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Mixed Strategies For Managers

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Dominant and dominated strategies Dominant strategy equilibrium Prisoners’ dilemma Nash equilibrium in pure strategies Games with multiple Nash equilibria Equilibrium selection Games with no pure strategy Nash equilibria Mixed strategy Nash equilibrium

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Games with no pure strategy Nash equilibrium Mixed Strategies What is the idea? How do we compute them? Mixed strategies in practice Examples Evidence from football penalty kicks Minimax strategies in zero-sum games

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Fiscal Authority Taxpayer Cheat Don’t cheat AuditDon’t audit pays low taxes gets punished pays low taxes pays high taxes costly auditing costly auditing (waste) low tax revenue high tax revenue Mixed strategies are strategies that involve randomization.

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Fiscal Authority Taxpayer Cheat Don’t cheat AuditDon’t audit pays low taxes gets punished pays low taxes pays high taxes costly auditing costly auditing (waste) low tax revenue high tax revenue

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Fiscal Authority Taxpayer Cheat Don’t cheat AuditDon’t audit pays low taxes gets punished pays low taxes pays high taxes costly auditing costly auditing (waste) low tax revenue high tax revenue

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Fiscal Authority Taxpayer Cheat Don’t cheat AuditDon’t audit pays low taxes gets punished pays low taxes pays high taxes costly auditing costly auditing (waste) low tax revenue high tax revenue No Nash equilibrium in pure strategies

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Players Employee Work Shirk Manager Monitor Do not monitor

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The employee Salary: $100K unless caught shirking Cost of effort: $50K The manager Value of the employee output: $200K Profit if the employee doesn’t work: $0 Cost of monitoring: $10K

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MonitorNo monitor Employee Manager Work Shirk MonitorNo Monitor

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MonitorNo monitor 50, 9050, 100 0, , -100 Employee Manager Work Shirk MonitorNo Monitor

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MonitorNo monitor 50, 9050, 100 0, , -100 Employee Manager Work Shirk MonitorNo Monitor

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MonitorNo monitor 50, 9050, 100 0, , -100 Employee Manager Work Shirk MonitorNo Monitor

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MonitorNo monitor 50, 9050, 100 0, , -100 Employee Manager Work Shirk MonitorNo Monitor

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(1) What is the idea? (2) How do we compute mixed strategies?

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The idea is to prevent the other player to anticipate my strategy. Randomizing “just right” takes away any ability to be taken advantage of. Just right: Making other player indifferent to her strategies. Mixed Strategies

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Manager MonitorNo monitor Employee Work50, 9050, 100 Shirk0, , -100 Suppose that: The employee chooses to work with probability p (and shirk with 1 p) The manager chooses to monitor with probability q (and no monitor with 1 q) Mixed Strategies q 1q1q p 1p1p

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1. Calculate the employee’s expected payoff. 2. Find out his best response to each possible strategy of the manager. Mixed Strategies

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Manager MonitorNo monitor Employee Work50, 9050, 100 Shirk0, , -100 Mixed Strategies Expected payoff from working: Expected payoff from shirking: q 1q1q (50 x q)+ (50 x (1 q))= 50 (0 x q) + (100 x (1 q))= 100 100q

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What is the employee’s best response for all possible strategies of the manager? The manager’s possible strategies: q=0, q=0.1, …, q=0.5,..., q=1 Technically, q [0,1] Mixed Strategies

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Expected payoff from working: 50 Expected payoff from shirking:100 100q E. P. working > E.P. of shirking 50 > 100 – 100q if q >1/2 E. P. working < E.P. of shirking 50 < 100 – 100q if q <1/2 E. P. working = E.P. of shirking if q =1/2 Recap:

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Best response to all q >1/2: Work Best response to all q <1/2: Shirk Best response to q=1/2: Work or Shirk (i.e., the employee is indifferent) If you want to keep the employee from shirking, you should set q >1/2 (i.e., monitor more than half of the time). Mixed Strategies

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All this was from the Manager’s perspective; she wants to determine the best q to induce the Employee not to shirk. To do so, she tried to figure out how the employee would respond to different q. Now look at things from the Employee’s perspective. The employee will also try to determine the best p. Mixed Strategies

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1. Calculate the manager’s expected payoff. 2. Find out her best response to each possible strategy of the employee. Mixed Strategies

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Manager MonitorNo monitor Employee Work50, 9050, 100 Shirk0, , -100 Mixed Strategies Expected payoff from monitoring: Expected payoff from not monitoring: p 1p1p (90 x p)+ (-10 x (1 p))= 100p 10 (100 x p) + (-100 x (1 p))= 200p 100

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What is the manager’s best response for all possible strategies of the employee? The employee’s possible strategies: p=0, p=0.1, …, p=0.5,..., p=1 Technically, p [0,1] Mixed Strategies

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Expected payoff from monitoring: 100p 10 Expected payoff from not monitoring:200p 100 E. P. of monitoring > E.P. of no monitoring 100p-10 > 200p – 100 if p <9/10 E. P. of monitoring < E.P. of no monitoring 100p-10 > 200p – 100 if p >9/10 E. P. of monitoring = E.P. of no monitoring if p =9/10 Recap:

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Best response to all p <9/10: Monitor Best response to all p >9/10: No monitor Best response to p=9/10: Monitor or No Monitor (i.e., the manager is indifferent) If you want keep the manager from monitoring, you should set p > 9/10 (work “most of the time”). Mixed Strategies

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The employer works with probability 9/10 and shirks with probability 1/10. The manager monitors with probability ½ and does not monitor with probability ½. Mixed Strategies

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0 1 1 q p Probability of monitoring Probability of working Can this be an equilibrium? 1/4 1/3

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0 1 1 q p Probability of monitoring Probability of working What is the employee’s best response to q =1/4? 1/4 1/3 Shirk! ( Shirk if q <1/2 )

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0 1 1 q p Probability of monitoring Probability of working 1/4 Can this be an equilibrium?

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0 1 1 q p Probability of monitoring Probability of working 1/4 What is the manager’s best response to p =0 (shirk)? Monitor! ( Monitor if p <9/10 )

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0 1 1 q p Probability of monitoring Probability of working Can this be an equilibrium?

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0 1 1 q p Probability of monitoring Probability of working 1/2 shirkwork

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0 1 1 q p Probability of monitoring Probability of working monitor no monitor 9/10

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0 1 1 q p Probability of monitoring Probability of working 1/2 shirkwork monitor no monitor 9/10 The employee is Indifferent between “work” and “shirk” The manager is Indifferent between “monitor” and “no monitor” Unique N.E. in mixed strategies

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Manager MonitorNo monitor Employee Work50, 9050, 100 Shirk0, , -100 Mixed Strategies Expected payoff from working: (50 x ½ ) + (50 x ½ ) = 50 Expected payoff from shirking: (0 x ½ ) + (100 x ½ ) = 50 Gets (50 x 9/10) + (50 x 1/10) = 50 9/10 1/10 1/2

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Manager MonitorNo monitor Employee Work50, 9050, 100 Shirk0, , -100 Mixed Strategies Expected payoff from monitoring: (90 x 9/10 ) + (-10 x 1/10) = 80 Expected payoff from no monitoring: (100 x 9/10 ) + (-100 x 1/10 ) = 80 Gets (80 x 1/2) + (80 x 1/2) = 80 9/10 1/10 1/2

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Initial Payoff Matrix Manager MonitorNo monitor Employee Work50, 9050, 100 Shirk0, , -100 Mixed Strategies New Payoff Matrix Manager MonitorNo monitor Employee Work50,...50, 100 Shirk0,...100,

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Mixed Strategies Which player’s equilibrium strategy will change? The employee’s equilibrium strategy: “Work with probability ½ and shirk with probability ½” (As opposed to “work with probability 9/10 …” with a less expensive monitoring technology) New Payoff Matrix Manager MonitorNo monitor Employee Work50, 5050, 100 Shirk0, , -100

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A player chooses his strategy so as to make his rival indifferent. As a player, you want to prevent others from exploiting any systematic behavior of yours. A player earns the same expected payoff for each pure strategy chosen with positive probability. When a player’s own payoff from a pure strategy changes (e.g., more costly monitoring), his mixture does not change but his opponent’s does. Mixed Strategies

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