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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

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G AME THEORY

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  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.

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