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.