Mixed Strategies

1
Mixed Strategies
2
Overview

• Principles of mixed strategy equilibria
• Wars of attrition
• All-pay auctions

3
Tennis Anyone
R
S
4
Serving
R
S
5
Serving
R
S
6
The Game of Tennis

• Server chooses to serve either left or right
• Receiver defends either left or right
• Better chance to get a good return if you defend
  in the area the server is serving to

7
Game Table
8
Game Table
For server Best response to defend left is to
serve right Best response to defend right is to
serve left For receiver Just the opposite
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Nash Equilibrium

• Notice that there are no mutual best responses in
  this game.
• This means there are no Nash equilibria in pure
  strategies
• But games like this always have at least one Nash
  equilibrium
• What are we missing?

10
Extended Game

• Suppose we allow each player to choose
  randomizing strategies
• For example, the server might serve left half the
  time and right half the time.
• In general, suppose the server serves left a
  fraction p of the time
• What is the receivers best response?

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Calculating Best Responses

• Clearly if p 1, then the receiver should defend
  to the left
• If p 0, the receiver should defend to the
  right.
• The expected payoff to the receiver is
• p x ¾ (1 p) x ¼ if defending left
• p x ¼ (1 p) x ¾ if defending right
• Therefore, she should defend left if
• p x ¾ (1 p) x ¼ gt p x ¼ (1 p) x ¾

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When to Defend Left

• We said to defend left whenever
• p x ¾ (1 p) x ¼ gt p x ¼ (1 p) x ¾
• Rewriting
• p gt 1 p
• Or
• p gt ½

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Receivers Best Response
Left
Right
p
½
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Servers Best Response

• Suppose that the receiver goes left with
  probability q.
• Clearly, if q 1, the server should serve right
• If q 0, the server should serve left.
• More generally, serve left if
• ¼ x q ¾ x (1 q) gt ¾ x q ¼ x (1 q)
• Simplifying, he should serve left if
• q lt ½

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Servers Best Response
q
½
Left
Right
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Putting Things Together
Rs best response
q
Ss best response
½
p
1/2
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Equilibrium
Rs best response
q
Mutual best responses
Ss best response
½
p
1/2
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Mixed Strategy Equilibrium

• A mixed strategy equilibrium is a pair of mixed
  strategies that are mutual best responses
• In the tennis example, this occurred when each
  player chose a 50-50 mixture of left and right.

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General Properties of Mixed Strategy Equilibria

• A player chooses his strategy so as to make his
  rival indifferent
• A player earns the same expected payoff for each
  pure strategy chosen with positive probability
• Funny property When a players own payoff from a
  pure strategy goes up (or down), his mixture does
  not change

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Generalized Tennis
Suppose c gt a, b gt d Suppose 1 a gt 1 b, 1 - d
gt 1 c (equivalently b gt a, c gt d)
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Receivers Best Response

• Suppose the sender plays left with probability p,
  then receiver should play left provided
• (1-a)p (1-c)(1-p) gt (1-b)p (1-d)(1-p)
• Or
• p gt (c d)/(c d b a)

22
Senders Best Response

• Same exercise only where the receiver plays left
  with probability q.
• The sender should serve left if
• aq b(1 q) gt cq d(1 q)
• Or
• q lt (b d)/(b d a b)

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Equilibrium

• In equilibrium, both sides are indifferent
  therefore
• p (c d)/(c d b a)
• q (b d)/(b d a b)

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Minmax Equilibrium

• Tennis is a constant sum game
• In such games, the mixed strategy equilibrium is
  also a minmax strategy
• That is, each player plays assuming his opponent
  is out to mimimize his payoff (which he is)
• and therefore, the best response is to maximize
  this minimum.

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Does Game Theory Work?

• Walker and Wooders (2002)
• Ten grand slam tennis finals
• Coded serves as left or right
• Determined who won each point
• Tests
• Equal probability of winning
• Pass
• Serial independence of choices
• Fail

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Battle of the Sexes
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Hawk-Dove
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Wars of Attrition

• Two sides are engaged in a costly conflict
• As long as neither side concedes, it costs each
  side 1 per period
• Once one side concedes, the other wins a prize
  worth V.
• V is a common value and is commonly known by both
  parties
• What advice can you give for this game?

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Pure Strategy Equilibria

• Suppose that player 1 will concede after t1
  periods and player 2 after t2 periods
• Where 0 lt t1 lt t2
• Is this an equilibrium?
• No 1 should concede immediately in that case
• This is true of any equilibrium of this type

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More Pure Strategy Equilibria

• Suppose 1 concedes immediately
• Suppose 2 never concedes
• This is an equilibrium though 2s strategy is not
  credible

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Symmetric Pure Strategy Equilibria

• Suppose 1 and 2 will concede at time t.
• Is this an equilibrium?
• No either can make more by waiting a split
  second longer to concede
• Or, if t is a really long time, better to concede
  immediately

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Symmetric Equilibrium

• There is a symmetric equilibrium in this game,
  but it is in mixed strategies
• Suppose each party concedes with probability p in
  each period
• For this to be an equilibrium, it must leave the
  other side indifferent between conceding and not

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When to concede

• Suppose up to time t, no one has conceded
• If I concede now, I earn t
• If I wait a split second to concede, I earn
• V t e if my rival concedes
• t e if not
• Notice the t term is irrelevant
• Indifference
• (V e) x (f/(1 F)) - e x (1 f/(1-F))
• f/(1 F) 1/V

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Hazard Rates

• The term f/(1 F) is called the hazard rate of a
  distribution
• In words, this is the probability that an event
  will happen in the next moment given that it has
  not happened up until that point
• Used a lot operations research to optimize
  fail/repair rates on processes

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Mixed Strategy Equilibrium

• The mixed strategy equilibrium says that the
  distribution of the probability of concession for
  each player has a constant hazard rate, 1/V
• There is only one distribution with this
  memoryless property of hazard rates
• That is the exponential distribution.
• Therefore, we conclude that concessions will come
  exponentially with parameter V.

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Observations

• Exponential distributions have no upper
  bound---in principle the war of attrition could
  go on forever
• Conditional on the war lasting until time t, the
  future expected duration of the war is exactly as
  long as it was when the war started
• The larger are the stakes (V), the longer the
  expected duration of the war

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Economic Costs of Wars of Attrition

• The expectation of an exponential distribution
  with parameter V is V.
• Since both firms pay their bids, it would seem
  that the economic costs of the war would be 2V
• Twice the value of the item????
• But this neglects the fact that the winner only
  has to pay until the loser concedes.
• One can show that the expected total cost if
  equal to V.

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Big Lesson

• There are no economic profits to be had in a war
  of attrition with a symmetric rival.
• Look for the warning signs of wars of attrition

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Wars of Attrition in Practice

• Patent races
• RD races
• Browser wars
• Costly negotiations
• Brinkmanship

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All-Pay Auctions

• Next consider a situation where expenditures must
  be decided up front
• No one gets back expenditures
• Biggest spender wins a price worth V.
• How much to spend?

41
Pure Strategies

• Suppose you project that your rival will spend
  exactly b lt V.
• Then you should bid just a bit higher
• Suppose you expect your rival will bid b gtV
• Then you should stay out of the auction
• But then it was not in the rivals interest to
  bid b gt v in the first place
• Therefore, there is no equilibrium in pure
  strategies

42
Mixed Strategies

• Suppose that I expect my rival will bid according
  to the distribution F.
• Then my expected payoffs when I bid B are
• V x Pr(Win) B
• I win when B gt rivals bid
• That is, Pr(Win) F(B)

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Best Responding

• My expected payoff is then
• VF(B) B
• Since Im supposed to be indifferent over all B,
  then
• VF(B) B k
• For some constant kgt0.
• This means
• F(B) (B k)/V

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Equilibrium Mixed Strategy

• Recall
• F(B) (B k)/V
• For this to be a real randomization, we need it
  to be zero at the bottom and 1 at the top.
• Zero at the bottom
• F(0) k/V, which means k 0
• One at the top
• F(B1) B1/V 1
• So B1 V

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Putting Things Together

• F(B) B/V on 0 , V.
• In words, this means that each side chooses its
  bid with equal probability from 0 to V.

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Properties of the All-Pay Auction

• The more valuable the prize, the higher the
  average bid
• The more valuable the prize, the more diffuse the
  bids
• More rivals leads to less aggressive bidding
• There is no economic surplus to firms competing
  in this auction
• Easy to see Average bid V/2
• Two firms each pay their bid
• Therefore, expected payment V, the total value
  of the prize.

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Big Lesson

• Wars of attrition and all-pay auctions are a kind
  of disguised form of Bertrand competition
• With equally matched opponents, they compete away
  all the economic surplus from the contest
• On the flipside, if selling an item or setting up
  competition among suppliers, wars of attrition
  and all-pay auctions are extremely attractive.