R&;D Equilibrium Strategies with Surfers

A typical assumption in the game-theoretic literature on research and development (R&D) is that all firms belonging to the industry under investigation pursue R&D activities. In this paper, we assume that the industry is composed of two groups; the first (the investors) is made of firms that have R&D facilities and are involved in this type of activity. The second group corresponds to firms that are inactive in R&D (the surfers). The latter group benefits from its competitors’ R&D efforts, thanks to involuntary spillovers. This division of the industry is in line with actual practice, where indeed not all firms are engaged in costly and risky R&D. We adopt a two-stage game formalism where, in the first stage investors decide on their levels of investment in R&D, and in the second stage all firms compete à la Cournot in the product market. We characterize and analyze the unique subgame perfect Nash equilibrium. Research supported by NSERC, Canada. F. Ben Abdelaziz is on leave at The College of Engineering, American University of Sharjah, UAE.     

Real R&;D options with time-to-learn and learning-by-doing

We model R&D efforts to enhance the value of a product or technology before final development. Such efforts may be directed towards improving quality, adding new features, or adopting technological innovations. They are implemented as optional, costly and interacting control actions expected to enhance value but with uncertain outcome. We examine the interesting issues of the optimal timing of R&D, the impact of lags in the realization of the R&D outcome, and the choice between accelerated versus staged (sequential) R&D. These issues are also especially interesting since the history of decisions affects future decisions and the distributions of asset prices and induces path-dependency. We show that the existence of optional R&D efforts enhances the investment option value significantly. The impact of a dividend-like payout rate or of project volatility on optimal R&D decisions may be different with R&D timing flexibility than without. The attractiveness of sequential strategies is enhanced in the presence of learning-by-doing and decreasing marginal reversibility of capital effects.     

Differential games results for military expenditures of unequal antagonists in the third world

In developing countries, resources are scarce and military expenditure can hinder development. Thus, allocation of resources between defense and civilian sectors is an important decision. Also, in developing countries, most defense spending is related to regional conflicts where one frequently encounters unequal participants (e.g., India—Pakistan, Iran—Iraq, China—Vietnam). In such situations, the bigger power aspires to be the regional leader, and allocates resources to the defense sector accordingly.

A model incorporating the two above mentioned stylized facts has been examined by Deger and Sen [1] in two different ways. They first look at the problem from a single country's point of view, which takes the other's reaction function as given, and show that in the long run, military expenditures and armament stock levels will exhibit stability. Next, they formulate the problem as a noncooperative differential game in which each country assumes knowledge of the other country's welfare function, and conjecture that stability will once again be exhibited. The present paper attempts to verify that conjecture using a simplifying assumption which is necessary to be able to qualitatively solve the problem. This simplifying assumption does not change the results for the optimal control models, however, the conjecture is not valid for the simplified differential game model.     

Licensing radical product innovations to speed up the diffusion

We present a differential game to study how companies can simultaneously license their innovations to other firms when launching a new product. The licensee may cannibalize licensor’s sales, albeit this can be compensated by gains from royalties. Nonetheless, patent royalties are generally so low that licensing is not an attractive strategy. In this paper we consider the role of licensing to speed up the product diffusion. Word of mouth by licensee’s customers and licensee’s advertising indirectly push forward sales of the licensing company, accelerating new product diffusion. We find evidence that licensing can be a potentially profitable strategy. However, we also find that a weak Intellectual Property Right (IPR) protection can easily diminish the financial attractiveness of licensing.     

Advertising Strategies in a Differential Game with Negative Competitor’s Interference

We consider a duopolistic industry where the current sales of each firm is proportional to its goodwill stock. The evolution of the latter depends positively on own advertising effort and negatively on competitor’s advertising. A standard assumption in the literature in differential games of advertising is that the players remain active throughout the whole (infinite) duration of the game. We relax this assumption and characterize the circumstances under which a firm finds it optimal to remain or exit the industry. Among other things, it is shown that, if both players are “strong”, then the unique Nash equilibrium is the same that one would obtain in the absence of interference from competitor’s advertising. Research supported by Ministry of University and Research of Italy, University of Padua and NSERC, Canada.     

A Differential R&D Game: Implications for Knowledge-Based Growth Models

This paper compares the outcomes of strategic and nonstrategic research and development (R&D). Firms undertake cost-reducing R&D in an environment where benefits accrue from their rivals' R&D via a common pool of knowledge. We demonstrate that multiple suboptimal equilibria exist; under these conditions, growth models which do not consider strategic interactions will overestimate or underestimate the growth rate.     

Applying the Leitmann-Stalford sufficient conditions to maximization control problems with non-concave Hamiltonian

The main result in this short note is that the integral form of the Leitmann-Stalford sufficiency conditions can be verified for a class of optimal control problems whose Hamiltonian is not concave with respect to the state variable. The main requirement for this class of problems is that the dynamics is sufficiently dissipative. An application to a Stackelberg differential game between a producer and a developer is exemplified. Using our result we show that the necessary conditions implied by Pontryagin’s maximum principle are also sufficient. This allows a complete characterization of the solution.     

Dynamic Model of R&D,Spillovers, and Efficiency of Bertrand and Cournot Equilibria

Using an infinite-horizon two-player differential game, we derive and compare Bertrand and Cournot equilibria for a differentiated duopoly engaging in the process of R&D competition. The main findings of this study are as follows. First, Bertrand competition is more efficient if either R&D productivity is low or products are very different. Second, Cournot competition is more efficient provided that R&D productivity is high, products are close substitutes, and spillovers are not close to zero. This last result is different from what has been obtained in the literature. Hence, this shows that considering a dynamic model and more general investment costs does have an impact on the efficiency results.     

Collaboration in R&D activities: Firm-specific decisions

In this paper, we study the strategic R&D collaboration by introducing a virtual player to reveal cooperative incentives and keeping investment share and market share independent of each other. Not consistently with the traditional opinions, we show that the superiority of the R&D cartel is due to the coexistence of cooperation and competition when spillovers are exogenous. Moreover, we conclude that high R&D input share must be reflected implicitly by high market share, and that firms’ R&D decisions vary with firms’ specific characteristics when spillovers is endogenous.     

具有年龄结构的线性周期种群动力系统的最优收获控制问题

研究一类具有年龄结构的线性周期种群动力系统的最优收获控制问题,即讨论了具有周期的生死率和周期变化的收获项的Lotka Mckendrick模型.利用Mazur's定理,作者证明了控制问题最优解的存在性,同时借助于法锥概念,还得到了控制问题最优解存在的必要条件。最后，在适当的假设下，得到了最优控制问题的唯一解。该文的结论推广了某些已有的结果.      

R&D pipeline management: Task interdependencies and risk management

Maintaining a rich research and development (R&D) pipeline is the key to remaining competitive in many industrial sectors. Due to its nature, R&D activities are subject to multiple sources of uncertainty, the modeling of which is compounded by the ability of the decision maker to alter the underlying process. In this paper, we present a multi-stage stochastic programming framework for R&D pipeline management, which demonstrates how essential considerations can be modeled in an efficient manner including: (i) the selection and scheduling of R&D tasks with general precedence constraints under pass/fail uncertainty, and (ii) resource planning decisions (expansion/contraction and outsourcing) for multiple resource types. Furthermore, we study interdependencies between tasks in terms of probability of success, resource usage and market impact. Finally, we explore risk management approaches, including novel formulations for value at risk and conditional value at risk.     

Existence and Structure of Optimal Solutions of Infinite-Dimensional Control Problems

In this work we analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval [0,τ] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t ∈ [0,τ] \backslash E , where E \subset [0,τ] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on τ . Accepted 26 July 2000. Online publication 13 November 2000.     

Dynamic common property resources and environmental problems

Problems of interacting common-property resources are set up as stochastic differential games. A class of models is solved where equilibrium closed-loop strategies keep harvest rates proportional to stocks. Corrective taxes, etc., are considered.Dedicated to G. LeitmannEarlier versions of this paper (Ref. 1) were presented at the International Conference on Economics and Control, Cambridge University, England, June 18–21, 1979, and at the Winter Econometric Society Meetings, Atlanta, Georgia, December 1978. Space limitations shortened this paper; a complete working paper is available from the authors. The authors would like to extend their appreciation to George Leitmann, who first introduced them to differential games.     

The Leitmann-Schmitendorf advertising game with n players and time discounting

The extension of the Leitmann-Schmitendorf advertising game to n players and positive time discounting is investigated. We show that the strong time consistency of the open-loop Nash equilibrium is preserved. As to optimal controls, while the boundary solution is unaffected by the number of firms as well as discounting, the inner solution depends on industry structure. The fully symmetric version of the game allows us to identify the parameter regions wherein both solutions are sustainable.     

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.     

Nonsmooth semipermeable barriers, Isaacs’ equation, and application to a differential game with one target and two players

We study the target problem which is a differential game where one of the players aims at reaching a target while the other player aims at avoiding this target forever. We characterize the victory domains of the players by means of geometric conditions and prove that the boundary of the victory domains is a nonsmooth semipermeable surface, i.e., is a solution (in a weak sense) of the Isaacs equation: sup _u inf _v 〈f (x, u, v),p〉 = 0, wheref is the dynamic of the system,u andv are the respective controls of the players, andp is a normal to the boundary of the victory domains at the pointx.     

Dynamic programming for some optimal control problems with hysteresis

We study an infinite horizon optimal control problem for a system with two state variables. One of them has the evolution governed by a controlled ordinary differential equation and the other one is related to the latter by a hysteresis relation, represented here by either a play operator or a Prandtl-Ishlinskii operator. By dynamic programming, we derive the corresponding (discontinuous) first order Hamilton-Jacobi equation, which in the first case is of finite dimension and in the second case is of infinite dimension. In both cases we prove that the value function is the only bounded uniformly continuous viscosity solution of the equation.     

Dynamic Cost-Benefit Analysis of Drug Substitution Programs

This paper studies optimal spending for drug substitution programs in the context of a dynamic epidemic model of both drug use and drug use-related infections. Two types of costs are considered in addition to control costs: social costs resulting from individuals being dependent on drugs; additional costs due to drug users being vulnerable to infections like hepatitis C or HIV. Analysis of the model demonstrates that the long-run equilibrium is not necessarily unique. Instead, there may be multiple equilibria. Which of these equilibria is optimal depends on the initial conditions for the number of drug addicts and the number of those who are infected. So, for a given set of epidemic parameters, it may be optimal to spend a lot on substitution programs that reduce the number of drug addicts or to spend little and to accept a high level of drug use.     

Optimal and adaptive control for a kind of 3D chaotic and 4D hyper-chaotic systems

This paper is concerned with the chaos control of two autonomous chaotic and hyper-chaotic systems. First, based on the Pontryagin minimum principle (PMP), an optimal control technique is presented. Next, we proposed Lyapunov stability to control of the autonomous chaotic and hyper-chaotic systems with unknown parameters by a feedback control approach. Matlab bvp4c and ode45 have been used for solving the autonomous chaotic systems and the extreme conditions obtained from the PMP. Numerical simulations on the chaotic and hyper-chaotic systems are illustrated to show the effectiveness of the analytical results.     

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.