International Game Theory Review

An advanced RF surface-to-air missile (SAM) system is accompanied by ground radars, missiles, and launchers, which are very vulnerable to air-to-surface missiles (ASMs). This paper studies the threat of ASMs and an IR homing anti-missile-missile (AMM) system which defends the SAM site. A supersonic highly maneuverable ASM is assumed and the performance required for this counterattack AMM is briefly studied. The interception problem between the ASM and the AMM is considered as a game and some features of the game are discussed and an approach is explained.

The paper deals with standard mathematical models for nonlinear affine control systems with two (vector-valued) inputs u (=control) and w (unknown except for a bound for the sup-norm). Interpretation of this scenario and its wide range of applications (in control and differential game theory): See the introduction. The main result of the paper is the presentation of a dissipation equality which seems to be new and which is the result of a special control strategy (discretized state feedback, see the introduction). Combination with Lyapunov techniques leads to a concrete proposal for stabilizing disturbed control system. The mathematical background — which is worked out in detail — amounts to an explicit integration of the Hamilton-Jacobi partial differential equation via the method of characteristics.

Examples with zero-sum linear differential games of fixed terminal time and a convex terminal payoff function depending on two components of the phase vector are considered. Such games can have an indifferent zone with constant value function. The level set of the value function associated with the indifferent zone is called the "critical" tube. In the selected examples, the critical tube and the neighboring level sets exhibit "narrow throats". Presence of such throats requires extremely precise computations for constructing the level sets. The paper presents different forms of critical tubes with narrow throats and indicates the combinations of problem parameters that can produce them.

This paper deals with the issue of shelf-space allocation and advertising decisions in marketing channels. We consider a network composed of a unique retailer offering the products of two competing manufacturers. The retailer controls the amount of shelf-space to allocate to both brands, while the manufacturers make advertising decisions in order to build their brand image (i.e. the goodwill stock). The demand for each brand is affected by its own goodwill level and the shelf-space allocated to the brand at retailer's store. The problem is formulated as a Stackelberg differential game played over an infinite horizon, with the manufacturers as leaders and the retailer as the follower. Stationary feedback equilibria are computed. Our main results indicate that the shelf-space allocated to each brand, manufacturers' advertising strategies at the equilibrium and channel members' value functions are affected by the goodwill levels of both products.

We present two different mathematical models that examine the role of cellular evolution in the development and treatment of cancer. The first is a Lotka-Volterra model of an invasive cancer subjected to chemotherapy that demonstrates rapid evolution of drug-resistant phenotypes. The second model represents a major refinement of the first one with the objective of producing a simple, but realistic model of carcinogenesis. It is a consumer-resource model (consumers = normal + mutuant cells, resource = glucose). We conclude that if mutant cells are not allowed to evolve, they will not ordinarily progress to cancer. However, if accumulating mutations allow cellular evolution and successful adaptation to proliferation constraints, normal cells will develop into invasive cancer. We find this progression is possible because normal cellular populations are not at an evolutionarily stable state and so are subject to invasion by more fit phenotypes.

Today artificial neural networks are very useful to solve complex dynamic games of various types, i.e., to approximate optimal strategies with sufficient accuracy. Exemplarily four synthesis approaches for the solution of zero-sum, noncooperative dynamic games are outlined and discussed. Either value function, adjoint vector components or optimal strategies can be synthesized as functions of the state variables. In principle all approaches enable the solution of dynamic games. Nevertheless every approach has advantages and disadvantages which are discussed. The neural network training usually is very difficult and computationally very expensive. The coarse-grained parallelization FAUN 1.0-HPC-PVM of the advanced neurosimulator FAUN uses PVM subroutines and runs on heterogeneous and decentralized networks interconnecting general-purpose workstations, PCs and also high-performance computers. Computing times of days, weeks or months can be cut down to hours. An enhanced cornered rat game — formulated and analyzed in 1993 — serves as an example. Optimal strategies for cat and rat are synthesized. For this purpose open-loop representations of optimal strategies on an equidistant grid in the state space are used. An important end game modification is presented.