Mixed Strategies
Mixed Strategies
Player 2
Head
Tail
Head
1, -1
-1, 1
Tail
-1, 1
1, -1
Player 1
Mixed Strategies
Player 2
Head
Tail
Head
1, -1
-1, 1
Tail
-1, 1
1, -1
Player 1
Mixed Strategies
Definition: A mixed strategy of a player in a
simultaneous move game is a probability
distribution over the player’s actions
In matching pennies a mixed strategy will be
ai = (ai(H), ai(T)), where 0 ≤ ai(.) ≤ 1
Mixed Strategies
Player 2
q
1-q
Head
Tail
Head
1, -1
-1, 1
Tail
-1, 1
1, -1
Player 1
Where 0  q  1
Mixed Strategies
Player 2
q
1-q
Head
Tail
Expected
Payoff
Head
1, -1
-1, 1
2q - 1
Tail
-1, 1
1, -1
1 – 2q
Player 1
Mixed Strategies
Player 2
q
1-q
Head
Tail
Expected
Payoff
Head
1, -1
-1, 1
2q - 1
Tail
-1, 1
1, -1
1 – 2q
Expected
Payoff
1 - 2p
2p-1
p
Player 1
1-p
Mixed Strategies
1 – 2q > 2q – 1 if and only if q < ½
 player 1’s Best pure-strategy response is:
- Tail if q < ½
- Head if q > ½
- Indifferent between H and T if q = ½
Mixed Strategies
Player 2
q
1-q
Head
Tail
p
Head
1, -1
-1, 1
1–p
Tail
-1, 1
1, -1
Player 1
Where 0  p  1
Mixed Strategies
E1(Payoff) = pq*1 + p(1 - q)*(-1) + (1 – p)q*(-1) + (1
– p)(1 – q) * 1
= (1 – 2q) + p(4q – 2)
Maximize E1(Payoff) choosing p.
If 4q – 2 < 0 [q < ½]  p = 0 (Tail) is best
response
If 4q – 2 > 0 [q > ½]  p = 1 (Head) is best
response
If 4q – 2 = 0 [q = ½]  any p in [0, 1] is a best
response
Mixed Strategies
E2(Payoff) = pq*(-1) + p(1 - q)*1 + (1 – p)q*1 +
(1 – p)(1 – q) *(-1)
= (2p - 1) + q(2 – 4p)
Maximize E2(Payoff) choosing q.
If 2 - 4p < 0 [p > ½]  q = 0 (Tail) is best
response
If 2 - 4p > 0 [p < ½]  q = 1 (Head) is best
response
If 2 - 4p = 0 [p = ½]  any q in [0, 1] is a best
response
Mixed Strategies
q
b2(p)
1
b1(q)
1/2
The unique Nash equilibrium
is in mixed-strategy:
(a1, a2) = ((1/2,1/2), (1/2,1/2))
1/2
1
p
Mixed Strategies
Definition: The mixed strategy profile a* in a
simultaneous-move game with VNM preferences
is a mixed strategy Nash equilibrium if, for each
player i and every mixed strategy ai of player i,
the expected payoff to player i of a* is at least as
large as the expected payoff to player i of (ai, a*-i)
according to a payoff function whose expected
value represents player i’s preferences over
lotteries.
Mixed Strategies
Equivalently, for each player i,
Ui(a*) ≥ Ui (ai, a*-i) for every mixed strategy profile
ai of player i,
Where Ui(a) is player i’s expected payoff to the
mixed strategy profile a
Mixed Strategies
Alternative definition: The mixed strategy
profile a* is a mixed strategy Nash equilibrium if
and only if a*i is in Bi(a*-i) for every player i.
Mixed Strategies
A player’s expected payoff to the mixed strategy
profile a is a weighted average of her expected
payoffs to all mixed strategy profiles of the type
(ai, a-i), where the weight attached to (ai, a-i) is
the probability ai(ai) assigned to ai by player i’s
mixed strategy ai
Ui a  aiai  Eiai, a  i 
ai Ai
Where Ai is player i’s set of actions (pure strategies)
Mixed Strategies
MSNE Proposition: A mixed strategy profile a* in a strategic
game in which each player has finitely many actions is a
mixed strategy Nash equilibrium if and only if, for each
player i,
• The expected payoff, given a*-i, to every action to which a*i
assigns positive probability is the same,
• The expected payoff, given a*-i, to every action to which a*i
assigns zero probability is at most the expected payoff to
any action to which a*i assigns positive probability.
(See page 116 in Osborne.)
 So actions which the player is mixing between must yield
the same expected payoff. Those that are not being mixed,
must not yield a higher expected payoff than those that are.
Mixed Strategies
American
q
1-q
Enter
Stay out
p
Enter
-50, -50
100, 0
1–p
Stay out
0, 100
0, 0
United
Mixed Strategies
•Suppose both airlines mix between both strategies.
•United’s expected payoff from entering and staying out must be the same:
-50q +100(1-q) = 0q + 0(1-q) --> q = 2/3
•American’s expected payoff from entering and staying out must be the same:
-50p +100(1-p) = 0p + 0(1-p) --> p = 2/3
•Symmetric expected payoffs are thus:
-50(2/3)(2/3) +100(2/3)(1/3) + 0(1/3)(2/3)+0(1/3)(1/3) = 0
•Note that equalizing the conditional expected payoffs gives you the interior solution
(if it exists) while maximizing the unconditional expected payoffs will give you ALL
NE.
•ALL NE are thus {((1,0),(0,1)); ((0,1),(1,0)); ((2/3,1/3),(2/3,1/3)) }
Mixed Strategies
Proposition: Every simultaneous-move game with
vNM preferences and a finite number of players
in which each player has finitely many actions
has at least one Nash equilibrium, possibly
involving mixed strategies.
Mixed Strategies
Asymmetric game
American
q
1-q
Enter
Stay out
p
Enter
-50, -50
150, 0
1–p
Stay out
0, 100
0, 0
United
Asymmetric United/American
Solution
Consider the unconditional expected payoff of United:
E[UU] = -50pq + 150p(1-q) + 0(1-p)q + 0(1-p)(1-q)
= -200pq + 150p = p(150-200q)
So United’s Best Response correspondence is:
•If 150-200q > 0 <=> q < 3/4 ==> p=1.
•If 150-200q < 0 <=> q > 3/4 ==> p=0.
•If 150-200q = 0 <=> q = 3/4 ==> p  [0,1].
Consider the unconditional expected payoff of American:
E[UA] = -50pq + 100q(1-p) + 0(1-q)p + 0(1-p)(1-q)
= -150pq + 100q = q(100-150p)
So American’s Best Response correspondence is:
•If 100-150p > 0 <=> p < 2/3 ==> p=1.
•If 100-150p < 0 <=> p > 2/3 ==> p=0.
•If 100-150p = 0 <=> p = 2/3 ==> p  [0,1].
Graph the BR correspondences (in p,q space) to find ALL NE.
Mixed Strategies
Asymmetric game
• Pure-strategy Nash equilibrium:
(Enter, Stay out)
(Stay out, Enter)
• Mixed-strategy Nash equilibrium:
(aU, aA) = ((2/3,1/3), (3/4,1/4))
Mixed Strategies
Definition: In a strategic game with vNM
preferences, player i’s mixed strategy ai strictly
dominates her action a’i if
Ui(ai, a-i) > ui(a’i, a-i) for every a-i
Mixed Strategies
T
L
1, .
R
1, .
M
4, .
0, .
B
0, .
3, .
Does this game have any dominated pure strategies? No, but if the row
player mixes equally between M and B, then if the column player plays L, row
gets 4(1/2)+0(1/2) = 2 if she mixes while just 1 if she plays T. If column
plays R, row gets 0(1/2)+3(1/2) = 3/2 if she mixes, while again just 1 by
playing T.
Thus T is strictly dominated by a mixed strategy.
Mixed Strategies
L
C
R
T
5, 5
20, 10
25, 3
M
10, 15
10, 10
15, 10
B
3, 25
15, 10
20, 15
What are the NE (pure and mixed) of this game?
Method of finding all mixed-strategy Nash
equilibrium
• For each player i, choose a subset Si of her set
Ai of actions.
• Check whether there exists a mixed strategy
profile a such that (1) the set of actions to which
each strategy ai assigns positive probability is Si
and (2) a satisfies the conditions in proposition
116.2 in Osborne.
• Repeat the analysis for every collection of
subsets of the players’ sets of actions
Mixed Strategies
B
S
X
B
4, 2
0, 0
0,1
S
0, 0
2, 4
1, 3
Mixed Strategies
• Potential types of equilibria:
– 1) Player one plays 1 strategy, Player two plays 1 strategy.
• These are pure strategy NE.
– 2) Player one plays 1 strategy, Player two plays 2 strategies.
• One plays a pure strategy, Two mixes on BS, BX, or SX
– 3) Player one plays 1 strategy, Player two plays 3 strategies.
• One plays a pure strategy, Two mixes on BSX
– 4) Player one plays 2 strategies, Player two plays 1 strategy.
• One mixes on BS, Two plays a pure strategy
– 5) Player one plays 2 strategies, Player two plays 2 strategies.
• One mixes on BS, Two plays BS, BX, or SX
– 6) Player one plays 2 strategies, Player two plays 3 strategies.
• One mixes on BS, Two mixes on BSX