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# Lesson 34: Introduction To Game Theory

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We define zero-sum games and show that they can be modeled with matrices. We find optimal strategies for two types of such games: (1) strictly determined games which have a saddle point, and (2) 2x2 non-strictly determined games, for which a calculus computation finds the optimal strategy

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### Lesson 34: Introduction To Game Theory

1. 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. 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. 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. 4. The Payoﬀ Matrix Lists each player’s C ’s outcomes outcomes versus 1 2345 6 the other’s 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 3210 -1 occur (a zero-sum 6 5 4321 0 game).
5. 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. 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. 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. 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. 9. Expected value of this game pAq 0 −1 −2 −3 −4 −5    1/6 −1 −2 −3 −4  1/6 1 0    0 −1 −2 −3  1/6 2 1 1/6 1/6 1/6 1/6 1/6 1/6  =   0 −1 −2  1/6 3 2 1    0 −1  1/6 4 3 2 1 54 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. 10. Expected value with an unfair die 1/10 1/10 1/5 1/5 1/5 1/5 Suppose p = . Then pAq 0 −1 −2 −3 −4 −5   1/6 1 0 −1 −2 −3 −4  1/6   0 −1 −2 −3 2 1  1/6 = 1/10 1/10 1/5 1/5 1/5 1/5   0 −1 −2 3 2 1  1/6   0 −1 4 3 2 1  1/6 54 3 2 1 0 1/6 −15    −9     −3  24 2 1 1 · 1 1 2 2 2 2 = =  60 = 5  10 6 3 9 15
11. 11. Strategies What if we could choose a die to be C ’s outcomes as biased as we 1 2345 6 wanted? 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 3210 -1 Clearly, we’d want 6 5 4321 0 to get a 6 all the time!
12. 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 1 2 Neither pure strategy is Vacc 1 0.85 0.70 the clear favorite 2 0.60 0.90 Is there a combination of vaccines (a mixed strategy) that maximizes total immunity of the population?
13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 arj qj ≥ E (er , q) = er Aq = ars qj = ars = E (er , es ) j=1 j=1 If p is a strategy for R, then m m pi ais ≤ E (er , es ) = pAes = 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. 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. 36. Finding equilibria by gravity If C chose strategy 2, and R knew it, R would   1 3 deﬁnitely choose 2   This would make C     choose strategy 1     but (2, 1) is an   2 4 equilibrium, a saddle point.
37. 37. Finding equilibria by gravity   2 3 Here (1, 1) is an equilibrium position; starting from there     neither player would want to     deviate from this.     1 4
38. 38. Finding equilibria by gravity   2 3   What about this one?           4 1
39. 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. 40. Two-by-two non-strictly-determined games Calculation In this case we can compute E (p, q) by hand in terms of p1 = p and q1 = q: E (p, q) = pa11 q + pa12 (1 − q) + (1 − p)a21 q + (1 − p)a22 (1 − q)
41. 41. Two-by-two non-strictly-determined games Calculation In this case we can compute E (p, q) by hand in terms of p1 = p and q1 = q: E (p, q) = pa11 q + pa12 (1 − q) + (1 − p)a21 q + (1 − p)a22 (1 − q) The critical points are when ∂E = a11 q + a12 (1 − q) − a21 q − a22 (1 − q) 0= ∂p ∂E = pa11 − pa12 + (1 − p)a21 − (1 − p)a22 0= ∂q
42. 42. Two-by-two non-strictly-determined games Calculation In this case we can compute E (p, q) by hand in terms of p1 = p and q1 = q: E (p, q) = pa11 q + pa12 (1 − q) + (1 − p)a21 q + (1 − p)a22 (1 − q) The critical points are when ∂E = a11 q + a12 (1 − q) − a21 q − a22 (1 − q) 0= ∂p ∂E = pa11 − pa12 + (1 − p)a21 − (1 − p)a22 0= ∂q So a22 − a12 a22 − a21 p= q= a11 + a22 − a21 − a22 a11 + a22 − a21 − a12 These are in between 0 and 1 if there are no saddle points in the matrix.
43. 43. Examples 13 , then p = 2 ? Doesn’t work because A has a If A = 0 24 saddle point. 23 3 If A = , p = 2 ? Again, doesn’t work. 14 2 3 , p = −3 = 3/4, while q = −4 = 1/2. So R −2 If A = −4 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. 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. 45. Example: Vaccination We have 0.9 − 0.6 2 p1 = = Strain 0.85 + 0.9 − 0.6 − 0.7 3 1 2 0.9 − 0.7 4 q1 = = Vacc 1 0.85 0.70 0.85 + 0.9 − 0.6 − 0.7 9 2 0.60 0.90 (0.85)(0.9) − (0.6)(0.7) ≈ 0.767 v= 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. 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. 47. Other Applications of GT War the Battle of the Bismarck Sea Business product introduction pricing Dating