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For finding a stationary min-max point of a scalar-valued function, we develop and investigate a family of gradient transformation differential equation algorithms. This family includes, as special cases: Min-Max Ascent, Newton’s method, and a Gradient Enhanced Min-Max (GEMM) algorithm that we develop. We apply these methods to a sharp-spined “Stingray” saddle function, in which Min-Max Ascent is globally asymptotically stable but stiff, and Newton’s method is not stiff, but does not yield global asymptotic stability. However, GEMM is both globally asymptotically stable and not stiff. Using the Stingray function we study the stiffness of the gradient transformation family in terms of Lyapunov exponent time histories. Starting from points where Min-Max Ascent, Newton‘s method, and the GEMM method do work, we show that Min-Max Ascent is very stiff. However, Newton’s method is not stiff and is approximately 60 to 440 times as fast as Min-Max Ascent. In contrast, the GEMM method is globally convergent, is not stiff, and is approximately 3 times faster than Newton’s method and approximately 175 to 1000 times faster than Min-Max Ascent.

This chapter examines trajectory following algorithms for differential games subject to simple bounds on player strategy variables. These algorithms are trajectory following in the sense that closed-loop player strategies are generated directly by the solutions to ordinary differential equations. Player strategy differential equations are based upon Lyapunov optimizing control techniques and represent a balance between the current penetration rate for an appropriate descent function and the current cost accumulation rate. This numerical strategy eliminates the need to solve 1) a min-max optimization problem at each point along the state trajectory and 2) nonlinear two-point boundary-value problems. Furthermore, we address “stiff” systems of differential equations that arise during the design process and seriously degrade algorithmic performance. We use standard singular perturbation methodology to produce a numerically tractable algorithm. This results in the Efficient Cost Descent (ECD) algorithm which possesses desirable characteristics unique to the trajectory following method. Equally important as a specification of a new trajectory following algorithm is the observation and resolution of several issues regarding the design and implementation of a trajectory following algorithm in a differential game setting.

This chapter deals with air-to-air missile guidance law design. We model the terminal engagement of a ramjet missile with radar seeker lock on a single target (generic aircraft). We consider a realistic interception simulation with measurement errors and, more particularly, radome aberration errors.We define an extended Kalman filter to compensate in line those errors and to stabilize the guidance loop. Then, to decrease the miss distance against manoeuverable targets we implement an optimized guidance law. The guidance law we propose is based on a linear quadratic differential game linearized around the collision course using the target acceleration estimation provided by the Kalman filter. In this game, the evader control is defined around the Kalman target acceleration estimation to take into account delays and lags due to the filters we apply. The Kalman target acceleration estimation (assumed constant) is a parameter of the differential game kinematics.

We report computer simulation experiments using our agent-based simulation tool to model the multi-person Chicken game. We simplify the agents according to the Pavlovian principle: their probability of taking a certain action changes by an amount proportional to the reward or penalty received from the environment for that action. The individual agents may cooperate with each other for the collective interest or may defect, i.e., pursue their selfish interests only. Their decisions to cooperate or defect accumulate over time to produce a resulting collective order that determines the success or failure of the public system. After a certain number of iterations, the proportion of cooperators stabilizes to either a constant value or oscillates around such a value. The payoff (reward/penalty) functions are given as two curves: one for the cooperators and another for the defectors. The payoff curves are functions of the ratio of cooperators to the total number of agents. The actual shapes of the payoff functions depend on many factors. Even if we assume linear payoff functions, there are four parameters that are determined by these factors. The payoff functions for a multi-agent Chicken game have the following properties. (1) Both payoff functions increase with an increasing number of cooperators. (2) In the region of low cooperation the cooperators have a higher reward than the defectors. (3) When the cooperation rate is high, there is a higher payoff for defecting behavior than for cooperating behavior. (4) As a consequence, the slope of the D function is greater than that of the C function and the two payoff functions intersect. (5) All agents receive a lower payoff if all defect than if all cooperate. We have investigated the behavior of the agents under a wide range of payoff functions. The results show that it is quite possible to achieve a situation where the enormous majority of the agents prefer cooperation to defection. The Chicken game of using cars or mass transportation in large cities is considered as a practical application of the simulation.

This chapter investigates a partially cooperative game in an extended form. The method of finding the optimal behavior of players and the value for such games are presented. During the course of the game, auxiliary games and Nash’s equilibrium situations are considered to define the distribution between players of a coalition. An example is also presented.