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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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overview
Overview
  • Principles of mixed strategy equilibria
  • Wars of attrition
  • All-pay auctions
the game of tennis
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 table8
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

nash equilibrium
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
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
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
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
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
Putting Things Together

R’s best

response

q

S’s best

response

½

1/2

p

equilibrium
Equilibrium

R’s best

response

q

Mutual best responses

S’s best

response

½

1/2

p

mixed strategy equilibrium
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
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
Generalized Tennis

Suppose c > a, b > d

Suppose 1 – a > 1 – b, 1 - d > 1 – c

(equivalently: b > a, c > d)

receiver s best response21
Receiver’s Best Response
  • 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
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)
equilibrium23
Equilibrium
  • In equilibrium, both sides are indifferent therefore:
    • p = (c – d)/(c – d + b – a)
    • q = (b – d)/(b – d + a – b)
minmax equilibrium
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
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
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
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
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
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
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
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
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 equilibrium35
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
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
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
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
Wars of Attrition in Practice
  • Patent races
  • R&D races
    • Browser wars
  • Costly negotiations
  • Brinkmanship
all pay auctions
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
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 strategies42
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
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
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 together45
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
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 lesson47
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.