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# Lesson34 Intro To Game Theory Slides

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• 1. Lesson 34 (KH, Section 11.4) Introduction to Game Theory Math 20 December 12, 2007 Announcements Pset 12 due December 17 (last day of class) next OH today 1&#x2013;3 (SC 323)
• 2. Outline Games and payo&#xFB00;s Matching dice Vaccination The theorem of the day Strictly determined games Example: Network programming Characteristics of an Equlibrium Two-by-two strictly-determined games Two-by-two non-strictly-determined games Calculation Example: Vaccination Other
• 3. A Game of Chance You and I each have a six-sided die We roll and the loser pays the winner the di&#xFB00;erence in the numbers shown If we play this a number of times, who&#x2019;s going to win?
• 4. The Payo&#xFB00; Matrix Lists each player&#x2019;s outcomes versus C &#x2019;s outcomes the other&#x2019;s 1 2 3 4 5 6 1 0 -1 -2 -3 -4 -5 Each aij represents R&#x2019;s outcomes 2 1 0 -1 -2 -3 -4 the payo&#xFB00; from C 3 2 1 0 -1 -2 -3 to R if outcomes i 4 3 2 1 0 -1 -2 for R and j for C 5 4 3 2 1 0 -1 occur (a zero-sum 6 5 4 3 2 1 0 game).
• 5. Expected Value Let the probabilities of R&#x2019;s outcomes and C &#x2019;s outcomes be given by probability vectors &#xF8EB; &#xF8F6; q1 &#xF8EC; q2 &#xF8F7; p = p1 p2 &#xB7; &#xB7; &#xB7; pn q=&#xF8EC;.&#xF8F7; &#xF8EC; &#xF8F7; &#xF8ED;.&#xF8F8;. qn
• 6. Expected Value Let the probabilities of R&#x2019;s outcomes and C &#x2019;s outcomes be given by probability vectors &#xF8EB; &#xF8F6; q1 &#xF8EC; q2 &#xF8F7; p = p1 p2 &#xB7; &#xB7; &#xB7; pn q=&#xF8EC;.&#xF8F7; &#xF8EC; &#xF8F7; &#xF8ED;.&#xF8F8;. qn The probability of R having outcome i and C having outcome j is therefore pi qj .
• 7. Expected Value Let the probabilities of R&#x2019;s outcomes and C &#x2019;s outcomes be given by probability vectors &#xF8EB; &#xF8F6; q1 &#xF8EC; q2 &#xF8F7; p = p1 p2 &#xB7; &#xB7; &#xB7; pn q=&#xF8EC;.&#xF8F7; &#xF8EC; &#xF8F7; &#xF8ED;.&#xF8F8;. qn The probability of R having outcome i and C having outcome j is therefore pi qj . The expected value of R&#x2019;s payo&#xFB00; is n E (p, q) = pi aij qj = pAq i,j=1
• 8. Expected Value Let the probabilities of R&#x2019;s outcomes and C &#x2019;s outcomes be given by probability vectors &#xF8EB; &#xF8F6; q1 &#xF8EC; q2 &#xF8F7; p = p1 p2 &#xB7; &#xB7; &#xB7; pn q=&#xF8EC;.&#xF8F7; &#xF8EC; &#xF8F7; &#xF8ED;.&#xF8F8;. qn The probability of R having outcome i and C having outcome j is therefore pi qj . The expected value of R&#x2019;s payo&#xFB00; is n E (p, q) = pi aij qj = pAq i,j=1 A &#x201C;fair game&#x201D; if the dice are fair.
• 9. Expected value of this game pAq 0 &#x2212;1 &#x2212;2 &#x2212;3 &#x2212;4 &#x2212;5 &#xF8EB; &#xF8F6;&#xF8EB; &#xF8F6; 1/6 &#xF8EC;1 0 &#xF8EC; &#x2212;1 &#x2212;2 &#x2212;3 &#x2212;4 &#xF8F7; &#xF8EC;1/6&#xF8F7; &#xF8F7;&#xF8EC; &#xF8F7; 1/6 1/6 1/6 1/6 1/6 &#xF8EC;2 1 1/6 &#xF8EC; 0 &#x2212;1 &#x2212;2 &#x2212;3 &#xF8F7; &#xF8EC;1/6&#xF8F7; = &#xF8F7;&#xF8EC; &#xF8F7; &#xF8EC;3 2 &#xF8EC; 1 0 &#x2212;1 &#x2212;2 &#xF8F7; &#xF8EC;1/6&#xF8F7; &#xF8F7;&#xF8EC; &#xF8F7; &#xF8ED;4 3 2 1 0 &#x2212;1 &#xF8F8; &#xF8ED;1/6&#xF8F8; 5 4 3 2 1 0 1/6 &#x2212;15/6 &#xF8EB; &#xF8F6; &#xF8EC; &#x2212;9/6 &#xF8F7; &#xF8EC; &#xF8F7; &#xF8EC; &#x2212;3/6 &#xF8F7; = 1/6 1/6 1/6 1/6 1/6 1/6 &#xF8EC; &#xF8F7; &#xF8EC; 3/6 &#xF8F7; &#xF8EC; &#xF8F7; &#xF8ED; 9/6 &#xF8F8; 15/6 =0
• 10. Expected value with an unfair die Suppose p = 1/10 1/10 1/5 1/5 1/5 1/5 . Then pAq 0 &#x2212;1 &#x2212;2 &#x2212;3 &#x2212;4 &#x2212;5 &#xF8EB; &#xF8F6;&#xF8EB; 1/6 &#xF8EC;1 0 &#x2212;1 &#x2212;2 &#x2212;3 &#x2212;4 &#xF8F7; &#xF8EC;1/6 &#xF8EC; &#xF8F7;&#xF8EC; 1/10 1/10 1/5 1/5 1/5 &#xF8EC;2 1 1/5 &#xF8EC; 0 &#x2212;1 &#x2212;2 &#x2212;3 &#xF8F7; &#xF8EC;1/6 = &#xF8F7;&#xF8EC; &#xF8EC;3 2 &#xF8EC; 1 0 &#x2212;1 &#x2212;2 &#xF8F7; &#xF8EC;1/6 &#xF8F7;&#xF8EC; &#xF8ED;4 3 2 1 0 &#x2212;1 &#xF8F8; &#xF8ED;1/6 5 4 3 2 1 0 1/6 &#x2212;15 &#xF8EB; &#xF8F6; &#xF8EC; &#x2212;9 &#xF8F7; &#xF8EC; &#xF8F7; 1 1 &#xF8EC; &#x2212;3 &#xF8F7; 24 2 = 10 &#xB7; 6 1 1 2 2 2 2 &#xF8EC; &#xF8EC; &#xF8F7;= &#xF8F7; 60 = 5 &#xF8EC; 3 &#xF8F7; &#xF8ED; 9 &#xF8F8; 15
• 11. Strategies What if we could choose a die to be as biased as we C &#x2019;s outcomes wanted? 1 2 3 4 5 6 1 0 -1 -2 -3 -4 -5 In other words, R&#x2019;s outcomes 2 1 0 -1 -2 -3 -4 what if we could 3 2 1 0 -1 -2 -3 choose a strategy 4 3 2 1 0 -1 -2 p for this game? 5 4 3 2 1 0 -1 Clearly, we&#x2019;d want 6 5 4 3 2 1 0 to get a 6 all the time!
• 12. Flu Vaccination Suppose there are two &#xFB02;u strains, and we have two &#xFB02;u vaccines to combat them. We don&#x2019;t know distribution of strains Strain Neither pure strategy is 1 2 Vacc the clear favorite 1 0.85 0.70 Is there a combination of 2 0.60 0.90 vaccines (a mixed strategy) that maximizes total immunity of the population?
• 13. Outline Games and payo&#xFB00;s Matching dice Vaccination The theorem of the day Strictly determined games Example: Network programming Characteristics of an Equlibrium Two-by-two strictly-determined games Two-by-two non-strictly-determined games Calculation Example: Vaccination Other
• 14. Theorem (Fundamental Theorem of Zero-Sum Games) There exist optimal strategies p&#x2217; for R and q&#x2217; for C such that for all strategies p and q: E (p&#x2217; , q) &#x2265; E (p&#x2217; , q&#x2217; ) &#x2265; E (p, q&#x2217; )
• 15. Theorem (Fundamental Theorem of Zero-Sum Games) There exist optimal strategies p&#x2217; for R and q&#x2217; for C such that for all strategies p and q: E (p&#x2217; , q) &#x2265; E (p&#x2217; , q&#x2217; ) &#x2265; E (p, q&#x2217; ) E (p&#x2217; , q&#x2217; ) is called the value v of the game.
• 16. Re&#xFB02;ect on the inequality E (p&#x2217; , q) &#x2265; E (p&#x2217; , q&#x2217; ) &#x2265; E (p, q&#x2217; ) In other words, E (p&#x2217; , q) &#x2265; E (p&#x2217; , q&#x2217; ): R can guarantee a lower bound on his/her payo&#xFB00; E (p&#x2217; , q&#x2217; ) &#x2265; E (p, q&#x2217; ): C can guarantee an upper bound on how much he/she loses This value could be negative in which case C has the advantage
• 17. Fundamental problem of zero-sum games Find the p&#x2217; and q&#x2217; ! The general case we&#x2019;ll look at next time (hard-ish) There are some games in which we can &#xFB01;nd optimal strategies now: Strictly-determined games 2 &#xD7; 2 non-strictly-determined games
• 18. Outline Games and payo&#xFB00;s Matching dice Vaccination The theorem of the day Strictly determined games Example: Network programming Characteristics of an Equlibrium Two-by-two strictly-determined games Two-by-two non-strictly-determined games Calculation Example: Vaccination Other
• 19. Example: Network programming Suppose we have two networks, NBC and CBS Each chooses which program to show in a certain time slot Viewer share varies depending on these combinations How can NBC get the most viewers?
• 20. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30
• 21. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is NBC&#x2019;s strategy?
• 22. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is NBC&#x2019;s strategy? NBC wants to maximize NBC&#x2019;s minimum share
• 23. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is NBC&#x2019;s strategy? NBC wants to maximize NBC&#x2019;s minimum share In airing Dateline, NBC&#x2019;s share is at least 45
• 24. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is NBC&#x2019;s strategy? NBC wants to maximize NBC&#x2019;s minimum share In airing Dateline, NBC&#x2019;s share is at least 45 This is a good strategy for NBC
• 25. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is CBS&#x2019;s strategy?
• 26. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is CBS&#x2019;s strategy? CBS wants to minimize NBC&#x2019;s maximum share
• 27. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is CBS&#x2019;s strategy? CBS wants to minimize NBC&#x2019;s maximum share In airing CSI, CBS keeps NBC&#x2019;s share no bigger than 45
• 28. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 What is CBS&#x2019;s strategy? CBS wants to minimize NBC&#x2019;s maximum share In airing CSI, CBS keeps NBC&#x2019;s share no bigger than 45 This is a good strategy for CBS
• 29. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 Equilibrium
• 30. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 Equilibrium (Dateline,CSI) is an equilibrium pair of strategies
• 31. The payo&#xFB00; matrix and strategies CBS es r ut ea CS r ivo in D M rv s, I Ye Su 60 My Name is Earl 60 20 30 55 NBC Dateline 50 75 45 60 Law &amp; Order 70 45 35 30 Equilibrium (Dateline,CSI) is an equilibrium pair of strategies Assuming NBC airs Dateline, CBS&#x2019;s best choice is to air CSI, and vice versa
• 32. Characteristics of an Equlibrium Let A be a payo&#xFB00; matrix. A saddle point is an entry ars which is the minimum entry in its row and the maximum entry in its column. A game whose payo&#xFB00; matrix has a saddle point is called strictly determined Payo&#xFB00; matrices can have multiple saddle points
• 33. Pure Strategies are optimal in Strictly-Determined Games Theorem Let A be a payo&#xFB00; matrix. If ars is a saddle point, then er is an optimal strategy for R and es is an optimal strategy for C.
• 34. Pure Strategies are optimal in Strictly-Determined Games Theorem Let A be a payo&#xFB00; matrix. If ars is a saddle point, then er is an optimal strategy for R and es is an optimal strategy for C. Proof. If q is a strategy for C, then n n E (er , q) = er Aq = arj qj &#x2265; ars qj = ars = E (er , es ) j=1 j=1 If p is a strategy for R, then m m E (er , es ) = pAes = pi ais &#x2264; pi ars = E (er , es ) i=1 i=1 So for any p and q, we have E (er , q) &#x2265; E (er , es ) &#x2265; E (er , es )
• 35. Outline Games and payo&#xFB00;s Matching dice Vaccination The theorem of the day Strictly determined games Example: Network programming Characteristics of an Equlibrium Two-by-two strictly-determined games Two-by-two non-strictly-determined games Calculation Example: Vaccination Other
• 36. Finding equilibria by gravity If C chose strategy 2, and R knew it, R would &#xF8EB; &#xF8F6; de&#xFB01;nitely choose 2 &#xF8EC; 1 3 &#xF8F7; &#xF8EC; &#xF8F7; This would make C &#xF8EC; &#xF8EC; &#xF8F7; &#xF8F7; choose strategy 1 &#xF8EC; &#xF8EC; &#xF8F7; &#xF8F7; but (2, 1) is an &#xF8ED; &#xF8F8; 2 4 equilibrium, a saddle point.
• 37. Finding equilibria by gravity &#xF8EB; &#xF8F6; Here (1, 1) is an equilibrium &#xF8EC; 2 3 &#xF8F7; position; starting from there &#xF8EC; &#xF8EC; &#xF8F7; &#xF8F7; neither player would want to &#xF8EC; &#xF8F7; &#xF8EC; &#xF8F7; deviate from this. &#xF8EC; &#xF8F7; &#xF8ED; &#xF8F8; 1 4
• 38. Finding equilibria by gravity &#xF8EB; &#xF8F6; &#xF8EC; 2 3 &#xF8F7; &#xF8EC; &#xF8F7; What about this one? &#xF8EC; &#xF8EC; &#xF8F7; &#xF8F7; &#xF8EC; &#xF8F7; &#xF8EC; &#xF8F7; &#xF8ED; &#xF8F8; 4 1
• 39. Outline Games and payo&#xFB00;s Matching dice Vaccination The theorem of the day Strictly determined games Example: Network programming Characteristics of an Equlibrium Two-by-two strictly-determined games Two-by-two non-strictly-determined games Calculation Example: Vaccination Other
• 40. Two-by-two non-strictly-determined games Calculation In this case we can compute E (p, q) by hand in terms of p1 and q1 : E (p, q) = p1 a11 q1 +p1 a12 (1&#x2212;q1 )+(1&#x2212;p1 )a21 q1 +(1&#x2212;p1 )a22 (1&#x2212;q1 )
• 41. Two-by-two non-strictly-determined games Calculation In this case we can compute E (p, q) by hand in terms of p1 and q1 : E (p, q) = p1 a11 q1 +p1 a12 (1&#x2212;q1 )+(1&#x2212;p1 )a21 q1 +(1&#x2212;p1 )a22 (1&#x2212;q1 ) The critical points are when &#x2202;E 0= = a11 q1 + a12 (1 &#x2212; q1 ) &#x2212; a21 q1 &#x2212; a22 (1 &#x2212; q1 ) &#x2202;p1 &#x2202;E 0= = p1 a11 &#x2212; p1 a12 + (1 &#x2212; p1 )a21 &#x2212; (1 &#x2212; p1 )a22 &#x2202;q
• 42. Two-by-two non-strictly-determined games Calculation In this case we can compute E (p, q) by hand in terms of p1 and q1 : E (p, q) = p1 a11 q1 +p1 a12 (1&#x2212;q1 )+(1&#x2212;p1 )a21 q1 +(1&#x2212;p1 )a22 (1&#x2212;q1 ) The critical points are when &#x2202;E 0= = a11 q1 + a12 (1 &#x2212; q1 ) &#x2212; a21 q1 &#x2212; a22 (1 &#x2212; q1 ) &#x2202;p1 &#x2202;E 0= = p1 a11 &#x2212; p1 a12 + (1 &#x2212; p1 )a21 &#x2212; (1 &#x2212; p1 )a22 &#x2202;q So a22 &#x2212; a12 a22 &#x2212; a21 q1 = p1 = a11 + a22 &#x2212; a21 &#x2212; a12 a11 + a22 &#x2212; a21 &#x2212; a12 These are in between 0 and 1 if there are no saddle points in the matrix.
• 43. Examples 1 3 If A = , then p1 = 2 ? Doesn&#x2019;t work because A has a 0 2 4 saddle point. 2 3 If A = , p1 = 3 ? Again, doesn&#x2019;t work. 2 1 4 2 3 If A = , p1 = &#x2212;3 = 3/4, while q1 = &#x2212;4 = 1/2. So R &#x2212;4 &#x2212;2 4 1 should pick 1 half the time and 2 the other half, while C should pick 1 3/4 of the time and 2 the rest.
• 44. Further Calculations Also &#x2202;2E &#x2202;2E =0 =0 &#x2202;p 2 &#x2202;q 2 So this is a saddle point! Finally, a11 a22 &#x2212; a12 a21 E (p, q) = a11 + a22 &#x2212; a21 &#x2212; a22
• 45. Example: Vaccination We have 0.9 &#x2212; 0.6 2 p1 = = Strain 0.85 + 0.9 &#x2212; 0.6 &#x2212; 0.7 3 0.9 &#x2212; 0.7 4 1 2 q1 = = Vacc 0.85 + 0.9 &#x2212; 0.6 &#x2212; 0.7 9 1 0.85 0.70 (0.85)(0.9) &#x2212; (0.6)(0.7) 2 0.60 0.90 v= &#x2248; 0.767 0.85 + 0.9 &#x2212; 0.6 &#x2212; 0.7 We should give 2/3 of the population vaccine 1 and the rest vacine 2 The worst case scenario is a 4 : 5 distribution of strains We&#x2019;ll still cover 76.7% of the population
• 46. Outline Games and payo&#xFB00;s Matching dice Vaccination The theorem of the day Strictly determined games Example: Network programming Characteristics of an Equlibrium Two-by-two strictly-determined games Two-by-two non-strictly-determined games Calculation Example: Vaccination Other
• 47. Other Applications of GT War the Battle of the Bismarck Sea Business product introduction pricing Dating