Game Theory “I used to think I was indecisive – but now I’m not so sure.” - Anonymous Topic 4 Mixed Strategies

Review Predicting likely outcome of a game Sequential games  Look forward and reason back Simultaneous games  Look for simultaneous best replies What if (seemingly) there are no equilibria? Mike Shor 2

Employee Monitoring Employees can work hard or shirk  Salary: $100K unless caught shirking  Cost of effort: $50K Managers can monitor or not  Value of employee output: $200K  Profit if employee doesn’t work: $0  Cost of monitoring: $10K Mike Shor 3

Employee Monitoring Best replies do not correspond No equilibrium in pure strategies What do the players do? Mike Shor 4 Manager MonitorNo Monitor Employee Work 50, 90 50, 100 Shirk 0, , -100

Employee Monitoring John Nash proved:  Every finite game has a Nash equilibrium So, if there is no equilibrium in pure strategies, we have to allow for mixing or randomization Mike Shor 5

Mixed Strategies Unreasonable predictors of one-time interaction Reasonable predictors of long-term proportions Mike Shor 6

Game Winning Goal Mike Shor 7

Soccer Penalty Kicks (Six Year Olds Version) Mike Shor 8 G O A L I E L R KICKERKICKER L R -1, 1 1, -1 -1, 1

Soccer Penalty Kicks There are no mutual best responses  Seemingly, no equilibria How would you play this game? What would you do if you know that the goalie jumps left 75% of the time? Mike Shor 9

Probabilistic Soccer Allow the goalie to randomize Suppose that the goalie jumps left p proportion of the time What is the kicker’s best response? If p=1, goalie always jumps left  we should kick right If p=0, goalie always jumps right  we should kick left Mike Shor 10

Probabilistic Soccer (continued) The kicker’s expected payoff is:  Kick left: -1 x p+1 x (1-p) = 1 – 2p  Kick right: 1 x p -1 x (1-p) = 2p – 1  should kick left if: p < ½ (1 – 2p > 2p – 1)  should kick right if: p > ½  is indifferent if: p = ½ What value of p is best for the goalie? Mike Shor 11

Probabilistic Soccer (continued) Mike Shor 12 Goalie’s p Kicker’s strategy Goalie’s Payoff L (p = 1)R R (p = 0)L p = 0.75R-0.5 p = 0.25L-0.5 p = 0.55R-0.1 p = 0.50Either 0

Probabilistic Soccer (continued) Mixed strategies:  If opponent knows what I will do, I will always lose!  Randomizing just right takes away any ability for the opponent to take advantage  If opponent has a preference for a particular action, that would mean that they had chosen the worst course from your perspective.  Make opponent indifferent between her strategies Mike Shor 13

Mixed Strategies Strange Implications  A player chooses his strategy so as to make her opponent indifferent  If done right, the other player earns the same payoff from either of her strategies Mike Shor 14

Mixed Strategies Mike Shor 15 COMMANDMENT Use the mixed strategy that keeps your opponents guessing.

Employee Monitoring Suppose:  Employee chooses (shirk, work) with probabilities (p,1-p)  Manager chooses (monitor, no monitor) with probabilities (q,1-q) Mike Shor 16 Manager MonitorNo Monitor Employee Work 50, 90 50, 100 Shirk 0, , -100

Keeping Employees from Shirking First, find employee’s expected payoff from each pure strategy If employee works: receives 50 Profit(work)= 50  q + 50  (1-q) = 50 If employee shirks: receives 0 or 100 Profit(shirk)= 0  q  (1-q) = 100 – 100q Mike Shor 17

Employee’s Best Response Next, calculate the best strategy for possible strategies of the opponent For q<1/2: SHIRK Profit (shirk) = q > 50 = Profit(work) For q>1/2: WORK Profit (shirk) = q < 50 = Profit(work) For q=1/2: INDIFFERENT Profit(shirk) = q = 50 = Profit(work) Mike Shor 18

Manager’s Equilibrium Strategy Employees will shirk if q<1/2 To keep employees from shirking, must monitor at least half of the time Note: Our monitoring strategy was obtained by using employees’ payoffs Mike Shor 19

Manager’s Best Response Monitor: 90  (1-p) - 10  p No monitor: 100  (1-p) -100  p For p<1/10: NO MONITOR monitor = p < p = no monitor For p>1/10: MONITOR monitor = p > p = no monitor For p=1/10: INDIFFERENT monitor = p = p = no monitor Mike Shor 20

Cycles Mike Shor 21 q 01 1/2 p 0 1/10 1 shirk work monitorno monitor

Mutual Best Responses Mike Shor 22 q 01 1/2 p 0 1/10 1 shirk work monitorno monitor

Equilibrium Payoffs Mike Shor 23 1/2 Monitor 1/2 No Monitor 9/10 Work 50, 9050, 100 1/10 Shirk 0, , -100

Solving Mixed Strategies Seeming random is too important to be left to chance!  Determine the probability mix for each player that makes the other player indifferent between her strategies Assign a probability to one strategy (e.g., p) Assign remaining probability to other strategy Calculate opponent’s expected payoff from each strategy Set them equal Mike Shor 24

New Scenario What if cost of monitoring were 50? Mike Shor 25 Manager MonitorNo Monitor Employee Work 50, 5050, 100 Shirk 0, , -100

New Scenario To make employee indifferent: Mike Shor 26

Real Life? Sports  Football  Tennis  Baseball Law Enforcement  Traffic tickets Price Discrimination  Airline stand-by policies Policy compliance  Random drug testing Mike Shor 27

IRS Audits 1997  Offshore evasion compliance study  Calibrated random audits 2002 IRS Commissioner Charles Rossotti:  Audits more expensive now than in ’97  Number of audits decreased slightly  Offshore evasion alone increased to $70 billion dollars! Recommendation: As audits get more expensive, need to increase budget to keep number of audits constant! Mike Shor 28

Law Enforcement Motivate compliance at lower monitoring cost  Audits  Drug Testing  Parking  Should punishment fit the crime? Mike Shor 29

Football You have a balanced offense Equilibrium:  run half of the time  defend the run half of the time Mike Shor 30 Defense RunPass Offense Run 0, 05, -5 Pass 5, -50, 0

Football You have a balanced offense The run now works better than before What is the equilibrium? Mike Shor 31 Defense RunPass Offense Run 1, -18, -8 Pass 5, -50, 0

Effects of Payoff Changes Direct Effect:  The player benefitted should take the better action more often Strategic Effect:  Opponent defends against my better strategy more often, so I should take the action less often Mike Shor 32

Mixed Strategy Examples Market entry Stopping to help All-pay auctions Mike Shor 33

Market Entry N potential entrants into market  Profit from staying out: 10  Profit from entry: 40 – 10 m  m is the number that enter Symmetric mixed strategy equilibrium:  Earn 10 if stay out. Must earn 10 if enter! Mike Shor 34

Stopping to Help N people pass a stranded motorist Cost of helping is 1 Benefit of helping is B > 1  i.e., if you are the only one who could help, you would, since net benefit is B-1 > 0 Symmetric Equilibria  p is the probability of stopping  Help:B-1  Don’t help:B x chance someone stops Mike Shor 35

Stopping to Help (continued) Don’t help:  B x chance someone stops =B x ( 1 – chance no one stops ) = B x ( 1 – (1-p) N-1 ) Set help = Don’t help  B x ( 1 – (1-p) N-1 ) = B – 1  p = 1 - (1/B) 1/(N-1) Mike Shor 36

Probability of Stopping Mike Shor 37 B=2

Probability of Someone Stopping Mike Shor 38

All-Pay Auctions Players decide how much to spend Expenditures are sunk Biggest spender wins a prize worth V How much would you spend? Mike Shor 39

Pure Strategy Equilibria? Suppose rival spends s < V  Then you should spend just a drop higher  Then rival will also spend a drop higher Suppose rival spends s ≥ V  Then you should spend 0  Then rival should spend a drop over 0 No equilibrium in pure strategies Mike Shor 40

Mixed Strategies We need a probability of each amount Use a distribution function F  F(s) is the probability of spending up to s Imagine I spend s  Profit: V x Pr{win} – s = V x F(s) – s Mike Shor 41 εε

Mixed Strategies For an equilibrium, I must be indifferent between all of my strategies V x F(s) – s = V x F(s’) – s’ for any s, s’ What about s=0?  Probability of winning = 0  So V x 0 – 0 = 0 V x F(s) – s = 0 F(s) = s/V Mike Shor 42

Mixed Strategies F(s) = s/V on [0,V]  This implies that every amount between 0 and V is equally likely Expected bid = V/2 Expected payment = V There is no economic surplus to firms competing in this auction Mike Shor 43

All-Pay Auctions Patent races Political contests Wars of attrition Lesson: With equally-matched opponents, all economic surplus is competed away If running the competition: all-pay auctions are very attractive Mike Shor 44

Mixed Strategies in Tennis Study:  Ten grand slam tennis finals  Coded serves as left or right  Determined who won each point Found:  All serves have equal probability of winning  But: serves are not temporally independent Mike Shor 45

What Random Means Study:  A fifteen percent chance of being stopped at an alcohol checkpoint will deter drinking and driving Implementation  Set up checkpoints one day a week (1 / 7 ≈ 14%)  How about Fridays? Mike Shor 46

Exploitable Patterns Mike Shor 47 CAVEAT Use the mixed strategy that keeps your opponents guessing. BUT Your probability of each action must be the same period to period.

Exploitable Patterns Manager’s strategy of monitoring half of the time must mean that there is a 50% chance of being monitored every day! Cannot just monitor every other day. Humans are very bad at this. Exploit patterns! Mike Shor 48