Monday, July 19, 2004

Be Game!
The old cupboard that carries n number of books in our house has always sprung me a few surprises. And this time the surprise came quite randomly, and I fouind myself gawking at the small collection of "Little Mathematics Library" series. The titles included "Pascal's triangles, and applications", "Differentiation simplified", etc. One book got my immediate attention was the following. "Elements of Game Theory". I was attracted towards this sect of mathematics when I first heard Dr Pavan from CEDT when he explained "the prisoner's dilemma" in one of his talks. I also had got hold of one of more involved books on Game Theory, but could not understand much that time around. This book proves just to be what I wanted. I'll go through some of the paragraphs from the introductory chapter on Game Theory.

When solving practical (economic, military or other) problems one often has to analyse situations in which there are two, or more quarrelling parties pursuing conflicting objectives, and where the outcome of each action of one party depends on the opponent's choice of a course of action. Such situations will be called "conflict situations". The need to analyse situations of this kind has brought about the development of special mathematical methods. The theory of games is in fact a mathematical theory of conflict situations. For a analysis of a situation to be possible, it is necessary to construct a simplified, formalized model of the situation. Such a model will be called "a game".

The author then continues explaining more game jargon, brings in stochatic considerations, and something called as "a strategy" to win the game. Actually, victory in such situations actually means least loss! By choosing one's strategy given the odds, one can always minimise the maximum loss possible in the game. That's how the principles of "minimax" and "maximin" come out.

After further bombardment of mathematical representations etc. come a real entertaining paragraph. Game theory proves that every game with perfect information has a saddle point and that consequently every game of this kind has a solution, i.e. there is a pair of optimal strategies, one for each player which gives an average gain equal to the value of the game (or in simple words, what the game is mathematically worth !). If a perfect-information game contains only personal moves (deterministic moves), then, if each player plays his optimal strategy, the game must always end with a definite outcome, namely the gain exactly equal to the value of the game.

The situation is observed in chess. Chess also has a saddle point and a solution which shows each of the players his optimal stratefy; but is far too difficult to be found. The number of combinations of possible moves in chess is too large for a game matrix to be constructed and a saddle point (giving optimal stratefy) to be found in it.

Now, do we see life as a mathematical game? And if so, can we ever imagine the number of possible moves to construct the matrix to find the optimal strategy? If you could, and if everybody agreed to play the optimal strategy, we could well consider the game won!

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