G AME THEORY MILJAN KNEŽEVIĆ FACULTY OF MATHEMATICS UNIVERSITY IN BELGRADE

MOTIVATION

max(S T - K, 0) exp(-rT)-c 1 max(K - S T, 0) exp(-rT)-c 0

MOTIVATION Black-Scholes model (Nobelova nagrada)

MOTIVATION

JOHN F. NASH 1994 Nobel Laureate in Economics

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

–1 +1 –1 +1 –1 +1 Player 1 H H H T TT Player 2 Terminal nodes Bacanje novčića B Player 1’s payoff: Player 2’s payoff:

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

GAME THEORY Moguće strategije: Bacanje novčića B Player 1: Play H Play T Player 2: s 1 : Play H if pl. 1 plays H, play H if pl. 1 plays T. s 2 : Play H if pl. 1 plays H, play T if pl. 1 plays T. s 3 : Play T if pl. 1 plays H, play H if pl. 1 plays T. s 4 : Play T if pl. 1 plays H, play T if pl. 1 plays T.

GAME THEORY –1, +1+1, –1 – 1, +1 Player 1 Player 2 H T HT Within each cell the payoffs are: (u 1 (s 1, s 2 ), u 2 (s 1, s 2 )) Bacanje novčića A

GAME THEORY –1, +1 +1, –1 -1, +1+1, –1–1, +1 Player 1 Player 2 H T HHHTTHTT Bacanje novčića B Within each cell the payoffs are: (u 1 (s 1, s 2 ), u 2 (s 1, s 2 ))

GAME THEORY A strategy s i is strictly dominant for player i if for all s i ’≠s i we have u i (s i, s –i ) > u i (s i ’, s –i ) for all the strategies s –i that player i’s rivals might play. A strategy s i is strictly dominated for player i if there exists another strategy s i ’≠s i such that u i (s i ’, s –i ) > u i (s i, 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.

GAME THEORY –2, –2–10, –1 –1, –10–5, –5 Z1Z1 Z2Z2 NP P is a strictly dominant strategy for both players. NP P P Zatvorenikova dilema A

GAME THEORY 1, –1–1, 1 1, –1 –2, 5–3, 2 Player 1 Player 2 L No strictly dominant strategies. U R M D D is dominated by both U and M for player 1.

GAME THEORY –2, –2–10, –1 –1, –10–5, –5 NP (P, P) is the unique outcome of the game if both players are rational (and do not cooperate among each other). NP P P Z1Z1 Z2 Zatvorenikova dilema A

GAME THEORY 0, –2–10, –1 –1, –10–5, –5 NP (P, P) is the unique outcome of the game if both players are rational and know that their opponent is rational (and do not cooperate among each other). NP P P Z1Z1 Z2 Zatvorenikova dilema B

GAME THEORY A strategy  i is strictly dominated for player i if there exists another strategy  i ’≠  i such that u i (  i ’,  –i ) > u i (  i,  –i ) for all the strategies  –i that player i’s rivals might play.

GAME THEORY 10, 10, 4 4, 24, 3 0, 510, 2 Player 1 Player 2 L No strictly dominated pure strategies for any of the players. U R M D M is dominated by 1 / 2 U+ 1 / 2 D for player 1 (payoff 5 rather than 4).

GAME THEORY A strategy  i is a best response for player i to his rivals’ strategies  –i if u i (  i,  –i ) ≥ u i (  i ’,  –i ) for all  i ’. A pure-strategy profile s = (s 1, …, s I ) constitutes a Nash equilibrium (NE) if for every player i = 1, …, I, u i (s i, s –i ) ≥ u i (s i ’, s –i ) for all s i ’. That is, a NE is a set of mutually best responses.

GAME THEORY 5, 30, 43, 5 4, 05, 54, 0 3, 50, 45, 3 Player 1 Player 2 l U m M D (M, m) is mutually best response for the players, hence (M, m) is a unique pure-strategy NE. r

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

GAME THEORY -1, +1+1, –1 –1, +1 Player 1 Player 2 H H T T What about mixed-strategy NE? No pure-strategy NE.

GAME THEORY -1, +1+1, –1 –1, +1 Player 1 Player 2 H H T T [p][p] [1–p] [q][q][1–q] 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) p = 1/2 q = 1/2

GAME THEORY 10, 10, 4 4, 24, 3 0, 510, 2 Player 1 Player 2 L U R M D [p][p] [1–p] [q][q][1–q] 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

GAME THEORY –5 –1 –10 –1 0 –2 Nature C Type IType II Prisoner 1  1–  DC Prisoner 2 –5 –11 –1 –10 –7 0 –2 CDC Prisoner 2 DC C C DC

GAME THEORY 0, –2–10, –1 –1, –10–5, –5 P1 P2, type I DC C C 0, –2–10, –7 –1, –10–5, –11 P1 P2, type II DC C C Normalna forma

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

GAME THEORY 0, –2–10, –1 –1, –10–5, –5 P1 P2, type I DC C C 0, –2–10, –7 –1, –10–5, –11 P1 P2, type II DC C C  1–  -10  +0(1-  ), -1  -2(1-  ) –5  -1(1-  ), -5  -10(1-  ) C if type I, DC if type II DC C

GAME THEORY 0, –2–10, –1 –1, –10–5, –5 P1 P2, type I DC C C 0, –2–10, –7 –1, –10–5, –11 P1 P2, type II DC C C  1–  -10 ,   -1, 5  -10 C if type I, DC if type II DC C

GAME THEORY -10 ,  -2 –4  -1, 5  -10 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  1/6 player indifferent between DC and C if –10  = –4  –1   = 1/6

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

GAME THEORY 0202 –3 – NOVA I U BP MONOPOLISTA Predatorske igre

GAME THEORY 0, 2 –3, –12, 1 NOVA MONOPOLISTA B if N plays U I P if N plays U U Two NE: (I, B if N plays U) (U, P if N plays U)

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

GAME THEORY 0202 –3 – NOVA I U BP MONOPOLISTA

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