Mathematical Modeling with Multidisciplinary Applications

CHAPTER 17

DECISION THEORY UNDER RISK AND APPLICATIONS IN SOCIAL SCIENCES: II. GAME THEORY

E. V. PETRACOU¹ AND A. N. YANNACOPOULOS²

¹ Department of Geography, University of the Aegean, Greece

² Department of Statistics, Athens University of Economics and Business, Greece

Obviously, you are not a golfer—The Big Lebowski.

17.1 INTRODUCTION

In this chapter we continue our investigation of decision theory and its use in the social sciences, by looking at the problem of decision making when there is more than one agent involved and furthermore, the actions of one agent may interfere with the gain (or loss) of the other. This is a more realistic situation and is clearly needed for a better understanding of social phenomena.

This leads us to the theory of games. The theory of games was introduced in the 1940’s by Von Neumann and Morgernstern as a model for economic behavior and has since become a dominant tool in the economic and social sciences. The mathematical content of the theory is very rich, and brilliant mathematicians such as, e.g., Von Neumann and Nash have provided the mathematical framework. Our approach is inspired by that of Aubin [1].

17.2 BEST REPLIES AND NASH EQUILIBRIA

The simplest possible situation we may envisage is that of two agents A (Athanasios) and E (Electra). Each of the players has a strategy set S_A = {s₁, ···, s_n} and S_E = {s₁, ···, s_m}. Let us denote by P_A(s_A, s_E) the payoff of agent A is he plays s_A s_A while E plays s_E S_E and by P_E(s_A, s_E) the payoff ...