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Mixed StrategiesWhy use Mixed Strategies?
More Examples
Lecture 13Mixed 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, INhttp://engineering.purdue.edu/delp
November 11, 2014
c©Jitesh H. Panchal Lecture 13 1 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
Lecture Outline
1 Mixed StrategiesDefinitionRelationship between Mixed Strategies and Pure Strategies
2 Why use Mixed Strategies?1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. 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.
c©Jitesh H. Panchal Lecture 13 2 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
DefinitionRelationship 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 Football2 Opera
Another possible (“Mixed”) strategy: Tossing a coin to decide Football orOpera!
pF =12
and pO =12
c©Jitesh H. Panchal Lecture 13 3 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship 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 forthis player is a probability distribution over his pure strategies; that is, it is a
probability vector (p1, p2, . . . , pM), with pk ≥ 0, k = 1, . . . ,M, andM∑
k=1pk = 1
c©Jitesh H. Panchal Lecture 13 4 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship 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”!
c©Jitesh H. Panchal Lecture 13 5 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
DefinitionRelationship 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
Say, Husband’s mixed strategy:(
23,
13
); Wife’s mixed strategy: (1, 0)
Likelihood that both spouses go to the football game:23
Probability of the husband going to opera by himself:13
Husband’s expected payoff:[23× 3]+
[13× 0]= 2
c©Jitesh H. Panchal Lecture 13 6 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
DefinitionRelationship 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
Say, Husband’s mixed strategy:(
23,
13
); Wife’s mixed strategy:
(12,
12
)Husband’s expected payoff:[
13× 3]+
[16× 0]+
[13× 0]+
[16× 1]=
76
c©Jitesh H. Panchal Lecture 13 7 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship 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 thatthe other players play the pure strategy s#
−i . Then the expected payoff toplayer i is equal to
p1 × πi(s1, s#−i) + p2 × πi(s2, s#
−i) + · · ·+ pM × πi(sM , s#−i)
c©Jitesh H. Panchal Lecture 13 8 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship 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; saythe 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
[p1q × πi(s1, s#−i) + · · ·+ pMq × πi(sM , s#
−i)]
+[p1(1 − q)× πi(s1, s#−i) + · · ·+ pM(1 − q)× πi(sM , s#
−i)]
c©Jitesh H. Panchal Lecture 13 9 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship between Mixed Strategies and Pure Strategies
Other Examples
Matching pennies
Player 1 / Player 2 Heads TailsHeads 1,−1 −1, 1
Tails −1, 1 1,−1
Find the expected payoff for the two players considering mixed strategy for
player 1:(
23,
13
)and pure strategy for player 2: Tails
c©Jitesh H. Panchal Lecture 13 10 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship 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 thathave a positive probability of getting played (in this strategy).
Note: The expected payoff to a mixed strategy is an average of thecomponent pure-strategy payoffs in the support of this mixed strategy.Deleting the pure strategies with lower payoffs reduces the expected payoff!
c©Jitesh H. Panchal Lecture 13 11 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship between Mixed Strategies and Pure Strategies
Mixed Strategy as a Best Response
Implications
1 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 .
2 In that case, any mixed strategy over this support will be a bestresponse.
c©Jitesh H. Panchal Lecture 13 12 / 24
Mixed StrategiesWhy use Mixed Strategies?
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DefinitionRelationship between Mixed Strategies and Pure Strategies
Mixed Strategy as a Best Response
The No-Name game:
Player 1 / Player 2 L M1 M2 RU 1, 0 4, 2 2, 4 3, 1M 2, 4 2, 0 2, 2 2, 1D 4, 2 1, 4 2, 0 3, 1
What are Player 1’s best responses to R?
Mixed strategies of the pure strategies?
c©Jitesh H. Panchal Lecture 13 13 / 24
Mixed StrategiesWhy use Mixed Strategies?
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1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. Mixed Strategies and Nash Equilibrium
Reasons for Using Mixed Strategies
1. A mixed strategy may dominate some pure strategies (that are themselvesundominated by other pure strategies).
2. The worst-case payoff of a mixed strategy may be better than theworst-case payoff of every pure strategy.
3. If we restrict ourselves to pure strategies, we may not be able to find aNash equilibrium to a game.
c©Jitesh H. Panchal Lecture 13 14 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. Mixed Strategies and Nash Equilibrium
1. Mixed Strategies can Dominate Some Pure Strategies
The No-Name game:
Player 1 / Player 2 L M1 M2 RU 1, 0 4, 2 2, 4 3, 1M 2, 4 2, 0 2, 2 2, 1D 4, 2 1, 4 2, 0 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
probabilities(
12,
12
)? Show that this mixed strategy dominates pure
strategy M.
For Player 2, show that mixing L,M1,M2 with equal probabilitiesdominates the pure strategy R.
c©Jitesh H. Panchal Lecture 13 15 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. Mixed Strategies and Nash Equilibrium
Key Points
If there is a pure strategy that dominates every other pure strategy, then itmust also dominate every other mixed strategy.
If there is no dominant strategy in pure strategies, there cannot be one inmixed strategies either.
However, in the IEDS solution concept, a game that has no IEDS solutionwhen only pure strategies are considered can have an IEDS solution in mixedstrategies (check for no-name game).
c©Jitesh H. Panchal Lecture 13 16 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. 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-casepayoff of every pure strategy.
c©Jitesh H. Panchal Lecture 13 17 / 24
Mixed StrategiesWhy use Mixed Strategies?
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1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. 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 TH 1,−1 −1, 1T −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,
H has a higher payoff than T iff p <12
If p =12
, then Eπ(T ) = Eπ(H). The best response is any mixedstrategy.
c©Jitesh H. Panchal Lecture 13 18 / 24
Mixed StrategiesWhy use Mixed Strategies?
More Examples
1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. Mixed Strategies and Nash Equilibrium
3. Mixed Strategies and Nash Equilibrium
In strategic form games, there is always a Nash equilibrium in mixedstrategies.
c©Jitesh H. Panchal Lecture 13 19 / 24
Mixed StrategiesWhy use Mixed Strategies?
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1. Game of Chicken
Player 1 / Player 2 Tough (T) Concede (C)Tough (T) a, a d , 0
Concede (C) 0, d b, b
Here, d > b > 0 > a. Two pure strategy equilibria. Can you find them?
Mixed strategy equilibrum: Each player plays T with probabilityd − b
d − b − a(Check!)
Find expected payoffs.
c©Jitesh H. Panchal Lecture 13 20 / 24
Mixed StrategiesWhy use Mixed Strategies?
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2. Natural Monopoly
Firm 1 / Firm 2 date 0 date 1 date 2date 0 0, 0 0, π 0, 2πdate 1 π, 0 −c,−c −c, π − cdate 2 2π, 0 π − c,−c −2c,−2c
c©Jitesh H. Panchal Lecture 13 21 / 24
Mixed StrategiesWhy 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 playingagainst.
The payoffs may be uncertain. If high and low payoffs are equally likely, it isas if the players are facing mixed strategies with equal probabilities.
Although each player actually plays a pure strategy, to the opponents–and anoutside observer–it appears as if mixed strategies are being played.
c©Jitesh H. Panchal Lecture 13 22 / 24
Mixed StrategiesWhy use Mixed Strategies?
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Summary
1 Mixed StrategiesDefinitionRelationship between Mixed Strategies and Pure Strategies
2 Why use Mixed Strategies?1. Mixed Strategies can Dominate Some Pure Strategies2. Mixed Strategies are Good for Bluffing3. Mixed Strategies and Nash Equilibrium
3 More Examples
c©Jitesh H. Panchal Lecture 13 23 / 24
Mixed StrategiesWhy use Mixed Strategies?
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References
1 Dutta, P.K. (1999). Strategies and Games: Theory and Practice.Cambridge, MA, The MIT Press. Chapter 8.
c©Jitesh H. Panchal Lecture 13 24 / 24