategies in Linear-State Differential Games

We characterize in this paper the credibility of incentive equilibrium strategies for the class of linear-state differential games. We derive a general condition for credibility and illustrate its use on two differential games taken from the literature of environmental economics and knowledge accumulation. We show that the proposed linear incentive strategies are not always credible. Further, we provide alternative nonlinear credible strategies which suggest that we should not stick only to linear incentive strategies, even in a simple class of differential games such as the linear-state one.This research was completed when the first author was visiting professor at GERAD, HEC, Montréal. The first author’s research was partially supported by MCYT under project BEC2002-02361 and by JCYL under project VA051/03, confinanced by FEDER funds. The second author’s research was supported by NSERC, Canada.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

Conditional Viability for Impulse Differential Games

We introduce in this paper the concept of “impulse evolutionary game”. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular “hybrid systems” as well as “qualitative systems”. The conditional viability kernel of a constrained set (with a target) is the set of initial states such that for all strategies (regarded as continuous feedbacks) played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the constraints until it hits the target. This paper characterizes the concept of conditional viability kernel for “qualitative games” and of conditional valuation function for “qualitative games” maximinimizing an intertemporal criterion. The theorems obtained so far about viability/capturability issues for evolutionary systems, conditional viability for differential games and about impulse and hybrid systems are used to provide characterizations of conditional viability under impulse evolutionary games.     

Exploitation of natural resources and pollution

This paper provides some differential game models of natural resource exploitation when environmental pollution takes place. The classical approach to determine optimal harvest rates of renewable resources utilizes Optimal Control models, i.e. there is either a monopolistic market structure or there is pure competition. In case of pollution, however, all agents can be put together, forming the groups of the resource harvesters on one side and of polluters on the other side. So differential games can be used to analyze environmental problems. The models introduced in this paper are put together in order to showdifferent problems that can all be analyzed using differential games.     

Maximum principle for differential games of forward–backward stochastic systems with applications

This paper is concerned with a maximum principle for both zero-sum and nonzero-sum games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by forward–backward stochastic differential equations. This kind of games is motivated by linear-quadratic differential game problems with generalized expectation. We give a necessary condition and a sufficient condition in the form of maximum principle for the foregoing games. Finally, an example of a nonzero-sum game is worked out to illustrate that the theories may find interesting applications in practice. In terms of the maximum principle, the explicit form of an equilibrium point is obtained.     

Dynamically stable cooperative solutions in randomly furcating differential games

The paradigm of randomly furcating differential games incorporates stochastic elements via randomly branching payoffs in differential games. This paper considers dynamically stable cooperative solutions in randomly furcating differential games. Analytically tractable payoff distribution procedures contingent upon specific random events are derived. This new approach widens the application of cooperative differential game theory to problems where future environments are not known with certainty.     

The core and related solution concepts for infinite assignment games

Assignment problems where both sets of agents that have to be matched are countably infinite, the so-called infinite assignment problems, are studied as well as the related cooperative assignment games. Further, several solution concepts for these assignment games are studied. The first one is the utopia payoff for games with an infinite value. In this solution each player receives the maximal amount he can think of with respect to the underlying assignment problem. This solution is contained in the core of the game. Second, we study two solutions for assignment games with a finite value. Our main result is the existence of core-elements of these games, although they are hard to calculate. Therefore another solution, the f-strong ε-core is studied. This particular solution takes into account that due to organisational limitations it seems reasonable that only finite groups of agents will eventually protest against unfair proposals of profit distributions. The f-strong ε-core is shown to be nonempty. These authors’ research is partially supported by the Generalitat Valenciana (Grant number GV-CTIDIA-2002-32) and by the Government of Spain (through a joint research grant Universidad Miguel Hernández — Università degli Studi di Genova HI2002-0032).     

War Gaming as a Technique in the Study of Operational Research Problems

The last 10 years have seen an immense amount of effort, both here and abroad, concentrated on the development of war games that can be used for the study of many of the important problems that arise in military operational research. The first part of this paper describes in general terms the background to such operational problems, their difficulties and complexities, and explains why war gaming is regarded as a particularly appropriate technique in their examination.On this basis, the fundamental conceptual development of a war game representing a land battle is described, and is illustrated by reference to the Army Operational Research Establishment (AORE) War Game. Practical experience with this and other hand-played games allows some of the advantages and disadvantages of this type of war game to be enumerated, and leads naturally to a discussion of computer-assisted games, and of the relationship between games and analytical models of combat. It is concluded that each of these approaches has its own particular advantages, and that it will usually be appropriate to use combinations of them in the study of major problems.     

Survey on applications of semi-tensor product method in networked evolutionary games

Semi-tensor product (STP) method of matrices has received more and more attention from the communities of both engineering and economics in recent years. This paper presents a comprehensive survey on the applications of STP method in the theory of networked evolutionary games. In the beginning, some preliminary results on STP method are recalled. Then, the applications of STP method in many kinds of networked evolutionary games, such as general networked evolutionary games, networked evolutionary games with finite memories, networked evolutionary games defined on finite networks, and random networked evolutionary games, are reviewed. Finally, several research problems in the future are predicted.     

On the Stackelberg strategy in nonzero-sum games

The properties of the Stackelberg solution in static and dynamic nonzero-sum two-player games are investigated, and necessary and sufficient conditions for its existence are derived. Several game problems, such as games where one of the two players does not know the other's performance criterion or games with different speeds in computing the strategies, are best modeled and solved within this solution concept. In the case of dynamic games, linear-quadratic problems are formulated and solved in a Hilbert space setting. As a special case, nonzero-sum linear-quadratic differential games are treated in detail, and the open-loop Stackelberg solution is obtained in terms of Riccati-like matrix differential equations. The results are applied to a simple nonzero-sum pursuit-evasion problem.This work was supported in part by the US Air Force under Grant No. AFOSR-68-1579D, in part by NSF under Grant No. GK-36276, and in part by the Joint Services Electronics Program under Contract No. DAAB-07-72-C-0259 with the Coordinated Science Laboratory, University of Illinois, Urbana, Illinois.     

Computing payoff allocations in the approximate core of linear programming games in a privacy-preserving manner

We investigate privacy-preserving ways of allocating payoffs among players participating in a joint venture, using tools from cooperative game theory and differential privacy. In particular, we examine linear programming games, an important class of cooperative games that model a myriad of payoff sharing problems, including those from logistics and network design. We show that we can compute a payoff allocation in the approximate core of these games in a way that satisfies joint differential privacy.     

Singular mean-field control games

This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Motivations are given as optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and mean-field singular investment games. In particular, a simple singular mean-field investment game is studied, where the Nash equilibrium exists but is not unique.     

Interconnected games and international environmental problems,II

Some features of international environmental problems are considered. A basic problem is to induce countries to adopt a cooperative approach. One of the instruments to induce countries to cooperate is an exchange of concessions in fields of relative strengths, such as swapping trade concessions for cooperation on international environmental problems. This instrument will be modelled in this paper with tensor games. Both tradeoff and non-tradeoff tensor games will be addressed, with emphasis on tradeoff tensor games with linear strict weights. The relationship between the Pareto equilibria of a non-tradeoff tensor game and the Nash equilibria of the associated tradeoff tensor games will be studied. Due to structural similarities between tensor games and repeated multiple objective games, some attention will also be paid to the latter. Relationships between objects related to Folk theorems for the tradeoff tensor game with completely additive weights and the corresponding objects for its constituting isolated games will be studied. Since many international environmental problems have prisoners' dilemma characteristics, it is analyzed how interconnection may enhance cooperation in prisoners' dilemma games.