Yale 9&10 Mixed Strategies in Theory and Tennis

Overview As I randomize the strategies, the expected payoff is a weighted average of the pure strategies. The mixed strategy payoff must lie “inside” of the other averages. The “support” for the mixed strategy is like a subteam I use. Each of the pure strategies in the mix, must themselves be best responses (to the mixed strategy of the other player). In particular, each must yield the same expected payoff. Otherwise, we would want MORE of the one which has a higher expected value. I know my opponent is mixing, if they weren’t I wouldn’t need to mix. I’d just pick the best for their choice.

How to mix Serena is row player Serena When Serena is mixing properly, the COLUMN player can’t distinguish between its choices. what is p? q(1-q) p50,5060,40 (1-p)90,1020,80

How to mix Serena is row player 50p + 10(1-p) (first column benefit) 40p + 80(1-p) (second column benefit) 50p+10-10p = 40p p 10+40p = 80-40p 80p = 70 p = 7/8 q(1-q) p50,5060,40 (1-p)90,1020,80 50q + 60(1-q) = 90q + 20(1-q) 60-10q = 20+70q 40=80q q=1/2

Serena is row player Serena q =.6 p=.7 Comparative static problem. We had a different table. We were at NE. Before we made changes, Venus was indifferent. After change, would not be indifferent. If we had not changed the way Serena played, Venus would have moved away from improvement. In fact, she would have ONLY shot one way – as it would be better. Thus, Serena had to change her strategy. q(1-q) p50,5080,20 (1-p)90,1020,80

Yale 10 q(1-q) p2,10,0 (1-p)0,01,2 q = 1/3 p = 2/3 Expected utility is 2/3. What is average utility for picking pure strategies? Why is the payoff so bad in mixed equilibrium? Failing to meet all the time. Interpretations of mixing 1.Strategies can be thought of as actually randomizing my choices. 2.Expressions of what people believe in equilibrium, rather than what they are literally doing. This is termed the empirical frequency. 3.Tells what a percent of the people are doing.

Tax audit q=2/3 p=2/7 Aim is to try to deter cheating. How would we change the fraction that cheats? Honest (q) Cheat (1-q) Audit (p)2,04, -10 Not Audit (1-p) 4,00,4

Tax audit Aim is to try to deter cheating. Suppose we increase the penalty for cheating. q=2/3 p=1/6 What did it change? Since we didn’t change the row players payoff, the column players equilibrium mix didn’t change. We did change the amount of auditing we did. We didn’t manage to change the compliance rate. Honest (q) Cheat (1-q) Audit (p)2,0 4, -20 Not Audit (1-p) 4,00,4

How could we increase compliance? Change payoffs for auditor. Make auditing less costly or catching people more reward. If I change the column players payoffs, it changes the row player’s mix. If we increased penalty for cheating and kept audit rate the same, or kept penalty the same but increased audit rate, would THAT increase compliance? Honest 2/3 Cheat 1/3 Audit 1/22,0 4, -20 Not Audit 1/2 4,00,4