Mixed Strategies
Why use Mixed Strategies?
More Examples
Lecture 13
Mixed Strategies
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)
School of Mechanical Engineering
Purdue University, West Lafayette, IN
http://engineering.purdue.edu/delp
November 11, 2014
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Lecture Outline
1
Mixed Strategies
Definition
Relationship between Mixed Strategies and Pure Strategies
2
Why use Mixed Strategies?
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
3
More Examples
Dutta, P.K. (1999). Strategies and Games: Theory and Practice. Cambridge, MA, The MIT Press. Chapter 8.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Mixed Strategies - Example
Battle of Sexes
Husband / Wife
Football (F)
Opera (O)
Football (F)
3, 1
0, 0
Opera (O)
0, 0
1, 3
“Pure” Strategies:
1
Football
2
Opera
Another possible (“Mixed”) strategy: Tossing a coin to decide Football or
Opera!
pF =
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1
1
and pO =
2
2
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Mixed Strategies - Definition
Definition (Mixed Strategies)
Suppose a player has M pure strategies, s1 , s2 , . . . , sM . A mixed strategy for
this player is a probability distribution over his pure strategies; that is, it is a
M
P
probability vector (p1 , p2 , . . . , pM ), with pk ≥ 0, k = 1, . . . , M, and
pk = 1
k =1
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Evaluating Payoff in Mixed Strategies
Using the expected utility theorem,
1
Weight the payoff to each pure strategy by the probability with which that
strategy is played.
2
Add up the weighted payoffs.
Mixed strategies are associated with “Expected payoff”!
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Evaluating Payoff in Mixed Strategies - Example
Example:
Husband / Wife Football (F) Opera (O)
Football (F)
3, 1
0, 0
Opera (O)
0, 0
1, 3
2 1
Say, Husband’s mixed strategy:
; Wife’s mixed strategy: (1, 0)
,
3 3
2
Likelihood that both spouses go to the football game:
3
1
Probability of the husband going to opera by himself:
3
Husband’s expected payoff:
2
1
×3 +
×0 =2
3
3
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Evaluating Payoff in Mixed Strategies - Example
Example:
Husband / Wife Football (F) Opera (O)
Football (F)
3, 1
0, 0
Opera (O)
0, 0
1, 3
2 1
1 1
Say, Husband’s mixed strategy:
,
; Wife’s mixed strategy:
,
3 3
2 2
Husband’s expected payoff:
1
1
1
1
7
×3 +
×0 +
×0 +
×1 =
3
6
3
6
6
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Expected Payoff – Formal Definition
Definition (Expected Payoff)
Suppose that player i plays a mixed strategy (p1 , p2 , . . . , pM ). Suppose that
#
the other players play the pure strategy s−i
. Then the expected payoff to
player i is equal to
#
#
#
) + p2 × πi (s2 , s−i
) + · · · + pM × πi (sM , s−i
)
p1 × πi (s1 , s−i
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Expected Payoff – Formal Definition (contd.)
Definition (Expected Payoff)
Now, suppose that the other players play a mixed strategy themselves; say
#
∗
the strategy s−i
is played with probability q while s−i
is played with probability
(1 − q). Then the expected payoff to player i is equal to
#
#
[p1 q × πi (s1 , s−i
) + · · · + pM q × πi (sM , s−i
)]
#
#
+[p1 (1 − q) × πi (s1 , s−i
) + · · · + pM (1 − q) × πi (sM , s−i
)]
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Other Examples
Matching pennies
Player 1 / Player 2
Heads
Tails
Heads
1, −1
−1, 1
Tails
−1, 1
1, −1
Find the expected
payoff for the two players considering mixed strategy for
2 1
player 1:
,
and pure strategy for player 2: Tails
3 3
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Support of a Mixed Strategy
Definition (Support of Mixed Strategy)
Consider a mixed strategy given by the probability vector (p1 , p2 , . . . , pM ).
The support of this mixed strategy is given by all those pure strategies that
have a positive probability of getting played (in this strategy).
Note: The expected payoff to a mixed strategy is an average of the
component pure-strategy payoffs in the support of this mixed strategy.
Deleting the pure strategies with lower payoffs reduces the expected payoff!
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Mixed Strategy as a Best Response
Implications
1
2
#
A mixed strategy (p1 , p2 , . . . , pM ) is a best response to s−i
if and only if
#
each of the pure strategies in its support is itself a best response to s−i
.
In that case, any mixed strategy over this support will be a best
response.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Definition
Relationship between Mixed Strategies and Pure Strategies
Mixed Strategy as a Best Response
The No-Name game:
Player 1 / Player 2
U
M
D
L
1, 0
2, 4
4, 2
M1
4, 2
2, 0
1, 4
M2
2, 4
2, 2
2, 0
R
3, 1
2, 1
3, 1
What are Player 1’s best responses to R?
Mixed strategies of the pure strategies?
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
Reasons for Using Mixed Strategies
1. A mixed strategy may dominate some pure strategies (that are themselves
undominated by other pure strategies).
2. The worst-case payoff of a mixed strategy may be better than the
worst-case payoff of every pure strategy.
3. If we restrict ourselves to pure strategies, we may not be able to find a
Nash equilibrium to a game.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
1. Mixed Strategies can Dominate Some Pure Strategies
The No-Name game:
Player 1 / Player 2
U
M
D
L
1, 0
2, 4
4, 2
M1
4, 2
2, 0
1, 4
M2
2, 4
2, 2
2, 0
R
3, 1
2, 1
3, 1
No pure strategy dominates any other pure strategy.
What is the payoff
for
Player 1’s mixed strategy of playing U and D with
1 1
probabilities
,
? Show that this mixed strategy dominates pure
2 2
strategy M.
For Player 2, show that mixing L, M1 , M2 with equal probabilities
dominates the pure strategy R.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
Key Points
If there is a pure strategy that dominates every other pure strategy, then it
must also dominate every other mixed strategy.
If there is no dominant strategy in pure strategies, there cannot be one in
mixed strategies either.
However, in the IEDS solution concept, a game that has no IEDS solution
when only pure strategies are considered can have an IEDS solution in mixed
strategies (check for no-name game).
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
2. Mixed Strategies are Good for Bluffing
The worst case payoff of a mixed strategy may be better than the worst-case
payoff of every pure strategy.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
3. Mixed Strategies and Nash Equilibrium
Without mixed strategies, Nash equilibria need not always exist.
Game of Matching Pennies (no pure strategy Nash equilibrium)
H
T
H
1, −1
−1, 1
T
−1, 1
1, −1
Suppose that Player 1 plays a mixed strategy: (H, p)
Player 2’s expected payoff from playing pure strategy H is
Eπ(H) = p(−1) + (1 − p)1 = (1 − 2p)
Similarly, Eπ(T ) = p(1) + (1 − p)(−1) = (2p − 1). Therefore,
1
H has a higher payoff than T iff p <
2
1
If p = , then Eπ(T ) = Eπ(H). The best response is any mixed
2
strategy.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
3. Mixed Strategies and Nash Equilibrium
In strategic form games, there is always a Nash equilibrium in mixed
strategies.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
1. Game of Chicken
Player 1 / Player 2
Tough (T)
Concede (C)
Tough (T)
a, a
0, d
Concede (C)
d, 0
b, b
Here, d > b > 0 > a. Two pure strategy equilibria. Can you find them?
Mixed strategy equilibrum: Each player plays T with probability
d −b
d −b−a
(Check!)
Find expected payoffs.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
2. Natural Monopoly
Firm 1 / Firm 2
date 0
date 1
date 2
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date 0
0, 0
π, 0
2π, 0
date 1
0, π
−c, −c
π − c, −c
Lecture 13
date 2
0, 2π
−c, π − c
−2c, −2c
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Mixed Strategies
Why use Mixed Strategies?
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Harsanyi’s Interpretation of Mixed Strategies
Assume that each player is unsure about exactly whom he/she is playing
against.
The payoffs may be uncertain. If high and low payoffs are equally likely, it is
as if the players are facing mixed strategies with equal probabilities.
Although each player actually plays a pure strategy, to the opponents–and an
outside observer–it appears as if mixed strategies are being played.
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Mixed Strategies
Why use Mixed Strategies?
More Examples
Summary
1
Mixed Strategies
Definition
Relationship between Mixed Strategies and Pure Strategies
2
Why use Mixed Strategies?
1. Mixed Strategies can Dominate Some Pure Strategies
2. Mixed Strategies are Good for Bluffing
3. Mixed Strategies and Nash Equilibrium
3
More Examples
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Mixed Strategies
Why use Mixed Strategies?
More Examples
References
1
Dutta, P.K. (1999). Strategies and Games: Theory and Practice.
Cambridge, MA, The MIT Press. Chapter 8.
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