1 / 38

G AME THEORY

G AME THEORY. MILJAN KNE Ž EVI Ć FACULTY OF MATHEMATICS UNIVERSITY IN BELGRADE. MOTIVATION. MOTIVATION. MOTIVATION. max(K - S T , 0) exp(-rT)-c 0. max(S T - K , 0) exp(-rT)-c 1. MOTIVATION. Black-Scholes model (Nobelova nagrada). MOTIVATION. MOTIVATION. JOHN F. NASH

tovi
Download Presentation

G AME THEORY

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. GAME THEORY MILJAN KNEŽEVIĆ FACULTY OF MATHEMATICS UNIVERSITYINBELGRADE

  2. MOTIVATION

  3. MOTIVATION

  4. MOTIVATION max(K - ST,0) exp(-rT)-c0 max(ST - K,0) exp(-rT)-c1

  5. MOTIVATION Black-Scholes model (Nobelova nagrada)

  6. MOTIVATION

  7. MOTIVATION JOHN F. NASH 1994 Nobel Laureate in Economics

  8. GAME THEORY Player 1 Bacanje novčića A H T Information set Player 2 H T H T Terminal nodes Player 1’s payoff: Player 2’s payoff: –1 +1 +1 –1 +1 –1 –1 +1

  9. GAME THEORY Player 1 Bacanje novčića B H T Player 2 Player 2 H T H T Terminal nodes Player 1’s payoff: Player 2’s payoff: –1 +1 +1 –1 +1 –1 –1 +1

  10. GAME THEORY Moguće strategije: Bacanje novčića A • Player 1: • Play H • Play T • Player 2: • Play H • Play T

  11. GAME THEORY Moguće strategije: Bacanje novčića B • Player 1: • Play H • Play T • Player2: • s1: Play H if pl. 1 plays H, play H if pl. 1 plays T. • s2: Play H if pl. 1 plays H, play T if pl. 1 plays T. • s3: Play T if pl. 1 plays H, play H if pl. 1 plays T. • s4: Play T if pl. 1 plays H, play T if pl. 1 plays T.

  12. GAME THEORY Bacanje novčića A Player 2 H T H Player 1 T Within each cell the payoffs are: (u1(s1,s2), u2(s1,s2))

  13. GAME THEORY Bacanje novčića B Player 2 HH HT TH TT H Player 1 T Within each cell the payoffs are: (u1(s1,s2), u2(s1,s2))

  14. GAME THEORY • A strategy siisstrictly dominant for player i if for all si’≠si we haveui(si, s–i) > ui(si’, s–i) for all the strategies s–i that player i’s rivals might play. • A strategy siisstrictly dominated for player i if there exists another strategy si’≠si such thatui(si’, s–i) > ui(si, s–i) for all the strategies s–i that player i’s rivals might play. • If you are rational, you would NEVER play a strictly dominated strategy.

  15. GAME THEORY Zatvorenikova dilema A Z2 NP P NP Z1 P P is a strictly dominant strategy for both players.

  16. GAME THEORY Player 2 L R U M Player 1 D • No strictly dominant strategies. • D is dominated by both U and M for player 1.

  17. GAME THEORY Zatvorenikova dilema A Z2 NP P NP Z1 P (P, P) is the unique outcome of the game if both players are rational (and do not cooperate among each other).

  18. GAME THEORY Zatvorenikova dilema B Z2 NP P NP Z1 P (P, P) is the unique outcome of the game if both players are rationaland know that their opponent is rational (and do not cooperate among each other).

  19. GAME THEORY • A strategy iisstrictly dominated for player i if there exists another strategy i’≠i such thatui(i’, –i) > ui(i, –i) for all the strategies –i that player i’s rivals might play.

  20. GAME THEORY Player 2 L R U M Player 1 D • No strictly dominatedpure strategies for any of the players. • M is dominated by 1/2U+1/2D for player 1 (payoff 5 rather than 4).

  21. GAME THEORY • A strategy iis a best response for player i to his rivals’ strategies –i if ui(i, –i) ≥ ui(i’, –i) for all i’. • A pure-strategy profiles = (s1, …, sI)constitutesa Nash equilibrium (NE)iffor every player i = 1, …, I, ui(si, s–i) ≥ ui(si’, s–i) for all si’. • That is, a NE is a set of mutually best responses.

  22. GAME THEORY Player 2 l m r U M Player 1 D • (M, m) is mutually best responsefor the players, hence (M, m) is a unique pure-strategy NE.

  23. GAME THEORY • A mixed-strategy profile= (1, …, I)constitutesa Nash equilibrium (NE)iffor every player i = 1, …, I, ui(i, –i) ≥ ui(i’, –i) for all i’. • That is, a NE is a set of mutually best responses (players allowed to ranomize).

  24. GAME THEORY Player 2 H T H Player 1 T • No pure-strategy NE. • What about mixed-strategy NE?

  25. GAME THEORY Player 2 H T [q] [1–q] p = 1/2 q = 1/2 H [p] Player 1 T [1–p] • Player 2 is indifferent between pl. 1 playing H and T if • 1*p -1*(1-p) = -1*p +1*(1-p) • Player 1 is indifferent between pl. 2 playing H and T if • -1*q +1*(1-q) = 1*q -1*(1-q)

  26. GAME THEORY Player 2 L R [q] [1–q] U [p] M Player 1 D [1–p] 10*q +0*(1-q) = 0*q +10*(1-q) => q = 1/2 1*p +5*(1-p) = 4*p +2*(1-p) => p = 1/2

  27. GAME THEORY Nature Type I Type II  1– Prisoner 1 C DC C DC Prisoner 2 Prisoner 2 C DC C DC C DC C DC –5 –5 –1 –10 –10 –1 0 –2 –5 –11 –1 –10 –10 –7 0 –2

  28. GAME THEORY Normalna forma P2, type I P2, type II DC C DC C DC DC P1 P1 C C

  29. GAME THEORY Possible pure strategies • Player 1: • C • DC • Player 2: • s1: C if type I, C if type II • s2: C if type I, DC if type II • s3: DC if type I, C if type II • s4: DC if type I, DC if type II

  30. GAME THEORY P2, type I P2, type II DC C DC C DC DC P1 P1 C C  1– C if type I, DC if type II DC C

  31. GAME THEORY GAME THEORY P2, type I P2, type II DC C DC C DC DC P1 P1 C C  1– C if type I, DC if type II DC C

  32. GAME THEORY C if type I, DC if type II DC C • For Prisoner 1, • DC dominates C if –10 > –4–1   < 1/6 • C dominates DC if –10 < –4–1   > 1/6 • player indifferent between DC and C if –10 = –4–1   = 1/6

  33. GAME THEORY • BNE: • s* = (DC, (C DC)), if  < 1/6 • s** = (C, (C DC)), if  > 1/6 • mixed, if  = 1/6

  34. GAME THEORY NOVA U I MONOPOLISTA 0 2 B P –3 –1 2 1 Predatorske igre

  35. GAME THEORY MONOPOLISTA Bif N plays U P if N plays U I NOVA U • Two NE: (I, B if N plays U) (U, P if N plays U)

  36. GAME THEORY • A player’s strategy should specify optimal actions at every point in the game tree.

  37. GAME THEORY NOVA U I MONOPOLISTA 0 2 B P –3 –1 2 1

  38. GAME THEORY NOVA Redukovana igra U I MONOPOLISTA P 0 2 2 1 • Sequentially rational NE:(U, P if N plays U) • This procedure is called backward induction.

More Related