Saddle-point Equilibrium Research Articles

Optimal Interception of Evasive Missile Warheads: Numerical Solution of the Differential Game

The problem of interception of a ballistic missile warhead by a defending missile is formulated as a differential game. Each missile is given a modest postlaunch capability to maneuver. Interception concludes the game and occurs if the interceptor reduces the distance between it and the warhead to a specified value. The objective of the warhead is to minimize the final distance to the target, which lies on the Earth's surface but is not necessarily collocated with the interceptor launch site. The objective of the interceptor is to maximize this same distance at the capture time. Saddle-point equilibrium solutions are found using a recently developed, direct numerical method that nevertheless uses the analytical necessary conditions to find the optimal control for one of the players. The method requires an initial guess of the solution, and this is provided by generating an approximate solution using genetic algorithms. For initial conditions yielding eventual capture, we derive the necessary condition that the saddle-point trajectories terminate on the usable part of the terminal hypersurface; this condition is shown to have an interesting physical interpretation. The numerical method successfully finds the saddle-point trajectories and is used to determine the sensitivity of the value of the game to the capability of the interceptor, for design purposes, and also to gauge the robustness of the numerical method for the solution of this dynamic game.

Partially observed semi-Markov zero-sum games with average payoff

We study a zero-sum partially observed semi-Markov game with average payoff on a countable state space. Under certain ergodicity conditions we show that a saddle point equilibrium exists. We achieve this by solving the corresponding average cost optimality equation using a span contraction method. The average value is shown to be the unique zero of a Lipschitz continuous function. A value iteration scheme is developed to compute the value.

A Class of Solvable Stopping Games

We consider a class of Dynkin games in the case where the underlying process evolves according to a one-dimensional but otherwise general diffusion. We establish general conditions under which both the value and the saddle point equilibrium exist and under which the exercise boundaries characterizing the saddle point strategy can be explicitly characterized in terms of a pair of standard first order necessary conditions for optimality. We also analyze those cases where an extremal pair of boundaries exists and investigate the overall impact of increased volatility on the equilibrium stopping strategies and their values.

Tourism Specialization and Environmental Sustainability in a Dynamic Economy

This study focuses on the dynamic behaviour of a small open economy specialized in tourism based on natural resources. The author analyses the steady-state properties in two scenarios, with and without public abatement expenditures, and a unique local saddle-point equilibrium is found for both cases. The analysis of the dynamics provides an alternative explanation for the observed positive growth performance of small open tourism-based economies and for the worldwide increases in tourist inflows, which are seen as transitional phenomena towards the path to the steady state. Moreover, in defining the conditions under which tourism development, improvements in environmental quality and economic growth can simultaneously occur, the model provides theoretical microfoundations for sustainable tourism. Finally, in both scenarios, the author analyses the issue of market failures, taking into account two different kinds of externality and finding the respective optimal tax rates that will induce private agents to replicate the social optimum.

A macroeconomic analysis of foreign assets and foreign liabilities

Although an increase in foreign assets and a decrease in foreign liabilities both increase a nation’s net foreign assets (NFA), they have alternative macroeconomic transmission mechanisms: while an increase in foreign assets is expansionary, the effect of a decrease in foreign liabilities is mixed due to the asymmetry between its income effect and wealth effect on aggregate demand. It is the relative strengths of the NFA’s wealth effect and income effect that determine the existence and natures of a saddle-point equilibrium in the NFA-real balance space as well as its comparative statics. The cointegration analysis suggests that in the 1990s, foreign liabilities bear more weight than foreign assets in the US NFA movement whereas the opposite holds for the case of Japan; therefore, correcting NFA imbalances calls for accelerated money growth and fiscal expenditure pruning in the U.S. but for the opposite policy responses in Japan.

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.

Zero-sum stochastic games with stopping and control

We study a zero-sum stochastic game where each player uses both control and stopping times. Under certain conditions we establish the existence of a saddle point equilibrium, and show that the value function of the game is the unique solution of certain dynamic programming inequalities with bilateral constraints.

New Generalized R-KKM Type Theorems in General Topological Spaces and Applications

In this paper, some new generalized R-KKM type theorems for generalized R-KKM mappings with finitely closed values and with finitely open values are established in noncompact topological spaces without any convexity structure under much weaker assumptions. As applications, some new minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds are established in general noncompact topological spaces. These theorems unify and generalize many known results in the literature.

Partially Observable Semi-Markov Games with Discounted Payoff

We study partially observable semi-Markov game with discounted payoff on a Borel state space. We study both zero sum and nonzero sum games. We establish saddle point equilibrium and Nash equilibrium for zero sum and nonzero sum cases, respectively.

Tourism Specialization and Sustainability: A Long-Run Policy Analysis

This study focuses on the dynamic evolution of a small open economy specialized in tourism based on natural resources when tourist services are supplied to foreign tourists who are crowding-averse and give positive value to the environmental quality. We analyse the steady-state properties and run several policy exercises in two versions of our model: in the first, private agents’ income is spent entirely on consumption while, in the second, agents are allowed to invest part of their income in pollution abatement technology (PAT) which artificially increases the rate of regeneration of the environmental asset. A unique locally saddle point equilibrium is found in both versions and for both the market and the centralized solution. Our main findings are that: 1) a corrective income tax raises steady state utility in both versions but is capable of leading the economy in its first-best dynamic path only when agents cannot invest in the PAT; 2) when the PAT is available to the government but not to agents, an income tax which finances abatement expenditures may increase steady state utility with respect to the market solution when the natural regeneration rate of the environment and the degree of crowding-aversion are both low enough; 3) when PAT is available, the market chooses to devote a higher fraction of income to abatement than the central planner but in both cases this fraction is positive only if the natural rate of regeneration is not too large; 4) when PAT is available an income pollution tax does not affect the dynamic path of the market economy.

GROUP DIFFERENTIAL GAMES FOR MULTIPARAMETER SINGULARLY PERTURBED SYSTEMS

In this paper, a group differential game problem is formulated using the system model of multiparameter singularly perturbed systems (MSPS). The case that there exist two groups of players with a conflict interest in the game is considered, and the players in each group must make their own decisions by taking into account the group interest. A method is proposed to find the approximate strategy for every player which will lead to an O(||μ||) near saddle–point equilibrium.

Stochastic<tex>$H_infty$</tex>Tracking With Preview for State-Multiplicative Systems

The problem of finite-horizon H/sub /spl infin// tracking for linear time-varying systems with stochastic parameter uncertainties is investigated. We consider three tracking patterns depending on the nature of the reference signal, i.e., whether it is perfectly known in advance, measured on line or previewed in a fixed time-interval ahead. The stochastic uncertainties appear in both the dynamic and measurement matrices of the system. For each of the above three cases a game theory approach is applied for the state-feedback case where, given a specific reference signal, the controller plays against nature which chooses the initial condition and the energy-bounded disturbance. The problems are solved using an expected value of the standard performance index over the stochastic parameters, where necessary and sufficient conditions are found for the existence of a saddle-point equilibrium. The infinite-horizon time-invariant tracking problem is also solved. The theory developed is demonstrated by a simple tracking example.

Monotonicity of solutions converging to a saddle point equilibrium

If a difference equation has a saddle point equilibrium, then there are solutions which converge to this equilibrium. For second order equations, conditions are given which imply that these solutions are monotone. These results are used to analyze a rational difference equation which possesses a saddle point equilibrium.

Monotonicity of solutions converging to a saddle point equilibrium

If a difference equation has a saddle point equilibrium, then there are solutions which converge to this equilibrium. For second order equations, conditions are given which imply that these solutions are monotone. These results are used to analyze a rational difference equation which possesses a saddle point equilibrium.

Existence of Value and Saddle Point in Infinite-Dimensional Differential Games

We study two-player zero-sum differential games of finite duration in a Hilbert space. Following the Berkovitz notion of strategies, we prove the existence of value and saddle-point equilibrium. We characterize the value as the unique viscosity solution of the associated Hamilton–Jacobi–Isaacs equation using dynamic programming inequalities.

Optimal investments with convex–concave revenue: a focus‐node distinction

AbstractThis paper considers a capital accumulation model in which revenue is a convex–concave function of the capital stock. While for certain capital values increasing returns to scale are reasonable, usually this property does not hold in general. In particular for large capital stock values it becomes increasingly difficult and thus expensive to produce more and more because of limitations of resources or infrastructure, lack of trained personnel in the region, etc. We give a complete classification under which parameter constellations a saddle point equilibrium is optimal, when it is optimal to close down by choosing zero investment and when history dependent equilibria occur. In the last scenario we distinguish between different types of the unstable equilibrium, which can each have their own implication for the firm's investment policy. Copyright © 2004 John Wiley & Sons, Ltd.

H∞ tracking of linear continuous‐time systems with stochastic uncertainties and preview

AbstractThe problem of finite‐horizon H∞ tracking for linear continuous time‐invariant systems with stochastic parameter uncertainties is investigated for both, the state‐feedback and the output‐feedback control problems. We consider three tracking patterns depending on the nature of the reference signal i.e. whether it is perfectly known in advance, measured on line or previewed in a fixed time‐interval ahead. The stochastic uncertainties appear in both the dynamic and measurement matrices of the system. In the state‐feedback case, for each of the above three cases a game theory approach is applied where, given a specific reference signal, the controller plays against nature which chooses the initial condition and the energy‐bounded disturbance. The problems are solved using the expected value of the standard performance index over the stochastic parameters, where, in the state‐feedback case, necessary and sufficient conditions are found for the existence of a saddle‐point equilibrium. The corresponding infinite‐horizon time‐invariant tracking problem is also solved for the latter case, where a dissipativity approach is considered. The output‐feedback control problem is solved as a max–min problem for the three tracking patterns, where necessary and sufficient condition are obtained for the solution. The theory developed is demonstrated by a simple example where we compare our solution with an alternative solution which models the tracking signal as a disturbance. Copyright © 2004 John Wiley & Sons, Ltd.

A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of aunique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.

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.

Quantitative and qualitative growth analysis

Quantitative growth of the economies has been at the focus of economic analysis. In modeling the quantitative economic growth process the quantitative or primal problem is focused on the rate of growth in the equilibrium, while in the dual formulation of the equilibrium problem the focus is on the combination of a real rate of interest and structural prices ensuring a sustained equilibrium solution. In the first formulation of this problem John von Neumann showed that the real rate of interest and the quantitative rate of growth are equal at the saddle point equilibrium. In this model as well as later reformulations the starting point is an economy operating in isolation from the rest of the world. In a globalized economy the role of the financial sector has to be introduced into the modeling effort. With such a reformulation not only growth of an existing economy but also births of new economies can be properly considered. In recent decades the role of endogenous technological change by research and development efforts has become increasingly important. Most of the studies of the impact of new knowledge upon the economy has focused the attention on improved methods of production. However, improvement of the products and their qualitative characteristics is a more fundamental aspect of long-term economic development. Increasing quality implies mostly an increasing recipe or blueprint complexity of the products. Such an increase of quality and complexity of the products tends to go hand-in-hand with an increasing willingness to pay for the products. A simplified modeling framework is used to show that it is possible to model qualitative and quantitative economic growth within a coherent theoretical framework.