Mixed Strategy Nash Equilibrium

1
Mixed Strategy Nash Equilibrium
2
Unpredictability
3
Equilibrium mix 5050
4
Chicken revisited the mixed strategy
1 2 q turn 1-q dont turn
p turn 0, 0 0, 4
1-p dont turn 4, 0 -4, -4
If (1) plays both turn and dont with positive
probability then (1) must be
indifferent between turn and dont

Note that in a mixed strategy NE - player i is
indifferent about the probable values he assigns
to the pure strategies that he uses
5
Unclear predictions
The only reason we pick the equilibrium mixed
strategy for player i is to ensure that all other
players are indifferent about the probabilities
assigned to the pure strategies that they use.
As a practical matter, it often isnt clear why
a player would bother to randomize given that the
game is a simultaneous move game that is played
only once. Note that if one of the players sets
one probability just a small amount different
from the equilibrium probability the whole
equilibrium would break down. Further, as a
practical matter, it is more difficult to get
clear predictions when mixed strategies are being
used as long as a pure strategy is played with
positive probability, then the actions in that
strategy can be taken, and the outcome associated
with those actions can result.

6
Existence of Nash Equilibrium
Mixed strategies can be important in ensuring
that equilibrium exists.

Note these are sufficient conditions, not
necessary. In applied work, existence is
important, but the main goal is to characterize
the equilibrium and describe its economic content
Punch Line under standard assumptions, a NE
exists