In this document we provide a couple of exercises that help understand the basic concepts of game theory, including pure and mixed strategies, zero sum and non-zero sum games, and saddle-point and Nash solutions. The definition for concepts and the ideas for these exercises were taken in part from [1, 2]. Before we start, let us introduce some notation. We are going to assume that we have only ...

A discretized game model of an anti-missile defense scenario is presented. Such an engagement is a pursuit of a "blind" evader, committed to satisfy a terminal constraint. It can be formulated as an incomplete information zero-sum pursuit-evasion game with state constraint. Such a game becomes non trivial if the evader has higher maneuverability than the pursuer. In this case the game admits a ...

We provide several tests to determine whether a game is a potential game or whether it is a zero-sum equivalent game—a game which is strategically equivalent to a zero-sum game in the same way that a potential game is strategically equivalent to a common interest game. We present a unified framework applicable for both potential and zero-sum equivalent games by deriving a simple but useful char...

We investigate the problem of learning to play the game of rock-paper-scissors. Each player attempts to improve her/his average score by adjusting the frequency of the three possible responses, using reinforcement learning. For the zero sum game the learning process displays Hamiltonian chaos. Thus, the learning trajectory can be simple or complex, depending on initial conditions. We also inves...

Cognitive radio has been recently proposed as a promising technology to achieve efficient use of spectrum resources. In this paper, we consider spectrum sharing in a MIMO system where several secondary users (SUs) share spectrum with a primary user (PU). A leakage-based beamforming algorithm is proposed via game theory. The objective is to maximize the sum throughput of SUs subject to the signa...

There is limited formal mathematical analysis of one type of games — dynamic sequential games with large, or even infinitely large, planning horizons, from the point view of system controls. In this paper, we use a zero-sum game theoretical approach to address the disturbance attenuation analysis of state feedback Nash strategies for Dynamic Linear Quadratic Sequential Games (LQSGs) with uncert...

A new treatment to determine the Pareto-optimal outcome for a non-zero-sum game is presented. An equilibrium point for any game is defined here as a set of strategy choices for the players, such that no change in the choice of any single player will increase the overall payoff of all the players. Determining equilibrium for multi-player games is a complex problem. An intuitive conceptual tool f...

In this paper, we investigate a two-person zero-sum game with fractional loss function, which we call the two-person zero-sum fractional game. We are interested in a game value and a saddle point of the two-person zero-sum game and observe that they exist under various conditions. But, in many cases, it seems to be difficult to search directly for a game value and a saddle point of the fraction...

We consider the problem of finding stationary Nash equilibria (NE) in a finite discounted general-sum stochastic game. We first generalize a non-linear optimization problem from [9] to a general N player game setting. Next, we break down the optimization problem into simpler sub-problems that ensure there is no Bellman error for a given state and an agent. We then provide a characterization of ...