January 24, 2001. 4 October 2005 Eric Rasmusen notes, erasmuse@indiana.edu http://rasmusen.org John Nash, ``Non-Cooperative Games,'' Annals of Mathematics, 54: 286-95 (September 1951). p. 11. Gothic-s and Gothic-t are strategy combinations, one strategy for each player. The semicolon notation (s;t_i) is very important. It means "Strategy combination s, except that for player i, s_i is replaced by t_i". p. 12. Equation (3) needs the words "for all i" for it to really be a sufficient condition. So does Equation (4). "Since a criterion (3)" should be "Since criterion (3)". Strong inequalities create open sets (x<3), whereas equalities create closed sets (x=3). The equation at the bottom of the page defines max(0, gain to player i from his deviation to strategy alpha) p. 13. The superscript notation indicates a permutation, not an exponent here. p. 14. Nash says, that "A non-cooperative game does not always have a solution, but when it does the solution is unique." And on p. 15 a solution is a set of equilibrium points. But on p. 16, Example 4 has, he says, a strong solution-- but every possible mixed and pure strategy combination is an equilibrium point. So what does he mean when he says that solutions are always unique? Surely not the literal meaning, which is that the *set* of equilibria is unique (that there is not more than one set of equilibrium points!). I think what he means is that if the game has a solution, then we can tell a player what pure strategies he should use in equilibrium. Thus, in Example 4, we can tell the ROman player to use a or b, and he has the same expected payoff from both. A game is SOLVABLE if when any equilibrium t remains an equilibrium when it has element t_i replaced by r_i, then every other equilibrium s leads to another equilibrium s' when s_i is replaced by that same r_i (Condition 5). Since it is OK for r_i to be the same as t_i, this would be saying that if the game is solvable and t is an equilibrium, then each of t's component strategies could be substituted into every other equilibrium s and the new s' would be an equilibrium. This tends to rule out multiple equilibria, except when every possible strategy combination is an equilibrium or when one player's strategy is irrelevant to his own or other people's payoffs. p. 15.STRONG SOLVABILITY says that in equilibrium s, if replacing s_i with r_i doesn't change i's payoff, then s with r_i must be an equilibrium too. This rules out games with unique equilibria in mixed strategies, games like Example 1 on page 16. In any mixed strategy, a player could switch to a different mixing probability without changing his payoff. So for the game to be strongly solvable, it must be that all possible mixtures are parts of equilibria, which is true only in degenerate games like Example 4. It also allows games with a unique strong equilibrium in pure strategies, like the Prisoner's Dilemma in Example 2. It would be better to say, "Gothic-S is the set of all equilibrium points How does a sub-solution differ from being the set of all equilibrium points? It seems it does PROOF. (a) is to prove convexity and (b) is to prove closedness. p. 16. Example 2 is a prisoner's dilemma.