2
3
4
Microeconomics aims to model economic activity as an interaction of individual economic agents pursuing their private interests ---M-W-G (1995) Game theory strategic interaction 5
2/3 n 0-100 x i TA TA x1 + x2 +... + xn x = n 100 x 2 x / 3 i 6
Game theory strategic interaction 7
A B C 1 2 3 A 2 0 1 B 1 2 0 C 0 1 2 A 1 B 1 C B 2 C 2 A C 3 A 3 B 8
1 (A,B) (X,C) (A,B) (A,C) C 1 2 3 A 2 0 1 B 1 2 0 C 0 1 2 9
1 (A,B) (X,C) (A,C) C; (B,C) B 1 B 3 1 A 1 2 3 A 2 0 1 B 1 2 0 C 0 1 2 10
Microeconomics aims to model economic activity as an interaction of individual economic agents pursuing their private interests ---M-W-G (1995) Game theory strategic interaction 11
: X B u(x) ( ) x X x * B x B x * x Max x B u(x) 12
Common Knowledge M M M 13
M 14
AWell Well-known Fable 100 100 99. 15
AWell Well-known Fable 16
n=2 i i j i=1,2; i j 2 > 1 17
n=3 i i j i=1,2,3; 123 i j 18
AWell Well-known Fable 99 100 19
i 20
21
Nobel Prize in Economics since 1969 1994 John F. Nash Jr., John C. Harsanyi, Reinhard Selten 1996 James A. Mirrlees, William Vickrey 2001 George A. Akerlof, A. Michael Spence, Joseph E. Stiglitz 2005 Robert J. Aumann, Thomas C. Schelling 2007 Leonid Hurwicz, Eric S. Maskin, Roger B. Myerson 22
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A Strategic g Game or Game in Strategic Form B (Dominated Strategy) 24
A. 4. 25
A. 1 2 (d) (c) (s 1, s 2 ) (c,c) c) (c,d), (d,c), (d,d) u 1 (c,c)= -1; u 1 (c,d)= -4; u 1 (d,c)= 0; u 1 (d,d)= -3; u 2 (c,c)= -1; u 2 (c,d)= 0; u 2 (d,c)= -4; u 2 (d,d)= -3; 26
G = ( S, u ) = A. ( ) N i i i 1 i=1,2,n I {1,2,N} i s i S i s=(s ( 1, s 2 2,,,s N ) S=S 1 S 2 S N u i (s) : S R i=1,2,n 27
I={1,2} S 1 {c d}; { S 2 {c d}; c: Quiet d:fink S S 1 S 2 {(c,c); (c,d); (d,c); (d,d) } u 1 (c,c)= -1; u 1 (c,d)= -4; u 1 (d,c)= 0; u 1 (d,d)= -3; u 2 (c,c)= c)= -1; u 2 (c,d)= 0; u 2 (d,c)= -4; u 2 (d,d)= d)= -3; (d,c) 1 (c,c) 1 (d,d) 1 (c, d) (c,d) 2 (c,c) c) 2 (d,d) d) 2 (d, c) 28
Suspect 1 Quiet (c) Fink (d) Suspect 2 Quiet t( (c) Fink (d) -1, -1-4, 0 0, -4-3, -3 29
Boxed Pig Game 2 10 1 4 4 3 press Small pig wait Big pig press wait (5, 1 ) (4, 4) (9, -1) (0, 0) 30
Bach or Stravinsky Bach, Stravinsky Stravinsky. 1 Bach 2 Stravinsky Player 2 Player 1 Bach Stravinsky Bach Stravinsky 2, 1 0, 0 0, 0 1, 2 31
the Stag Hunt stag Hunter 2 hare Hunter 1 stag hare (2, 2) (0, 1) (1, 0) (1, 1) 32
.. 2 1 Player 2 Player 1 Head Tail Head Tail -1, 1 1, -1 1, -1-1, 1 33
100 s i, s 1 + s 2 100 s 1 + s 2 >100, 0 I={1,2} s i [0, 100], i=1,2 s=(s 1, s 2 ) s if s +s 100 = i = 12 0 if +s >100 1, 1 1 2 ui () s s1 2 34
2/3 I={1,2,,N} x i [0, [, 100], i=1,2,n,, x=(x 1, x 2,,x N ) u ( x) = 100 x 2 x/ 3 i x 1+ x2 +... + xn x = n i 35
: N I i, i=1,2,n U i =(1-t) F i + λ C i λ 0 C i S tf i I i α β F ( ) i = A ρi ki I α > 0, β > 0, α + β < 1 i Fi (1 t) = r kk i N N * k = K = k i i= 1 i= 1 i 36
1 u 1 (d,c)= 0 > u 1 (c,c)= -1; u 1 (d,d)= -3 > u 1 (c,d)= -4; 2 u 2 (c, d)= 0 > u 2 (c, c)= -1 u 2 2( (d, d)= ) -3 > u 2 2( (d, c)= ) -4; d, d Suspect 1 Quiet (c) Suspect 2 Fink (d) Quiet (c) -1, -1-4, 0 Fink (d) 0, -4-3, -3 37
Dominant strategy {S 1, S 2,..., S n, u 1, u 2,..., u n }, s * i* s -i S -i. s i S i. u i (s i *, s -i ) u i (s i, s -i ) 38
(Vickrey ) N N i v i v i (private value) i b b i 39
(Vickrey ) SPA(c, r) c r b i (v ) { No} [ r, ) v [0, 1] 0 0 v - z - c if b > z ub ( i, zi, c) = pb ( i)( vi zi) c if bi = zi - c if bi < zi z = Max {b -i } p(b i ) i i i i i Jijun Xia, Spring 2009 40
(Vickrey ) 1 c=0, r=0, b(v)=v z z < v u(b=v)=v-z ) u(b v) ) z > v u(b=v)=0 u(b v) z = v u(b=v)=0 =u(b v) w 2 = v (2) v-z r z v b r v-z v z b Jijun Xia, Spring 2009 41
(Vickrey ) 2 SPA(c=0, r>0) v-z r v if v r bv () = No if v < r z v b v-z v r z b SPA(r>0, c=0) i Jijun Xia, Spring 2009 42
(Vickrey ) 3 SPA(c>0, SPA( 0 r 0) b(v)=v c+r 1 v 0 v 0 0 (v 0 -r) G(v 0 )-c=0 bv () v if v v0 ( r, c) = No if v < v0 ( r, c) Jijun Xia, Spring 2009 43
( ) ) button auction b i (v ) { No} [ r, ) v [0, 1] c=0, r=0, b(v)=v w E = v (2) =w 2 E[w E ]= E[w 2 ] Jijun Xia, Spring 2008 44
( ) ) 1 c=0, r=0, b(v)=v z z < v u(b=v)=v-z ) u(b v) ) z > v u(b=v)=0 u(b v) z = v u(b=v)=0 =u(b v) w E = v (2) v-z r z v b r v-z v z b Jijun Xia, Spring 2009 45
u S (p,w)= 4 > u S (p, p)= 1; u S (w,w)= w)= 0 > u S (w,p)= -1; 1 Press wait 2 Press Press press Small pig wait Big pig press wait (5, 1 ) (4, 4) (9,-1) (0, 0) 1 r 46
2 u 2 (L,B)= 2 > u 2 (L, T)= 1; u 2 (R,B)= 1 > u 2 (R, T)= 0; Player 2 T M B 0, 1 2, 3 1, 2 Player 1 L R 3, 0 1, 0 0, 1 47
{S 1, S 2,..., S n, u 1, u 2,..., u n }, s i S i i s s i s -i S -i. s i u i (s i ', s -i ) < u i (s i, s -i ) Player 1 L R Player 2 T M B 0, 1 2, 3 1, 2 3, 0 1, 0 0, 1 48
T 2 T 2 2 T R 1 R 1 2 2 Player 2 T M B 0, 1 2, 3 1, 2 Player 1 L R 2 r 3, 0 1, 0 0, 1 1 r 49
T 2 T 2 T R 1 R R 1 R B 2 B B 2 1 2 2 1 2 : (L, M) Player 2 T M 0, 1 2, 3 1, 2 3, 0 1, 0 0, 1 2 Player 1 L R 2 r B 1 r 3r 50
51
: (press, wait) ) press Small pig wait Big pig press wait (5, 1 ) (4, 4) (9,-1) (0, 0) 2 r 1 r 52
(d, d) ) Suspect 2 Quiet (c) Fink (d) Suspect 1 Quiet (c) -1, -1-4, 0 1 r Fink (d) 0, -4-3, -3 1 r 53
2/3 2/3 ui( x) = 100 xi 2 x/ 3 x -i 2/3 200/3 x i = 200/3, 100>x i > 200/3 i, i 54
{S 1, S 2,..., S n, u 1, u 2,..., u n }, s i, s i " S i i s i ', s i u i (s i ', s -i ) u i (s i ", s -i ) for all s -i S -i. u i (s i ', s -i ) < u i (s i ", s -i ) for some s -i S -i Player 1 L R Player 2 T M B 0, 1 2, 3 1, 2 3, 1 1, 0 0, 1 55
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Player 2 1 L M R Player 1 U 2, 1 1, 1 0, 0 C 1, 2 3, 1 2, 1 D 2, -2 1, -1-1,-1 1 2 r 2 r 2 L M R U 2, 1 1, 1 0, 0 3 r 1 r Player 1 C 1, 2 3, 1 2, 1 D 2, -2 1, -1-1,-1 1 r 57
Player 2 no solution Bach Stravinsky Player 1 Bach Stravinsky 2, 1 0, 0 0, 0 1, 2 Hunter 2 stag hare Hunter 1 stag hare (2, 2) (0, 1) (1, 0) (1, 1) 58