(90, 90) (86, 92) (92, 86) (88, 88) Figure 1 Exam or presentation? a (players) k N = (1, 2,..., k) 2 k = 2 b (strategies) i S i, i = 1, 2 1

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1 in progress Nash Equilibrium : The Hawk-Dove Game Mixed Strategies ( ) ; 92 1

2 (90, 90) (86, 92) (92, 86) (88, 88) Figure 1 Exam or presentation? a (players) k N = (1, 2,..., k) 2 k = 2 b (strategies) i S i, i = 1, 2 1 n S 1 = {(s 1 )} = (s 1 1, s 2 1,..., s n 1 ) 2 m S 2 = {(s 2 )} = (s 1 2, s 2 2,..., s m 2 ) n m c payoff) (payoff matrix) i π i = π i (s 1, s 2 )

3 * 1 3. (rationality) the Exam-or-Presentation Game i ii (strictly dominant strategy) (88, 88) (90, 90) The Prisoner s Dilemma 2.1 ( ) (confess) (not confesss) 10 *1 John Harsanyi

4 1 2 NC 1 NC ( 1, 1) ( 10, 0) C (0, 10) ( 4, 4) Figure 2 Prisoner s Dilemma (C) ( 4, 4) C the Prisoner s Dilemma 2.2 Athlete 2 Don t use drugs Use drugs Athlete 1 Don t use drugs (3, 3) (1, 4) Use drugs (4, 1) (2, 2) Figure 3 Performance -enhancing drugs (2, 2) (arms race) 2.2 G = {S 1, S 2 ; π 1, π 2 }, S 1 = {(s 1 )} = {a 1, a 2,..., a n }, S 2 = {(s 2 )} = {b 1, b 2,..., b m } s 2 1 s 1 s 1 1 (best response) 1 a i S 1 π 1 (s 1, s 2 ) π 1 (a i, s 2 ), i = 1, 2,..., n 2 s 2 S 2 s 1 1 π 1 (s 1, s 2 ) > π 1 (a i, s 2 ), i = 1, 2,..., n (strictly best response) 2 4

5 s 2 S (dominant strategy) 1 1 (strictly dominant strategy) 2.3 (1) (2) (3) (4) = = 0.12 (5) = = 0.08 Firm 2 Low-priced Upscale Firm 1 Low-priced (.48,.12) (.60,.40) Upscale (.40,.60) (.32,.08) Figure 5 Marketing strategy 1 low-priced 1 low-priced upscale 1 upscale low-priced low-priced 2 upsclae (0.60,0.40) 2 (common knowledge) 5

6 3 Nash Equilibrium (A Three-Client Game) 2 A,B,C A,B,C (1) (2) 1 2 (3) 2 B C (4) A 8 B C (Nash Equilibrium) Firm 2 A B C Firm 1 A (4, 4) (0, 2) (0, 2) B (0, 0) (1, 1) (0, 2) C (0, 0) (0, 2) (1, 1) Figure 6 Three-client game 1 s 1 2 s 2 s 1 s 2 s 2 s 1 (s 1, s 2) * 2 Three-client 2 A 1 A 2 B(C) 1 B(C) 1 A 2 A; 1 B(C) 2 C(B) A,A) * John Nash 6

7 G = {S 1, S 2 ; π 1, π 2 }, S 1 = {(s 1 )} = {a 1, a 2,..., a n }, S 2 = {(s 2 )} = {b 1, b 2,..., b m } (1) 2 s 2 s 1 π 1 (s 1, s 2) π 1 (a i, s 2), i = 1, 2,..., n (2) s 1 s 2 π 2 (s 1, s 2) π 2 (s i, b j ), j = 1, 2,..., m (s 1, s 2) 1 2 s 2 s 1 2 s 1 s A Coordination Game 3.2 ( ) MS PowerPoint Apple Keynote PowerPoint Keynote PowerPoint (1, 1) (0, 0) Keynote (0, 0) (1, 1) Figure 7 Coordination game 7

8 *3 2 (social convention) 3.3 (the Battle of Sexes) 3.4 (Stag Hunt Game) wife romance action husband romance (1, 2) (0, 0) action (0, 0) (2, 1) Figure 9 Battle of the sexes (stag) (hare) hunter 2 hunt stag hunt hare hunter 1 hunt stag (4, 4) (0, 3) hunt hare (3, 0) (3, 3) Figure 10 Stag Hunt game (hunt stag, hunt stag) (hunt hare, hunt hare) 3.3 : The Hawk-Dove Game 3.5 ( Hawk-dove game) hawk dove *3 Thomas Schelling focal point

9 animal 2 dove hawk animal 1 dove (3, 3) (1, 5) hawk (5, 1) (0, 0) Figure 12 Hawk-Dove game (D,H) (H,D) Hawk-dove (90, 90) (86, 92) (92, 86) (76, 76) Figure 13 Exam or presentation game hawk-dove hawk-dove 4 Mixed Strategies (mixed strategies) 4.1 ( ) 4.1 (attack-defense games) A,B A B attack-defense Matching Pennies game 4.2 ( ) 1 (head) (tail) 1 2 9

10 player 2 Head Tail player 1 Head ( 1, +1) (+1, 1) Tail (+1, 1) ( 1, +1) Figure 14 Matching Pennies game Mixed Strategies 1 H p, 0 p 1 T 1 p 2 H q, 0 q 1 T 1 q p = 1 1 H q H 1 q T 1 H ( 1)q + (+1)(1 q) = 1 2q 1 T (+1)q + ( 1)(1 q) = 1 + 2q 1 2q > 1 + 2q 1 H 1 2q < 1 + 2q 1 T 1 2q = 1 + 2q q = 1/2 1 2 p = 1/2 2 q = 1/2 H 1 p = 1/2 H (p, q) = (1/2, 1/2) 1 2 p q H 1 2 1/2 H 1 2q < 1 + 2q T 10

11 2 2 1/2 H 2 H /2 T 2 q = 1/2 1 H T p = 1/2 H T p = q = 1/2 G = {S 1, S 2 ; π 1, π 2 }, S 1 = {(s 1 )} = {a 1, a 2,..., a n }, S 2 = {(s 2 )} = {b 1, b 2,..., b m } i(i = 1, 2) α i α 1 = (α 1 (H), α 1 (T )) = (p, 1 p), α 2 = (α 2 (H), α 2 (T )) = (q, 1 q) α 2 1 H E 1 (H, α 2 ) E 1 (H, α 2 ) = α 2 (H)π 1 (H, H) + α 2 (T )π 1 (H, T ) 1 T E 1 (T, α 2 ) = α 2 (H)π 1 (T, H) + α 2 (T )π 1 (T, T ) 1 α 1 2 E 2 (H, α 1 ), E 2 (T, α 1 ) α 2 1 u 1 (α 1, α 2 ) n u 1 (α 1, α 2 ) = α 1 (a i )E 1 (a i, α 2 ) i=1 2 u 2 (α 2, α 1 ) = 4.2 ( ) m α 2 (b i )E 2 (b i, α 1 ) j=1 i α i i α i i α i α = (α i, α i ) i α α i u i (αi, α i) u i (α i, α i), α 1 E 1 (a 1 ) > E 1 (a 2 ) a 1 11

12 4.3 ( ) defense defense pass defense run offense pass (0, 0) (10, 10) run (5, 5) (0, 0) Figure 15 Run-Pass game pass q pass run 0 q + 10 (1 q) = 10 10q 5 q + 0 (1 q) = 5q pass run q = 2/3 pass p pass 0 p 5 (1 p) = 5 + 5p run 10 p + 0 (1 p) = 10p p = 1/3 (1/3, 2/3) 1/3 2/3 10/3 10/3 1/3 (q 2/3) 4.4 (Penalty-Kick games) Goalie L Kicher L (0.58, 0.58) (0.95, 0.95) R (0.93, 0.93) (0.70, 0.70) Figure 16 Penalty-Kick game R 12

13 5 5.1 (Incumbent) (Challenger) (In) (Out) (Fight) (Acquiesce) Fig.5.1 (terminal history) (player function) k {a 1, a 2,..., a k } (subhistories) {a 1, a 2,..., a m, 1 m k} (perfect information) 13

14 3 h = {(In, Acquiesce), (In, F ight), (Out)} π 1 (In, Acquiesce) = 2, π 1 (In, F ight) = 0, π 1 (Out) = 1 π 2 (In, Acquiesce) = 1, π 2 (In, F ight) = 0, π 2 (Out) = 2 {In, } P l( ) = Challenger, P l(in) = Incumbent (Acquiesce) 0 1 (In) (backward induction) h a (h, a) h P l(h) (h, a) a S(h) = {a : (h, a) } 5.1 (2 ) 1 P l( ) = 1 1 S( ) = {C, D} 2 h = {C, D} P l(c) = P l(d) = 2 S(C) = {E, F } S(D) = {G, H} Fig

15 3 P l( ) = P l(c, E) = 1, P l(c) = 2 h = {(C, E, G), (C, E, H), (C, F ), D} 4 Fig i h i(p l(h) = i) P l( ) P l( ) S P l( ) S P l( ) ( ) a 1 = S P l( ) ( ) P l(a 1 ) P l(a 1 ) S P l(a1 ) S P l(a1 )(a 1 ) a 2 = S P l(a1 )(a 1 ) (a 1, a 2 ) P l(a 1, a 2 ) 5.2 h h h h (subgame) 2 15

16 Fig Fig s (subgame perfect equilibrium) j i, j N s j i s i Martin J. Osborne, An Introduction to GAME THEORY, 2009, Oxford University Press. 16

17 Fig h = (C) 2 (E, F ) 2 2 E h = (D) 2 (G, H) 2 2 H 1 P l( ) = 1 S P l( ) = {C, D} 1 2 C (2, 1) D (1, 3) 1 C (C, E) Fig h = (C, E) 1 P l(c, E) = 1 S P l(c,e) = {G, H} G 1 P l(c) = 2 S P l(c) = {E, F } E 2 2 F 1 E 2 P l( ) = 1 S P l( ) = {C, D} (D, E, G) h = (E, G) 5.2 ( ) Fig Fig

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