Mixed Strategies Overview n Principles of mixed strategy

Mixed Strategies

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

Tennis Anyone R S

Serving R S

Serving R S

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

Game Table Receiver Server Left Right Left ¼ ¾ Right ¾ ¼

Game Table Receiver Server Left Right Left ¼ ¾ Right ¾ ¼ For server: For receiver: Best response to defend left is to serve right Best response to defend right is to serve left Just the opposite

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

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

Calculating Best Responses n Clearly if p = 1, then the receiver should defend to the left n If p = 0, the receiver should defend to the right. n The expected payoff to the receiver is: p x ¾ + (1 – p) x ¼ if defending left n p x ¼ + (1 – p) x ¾ if defending right n n Therefore, she should defend left if n p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾

When to Defend Left n We said to defend left whenever: n p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾ n Rewriting np>1–p n Or n p>½

Receiver’s Best Response Left Right ½ p

Server’s Best Response n Suppose that the receiver goes left with probability q. n Clearly, if q = 1, the server should serve right n If q = 0, the server should serve left. n More generally, serve left if n ¼ x q + ¾ x (1 – q) > ¾ x q + ¼ x (1 – q) n Simplifying, he should serve left if n 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 n A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses n In the tennis example, this occurred when each player chose a 50 -50 mixture of left and right.

General Properties of Mixed Strategy Equilibria n A player chooses his strategy so as to make his rival indifferent n A player earns the same expected payoff for each pure strategy chosen with positive probability n Funny property: When a player’s own payoff from a pure strategy goes up (or down), his mixture does not change

Generalized Tennis Receiver Server Left Right Left a, 1 -a b, 1 -b Right c, 1 -c d, 1 -d Suppose c > a, b > d Suppose 1 – a > 1 – b, 1 - d > 1 – c (equivalently: b > a, c > d)

Receiver’s Best Response n Suppose the sender plays left with probability p, then receiver should play left provided: n (1 -a)p + (1 -c)(1 -p) > (1 -b)p + (1 -d)(1 -p) n Or: n p >= (c – d)/(c – d + b – a)

Sender’s Best Response n Same exercise only where the receiver plays left with probability q. n The sender should serve left if n aq + b(1 – q) > cq + d(1 – q) n Or: n q <= (b – d)/(b – d + a – b)

Equilibrium n In equilibrium, both sides are indifferent therefore: p = (c – d)/(c – d + b – a) n q = (b – d)/(b – d + a – b) n

Minmax Equilibrium n Tennis is a constant sum game n 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) n and therefore, the best response is to maximize this minimum. n

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

Battle of the Sexes Chris Pat Opera Fights Opera 3, 1 0, 0 Fights 0, 0 1, 3

Hawk-Dove Krushchev Kennedy Hawk Dove Hawk 0, 0 4, 1 Dove 1, 4 2, 2

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

Pure Strategy Equilibria n Suppose that player 1 will concede after t 1 periods and player 2 after t 2 periods n Where 0 < t 1 < t 2 n Is this an equilibrium? n No: 1 should concede immediately in that case n This is true of any equilibrium of this type

More Pure Strategy Equilibria n Suppose 1 concedes immediately n Suppose 2 never concedes n This is an equilibrium though 2’s strategy is not credible

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

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

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

Hazard Rates n The term f/(1 – F) is called the hazard rate of a distribution n 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 n Used a lot operations research to optimize fail/repair rates on processes

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

Observations n Exponential distributions have no upper bound---in principle the war of attrition could go on forever n 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 n The larger are the stakes (V), the longer the expected duration of the war

Economic Costs of Wars of Attrition n 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 2 V n Twice the value of the item? ? n n But this neglects the fact that the winner only has to pay until the loser concedes. n One can show that the expected total cost if equal to V.

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

Wars of Attrition in Practice n Patent races n R&D races n Browser wars n Costly negotiations n Brinkmanship

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

Pure Strategies n Suppose you project that your rival will spend exactly b < V. n Then you should bid just a bit higher n Suppose you expect your rival will bid b >=V n Then you should stay out of the auction n But then it was not in the rival’s interest to bid b >= v in the first place n Therefore, there is no equilibrium in pure strategies

Mixed Strategies n Suppose that I expect my rival will bid according to the distribution F. n Then my expected payoffs when I bid B are n V x Pr(Win) – B n I win when B > rival’s bid n That is, Pr(Win) = F(B)

Best Responding n My expected payoff is then: n VF(B) – B n Since I’m supposed to be indifferent over all B, then VF(B) – B = k n For some constant k>=0. n n This means n F(B) = (B + k)/V

Equilibrium Mixed Strategy n Recall n F(B) = (B + k)/V n For this to be a real randomization, we need it to be zero at the bottom and 1 at the top. n Zero at the bottom: n F(0) = k/V, which means k = 0 n One at the top: n F(B 1) = B 1/V = 1 n So B 1 = V

Putting Things Together n F(B) = B/V on [0 , V]. n In words, this means that each side chooses its bid with equal probability from 0 to V.

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

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