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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 1–3 (SC 323)
- 2. Outline Games and payoﬀs 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 diﬀerence in the numbers shown If we play this a number of times, who’s going to win?
- 4. The Payoﬀ Matrix Lists each player’s outcomes versus C ’s outcomes the other’s 1 2 3 4 5 6 1 0 -1 -2 -3 -4 -5 Each aij represents R’s outcomes 2 1 0 -1 -2 -3 -4 the payoﬀ 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’s outcomes and C ’s outcomes be given by probability vectors q1 q2 p = p1 p2 · · · pn q=. .. qn
- 6. Expected Value Let the probabilities of R’s outcomes and C ’s outcomes be given by probability vectors q1 q2 p = p1 p2 · · · pn q=. .. 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’s outcomes and C ’s outcomes be given by probability vectors q1 q2 p = p1 p2 · · · pn q=. .. qn The probability of R having outcome i and C having outcome j is therefore pi qj . The expected value of R’s payoﬀ is n E (p, q) = pi aij qj = pAq i,j=1
- 8. Expected Value Let the probabilities of R’s outcomes and C ’s outcomes be given by probability vectors q1 q2 p = p1 p2 · · · pn q=. .. qn The probability of R having outcome i and C having outcome j is therefore pi qj . The expected value of R’s payoﬀ is n E (p, q) = pi aij qj = pAq i,j=1 A “fair game” if the dice are fair.
- 9. Expected value of this game pAq 0 −1 −2 −3 −4 −5 1/6 1 0 −1 −2 −3 −4 1/6 1/6 1/6 1/6 1/6 1/6 2 1 1/6 0 −1 −2 −3 1/6 = 3 2 1 0 −1 −2 1/6 4 3 2 1 0 −1 1/6 5 4 3 2 1 0 1/6 −15/6 −9/6 −3/6 = 1/6 1/6 1/6 1/6 1/6 1/6 3/6 9/6 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 −1 −2 −3 −4 −5 1/6 1 0 −1 −2 −3 −4 1/6 1/10 1/10 1/5 1/5 1/5 2 1 1/5 0 −1 −2 −3 1/6 = 3 2 1 0 −1 −2 1/6 4 3 2 1 0 −1 1/6 5 4 3 2 1 0 1/6 −15 −9 1 1 −3 24 2 = 10 · 6 1 1 2 2 2 2 = 60 = 5 3 9 15
- 11. Strategies What if we could choose a die to be as biased as we C ’s outcomes wanted? 1 2 3 4 5 6 1 0 -1 -2 -3 -4 -5 In other words, R’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’d want 6 5 4 3 2 1 0 to get a 6 all the time!
- 12. Flu Vaccination Suppose there are two ﬂu strains, and we have two ﬂu vaccines to combat them. We don’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 payoﬀs 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∗ for R and q∗ for C such that for all strategies p and q: E (p∗ , q) ≥ E (p∗ , q∗ ) ≥ E (p, q∗ )
- 15. Theorem (Fundamental Theorem of Zero-Sum Games) There exist optimal strategies p∗ for R and q∗ for C such that for all strategies p and q: E (p∗ , q) ≥ E (p∗ , q∗ ) ≥ E (p, q∗ ) E (p∗ , q∗ ) is called the value v of the game.
- 16. Reﬂect on the inequality E (p∗ , q) ≥ E (p∗ , q∗ ) ≥ E (p, q∗ ) In other words, E (p∗ , q) ≥ E (p∗ , q∗ ): R can guarantee a lower bound on his/her payoﬀ E (p∗ , q∗ ) ≥ E (p, q∗ ): 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∗ and q∗ ! The general case we’ll look at next time (hard-ish) There are some games in which we can ﬁnd optimal strategies now: Strictly-determined games 2 × 2 non-strictly-determined games
- 18. Outline Games and payoﬀs 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 payoﬀ 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 & Order 70 45 35 30
- 21. The payoﬀ 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 & Order 70 45 35 30 What is NBC’s strategy?
- 22. The payoﬀ 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 & Order 70 45 35 30 What is NBC’s strategy? NBC wants to maximize NBC’s minimum share
- 23. The payoﬀ 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 & Order 70 45 35 30 What is NBC’s strategy? NBC wants to maximize NBC’s minimum share In airing Dateline, NBC’s share is at least 45
- 24. The payoﬀ 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 & Order 70 45 35 30 What is NBC’s strategy? NBC wants to maximize NBC’s minimum share In airing Dateline, NBC’s share is at least 45 This is a good strategy for NBC
- 25. The payoﬀ 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 & Order 70 45 35 30 What is CBS’s strategy?
- 26. The payoﬀ 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 & Order 70 45 35 30 What is CBS’s strategy? CBS wants to minimize NBC’s maximum share
- 27. The payoﬀ 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 & Order 70 45 35 30 What is CBS’s strategy? CBS wants to minimize NBC’s maximum share In airing CSI, CBS keeps NBC’s share no bigger than 45
- 28. The payoﬀ 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 & Order 70 45 35 30 What is CBS’s strategy? CBS wants to minimize NBC’s maximum share In airing CSI, CBS keeps NBC’s share no bigger than 45 This is a good strategy for CBS
- 29. The payoﬀ 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 & Order 70 45 35 30 Equilibrium
- 30. The payoﬀ 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 & Order 70 45 35 30 Equilibrium (Dateline,CSI) is an equilibrium pair of strategies
- 31. The payoﬀ 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 & Order 70 45 35 30 Equilibrium (Dateline,CSI) is an equilibrium pair of strategies Assuming NBC airs Dateline, CBS’s best choice is to air CSI, and vice versa
- 32. Characteristics of an Equlibrium Let A be a payoﬀ 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 payoﬀ matrix has a saddle point is called strictly determined Payoﬀ matrices can have multiple saddle points
- 33. Pure Strategies are optimal in Strictly-Determined Games Theorem Let A be a payoﬀ 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 payoﬀ 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 ≥ 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 ≤ pi ars = E (er , es ) i=1 i=1 So for any p and q, we have E (er , q) ≥ E (er , es ) ≥ E (er , es )
- 35. Outline Games and payoﬀs 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 deﬁnitely choose 2 1 3 This would make C choose strategy 1 but (2, 1) is an 2 4 equilibrium, a saddle point.
- 37. Finding equilibria by gravity Here (1, 1) is an equilibrium 2 3 position; starting from there neither player would want to deviate from this. 1 4
- 38. Finding equilibria by gravity 2 3 What about this one? 4 1
- 39. Outline Games and payoﬀs 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−q1 )+(1−p1 )a21 q1 +(1−p1 )a22 (1−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−q1 )+(1−p1 )a21 q1 +(1−p1 )a22 (1−q1 ) The critical points are when ∂E 0= = a11 q1 + a12 (1 − q1 ) − a21 q1 − a22 (1 − q1 ) ∂p1 ∂E 0= = p1 a11 − p1 a12 + (1 − p1 )a21 − (1 − p1 )a22 ∂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−q1 )+(1−p1 )a21 q1 +(1−p1 )a22 (1−q1 ) The critical points are when ∂E 0= = a11 q1 + a12 (1 − q1 ) − a21 q1 − a22 (1 − q1 ) ∂p1 ∂E 0= = p1 a11 − p1 a12 + (1 − p1 )a21 − (1 − p1 )a22 ∂q So a22 − a12 a22 − a21 q1 = p1 = a11 + a22 − a21 − a12 a11 + a22 − a21 − 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’t work because A has a 0 2 4 saddle point. 2 3 If A = , p1 = 3 ? Again, doesn’t work. 2 1 4 2 3 If A = , p1 = −3 = 3/4, while q1 = −4 = 1/2. So R −4 −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 ∂2E ∂2E =0 =0 ∂p 2 ∂q 2 So this is a saddle point! Finally, a11 a22 − a12 a21 E (p, q) = a11 + a22 − a21 − a22
- 45. Example: Vaccination We have 0.9 − 0.6 2 p1 = = Strain 0.85 + 0.9 − 0.6 − 0.7 3 0.9 − 0.7 4 1 2 q1 = = Vacc 0.85 + 0.9 − 0.6 − 0.7 9 1 0.85 0.70 (0.85)(0.9) − (0.6)(0.7) 2 0.60 0.90 v= ≈ 0.767 0.85 + 0.9 − 0.6 − 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’ll still cover 76.7% of the population
- 46. Outline Games and payoﬀs 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

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