Mixed Strategies

Mixed Strategies

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

Tennis Anyone R S

Serving R S

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

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 • 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

Receiver’s Best Response Left Right ½ p

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

Server’s Best Response q ½ Right Left

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)

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 • 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

Battle of the Sexes

Hawk-Dove

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.

Mixed Strategies