An empirical study on the Lanchester model of combat for competitive advertising decisions

This paper presents an empirical study on the Lanchester model of combat for competitive advertising decisions. Three issues are evaluated: (i) the specification of the market share response model; (ii) the effect of inflation on the estimation of the response model; and (iii) the performance of competitive strategies. It is shown that (a) the square root function that is used in previous studies is often inappropriate to characterize the market share response model; (b) market share variations are more responsive to current advertising expenditures; (c) closed-loop Nash equilibrium strategies are better competitive advertising strategies for firms to maximize profits than open-loop Nash equilibrium strategies; and (d), finally, general perfect equilibria Nash equilibrium strategies developed by Case are usually not good competitive advertising strategies for firms to maximize profits.     

A duopolistic model of dynamic competitive advertising

This paper proposes a differential game model of competitive advertising decisions for non-durable products by extending the Lanchester model and the Deal model of competitive advertising in the literature. The proposed model is compared empirically with the Lanchester model for model fitting and forecast accuracy. It is shown that the model is suitable for an actual market and out-performs the Lanchester model in forecast accuracy. The model provides a sensible modeling alternative to the Lanchester model for the study of dynamic competitive advertising decisions. Necessary and sufficient conditions for open-loop and closed-loop Nash equilibrium solutions to the model are discussed. A numerical algorithm for open-loop and closed-loop Nash strategies to the model is developed.     

N-person differential games governed by semilinear stochastic evolution systems

In this paper the problem ofN-person infinite-dimensional stochastic differential games governed by semilinear stochastic evolution control systems is discussed. First the minimax principle which is the necessary condition for the existence of open-loop Nash equilibrium is proved. Then the necessary and sufficient conditions of open-loop and closed-loop Nash equilibrium for linear quadratic infinite-dimensional stochastic differential games are derived.     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.     

On a class of nonzero-sum,linear-quadratic differential games

A class of two-player, nonzero-sum, linear-quadratic differential games is investigated for Nash equilibrium solutions when both players use closed-loop control and when one or both of the players are required to use open-loop control. For three formulations of the game, necessary and sufficient conditions are obtained for a particular strategy set to be a Nash equilibrium strategy set. For a fourth formulation of the game, where both players use open-loop control, necessary and sufficient conditions for the existence of a Nash equilibrium strategy set are developed. Several examples are presented in order to illustrate the differences between this class of differential games and its zero-sum analog.This research was supported by the National Science Foundation under Grant No. GK-3341.     

An exponential differential game which admits a simple Nash solution

This paper deals with a class ofN-person nonzero-sum differential games where the control variables enter into the state equations as well as the payoff functionals in an exponential way. Due to the structure of the game, Nash-optimal controls are easily determined. The equilibrium in open-loop controls is also a closed-loop equilibrium. An example of optimal exploitation of an exhaustible resource is presented.The helpful comments of Professor Y. C. Ho and Dipl. Ing. E. Dockner are gratefully acknowledged.     

Advertising in a Differential Oligopoly Game

We illustrate a differential oligopoly game where firms compete à la Cournot in homogeneous goods in the market phase and invest in advertising activities aimed at increasing the consumers reservation price. Such investments produce external effects, characterizing the advertising activity as a public good. We derive the open-loop and closed-loop Nash equilibria, and show that the properties of the equilibria depend on the curvature of the market demand function. The comparative assessment of these equilibria shows that the firms advertising efforts are larger in the open-loop equilibrium than in the closed-loop equilibrium. We also show that a cartel involving all the firms, setting both output levels and advertising efforts, may produce a steady state where the social welfare level is higher than the social welfare levels associated with both open-loop and closed-loop noncooperative settings.     

Identification of classes of differential games for which the open loop is a degenerate feedback Nash equilibrium

In general, it is clear that open-loop Nash equilibrium and feedback Nash equilibrium do not coincide. In this paper, we study the structure of differential games and develop a technique using which we can identify classes of games for which the open-loop Nash equilibrium is a degenerate feedback equilibrium. This technique clarifies the relationship between the assumptions made on the structure of the game and the resultant equilibrium.The author would like to thank E. Dockner, A. Mehlmann, and an anonymous referee for helpful comments.     

N-person differential games governed by infinite-dimensional systems

This paper discussesN-person differential games governed by infinite-dimensional systems. The minimax principle, which is a necessary condition for the existence of open-loop equilibrium strategies, is proved. For linear-quadraticN-person differential games, global necessary and sufficient conditions for the existence of open-loop and closed-loop equilibrium strategies are derived.This work was supported by the Science Fund of the Chinese Academy of Sciences and the Research Foundation of Purdue University.The problems discussed in this paper were proposed by Professor G. Chen, during the author's visit to Pensylvania State University, and were completed at Purdue University. The author would like to thank Professors L. D. Berkovitz and G. Chen for their hospitality.     

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.     

State transformations and the derivation of Nash closed-loop equilibria for non-zero-sum differential games

In this paper the usefulness of state transformations in differential games is demonstrated. It is shown that different (admissible) state transformations give rise to different closed-loop Nash equilibrium candidates, which may all be found by solving systems of ordinary differential equations. A linear-quadratic duopoly differential game is solved to illustrate the results.     

Systemic Risk and Stochastic Games with Delay

We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is a finite-player linear–quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced-backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a closed-loop Nash equilibrium using a Hamilton–Jacobi–Bellman partial differential equation approach.     

Strategic new product pricing when demand obeys saturation effects

We analyze dynamic pricing strategies for new products over an infinite planning horizon in a duopolistic market. The sales dynamic is modelled as a linear demand function with saturation effects, marginal costs are assumed to be constant. The optimal pricing strategies are obtained as (degenerate) closed-loop Nash solutions. It is shown that the optimal dynamic prices are greater than the static ones. In the case of no discounting there is in addition to the constant solution also an equilibrium with monotonically increasing prices.     

Necessary conditions in stochastic linear quadratic problems and their applications

Stochastic linear quadratic optimal control problems are considered. A unified approach is proposed to treat the necessary optimality conditions of closed-loop optimal strategies and open-loop optimal controls. Notice that the former notion does not rely on initial wealth, while the later one does. Our conclusions of closed-loop optimal strategies are directly derived by suitable variational methods, the approach to which is different from [12], [11]. Moreover, the necessary conditions for closed-loop optimal strategies happen to be sufficient which takes us by surprise. Finally, two applications are given as illustration.     

Quantity strategies in economic networks

We study the optimal quantity strategies of the firm which owns many subsidiaries embedded in an economic network. A key feature of our model is that subsidiaries experience a negative local network effect. First, we show that there exists a unique Nash equilibrium in the game. Second, we characterize the equilibrium strategies by considering some specific network structures. Then, we identify how changes in the payoff parameters affect equilibrium play. Finally, we also analyze the strategy features of different models through two simple examples.     

Production capacity buildup and double marginalization mitigation in a dynamic supply chain

The paper investigates the extent to which capacity investment considerations interact with the double marginalization effect in a simple supply chain governed by a wholesale price contract. To do so, a non-cooperative differential game model is formulated to study the pricing and capacity investment decisions in a supply chain, which consists of a supplier and a manufacturer. In such a game, there are different decision rules—open-loop, closed-loop, feedback—that are available to the supply chain participants, depending on the observability of the current state of the supply chain. While closed-loop and feedback equilibrium strategies involve the observability of other chain member’s production capacity, open-loop equilibrium strategies do not have such requirement. We examine how the supplier and the manufacturer determine, with the different decision rules, their production capacities and pricing policies to maximize their profits over an infinite planning horizon, and determine how the observability of other supply chain’s members’ production capacity affects the magnitude of the double marginalization effect. Our study suggests that the observability of other chain member’s current production capacity entails a lower production efficiency that results in a greater double marginalization effect. This allows us to conclude that observability of other chain member’s current production capacity is associated with a greater double marginalization effect.     

Dynamic Game Model of Endogenous Growth with Consumption Externalities

This paper introduces consumption externalities into an endogenous growth model of common capital accumulation and characterizes balanced growth equilibria. Contrary to the standard argument in previous studies, we show that the growth rate in a feedback Nash equilibrium can be higher than that in an open-loop Nash equilibrium if agents strongly admire the consumption of others. This result is irrelevant to whether preferences exhibit “keeping up with the Joneses” or “running away from the Joneses”.     

On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games

The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAG-29-78-C-0016, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division, under Contract No. US-ERDA-EX-76-C-01-2088.     

Process Spillovers and Growth: A Comment on the Existence of a Stable Closed-Loop Nash Equilibrium

In a recent paper Luckraz (J. Optim. Theory Appl. 139:315–335, 2008) develops a differential game model to show that a lack of intellectual property protection can improve economic growth in economies, where large technology gaps prevail. However, this conclusion depends crucially on the fact that the closed-loop equilibrium of the model degenerates to the open-loop equilibrium. This note generalizes the results of Luckraz by formulating a setup in which the closed-loop equilibrium does not degenerate to the open-loop equilibrium of the differential game.