lecture 13 mixed strategies - purdue university · 2014-11-11 · lecture 13 mixed strategies...

Mixed StrategiesWhy use Mixed Strategies?

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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

http://engineering.purdue.edu/delp


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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.



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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



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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



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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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Evaluating Payoff in Mixed Strategies - Example

Example:


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



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Evaluating Payoff in Mixed Strategies - Example

Example:


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



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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)



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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)]



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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



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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!



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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.



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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?



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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.



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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.



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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).



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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.



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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.



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3. Mixed Strategies and Nash Equilibrium

In strategic form games, there is always a Nash equilibrium in mixedstrategies.



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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.



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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



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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.



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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



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References

1 Dutta, P.K. (1999). Strategies and Games: Theory and Practice.Cambridge, MA, The MIT Press. Chapter 8.


lecture 13 mixed strategies - purdue university · 2014-11-11 · lecture 13 mixed strategies...

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