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Applied Game Theory

Applied Game Theory. Lecture 8 Chapter 7-Mixed Strategies in 2x2 Games. Last Time. Relationship between Sequential and Simultaneous Move Games Simultaneous Game as a Sequential game Information Set Sequential Game as a Simultaneous game Undesirable Nash Equilibrium

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Applied Game Theory

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  1. Applied Game Theory Lecture 8 Chapter 7-Mixed Strategies in 2x2 Games

  2. Last Time • Relationship between • Sequential and Simultaneous Move Games • Simultaneous Game as a Sequential game • Information Set • Sequential Game as a Simultaneous game • Undesirable Nash Equilibrium • Rollback is a “refinement” of Nash Equilibrium • Picks most “reasonable” Nash Equilibrium

  3. Last Time Simultaneous Battle of the Sexes Game Nash Equilibria 2 B O B 1 O

  4. Last Time Nash Equilibria of Sequential game (with 1 going first) 2 B if B O if O O if B B if O O if B O if O B if B B if O B 1 O

  5. Last Time Nash Equilibria of Sequential game (with 1 going first) 2 B if B O if O O if B B if O O if B O if O B if B B if O B 1 O

  6. Last Time Nash Equilibria of Sequential game (with 1 going first) 2 B if B O if O O if B B if O O if B O if O B if B B if O B 1 O

  7. Last Time “Best” Nash Equilibrium 1: B 2: B if B O if O 2,1 B 2 B 0,0 O 1 B 0,0 O 1,2 O

  8. Last Time 2 not optimal if 1 plays O 1: B 2: B if B B if O 2,1 B 2 B 0,0 O 1 B 0,0 O 1,2 O

  9. Last Time 2 not optimal if 1 plays B 1: O 2: O if B O if O 2,1 B 2 B 0,0 O 1 B 0,0 O 1,2 O

  10. Last Time • Strategic form of Sequential Battle of Sexes • 3 Nash Equilibria • Two were sustained by suboptimality off-the-equilibrium path • Rollback picked the “best” Nash Equilibrium

  11. Mixed Strategies and the Simpsons • Lisa and Bart both want to be the first author • Rock, Scissors, Paper deemed to be best way of determining • Constant- (or Zero-) sum game • One person will be first • One person will be second • Bart’s poor use of mixed strategies hurts him • If you have any objections to violent cartoons • Feel free to step outside

  12. What is a Mixed Strategy • Perhaps if Bart will play Rock, Scissors and Paper • with some probability • can keep Lisa from systematically beating him • Perhaps if Evert and Nav. play each strategy DL or CC • with some probability • there will be a mutual best response

  13. Mixed Strategies • Only Pure Strategies to this point • Playing a particular strategy with certainty • Today we study Mixed Strategies • Playing a particular strategy with some probability • Solve for Nash Equilibrium in Mixed Strategies • Interpretation of the equilibrium

  14. What is a Mixed Strategy • Some games have no Nash Equilibrium using the tools we have developed Nav. DL CC DL Evert CC

  15. What is a Mixed Strategy • Some games have no Nash Equilibrium using the tools we have developed No Nash Equilibrium in Pure Strategies!!! Perhaps if we play strategies with some probability? Nav. DL CC DL Evert CC

  16. What is a Mixed Strategy Evert might play DL with probability p Evert might play CC with probability 1-p Nav. DL CC p DL Evert CC 1-p

  17. What is a Mixed Strategy Nav. might play DL with probability q Nav. might play CC with probability 1-q Nav. q 1-q DL CC p DL Evert CC 1-p

  18. What is a Mixed Strategy p and q are between 0 and 1 The values at exactly 0 or 1 represent pure strategies Nav. q 1-q DL CC p DL Evert CC 1-p

  19. Expected Payoffs • Since we are allowing players to select a strategy with some probability • important to recall expected utility • Suppose that Nav. plays DL with certainty • q=1 • Evert’s payoff for selecting p=0.5 is: • 0.5(50)+0.5(90) • =25+45 • =70 Nav. q 1-q DL CC p 0.5 DL Evert 0.5 CC 1-p

  20. Nash Equilibrium • Each player selects strategy • possibly with some probability • which is a mutual best response • At least one Nash Equilibrium • will exist in any game • Solving will be somewhat similar to the continuous strategy games • Allowing for mixed strategies • can find Nash Equilibrium in tennis game

  21. Best Response Analysis Given Evert’s p-mix Nav’s Payoffs are: Nav’s Payoffs given Nav’s actions DL: 50p+10(1-p) CC 80 CC: 20p+80(1-p) DL 50 Nav. 20 10 DL CC p p=0 p=1 DL Evert CC 1-p

  22. Best Response Analysis Given Evert’s p-mix Nav’s Payoffs are: Nav’s Payoffs given Nav’s actions DL: 50p+10(1-p) CC 80 CC: 20p+80(1-p) DL 50 If Nav mixes then between CC and DL line Nav. 20 10 DL CC p p=0 p=1 DL Evert CC 1-p

  23. Best Response Analysis Nav’s Payoffs given Nav’s actions Where do these intersect? CC 50p+10(1-p) =20p+80(1-p) 80 DL 30p=70(1-p) 50 30p=70-70p 20 100p=70 10 p=0.70 p=0 p=0.7 p=1

  24. Best Response Analysis Nav’s Payoffs given Nav’s actions Nav’s Payoffs? CC DL: 50p+10(1-p) 80 =50(0.7)+10(0.3) DL =35+3 50 =38 38 20 10 CC: 20p+80(1-p) =20(0.7)+80(0.3) p=0 p=0.7 p=1 =14+24 =38

  25. Best Response Analysis Given p what is best for Nav? Nav’s Payoffs given Nav’s actions CC then pure CC If p<0.7 80 DL 50 38 If p>0.7 then pure DL 20 10 then DL and CC do just as well If p=0.7 p=0 p=0.7 p=1

  26. Evert’s Payoffs given Nav’s Actions and Evert’s p-mix Best Response Analysis Everts Payoffs given Nav’s actions Nav plays DL, Evert gets: DL CC 90 50p+90(1-p) 80 Nav plays CC, Evert gets: 80p+20(1-p) 50 Nav. 20 DL CC p p=0 p=1 DL Evert 1-p CC

  27. Evert’s Payoffs given Nav’s Actions and Evert’s p-mix Best Response Analysis Everts Payoffs given Nav’s actions Where are they equal? DL CC 90 50p+90(1-p) =80p+20(1-p) 80 70(1-p)=30p 70-70p=30p 50 70=100p 20 p=0.7 p=0 p=0.7 p=1

  28. Best Response Analysis Evert’s Payoffs from p=0.7 if Nav plays DL Everts Payoffs given Nav’s actions DL: 50p+90(1-p) =50(0.7)+90(0.3) DL CC 90 =35+27 80 =62 Evert’s Payoffs from p=0.7 if Nav plays CC 62 50 CC: 80p+20(1-p) 20 =80(0.7)+20(0.3) =56+6 p=0 p=0.7 p=1 =62 Note 62 is 100-38 Constant-Sum Game

  29. Best Response Analysis Given Nav’s q-mix Evert’s Payoffs are: Evert’s Payoffs given Evert’s actions DL: 50q+80(1-q) CC DL 90 CC: 90q+20(1-q) 80 50 Nav. 20 q 1-q DL CC q=0 q=1 DL Evert CC

  30. Best Response Analysis Given Nav’s q-mix Evert’s Payoffs are: Evert’s Payoffs given Evert’s actions DL: 50q+80(1-q) CC DL 90 CC: 90q+20(1-q) 80 If Evert mixes then between CC and DL line 50 Nav. 20 q 1-q DL CC q=0 q=1 DL Evert CC

  31. Best Response Analysis Where do they intersect? Evert’s Payoffs given Evert’s actions CC 50q+80(1-q) =90q+20(1-q) DL 90 60(1-q)=40q 80 60-60q=40q 60=100q 50 20 q=0.6 q=0 q=1 q=0.6

  32. Best Response Analysis Given q what is best for Evert? Evert’s Payoffs given Evert’s actions CC If q>0.6 then pure CC DL 90 80 If q<0.6 then pure DL then DL and CC do just as well If q=0.6 50 20 q=0 q=1 q=0.6

  33. Best Response Analysis Nav’s Best Response If p<0.7 then pure CC q Nav’s BR: q=0 If p>0.7 then pure DL Nav’s BR: q=1 If p=0.7 then DL and CC do just as well p p=0.7

  34. Best Response Analysis Evert’s Best Response If q>0.6 then pure CC q Evert’s BR: p=0 If q<0.6 then pure DL q=0.6 Evert’s BR: p=1 If q=0.6 then DL and CC do just as well p

  35. Best Response Analysis Nash Equilibrium is where Nav’s Best Response q Evert hits the ball Down the Line with probability 0.7 q=0.6 and Nav. Defends the Down the Line shot with probability 0.6 Evert’s Best Response No pure strategy Nash Equilibrium p p=0.7

  36. Keep Opponent Guessing • The Nash Equilibrium is where • each player makes their opponent indifferent between strategies • If Evert picked any other strategy • then Nav. would no longer be indifferent between strategies • Only equilibrium is where • opponents keep each other guessing

  37. Minimax Method • Since tennis game zero-sum • when one player is indifferent • the other plays is also • Gain for one player • is loss for the other • Doing the best for one player • equivalent to doing the worst for the other • In Lecture 5 • if maximin is different from minimax • then no Nash Equilibrium in pure strategies

  38. Maximin Nav. Minimizes Evert’s action Evert picks the largest of these minimizations Maximin Nav. DL CC DL Min=50 Evert Min=20 CC

  39. Minimax Evert Maximizes Nav’s action Nav. picks the smallest of these maximizations Nav. DL CC DL Evert CC Max =80 Max =90 Minimax

  40. Minimax Since the Maximin does not equal the Minimax there is no Nash Equilibrium in Pure Strategies Maximin Nav. DL CC DL Min=50 Evert Min=20 CC Max =80 Max =90 Minimax

  41. Maximin For any p selected by Evert, Everts Payoffs given Nav’s actions Nav will select the strategy which does the worst for Evert DL CC 90 For p<0.7, 80 Nav will select CC For p>0.7, 50 Nav will select DL 20 Given these minimizations, Evert will pick the strategy which gives the highest payoff p=0 p=0.7 p=1 This is where p=0.7

  42. Minimax For any q selected by Nav, Evert will select the strategy which does the best Evert Payoffs given Evert’s actions CC DL 90 For q<0.6, 80 Evert will select DL For q>0.6, 50 Evert will select CC 20 Given these maximizations, Nav will pick the strategy which gives the lowest payoff to Evert q=0 q=1 q=0.6 This is where q=0.6

  43. Mixed Strategies Given Evert’s p-mix Nav’s Payoffs are: Where is Nav indifferent? DL: 50p+10(1-p) 50p+10(1-p) =20p+80(1-p) CC: 20p+80(1-p) 30p=70(1-p) p=0.7 Nav. DL CC p DL Evert CC 1-p

  44. Mixed Strategies Given Nav’s q-mix Evert’s Payoffs are: Where is Evert indifferent? DL: 50q+80(1-q) 50q+80(1-q) =90q+20(1-q) CC: 90q+20(1-q) 40q=60(1-q) q=0.6 Nav. q 1-q Nash Equilibrium is where p=0.7 and q=0.6 DL CC DL Evert CC

  45. More on Mixed Strategies Start Chapter 8 Problem Set #2 Posted on Thurs. Feb 23 Attendance 5% of grade Tues. Feb 28 Office Hours Today 12:30-1:30 Next Time

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