Decision Theory III - Game Theory

We briefly introduce and discuss two two-player non-zero-sum games, mainly to illustrate that such games are substantially more complex to investigate than zero-sum games. Actually, the key aspect of such games is that the sum of the pay-offs to the two players is not constant over all pairs of strategies, as constant sum games are equivalent to zero-sum games (think about this; one can imagine two players having to jointly pay a constant value in order to play a zero-sum game, this would not affect the solution method). Both games we discuss below have triggered considerable further contributions to the literature, with many variations studied (material on this is easily found on-line). John Nash proposed an equilibrium solution for such games, which has the advantage that, if either player plays the corresponding strategy, the other player cannot do better than also play their corresponding strategy. Hence, as for two-person zero-sum games, this could possibly again be considered as a sensible solution if one wishes to eliminate superior skills your opponent may have. In these games, however, it may be possible for both players to do better if they take a ‘cooperative view’ there is no discussion between the players, but sometimes doing so may be quite natural. (We have already seen this in the Prisoner’s Dilemma, so here just two further examples, and we formally formulate the Nash Equilibrium for such games.)