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Euphoriant asked, "Can anyone explain the Nash Equilibrium to me in simple terms?"

I will be glad to try. If my answer is not clear, try reading the extended article on Game Theory in the Encyclopedia Britannica.

The Nash Equilibrium appears in the literature on two-person non-zero-sum games. "Non-zero-sum" means that the two players may be able to cooperate with each other so as to both receive a positive payoff. Zero-sum games are never cooperative, and are much easier to analyze. Nash found a way to predict the cooperative solution to a broad category of such games, given only six reasonable assumptions (the Nash Axioms) and knowledge of each player's utility function.

For example, suppose a very rich man and a poor man are told that they will receive a total of $10,000, if they can agree on how it is to be divided between them. If they cannot agree, then they receive nothing. As the two men negotiate, each can threaten the other with breaking off the negotiations. The Nash Equilibirum is, essentially, an agreement that maximizes the product of the two utilities. The exact amounts to be distributed to the two men depend, of course, on the utility functions for the two men.

The Nash Equilibrium is remarkable and mathematically interesting because Nash was able to show that his six axioms always give a unique solution. Nobody had imagined that such a result was possible, hence the fame of his theorem. The actual equilibrium itself is also sometimes surprising. In the above example, if the rich man has a utility function which is proportional to the dollar amount while the poor man has a utility function which is proportional to the square root of the dollar amount, then the poor man will end up with only one-third of the cash (see below for details). In effect, the poor man has less bargaining power because small amounts of money have much more utility for him than for the rich man.

Details of the calculation: we need to maximize g(x) = (ax)( b sqrt(10000-x) ). Differentiating g with respect to x and setting the result equal to zero, we obtain:

0 = ab sqrt(10000-x) - (1/2)abx / sqrt(10000-x).

This has the solution x = $6666.67 for the rich man and $3333.33 for the poor man. Notice that we get these exact figures even without knowing the proportionality constants a and b!

I hope this helps somewhat.
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