7. Mixed Strategies

By the same token, when 2p < 1-p, that is p < 1/3, the best response of Column is his pure strategy W.
Finally, when p = 1/3 then all possible strategies of Column give him the same payoff—all are best response strategies.
Mixed Strategies in Normal and Sequential Form Games
3
How to calculate payoffs to a probabilistic mix of Row’s E and W against, for instance, Column’s E?
1 n p1s1 .... p n s1
a
……
a*
……
Mixed Strategies in Normal and Sequential Form Games
10
Battle of the Sexes Example
G
1 G 2 0 0
T
0
2 1
T
0
Identify all Nash equilibria in this game.
0 0 0 0 0 0 0 0 -3 3 4 0 -4 0 0 -4 0 3 4 -3
(1,2)
(2,1)
(2,2)
(1,1) 0 (1,1) 0 2 (1,2) -2 -3 (2,1) 3 (2,2) 0 2
(1,2) -2 -3 0
(2,1) 3
(2,2)
Mixed Strategies in Normal and Sequential Form Games
(1,1)
(1,2)
(2,1)
(2,2)
(1,1)
Two-Finger Morra Game
Two players simultaneously raise both hands with one or two fingers extended on each hand. Left hand is for “number shown,” right hand for “number guessed.” The guess is about how many fingers the other player is going to show―with his left hand, of course.
Japan 1 1 US 1/3 0 1/3 0
E -2
qE(1-q)W
best response of Column to Row’s choice of strategy “p” W q=0
E q=1
E p=1
1/3E2/3W q=1/3
best response of Row to Column’s choice of strategy “q”
p E
For notational convenience instead of
1-p W
I will write pE (1-p)W
stands for a “lottery”
Mixed Strategies in Normal and Sequential Form Games
5
Japan E W 2 E 0 0 0 W 0 -1 1 -2
Mixed Strategies in Normal and Sequential Form Games
2
The South Pacific Example with Different Payoffs
The original game
Japan
E
2 E US -2 0 5
W
0
US W E -2
Japan E W 2 0 0 0 -1 1 0
Mixed Strategies in Normal and Sequential Form Games 7
Game with pure strategies only
Japan E W 2 US 0 0 0 W 0 -1 1
A way to think about the same game with mixed strategies
uJ (0.6E 0.4W, W) = 0.6 uJ (E, W) + 0.4 uJ (W, W) = 0.6•0 + 0.4•1 = 0.4 uUS (W, 0.7E 0.3W) = 0.7 uUS (W, E) + 0.3 uUS (W, W) = 0.7•0 + 0.3•(-1) = -0.3 uJ (0.2E 0.8W, 0.3E 0.7W) = 0.2 uJ (E, 0.3E 0.7W ) + 0.8 uJ (W, 0.3E 0.7W) = 0.2•0.3 uJ (E, E) + 0.2•0.7 uJ (E, W) + 0.8•0.3 uJ (W, E) + 0.8•0.7 uJ (W, W) = 0.06•2 + 0.14•0 + 0.24•0 + 0.56•1 = 0.68
Mixed Strategies in Normal and Sequential Form Games 9
THEOREM
1 2 n Denote p1s1 p 2s1 ...... p n s1 s 2 m q1s1 q 2s 2 ...... q ms 2 s* 2
Japan
E 2 E W 0 0 0 W 0 -1 1
US
-2
p E
uUS (
,E)=?
1-p W
Mixed Strategies in Normal and Sequential Form Games 4
Japan
E
2
W
0 0 0 1 -1
p E
p (E, E)
E
uUS (
,E)=
1-p W
uUS (
pE(1-p)W
1/3E2/3W p=1/3
Nash equilibrium: each is the best response to the other
2 1 2 1 E W, E W 3 3 3 3
W p=0
Mixed Strategies in Normal and Sequential Form Games 8
Mixed Strategies in Normal and Sequential Form Games
11
G
T
qG(1-q)T
1
G 2 0 T 0 pG(1-p)T p 1 0
0
2q 2 1-q 2-2p
The value of p for which the payoffs to Column are the same: p = 2-2p or p = 2/3 The value of q for which the payoffs to Row are the same: 2q = 1-q or q = 1/3
Mixed Strategies in Normal and Sequential Form Games
6
q
1-q
E
2 p E 1-p W 0 pE(1-p)W 2p -2 0 0
W
0
qE(1-q)W
p•2 + (1-p)•0
-2q 1
-1 1-p
q-1
p•0 + (1-p)•1
Just like we were looking for Nash equilibria in pure strategies we now need to do the same with mixed strategies. The difference here is that since we have infinitely many mixed strategies we can no longer draw a payoff table with rows representing choices of the Row player. Note that when 2p > 1-p, or equivalently when p > 1/3, the best response of Column is his pure strategy E since mixing in W will only lower his payoff.
1-p (W, E)
)=
US W
-2 0
here we need expected utility theory
= p uUS (E, E) + (1-p) uUS (W, E) = p•(-2) + (1-p)•0 = -2p
Thus with mixed strategies we have to switch from ordinal utility assumption (preference theory) to cardinal utility assumption (expected utility theory)
s1 2
p1
……
sm 2
……
s1 1
If two proper mixes (s, s* )
……
are in Nash equilibrium then
m u 2 (s1 , s) = ... = u 2 (s 2 , s) and 2 1 n u1 (s1 , s* ) = ... = u1 (s1 , s* ).