2

» /09031 -, 2012

7 1. 9 2. 17 2.1 17 2.2 19 2.3 21 2.4 25 2.5 25 2.6 Hawk Dove 27 2.7 33 2.8 Hawk - Dove 34 2.9 35 2.10 35 3. 37 3.1 37 4. 39 4.1 39 4.2 41 5. 46 5.1 47 5.2 55 5.3 56 6. 59 6.1 60 6.2 62 7. 65 7.1 66 7.2 68 7.3 70 8. 72 8.1 72 9. 80 9.1 80 9.2 82 9.3 83 9.4 84 10. 87 11. 97 3

.,,...,,,. 5

,,.,,..,,,.,.,,,,,. 7

1 H,,.,. ( ) ( ) ),, ).,,,.,,,,,,,,.,. H,,.,.,,., 9

., - -,,.,,,.,.,..,,.,.,,.,,.,,,, ; ( ).,,.,,.,,..,. 10

..,.,,,, ( )...,,,.,,,.,.... 11

,,.,,..,.,,,. ).. 8.,..,.,. 8.,,,.,,. 12

, 2.,...,,.,,.,,,,,..,,,.,,,,,. : U A ER A U A = ER A + v A * pay off ] 13

v A,. = 1 + * 2 + * 3,..,...,,,.,...,.,, 14

,,.,,,.,,.,,.,.,,.,,,,.,.,,. ( ). ), ( ). :, 15

.,,,.,,.,,,. 16

2 H 40,.,.,,,,. ( ) ( ) ( ) ),,, ).,,,.,,. 2.1,,,., 17

,. : ).,,,,,,. ),,. ).,,.,,,,,,. H,,.,.,,.,.. 18

2.2.»,. : i., : ii. ), 1. ), 5,., : ) 3. ) 1., : ) ) ) ) : 1,1 5,0 0,5 3,3 19 pay off., pay off : 0 5., ( )..

20 1,1 5,0 0,5 3,3 :, 1, 0.. 1,1 5,0 + 0,5 3,3, pay off 3 5. 1,1 5,0 + 0,5 + 3,3 :.,.,. 1,1 5,0 - + 0,5 + 3,3 -

,. pay offs 5 3, 3,. : ( ), ( ). 2.3. i. pay offs, ii. iii. iv. pay off., pay off. : dominated. dominant. : dominated dominant Rational player: dominant )., Rational players. Rational players. Zero-Order CKR (Common Knowledge Rationality)., Rational player. 1 st Order CKR. 21

v. Rational player:,.. ( pay offs ) 1 1 2 1 1 2 2 1 1 2 1 10,4 1,5 2 9,9 0,3 1 1 2 + 10,4 1,5 2 9,9 0,3 1 1 + 10,4 2 + 1,5 2 9,9 0,3 1 1 + 10,4 2 + 1,5-2 9,9 0,3 1 1 + 10,4 2 + 1,5-2 9,9-0,3 22

i. dominant ( 1) ii. iii. 0-order CKR. 1 st order CKR, a) A1, b) 2 (dominant ) 1 st order CKR. ( 1, 2) «dominant strategy equilibrium».,, 1 st order CKR.( 1 st order CKR) n-order CKR. B1 B2 B3 B4 A1 5,10 0,11 1,20 10,10 A2 4,0 1,1 2,0 20,0 A3 3,2 0,4 4,3 50,1 A4 2,93 0,92 0,91 100,90 / 1 1 2 2 3 3 4 4 A1 A2 4,0 B1 B2 B3 B4 + 5,10 0,11 1,20-10,10 + 1,1-2,0 20,0 A3 3,2 0,4 - + 4,3 50,1 A4 2,93-0,92 0,91 + 100,90 23 4 pay offs 1 3 2 2 3 2 4 1

0 -order CKR : 4( dominated) 1 st -order CKR : 4, 2 nd -order CKR : 4, 1 3 rd -order CKR : 1, 1 24 4 4 th -order CKR : 1, 3 2 5 th -order CKR : 2, 2. : ( 2, 2) pay offs (1,1).. 2 2 : 2 2» ( 2, 2) NASH., NASH.. NASH (+) (-)., Nash., Nash.

( 3x3 ) 1 2-1,-1 1 2 3 + 5,0 - -1,-1 25 + 172,-2 + 0,5 - -2,-2 3-2,172 - -2,-2 100,100, 3 3 dominated,.,, 3, : : 3 0< <1 : 1 0< <1 : 2 (1- ) pay offs(expected returns) : ER( 1) = 5 + (-1)(1- ) + 172 = 6 + 173 1 ER( 2) =- 2 3 3 dominated) : : 1 ER( 1)>ER( 2) 1 : 2 ER( 1)<ER( 2) 1 2-1,-1 2 + 5,0 - -1,-1 + 0,5 -

, ( ), Nash.. 2.4., ( ),.. (backward induction),. 2.5. (externalities) 100 euro, : - 100,0 100,0-26

(-) ], 0 100 (utility)., utilities (.,,.).,, :, utilities,,., ( )., Nash. 2.6. Hawk Dove, Hawk ( ) Dove ( ).,.,., - 100+x,0-y 100,0 - hawk dove hawk -2,-2 2,0 dove 0,2 1,1

: Nash (d,h) (h,d). d, d 28 ( :d) ( :d) [ ], :d. : h, : h., (h,h) (d,d). Nash fairness equilibriums. utilities pay-offs( pay-offs), (utility functions). : u A = v A *ER A + u A. v A pay-off (cf) hawk hawk -2,-2 dove ER A pay-off dove + 2,0 - + 0,2-1,1

: u = v *ER + kindness functions : f A : f B : : 1 f A, f B 1 f A =1 : f A =0 : f A =-1 : : u A = v A *ER A + f B (1+f A ) Mathew-Rabin]. Rabin: f =0 u A pay-offs( ) f =1 u A f u A. 29 f =-1 u A f A =[ (s,s B ) E B (s B ) H B (s B ) - L B (s B )], f u A s s

f A =[ (s,s ) E B (s ) H B (s ) - L B (s )], f A =[ (s,s B ) E B (s B ) H B (s B ) - L B (s B )], f A =[ (s,s ) E B (s ) H B (s ) - L B (s )], : (j,k) : :j :k s s s s s s E B (j) : j (entitlement). pay-offs. H B (j) : :j L B (j) : :j H f A, : (s,s ) E B (s ) H B (s )-L B (s ) (s,s B ) E B (s B ) H B (s B )-L B (s B ) (s,s ) E B (s ) H B (s )-L B (s ) (s,s B ) E B (s B ) H B (s B ) - L B (s B ) f B. A (s,s ) E B (s ) H B (s )-L B (s ) A (s,s B ) E A (s B ) H B (s A )-L B (s A ) A (s,s ) E B (s ) H B (s B )-L B (s B ) (s,s B ) E B (s B ) H B (s B ) - L B (s B ) 30

(j): pay off pay off. hawk-dove (h,h) pay off (- 2,-2). : (s A ) : s B 2 ( hawk) (s B ) : s B (0+1)/2 = 0,5 ( s dove) (s A )=2 (s )=(0+1)/2= 0,5 : (s,s ) = -2, (s,s B ) = 0 (s,s ) = 2, (s,s B ) = 1 ) (s,s ) = -2, (s,s B ) = 2 (s,s ) = 0, (s,s B ) = 1 H B (s ) = 1, pay off s B dove H B (s ) = 2, pay off s B hawk L B (s ) = 0, pay off s B dove L B (s ) = -2, pay off s B hawk H (s ) = 2, pay off s dove H (s ) = 1, pay off s hawk 31 s A

L (s ) = -2, pay off s dove L (s ) = 0, pay off s hawk f A,f B : f A = f B = : u A = v A *ER A + f B (1+f A ) u A 2x2 ( pay offs ) v A f B (1+f A ) 1= u A = v A *ER A + f B (1+f A ) = 1 1 1 1-1 0 0,5 0,5-1 -0,5 0 0,5 u = v *ER + f (1+f ) = 32-2 v A 2 v A -0,5 v A +0,75-2v -0,5 2v v + 0,75

: hawk dove hawk -2 v A, -2v 2 v A, -0,5 dove -0,5, 2v v A +0,75,v +0,75. : : H : D : -2 v A >-0,5, v A <0,25 : D v A >0,25 : 2 v A > v A +0,75, v A >0,75 : D v A <0,75 Nash v A,v,. 2.7. (evolutionary theory- ) p, : ER(1) = p 11 + (1-p) 12 ER(2) = p 12 + (1-p) 22 d 12 (p) =ER(1) ER(2), 1 2 1 11 12 2 12 22 33

, d 12 (p). 2.8. hawk-dove d 12 (p) = -2p + 2(1-p) (1-p) = [-3p + 1] p<1/3 d 12 (1/3)>0, hawk p p>1/3 d 12 (1/3)<0, dove p d 1 0 0 0.333 0.666 0.999-1 p Nash.. ij units of evolutionary fitness,. Nash,. hawk dove hawk -2 2 dove 0 1 Nash - 34

, Nash. -. 2.9. ( Repeated games and reputation building)..,. : (finite,kind of player).,,. 2.10.,.,,,,,,.,,,,.,, 35

,..,,,.,.,,,.,,,.,,.,. 36

3.,. : -,, purification,.. 3.1..,.,,.,,.. textbooks. 37

,...,...,.,..,.,.. 38

4 :.,,,.,. subgame (Selten (1965)). 4.1, 1 2, 2,.,, k i, i = 1, 2., ki, x i, i = 1, 2. i i (x i, x j, k i ), i = 1, 2, j i. i, [k i, x i (k i, k j )], ki xi, ki. subgame perfect equilibrium (SPE), [ki*, xi* (ki, kj)], i = 1, 2, ( ) (k i, k j ), x i * (k i, k j ) = arg max x i [x, x j * (k j, k i ); k i ], ( ) k i * = arg max ki i [x i * (k i, k j *), x j (k j *, k i ); k i ]., x i,, k i., k i xi. 39

, k i,,, subgame. k i x i SPE, k i x i.,, k i x j k i. k i,,.. i, i =1,2,, i / x j 0, SPE, x i * (k i,k j ). : x j * (k j *,k i *) / k i 0, i [ x i * (k i *,k j *), x j * (k j *,k i *); k i * ] / k i 0. : SPE [ x i * (k i *, k j *), k i * ], i = 1,2. ( ) d i / dk i = 0, i / dk i ( i / x i )( x i * / k i ) + ( i / x j )( x j * / k i ) + ( i / k i ) = 0. ( ), i / x i = 0, i / dk i = ( i / x j )( x j * / k i ) + ( i / k i ) = 0.,, (k i, x i ), i = 1,2,. Nash, i = 1,2, = : k i k i *. : 40

,. 41,, SPE.,,.,., :,.,.,,. 4.2, (Brander & Spencer (1985))., 1 2,., 0 p = 1 - Q, Q.., ( ), t i,., Cournot : outputs q i p = 1 q 1 q 2.

c i (q i ;t i ) = t i q i. i,,,,.,,. ( ) ( ).. ( G i q i t i i, x i k i ). SPE : t 1 = t 2 = -1/5.,,. \ 4.1 42

4.1... R1 R2,...,,..,.,..,, :. 4.2 43

. (Eaton Grossman (1986)), Bertrand: p i q i (p i,p j ) = 1 p i + ap j, 0 < a < 1., ( p i x i, : 4.2.,,.,,. ( ),,.,,.,.,, Cournot.. Cournot Bertrand. 44

,..,,.,,.,,. Cournot Bertrand. ( ),. Cournot, Bertrand. Cournot.. Cournot Bertrand (., ),.,, (.,, ). Cournot Bertrand.,. 45

5...,...,, -.,.. :. /,. : 46

( ).., ( ),,.,,. 5.1 Milgrom and Roberts (1982).., :,., Milgrom and Roberts -. :,.., -., ( )., 47

.,... :,. :,c (t) c (L) < c(h) L. -,., p,. : 48 D.., 0 m (t),, d (t) e (t)., m (t) > d (t) e (H) > 0 > e (L). m (t) : (p,t) /... t {L,H} b 0 t b 0 L + b 0 H = 1. P : {L,H} [0, ). b t : [0, ) [0,1] t, p., p, b L (p) + b H (p) = 1. : E: [0, ) {0,1}

, «1» «0»., :,. b (b L, b H ): (Kreps and Wilson 1982) (Cho and Kreps 1987)., {P, E, b} : ( 1) 2) 49

3) ayes 3 2. (P (L) = P(H)),, (P(L) P(H)),., p P(H)}. 50 {P(L),.,.» : 4) p,,.,

,. {P, E, b} ( 1)- ( 2) ( 4).,.,. 1) 2) ( 2).,.. /,., p, p (0,p) p < p`, ( 2) single-crossing property SCP 51

(SCP), two price-entry pairs, ( ).,.,,, (SCP).. t., : p m L < p m H. p p.,.. 5.1. 52

(ii) p m L p {P, E, b} p = P(L) < P(H) = p m H E(P(L)) = 0 < 1 = E(P(H)). p m L < p {P, E, b} p m L = P(L) < P(H) = p m H E(P(L)) = 0 < 1 = E(P(H)). (iii) p m L p b 0 L b L P(L) = P(H) = p. 53 (iv) E(P(L)) = 0..., p m H., p m H. p m L p. p m L. p m L < p p m L. b 0 L b L. b 0 L < b L p m t.,, P(L) < p m L., (

)..., P(H) < p m H, P(H) < p m H p m L.,.,.....,,. Pareto. ( (iii) ),.. p m L p 54.

5.2,...,..., ( ).,,.,,..,.. SPE,.,,.,. 3.1,.,. 55

,.,. (Roberts 1985),,,. 5.3..,.,, -., -...,,.,.,. 56

..,,.,. ; ( ) ;... US.,. :?, ;.,.,. (Mc Gee 1958 Areeda- Turner Rule 1975).,.,., 57

, Areeda - Turner:.. 58

6 : ( ) :.,.,,,..,., :...,.,.,,,.. 59

..,, (. Stigler (1964)).,,.,,,,...,,.,. (Rotemberg Saloner (1986)).,,. 6.1 Bertrand. t 1 a t, 60

., : a t., 61., t i (p t i, p t j).,. :, t s (s 1 i, s 2 i, ), t. s = (s i, s j ). (0,1). SPE., SPE. SPE SPE. 6.1: SPE..

, 1.,, 1. Rotemberg Saloner....,.,.,,.. 6.2.. Rotemberg Saloner.,,.. (, ). ( ), 62

.,., Rotemberg-Saloner,....,.,,.,.. ( )....,,. : SPE, (,, ), 63

.. :.,.. 64

7 :.,,...,,.. Varian (1980)..,. Varian,.. 65

.,,. Harsanyi (1973) Nash.,., Varian.. 7.1. > 2 c. v, v > c > 0...,..,. i p - i i, i : 66

[c, v]. i F i F_ i, i, {p1,...,pn} ash, i 67, (F1,...FN) ash, i Fi Nash (F 1, F N ) Fi = F, i = 1,...N. 7.1 : ) Nash. (B) F Nash. :. -

.,,.,,.,,. ;, ;,, ;,. 7.2 arsanyi (1973).,,.. i t i E [0, 1].,. i.i.d [0, 1]. : i t i c (t i ), c 0 < c(0) < c(1) < u., 68

., Bayesian [0, 1]., c., i i (t i) [0, 1] [c (0), v ]. [ P 1,...,F N ], P -i i P -i (t -i ) 1) t -t. i t i p i t -i i (p i, F i (t -i ), t i ), i (5.1), c c(t i ). T [ P 1,..., F N ) Nash, i t, Nash P i (t i ) = P (t i ), i t i. H 7.2 (i) Nash,. (ii) c E (0, v) > 0, > 0, [ c (t) c / < t, c.., Nash, 69

c t, c. Varian. 7.3.,.,,..,,,.,.,., 70 SPE.,,,.,.,,.

.,.,. 71

8.,.,.,. 8.1,,. SPE. ),,.,., Pareto. Rotemberg Saloner (1986),,. Green Porter (1984). 72

Cournot ( Green Porter), shock.., shock. Cournot,.,,. /. Green Porter ( shock ) Cournot. Bertrand.. :, (1- ) 73.,,.,.,

p,.,., t = 1,2,.., t x t (0,1). «sunspot»,. t h t = (a 1,, a t-1 ), (i) a r = (p,x) r,, p > 0 x, (ii) a r = (~,x) 0 r. i t. (SE) ( ) i j, I j = 1, 2,., Bertrand: p i = 0, I = 1, 2. SE.,, SPE p 1 = p 2 = 1. SE. SE, p i = 1, i p i = 1. j i.. Green Porter, 74

.,. p i = 1 p i = 0.. t a t-1 = (~,x), x <,. SE,. ( ) G (T) (i) (ii) (1,x) x (iii) (1,x) k ( +1) (~,x) k. f T,, x <. V T,, G (T) 75 (6.1) RHS,,, x <,, 1 - = (1- ) + (1- ), x,. 1 -.

, f T, (h) = 1 ( p1 = p2 = 0 Nash )., LHS. 2. (~,x) (1 ), x V T,, x < +1 V,. (6,3) 8.1 i. ( ), ii., : ( ) = min {T (1 ) / (1 2 ) (1 1}. (i) (6,2) LHS 6,4 (, ) 76

RHS,, = = 1., (6,7). ii. ( ) V. `, ` = (1 ) / (1 `) ( `, `) V., ` 1. (6.6) (, ), ( `, `),., V, = V (6.2) (, ) ( `, `)., = V., (6,2) V, V (6,7) =, = (1 ) / (1 2. (6.2), < V, = / (1 ). 77 ).,, :. (ii) -. Abreu, Pearce Stacchetti (1986),, ( ).,. =,.., Nash

. Abreu, Pearce Stacchetti (1986)., Cournot,,.. : ( Cournot)., Green Porter.. Fudenberg, Levine Maskin (1993) 1, ( ) Pareto.,..,,. 78

9. /, -., :.. Cournot, Bertrand Stackelberg, " " /.,,. 9.1 :, /.,, 1 2, q 1 q 2, P(q 1 +q 2 ) i (q i, q j ) = q i P(q 1 +q 2 )-c i (q i ). i, qi C, j., i q i,. q i C = v i (q i ). q i *, i=1.2, : 80

q i * = argmaxq i i [q i, v i (q i )]and v i (q i *)=q j * i=1.2.,,., vi q i ( ) v., v / / : v = 1, 0-1,, Cournot,, v,.,,.,,. / (., ). " ". " " («reduced form»).,..,, Nash,., Nash " "., 81

.,. 9.2 :,. :, /.,, /., /, /., /. /. / ).,, /. /,., ( )., 82

-.,, /., / /.,.,, Green - Porter.. 9.3 /,.,,.., 4 6.,,, (a posteriori).,,, Nash.. 2. / ( ), SPE..,., 83

. Nash..,.,.. ',.,.,.,,,. 9.4, ;,..,,.,..,.,,. 84

,.,, ( ).,,. /., 2, ) (Cournot vs Bertrand)., ', /.,., 7,.,.,.,..., ;. 85

,..,.,... 86

10 3.1: (i) (ii) p m L p {P, E, b} p = P(L) < P(H) = p m H E(P(L)) = 0 < 1 = E(P(H)). p m L < p {P, E, b} p m L = P(L) < P(H) = p m H E(P(L)) = 0 < 1 = E(P(H)). (iii) p m L p b 0 L b L p [p,p m L] P(L) = P(H) = p. (iv) P(L) = P(H) [p,p m L] E(P(L)) = 0. (i) p m L < p {P, E, b} : P (ii), E(p) = 1 p p m L, b L (p) = 0, p p m L b L (p m L) = 1. (E1) E(4). p m L p {P, E, b} : P (ii), E(p) = 1 p p m L, b L (p) = 0, p p m L b L (p m L) = 1. ( 1) t = H t=l p m L = p. (E2) (E4). 1 = L p m L > p. p E(p) = 0, SCP (ii) : V (p, 0, L) > V (p, 0, L). p E(p) = 1, 87

p m L. V (p, 0, L) > V (p m L, 1, L). p V (p, 0, L) > V (p m L, 1, L). (iii) {P, E, b}., ( 2) 3) E(P(L)) = 0 < 1 = E(P(H)). ) p m H P(H) p m H V (P(H), 1, H) < V (p m H, 1, H) < V (p m H, E(p m H), H). 1., P (L) V(P(L), 0, H) V (p m H, 1, H) P(L) (p,p). p m L p. V p V (p, 0, L) > V (p, 0, L) p V (p, 0, L) p > p. p p SCP V (p, 0, L) > V (p, 0, L)., P(L) [p,p), > u V (p, 0, L) > V (P(L), 0, L) V (p, 0, H) < V (p m H, 1, H). 4 b L (p ) = 1. (p ) = 0 V (p, 0, L) > V (P(L), 0, L) 1., P(L) [p,p) P(L) (p,p), P(L) = p. p m L V (P(L), 0, L) V (p m L, 0, H) < V (p m H, 1, H)., 4 b L (p m L ) = 1 p m L, P(L) = p m L. (iii) p m L p b L b L. p [p, p m L ] {P, E, b} : P(L) = P(H) = p E(p) = 0 p p E(p) = 1 p > p b L (p) = b 0 L p p b L (p) = 0 p > p. {P, E, b} (E1) (E3). b 4 p < p 4. 88

p V (p, 0, H) = V (p, 0, H). p (p,p ], V (p, 0, H) V (p, 0, H), b L (p) = 0 4. p > p SCP V (p, 0, L) < V (p, 0, L) II p p > p m L., V (p, 0, L) < V (p, 0, L) V (p, 0, L) < V (p, 0, L) b L (p) = 0 4. {P, E, b}. (iv) {P, E, b}. p. E(p ) 0, t {L,H} p m L p t p m t. u(l) V (p, 0, L) u(h) V (p, 0, H)., p p p m H. p p m H... p p m H p u(l), 2 4 E(p ) = 0 > 0. V (p, E(p ), L) > u(l) 1., > 0 p > p m L. V (p, 0, H) < u(h) V (p, 0, L) > u(l) 4 2 E(p ) = 0., V (p, E(p ), L) > u(l) 1 L. p p m L p [p, p m L ]. 4.1. SPE. p t i = p t i = p(a t ) 89

SPE. : p i I = L, H,, 0.., : p(h) p(l) L. p(h) : {p(h)h + [wp(h)h + (1 w) p(l)l] / (1 )} / 2 p(h)h LHS RHS ( )., p(l) {p(l)l + [wp(h)h + (1 w)p(l)l] / (1 )} / 2 p(l)l, p i s. SPE., SPE SPE., SPE p i p j { p i, p j }. t SPE p i < p j j, SPE. i. t SPE p i = p j, SPE., V 90

SPE ( SPE, ). V ( ). V t a t, ( 1, p i 1, p j 1 ),, (a t-1, p i t-1, p j t-1 ). t, V.,.,, V., a t (a t ) = argmax { p a t s.t. (p a t + V) / 2 p a t p 1 (8.1), t., V = [w (H)H + (1 w) (L)L] / (1 ). V 8.1 (x) = p(x), x = L, H p(x). 5.1 ) ash. ) Nash f. (i) p(f) = u (ii) [p(f) c](u / N + I) = [u c] (U / N) (iii) [p c](u/n + (1 F(p)) N-1 I) = [u c](u/n) p {p(f), p(f)}. ) k p 2 k N. p > c, p, [p c](u/n + I) > [p c](u/n + I/k)., p = c, p > p (p c)(u/n) > 0. k = 1. 91

p + p +, [p + c](u/n + I) > [p c](u/n + I). ) Nash (i) (iii)., p(f) > c., F. p F,, p -. p -. p p p(f) > c. p(f) < u.,., p(f),., [p(f), p(f)]., p(f) = u.,,., F, [p(f) c](u/n + I) = [u c](u/n). F (p(f),p(f)). (p1, p2) p(f) p 2 F(p 1 ) = F(p 2 ). (p 1, p 2 ) 0., [p 1, p 1 ] p 2 -. (p 1, p 2 ),. [p(f), p(f)]. [p(f), p(f)] [u c](u/n). 92

[1 F(p)] N-1., [p c](u/n + (1 F(p)) N-1 I) = [u c])u/n) p [p(f), p(f)]. (i) (iii) Nash, (i) (iii) Nash. (iii) [1 F(p)] N-1 = (u p)u/n (p c)i, p [p(f), p(f)] RHS 0 1, F(p) (0,1). (i) (iii) F(p(F)) = 0 < 1 = F(p(F)) F (p) > 0 p [p(f), p(f)] F. F ash -1 F (p) (i) - (iii). [p(f), p(f)] F., p(f) p(f) p(f). F, F. 5.2 (i), Nash. (ii) c (0, u) > 0 > 0 c(t) c < t, P, P -1 (x) F c c. (i) P: [0,1] [c(0), u] 93

, P P(t) > c(t) P (t) > 0 t. Nash. (t,t) T P(t) P 1 (t,t) = [P(t) c(t)]{u/n + [1 t] N-1 I} [1 t] N-1 P(t). P(t) t, P(t) p > u. Nash - (t, t) (t,t) t,t [0,1] (8.4) 2(t,t) = -[P(t) c(t)][n 1][1 t] N-2 I + {U/N + [1 t] N-1 I}P (t) (8.5) 8.2 2 (t, t) = 0 t [0,1] 2 (x, x) = 0 12 (y, x) (8.5)., (8.4) 94

Nash. t P(t). (ii), c (0, u) P c c. P c P c (t) = F -1 C(t) t [0,1] P c. P c (8.2) (8.3) c(t) c. P c (1) = F -1 c(1) = i., (iii) 5.1 8.6. P c (t) p = P c (t), F = F c t = F c (P c (t)) P c (8.2) (8.2) (8.3).P: [0,1] [0, u]. c( ) P( ) = (c( )) c( ) c( ) c, (c)= P c. > 0 > 0, c( ) c <, (c( ))) P c <. Nash 95

[1] Abreu, D., D. Pearce and E. Stacchetti (1985), "Optimal Cartel Equilibria with Imperfect Monitoring," Journal of Economic Theory, 39, 251-269. [2] Areeda, P. and D. Turner, "Predatory Pricing and Related Practices Under Section 2 of the Sherman Act," Harvard Law Review, 88 (December 1975), 697-733. [3] Bagwell, K. and G. Ramey (1988), "Advertising and Limit Pricing," Rand Journal of Economics, 19, 59-71. [4] Bagwell, K. and G. Ramey (1991), "Oligopoly Limit Pricing," Rand Journal of Economics, 22, 155-72. [5] Bagwell, K and R. Staiger (1997), "Collusion over the Business Cycle," Rand Journal of Economics, 28, 82-106. [6] Bain, J. (1949), "A Note on Pricing in Monopoly and Oligopoly," American Economic Review, 39: 448-64. [7] Borenstein, S. and A. Shephard (1996), "Dynamic Pricing in Retail Gasoline Markets," Rand Journal of Economics, 27, 429-451. [8] Brander J. and B. Spencer (1985), "Export Subsidies and International Market Share Rivalry," Journal of International Economics, 18, 83-100. [9] Bulow, J., J. Geanakoplos and P. Klemperer (1985), "Multimarket Oligopoly: Strategic Substitutes and Complements," Journal of Political Economy, 93 (June 1985), 488-511. [10] Cho, I-K. and D. Kreps (1987), "Signalling Games and Stable Equilibria," Quarterly Journal of Economics, 102: 179-221. [11] Dixit, A. (1980), "The Role of Investment in Entry Deterrence," Economic Journal, 90, 95-106. [12] Eaton, J. and G. Grossman (1986), "Optimal Trade and Industrial Policy Under Oligopoly," Quarterly Journal of Economics, 101, 383-406. [13] Fellner, W. (1949), Competition Among the Few, New York: Knopf. [14] Friedman, J. (1990), Game Theory with Applications to Economics, Oxford University Press. [15] Friedman, J. (1971), "A Non-cooperative Equilibrium in Supergames," Review of Economic Studies, 38. 97

[16] Fudenberg, D. and J. Tirole (1984), "The Fat Cat Effect, the Puppy Dog Ploy and the Lean and Hungry Look," American Economic Review, Papers and proceedings, 74, 361-368. [17] Fudenberg, D. and J. Tirole (1986), Dynamic Models of Oligopoly. Chur: Harwood Academic Publishers. [18] Fudenberg, D. and J. Tirole (1987), "Understanding Rent Dissipation: On the Use of Game Theory in Industrial Organization," American Economic Review, Papers and proceedings, 77, 176-183. [19] Fudenberg, D., D. Levine and E. Maskin (1994), "The Folk Theorem with Imperfect Public Information," Econometrica, 62, 997-1039. [20] Green, E. and R. Porter, "Noncooperative Collusion Under Imperfect Price Information," Econometrica, 52 (January 1984), 87-100. [21] Haltiwanger, J. and J. Harrington (1991), "The Impact of Cyclical Demand Movements on Collusive Behavior," Rand Journal of Economics, 22, 89-106. [22] Harrington, J. (1986), "Limit Pricing when the Potential Entrant is Uncertain of Its Cost Function," Econometrica, 54: 429-37. [23] Harsanyi, J. (1967-68), "Games with Incomplete Information Played by 'Bayesian' Players," Parts I, II, and III, Management Science, 14, 159-82, 320-24, 486-502. [24] Harsanyi, J. (1973), "Games with Randomly Disturbed Payoffs: A New Rationale for Mixed Strategy Equilibrium Points," International Journal oj Game Theory, 2: 1-23. [25] Kadiyali, V., "Entry, Its Deterrence, and Its Accommodation: A Study of the U.S. Photographic Film Industry," Rand Journal of Economics, 27.3 (Autumn 1996), 452-478. [26] Kreps, D. and J. Scheinkman, "Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes," Bell Journal of Economics, 14 (Autumn 1983), 326-337. [27] Kreps, D. and R. Wilson (1982a), "Sequential Equilibria," Econometrica, 50: 863-894. [28] Kreps, D. and R. Wilson (1982b), "Reputation and Incomplete Information," Journal of Economic Theory, 27: 253-79. [29] McGee, J. (1958), "Predatory Price Cutting: The Standard Oil (N.J.) Case," Journal of Law and Economics, 1, 137-169. [30] Milgrom, P. and J. Roberts (1982a), "Limit Pricing and Entry under Incomplete Information: An Equilibrium Analysis," Econometrica, 50: 443-59. 98

[31] Milgrom, P. and J. Roberts (1982b), "Predation, Reputation and Entry Deterrence," Journal of Economic Theory, 27: 280-312. [32] Modigliani, F. (1958), "New Developments on the Oligopoly Front," Journal of Political Economy, 66, 215-232. [33] Nash, J. (1950), "Equilibrium Points in n-person Games," Proceedings of the National Academy of Sciences, 36: 48-49. [34] Porter, R. (1983), "A Study of Cartel Stability: The Joint Executive Committee, 1880-1886" Bell Journal of Economics, 14, 301-314. [35] Radner, R. (1981), "Monitoring Cooperative Agreements in a Repeated Principal-Agent Relationship," Econometrica, vol 49, No. 5, 1127-1148. [36] Roberts, J. (1985), "A Signaling Model of Predatory Pricing," Oxford Economic Papers, Supplement, 38, 75-93. [37] Rosenthal, R. (1980), "A Model in Which an Increase in the Number of Sellers Leads to a Higher Price," Econometrica, 48, 1575-1580. [38] Rotemberg J. and G. Saloner, "A Supergame-Theoretic Model of Business Cycles and Price Wars During Booms," American Economic Review, 76 (June 1986), 390-407. [39] Rubinstein, A. (1979), "Offenses that May Have Been Committed by- Accident An Optimal Policy of Retribution," in S. Brams, A. Schotter and G. Schwodiauer (eds), Applied Game Theory: 236-53. Wurzburg, Vienna: Physica-Verlag. [40] Selten, R. (1965), "Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit," Zeitschrift fur die gesamte Staatswissenschaft, 12, 301-324. [41] Shilony, Y. (1977), "Mixed Pricing in Oligopoly," Journal of Economic Theory, 14: 373-88. [42] Spence, M. (1977), "Entry, Capacity, Investment and Oligopolistic Pricing," Bell Journal of Economics, 8, 534-544. [43] Stigler, G. (1964), "A Theory of Oligopoly," Journal of Political Economy, 72, 44-61. [44] Tirole, J. (1988), The Theory of Industrial Organization, Cambridge: MIT Press. [45] Varian, H. (1980), "A Model of Sales," American Economic Review, 70, 651-659. [46] Villas-Boas, J. Miguel (1995), "Models of Competitive Price Promotions: Some Empirical Evidence from the Coffee and Saltine Crackers Markets," Journal of Economics and Management Strategy, 4: 85-107. 99