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**Mixed Strategies For Managers**

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Overview 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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Outline 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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**costly auditing (waste)**

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

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**Best response of the taxpayer**

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

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**Best response of the Fiscal Authorithy**

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

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**Best responses do not coincide:**

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

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**Example: to work or not to work…**

Players Employee Work Shirk Manager Monitor Do not monitor

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Payoffs 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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**The payoff matrix Manager Employee Monitor No monitor Monitor**

Work Employee Shirk

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**The payoff matrix Manager Employee Monitor No monitor 50 , 90 50 , 100**

0 , -10 100 , -100 Manager Monitor No Monitor Work Employee Shirk

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**Best response of the employee**

Monitor No monitor 50 , 90 50 , 100 0 , -10 100 , -100 Manager Monitor No Monitor Work Employee Shirk

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**Best response of the manager**

Monitor No monitor 50 , 90 50 , 100 0 , -10 100 , -100 Manager Monitor No Monitor Work Employee Shirk

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**No Nash equilibrium in pure strategies**

Monitor No monitor 50 , 90 50 , 100 0 , -10 100 , -100 Manager Monitor No Monitor Work Employee Shirk

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

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**The Idea Mixed Strategies**

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.

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**Computing mixed strategies**

Manager Monitor No monitor Employee Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 q 1q p 1p Suppose that: The employee chooses to work with probability p (and shirk with 1p) The manager chooses to monitor with probability q (and no monitor with 1q)

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**The manager’s perspective: how can I avoid shirking?**

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

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**1. Expected payoff of the employee**

Mixed Strategies Manager Monitor No monitor Employee Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 q 1q Expected payoff from working: Expected payoff from shirking: (50 x q) + (50 x (1q))= 50 (0 x q) + (100 x (1q))= 100100q

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**2. The employee’s best response**

Mixed Strategies 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]

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**2. The employee’s best response**

Expected payoff from working: 50 Expected payoff from shirking:100100q Recap: 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

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**2. The employee’s best response**

Mixed Strategies 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).

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**Not done yet… Mixed Strategies**

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.

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**The employee’s perspective: follow the same steps**

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

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**1. Expected payoff of the manager**

Mixed Strategies Manager Monitor No monitor Employee Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 p 1p Expected payoff from monitoring: Expected payoff from not monitoring: (90 x p) + (-10 x (1p))= 100p 10 (100 x p) + (-100 x (1p))= 200p100

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**2. The manager’s best response**

Mixed Strategies 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]

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**2. The manager’s best response**

Expected payoff from monitoring: 100p 10 Expected payoff from not monitoring:200p100 Recap: 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 E. P. of monitoring = E.P. of no monitoring if p =9/10

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**2. The manager’s best response**

Mixed Strategies 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”).

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**Nash equilibrium in mixed strategies**

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.

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**Nash equilibrium in mixed strategies**

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

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**Nash equilibrium in mixed strategies**

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

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**Nash equilibrium in mixed strategies**

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

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**Nash equilibrium in mixed strategies**

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

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**Nash equilibrium in mixed strategies**

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

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**Nash equilibrium in mixed strategies**

1 shirk work p Probability of working 1/2 1 q Probability of monitoring

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**Nash equilibrium in mixed strategies**

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

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**Nash equilibrium in mixed strategies**

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

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**Equilibrium Payoffs: the employee**

Mixed Strategies Manager Monitor No monitor Employee Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 1/2 1/2 9/10 1/10 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

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**Equilibrium Payoffs: the manager**

Mixed Strategies Manager Monitor No monitor Employee Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 1/2 1/2 9/10 1/10 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

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**What if cost of monitoring was 50 (instead of 10)?**

Mixed Strategies Initial Payoff Matrix Manager Monitor No monitor Employee Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 New Payoff Matrix Manager Monitor No monitor Employee Work 50 , . . . 50 , 100 Shirk 0 , . . . 100 , -100 50 -50

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**A change in the manager’s payoffs**

Mixed Strategies New Payoff Matrix Manager Monitor No monitor Employee Work 50 , 50 50 , 100 Shirk 0 , -50 100 , -100 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)

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**Properties of mixed strategy equilibria**

Mixed Strategies 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.

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