Mixed Strategies

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# Mixed Strategies - PowerPoint PPT Presentation

Mixed Strategies. Overview. Principles of mixed strategy equilibria Wars of attrition All-pay auctions. Tennis Anyone. R. S. Serving. R. S. Serving. R. S. The Game of Tennis. Server chooses to serve either left or right Receiver defends either left or right

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### Mixed Strategies

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

For server: Best response to defend left is to serve right

Best response to defend right is to serve left

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?
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 receiver’s best response?
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 > p x ¼ + (1 – p) x ¾
When to Defend Left
• We said to defend left whenever:
• p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾
• Rewriting
• p > 1 – p
• Or
• p > ½
Server’s 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) > ¾ x q + ¼ x (1 – q)
• Simplifying, he should serve left if
• q < ½
Putting Things Together

R’s best

response

q

S’s best

response

½

1/2

p

Equilibrium

R’s best

response

q

Mutual best responses

S’s best

response

½

1/2

p

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.
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 player’s own payoff from a pure strategy goes up (or down), his mixture does not change
Generalized Tennis

Suppose c > a, b > d

Suppose 1 – a > 1 – b, 1 - d > 1 – c

(equivalently: b > a, c > d)

• Suppose the sender plays left with probability p, then receiver should play left provided:
• (1-a)p + (1-c)(1-p) > (1-b)p + (1-d)(1-p)
• Or:
• p >= (c – d)/(c – d + b – a)
Sender’s Best Response
• Same exercise only where the receiver plays left with probability q.
• The sender should serve left if
• aq + b(1 – q) > cq + d(1 – q)
• Or:
• q <= (b – d)/(b – d + a – b)
Equilibrium
• In equilibrium, both sides are indifferent therefore:
• p = (c – d)/(c – d + b – a)
• q = (b – d)/(b – d + a – b)
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.
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
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?
Pure Strategy Equilibria
• Suppose that player 1 will concede after t1 periods and player 2 after t2 periods
• Where 0 < t1 < t2
• Is this an equilibrium?
• No: 1 should concede immediately in that case
• This is true of any equilibrium of this type
More Pure Strategy Equilibria
• Suppose 1 concedes immediately
• Suppose 2 never concedes
• This is an equilibrium though 2’s strategy is not credible
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
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
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
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
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.
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
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.
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
Wars of Attrition in Practice
• Patent races
• R&D races
• Browser wars
• Costly negotiations
• Brinkmanship
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?
Pure Strategies
• Suppose you project that your rival will spend exactly b < V.
• Then you should bid just a bit higher
• Suppose you expect your rival will bid b >=V
• Then you should stay out of the auction
• But then it was not in the rival’s interest to bid b >= v in the first place
• Therefore, there is no equilibrium in pure strategies
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 > rival’s bid
• That is, Pr(Win) = F(B)
Best Responding
• My expected payoff is then:
• VF(B) – B
• Since I’m supposed to be indifferent over all B, then
• VF(B) – B = k
• For some constant k>=0.
• This means
• F(B) = (B + k)/V
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
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.
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.
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.