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LECTURE 3 Mixed strategy Nash equilibrium

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Review The Nash equilibrium is the likely outcome of simultaneous games, both for discrete and continuous sets of actions. Derive the best response functions, find where they intersect. We have considered NE where players select one action with probability 100%  Pure strategies For each action of the Player 2, the best response of Player 1 is a deterministic (i.e. non random) action For each action of the Player 1, the best response of Player 2 is a deterministic action

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Review A Nash equilibrium in which every player plays a pure strategy is a pure strategy Nash equilibrium At the equilibrium, each player plays only one action with probability 1.

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Overview Pure strategy NE is just one type of NE, another type is mixed strategy NE. A player plays a mixed strategy when he chooses randomly between several actions. Some games do not have a pure strategy NE, but have a mixed strategy NE. Other games have both pure strategy NE and mixed strategy NE.

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Employee Monitoring Consider a company where: Employees can work hard or shirk Salary: $100K unless caught shirking Cost of effort: $50K The manager can monitor or not An employee caught shirking is fired Value of employee output: $200K Profit if employee doesn’t work: $0 Cost of monitoring: $10K

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Employee Monitoring No equilibrium in pure strategies What is the likely outcome?

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Football penalty shooting

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Football penalty shooting

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Football penalty shooting No equilibrium in pure strategies Similar to the employee/manager game How would you play this game? Players must make their actions unpredictable Suppose that the goal keeper jumps left with probability p, and jumps right with probability 1-p. What is the kicker’s best response?

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Football penalty shooting If p=1, i.e. if goal keeper always jumps left then we should kick right If p=0, i.e. if goal keeper always jumps right then we should kick left The kicker’s expected payoff is: π(left): -1 x p+1 x (1-p) = 1 – 2p π(right): 1 x p – 1 x (1-p) = 2p – 1  π(left) > π(right) if p

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Football penalty shooting 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 goal keeper?

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Football penalty shooting Mixed strategy: It makes sense for the goal keeper and the kicker to randomize their actions. If opponent knows what I will do, I will always lose! Players try to make themselves unpredictable. Implications: A player chooses his strategy so as to prevent his opponent from having a winning strategy. The opponent has to be made indifferent between his possible actions.

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Employee Monitoring Employee chooses (shirk, work) with probabilities (p,1-p) Manager chooses (monitor, no monitor) with probabilities (q,1-q)

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

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Employee’s Best Response Next, calculate the best strategy for possible strategies of the opponent For q 50 = π (work) For q>1/2: WORK π (shirk) = 100-100q < 50 = π (work) For q=1/2: INDIFFERENT π (shirk) = 100-100q = 50 = π (work) The manager has to monitor just often enough to make the employee work (q=1/2). No need to monitor more than that.

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Manager’s Best Response Manager’s payoff: Monitor: 90(1-p)- 10p=90-100p No monitor: 100(1-p)-100p=100-200p For p1/10: MONITOR π(monitor) = 90-100p > 100-200p = π(no monitor) For p=1/10: INDIFFERENT π(monitor) = 90-100p = 100-200p = π(no monitor) The employee has to work just enough to make the manager not monitor (p=1/10). No need to work more than that.

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Best responses

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Mutual Best Responses

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Equilibrium strategies

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Equilibrium payoffs Employee π (shirk)=0+100x0.5=50 π (work)=50 Manager π (monitor)=0.9x90-0.1x10=80 π (no monitor)=0.9x100-0.1x100=80

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Theorems If there are no pure strategy equilibria, there must be a unique mixed strategy equilibrium. However, it is possible for pure strategy and mixed strategy Nash equilibria to coexist. (for example coordination games)

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New Scenario What if cost of monitoring is 50, (instead of 10)?

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New Scenario To make employee indifferent: π(work)= π(shirk) implies 50=100 – 100q q=1/2 To make manager indifferent π(monitor)= π(no monitor) implies 50-100p = 100-200p p=1/2

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New Scenario Equilibrium: q=1/2, unchanged p=1/2, instead of 1/10 Why does q remain unchanged? Payoff of “shirk” unchanged: the manager must maintain a 50% probability of monitoring to prevent shirking. If q=49%, employees always shirk. Cost of monitoring higher, thus employees can afford to shirk more.  One player’s equilibrium mixture probabilities depend only on the other player’s payoff

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Application: Tax audits Mix strategy to prevent tax evasion: Random audits, just enough to induce people to pay their taxes. In 2002, IRS Commissioner noticed that: Audits have become more expensive Number of audits decreased slightly Offshore evasion increased by $70 billion dollars Recommendation: As audits get more expensive, need to increase budget to keep number of audits constant!

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Do players select the MSNE? Mixed strategies in football Economist Palacios-Huerta analyzed 1,417 penalty kicks. Success matrix: Equilibrium: Kicker: 58q+95(1-q)=93q+70(1-q)  q=42% Goalie: 42p+7(1-p)=5p+30(1-p)  p=38%

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Do players select the MSNE? Mixed strategies in football Observed behavior for the 1,417 penalty kicks: Kickers choose left with probability 40% Prediction was 38% Goalies jump to the left with probability 42% Prediction was 42% Players have the ability to randomize!

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Entry Coordination game Two firms are deciding which new market to enter. Market A is more profitable than market B Coordination game: 2 PSNE, where players enter a different market.

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Entry Coordination game Both player prefer choosing market A and let the other player choose market B. Two PSNE. Expected payoff for Firm 1 when playing A π(A)=2q+4(1-q)=4-2q If it plays B: π(B)=3q+(1-q)=1+2q  π(A)= π(B) if q=3/4

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Entry Coordination game For Firm 2: π(A)= π(B)  p=3/4 Equilibrium in mixed strategies: p=q=3/4 Expected payoff: Firm 1: Same for Firm 2. Expected payoff is 2.5 for both firms Lower than 3 or 4 In this example, pure strategy NE yields a higher payoff. There is a risk of miscoordination where both firms choose the same market.

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In what types of games are mixed strategies most useful? For games of cooperation, there is 1 PSNE, and no MSNE. For games with no PSNE (e.g. shirk/monitor game), there is one MSNE, which is the most likely outcome. For coordination games (e.g. the entry game), there are 2 PSNE and 1 MSNE. Theoretically, all equilibria are possible outcomes, but the difference in expected payoff may induce players to coordinate.

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Weak sense of equilibrium Mixed strategy NE are NE in a weak sense Players have no incentive to change action, but they would not be worse off if they did π(shirk)= π(work) Why should a player choose the equilibrium mixture when the other one is choosing his own?

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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?

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Summary Games may not have a PSNE, and mixed strategies help predict the likely outcome in those situations, e.g. shirk/monitor game. Mixed strategies are also relevant in games with multiple PSNE, e.g. coordination games. Randomization. Make the other player indifferent between his strategies.