A Nash equilibrium refers to the case where each person has adopted a
strategy such that neither person can improve their play by changing
their own strategy. In other words, if Player 1 keeps doing the same
thing, then there is nothing Player 2 can change about his strategy to
do any better and vice versa.
In a two-person zero-sum game, the outcome can generally be described
as a single value v (such as the reward to person 1), where one player
tries to maximize this value, and the other tries to minimize it (ie.
minimize their opponent's reward). The term saddle point refers to
the situation where, if the maximizing person changes their strategy
in either direction, v will decrease, and if the minimizing person
changes their strategy in either direction, v will increase.
Therefore, if (say) Player 1 changes their strategy to try to do
better, they run the risk that Player 2 will adjust in response, and
Player 1 will wind up worse off than they were at the saddle point.
Therefore, it is an example of a Nash equilibrium, as applied to this
type of game.
If you think about graphing v as a function of the strategies of each
player, it curves down in either direction along one axis, and up in
either direction along the other. This is similar to the shape of a
saddle, which curves down to the left or right, and up to the front or
back.
However, depending on the rules of the game, there may be a Nash
equilibrium which is not a saddle point. For example, Player 1 may be
able to choose a strategy whereby he always gets the maximum reward,
regardless of what Player 2 does. In this case, there isn't really a
single saddle point, since Player 2 wouldn't wind up any worse by
changing his strategy. However, since each player is doing the best
they can, given the other persons strategy, it's a Nash equilibrium.
I hope that all makes some sense. |
