Minimax Analysis: Introduction To The Game Theory

Introduction to the minimax analysis.
Over the last years, popularity of gambling has increased and man started thinking about strategies that would be more advantageous and make winning more probable. This has led to the development of the game theory. With the improvement of mathematics, it became possible to devise a rigorous theory based on statistics and probability theory. The main purpose of this paper is to define basic notions, to introduce the reader to the theory of games and minimax analysis using one model of this theory – zero-sum two-player games. First, some basic definitions will be addressed. Then examples will be examined. In the main part, two methods of finding the minimax solution of the game with some theorems required

If we look at example 1, there are no optimal strategies among those in the columns and rows of loss matrix. For this reason, randomized strategies are used. The only strategy for this game to avoid losses is to play the equilibrium. It means that both players should choose head or tails randomly with the same probability equaling 1/2. In this case, the minimax strategy is δ^M= 1/2 and the minimax value is equal to 0 as well as maximin value. Such games are called strictly determined. Their main property is the fact that the optimal strategies and expected outcome can be known before the beginning of the

Although there are no deterministic methods known for solving all the games, there are some of them in special cases. One of nondeterministic techniques is to guess a solution, but here we have a new problem – how to check whether the strategy is minimax or not. The book by James O. Berger provides a theorem stating that if there are strategies π_0 and δ_0 that for all θ and a: L(θ,δ_0)≤L(π_0,a) then the game has value L(π_0,δ_0) and is strictly determined, with π_0 and δ_0 being maximin and minimax strategies respectively. One of the most important achievements in this theory is the guess method of solving games consisting of eliminating bad strategies in the beginning, trying to guess the optimal strategy then using special theorem which can be found in “Statistical Decision Theory and Bayesian Analysis” by James O. Berger (p. 319) checking whether this strategy is minimax or