A differential oligopoly game with differentiated goods and sticky prices

We investigate a dynamic oligopoly game where goods are differentiated and prices are sticky. We study the open-loop and the closed-loop memoryless Nash equilibrium, and show that the latter equilibrium entails a larger level of steady state production as compared to the former; both equilibria entail a larger level of production in steady state than the static game. We also study the effects of price stickiness and product differentiation upon the steady state equilibrium allocation and profits. The per-firm equilibrium output is increasing in both product differentiation and price stickiness, while profits are increasing in both product differentiation and the speed of price adjustment. The steady state social welfare monotonically increases in the speed of price adjustment, and the overproduction entailed by dynamic competition has beneficial effect from a social standpoint.     

Sufficient conditions for optimality of threat strategies in a differential game

The problem of defining threat strategies in nonzero-sum games is considered, and a definition of optimal threat strategies is proposed in the static case. This definition is then extended to differential games, and sufficient conditions for optimality of threat strategies are derived. These are then applied to a simple example. The definition proposed here is then compared with the definition of threat strategies given by Nash.     

A differential game without a qualitative saddle point

Saddle points are defined for two-person differential games in which the players have opposing preference orderings over lotteries on a set of qualitative objectives, rather than numerical payoff functions. A simple example is then given of a game without such a qualitative saddle point.     

Persuasive advertising under Bertrand competition: A differential game

We investigate a linear state differential game of advertising, under Cournot and Bertrand competition. A unique saddlepoint equilibrium exists if the marginal cost of advertising is sufficiently low. Bertrand competition entails more intense advertising than Cournot competition, since increasing market size is more important to firms when competition is tough.     

A differential game of industrial pollution management

This paper explores a differential game between a policy maker and a profit maximizing entrepreneur in which production generates pollution. The government levies a pollution tax on output and uses the tax received for pollution abatement. The entrepreneur determines the level of output. A feedback Nash equilibrium is derived. Using more specific functional forms, the game is extended to cover the multiple firm case.     

Sharing costs in highways: A game theoretic approach

This paper introduces a new class of games, highway games, which arise from situations where there is a common resource that agents will jointly use. That resource is an ordered set of several indivisible sections, where each section has an associated fixed cost and each agent requires some consecutive sections. We present an easy formula to calculate the Shapley value, and we present an efficient procedure to calculate the nucleolus for this class of games.     

Optimal allocation of drug control efforts: A differential game analysis

The present paper considers a dynamic nonzero-sum game between drug dealers and the authorities. Although the game is neither linear-quadratic nor degenerate, in the sense that the closed-loop equilibria coincide with the open-loop equilibria, we are able to calculate explicitly a stationary feedback Nash equilibrium of that game. In a numerical example, we determine the optimal allocation of governmental efforts between treatment and law enforcement minimizing the total discounted cost stream in the equilibrium. Moreover, we provide sensitivity analyses with respect to the efficiency parameters of both competitors. Our results show that a farsighted authority should attack the drug problem from the demand side and put much effort in treatment measures and the improvement of the efficiency of the treatment.This research was supported by the Austrian Science Foundation under Contract P9112-SOZ. We would like to thank A. Luhmer and E. J. Dockner for helpful comments.     

A zero sum differential game in a Hilbert space

We study a zero sum differential game of fixed duration in a separable Hilbert space. We prove a minimax principle and establish the equivalence between the dynamic programming principle and the existence of a saddle point equilibrium. We also prove sufficient conditions for optimality.     

A differential game model of a market of substitutable products

Modern industrial organization often classifies groups of differentiated products that are fairly good substitutes to belong to the same market. This paper develops a differential game model of a market of substitutable products. To avoid the problem of time-inconsistency, we solve a feedback Nash equilibrium solution for the game. A set of state-dependent equilibrium strategies is derived. Extensions of the model to a stochastic formulation and to an infinite time horizon specification are also provided.     

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.     

A Pareto-optimal solution of a maintenance-production differential game

This paper deals with a maintenance-production problem, modelled as a two-player nonzero-sum differential game. In a firm, the maintenance department is responsible for the maintenance of the machines used by the production department. Maintenance expenditures improve the quality of the machines but are costly. Production, on the other hand, yields revenue to the firm, but reduces the quality of the machines.Assuming that the two departments will make a decision on maintenance expenditures and production rate in a cooperative mood, we are looking for a Pareto-optimal solution. Using phase diagram analysis it turns out that the equilibrium in state-costate plane is a saddle point, implying (in the most realistic cases) decresing maintenance expenditures and increasing production rate.It is interesting to note that the basic features of the Pareto solution are the same as in the one player solution (i.e. the optimal control problem). Also, some similarities with the non-cooperative Nash solution are pointed out.     

Generalization of the main equation of differential game theory

We consider feedback, two-person, zero-sum differential games. We obtain two inequalities for the directional derivatives of the nonsmooth value function. We show that these inequalities, together with the boundary conditions, constitute necessary and sufficient conditions which the value function must satisfy. In the region where the value function is differentiable, the inequalities become the well-known main equation of differential game theory (Isaacs-Bellman equation). The results obtained here may be useful in the approximation of the value function by piecewise smooth splines and also in the classification of singular surfaces.The author would like to thank Academician N. N. Krasovskii for his valuable advice and encouragement.     

On a class of three-player differential games

A class of differential games consisting of three players is presented. It is assumed that two of them, forming a coalition and working together, oppose the other. Under some additional assumptions, an optimality criterion for the players who form a coalition is proposed.     

A sufficient condition for the evadability of differential evasion games

A sufficient condition for the strict evadability of nonlinear differential evasion games is obtained. The result complements, in some sense, the relevant results obtained by the author in a previous paper. An illustrative example is discussed as well. The author thanks Professor L. D. Berkovitz for some discussions.     

A natural selection from the core of a TU game: the core-center

We present a new allocation rule for the class of games with a nonempty core: the core-center. This allocation rule selects a centrally located point within the core of any such game. We provide a deep discussion of its main properties.     

Location of terror response facilities: A game between state and terrorist

We study a leader follower game with two players: a terrorist and a state where the later one installs facilities that provide support in case of a terrorist attack. While the Terrorist attacks one of the metropolitan areas to maximize his utility, the State, which acts as a leader, installs the facilities such that the metropolitan area attacked is the one that minimizes her disutility (i.e., minimizes ‘loss’). We solve the problem efficiently for one facility and we formulate it as a mathematical programming problem for a general number of facilities. We demonstrate the problem via a case study of the 20 largest metropolitan areas in the United States.     

Agent‐based simulation of an N‐person game with parabolic payoff functions

We report computer simulation experiments based on our agent‐based simulation tool to model a new N‐person game based on John Conway's Game of Life. The individual agents may choose between two behavior options: cooperation or defection. The payoff (reward/penalty) functions are given as two parabolas: one for each option. After a certain number of iterations, the behavior of the agents stabilizes to either a constant value or oscillates around such a value. The simulation's goal is to investigate the effects of intermediate behavior on a society of agents. We have performed a systematic investigation of this game for all six possible cases of the mutual positions of parabolic payoff functions crossing each other at two points: x = 0.3 and 0.7 where x is the ratio of the cooperation choice to the total number of agents in the agent's neighborhood. The global ratios X(t) of the total number of cooperators in the entire array of agents as functions of time (iterations) and the solutions of the game X_final as functions of X₀ were observed for each case for Pavlovian, greedy, and conformist agents. The solutions have predictable tendencies only when the neighborhood is the entire array of greedy or conformist agents. In all other cases unexpected properties emerge. © 2009 Wiley Periodicals, Inc. Complexity, 2010     

Determination of optimal penalties for antitrust violations in a dynamic setting

We analyze a differential game describing the interactions between a firm that might be violating competition law and the antitrust authority. The objective of the authority is to minimize social costs (loss in consumer surplus) induced by an increase in prices above marginal costs. It turns out that the penalty schemes which are used now in EU and US legislation appear not to be as efficient as desired from the point of view of minimization of consumer loss from price-fixing activities of the firm. In particular, we prove that full compliance behavior is not sustainable as a Nash Equilibrium in Markovian strategies over the whole planning period, and, moreover, that it will never arise as the long-run steady-state equilibrium of the model. We also investigate the question which penalty system enables us to completely deter cartel formation in a dynamic setting. We found that this socially desirable outcome can be achieved in case the penalty is an increasing function of the degree of offence and is negatively related to the probability of law enforcement.     

Stability,structural stability,and games: Toward a theory of differential hypergames

Differential games (DG's) are investigated from a stability point of view. Several resemblances between the theory of optimal control and that of structural stability suggest a differential game approach in which the operators have conflicting interests regarding the stability of the system only. This qualitative approach adds several interesting new features. The solution of a differential game is defined to be the equilibrium position of a dynamical system in the framework of a given stability theory: this is the differential hypergame (DHG). Three types of DHG are discussed: abstract structural DHG, Liapunov DHG, and Popov DHG. The first makes the connection between DG and the catastrophe theory of Thom; the second makes the connection between the value function approach and Liapunov theory; and the third provides invariant properties for DG's. To illustrate the fact that the theory sketched here may find interesting applications, the up-to-date problem of the world economy is outlined.This research was supported by the National Research Council of Canada.     

Differential game model of the dynastic cycle: 3D-canonical system with a stable limit cycle

Ancient Chinese history reveals many examples of a cyclical pattern of social development connected with the rise and the decline of dynasties. In this paper, a possible explanation of the periodic alternation between despotism and anarchy by a dynamic game between the rulers and the bandits is offered. The third part of the society, the farmers, are dealt with as a renewable resource which is exploited by both players in a different manner. It is shown that the Nash solution of this one-state differential game may be a persistent cycle. Although we restrict the analysis to open-loop solutions, this result is of interest for at least two reasons. First, it provides one of the few existing dynamic economic games with periodic solutions. Second, and more important, the model is an example of a three-dimensional canonical system (one state, two costates) with a stable limit cycle as solution. As far as we see, our model provides up to now the simplest (i.e., lowest dimensional) case of a persistent periodic solution of an intertemporal decision problem.This research was partially supported by the Austrian Science Foundation under Contract No. P7783-PHY.Helpful comments of T. Basar, E. J. Dockner, R. F. Hartl, A. Mehlmann, G. Sorger, and F. Wirl as well as the generous help of George Leitmann are gratefully acknowledged.