Dynamics and Stability of Constitutions, Coalition

7/30/2019 Dynamics and Stability of Constitutions, Coalition

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Aeria Ei Review 2012, 102(4): 14461476hp://x.i.r/10.1257/aer.102.4.1446

1446

C h b h . N, h

w w f h . Y,

h b h w-k , h - h b . F ,

h b h j h

h b z -

. I , h h j b h

w h . M b , b

h, , z , z

h b h h.

W b wk h -

. C b .

I sae. A wk bh . I , ( ,b h ) hw h (.., whh b h h

Dynamics and Stability of Constitutions,

Coalitions, and Clubs

By Daron Acemoglu, Georgy Egorov, and Konstantin Sonin*

I yai lleive eisi aki, urre eisis eeriehe uure isribui pliial pwer a iuee uure ei-sis. We evelp a eeral raewrk suy his lass prb-les. Uer ayliiy, we haraerize yaially sable saes as

uis he iiial sae a bai w eeral isihs. Firs,a sial arraee is ae sable by he isabiliy alera-ive arraees ha are preerre by sufiely pwerul rups.Se, efiey-ehai haes ay be resise beause

urher haes hey will eeer. We use his raewrk aa-lyze yais pliial rihs i a siey wih iere ypes exreis views. (JEL D71, D72, K10)

* A: D E, Mh I Th, 50 M D, B E52,R 380B, Cb, MA 02142-1347 (-: @.); E: D M E D S, K Sh M, Nhw U, 2001 Sh R., Jb C,R 539, E, IL 60208 (-: [email protected].); S: Nw E Sh, 47Nkhk ., O 922, Mw, 117418, R (-: k@.). W hk h , Jh D, Mhw Jk, Rb Pw, h ASSA 2008(Nw O), h 2008 G Th W C, Bk, Cb, C I Th,H, P, Nhw, S, Y . A kw - h NSF, AFOSR, h C I A Rh.

T w , h h://../10.1257/.102.4.1446.
http://dx.doi.com/10.1257/aer.102.4.1446http://dx.doi.com/10.1257/aer.102.4.1446


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1447AcEmogLU Et AL .: conStItUtIonS, coALItIonS, And cLUBSVoL. 102 no. 4

; E 1 2). O qb wh w-k. U

h C- , w h hz h

(yaially) sable saes. A qb b

whh h b (s0) h s0. W hw h h b h .

Ahh h h -

b, bh wh h-

z b . Th hz h b h

w-k wh h w

() h h h, (sabiliy axi). W w h h - h h h w . W hz h

, , h wh h h whh h q b (Th 1). W hwh wh , h b b w1,2, .

O Th 2. U h h () h 1, () h () h , h qb -

. C, ,h h h h qb -

b .Bh hh b hhz h b , hh h b

.1 Th hh

b whh w ,

w, b h b h wh h w (whh -q k , bw). W b h j h z

.2 W h , hw, h h h

b h hh

( h, qb, q wh hw h b wh h ; bw).

A h h h k ie

-k bh h -k w

1Th h h wh h h w h h h h.S w w ( , bw ), h b wh w-k wh ( E 3, 4, 5 h A).

2O h b h h

h , , h, b A Rb (2000, 2006) Lzz P (2004). Th wk h wh b, , h h, b (hw, w h b ). Th b h h h h (.., , h bk bk wh ).


3/31

1448 tHE AmERIcAn EconomIc REVIEW JUnE 2012

h h h w

h h k. Th k w -

. F, (, , b) b b h b w h h

, b b h b sable h b w . T wh

b, w h h b h h w

h w h. S, b b

.., h b P b h h (Th 3).

O , Th 4, h

( Th 1 2) h wh b (.., b ). I , hw h h wh () - -k ( h w ). Th h j ,

b h h A. Th

4 hw h bh b w

; , Th 1 2

whh b .

Bw, w w h h h-

. W wh h h bw

w h hh -

. W h b , whh w S V.

Example 1: C h w ,E, h ,

m, h . Th h wh b

w: (1) b h a, whhE, wh hm; (2) h , whhmh w ; (3) , whmb f E. S

wE() < wE(a) < wE() , wm(a) < wm() < wm() .

Th hEh hh h h

b h (.., b bm ).O h h h, m h

w- b h. Bh h

(0, 1). S a,, , b . I b h,E whh w w;

bh ,m .

U , b , () = . I , b , , h w b

h, () = . Th, , a,Eh , hw h w , h E

UE(rer) = wE() + wE()_1

.


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IE a , UE( rer) = wE(a)/(1 ).I , h UE( rer) < UE(rer), k. Wh w-k, hw ( ), hUE( rer) > UE(rer). I h , (a) = a. Th bh

. F, a b b h b h ,, whh b h wh w a. S, bhEmw

b b h a, h b a P -

. I h h b wh w-

k (wh , b; wh , bh a b).

Example 2: C h h hw h b w -

w. A wh

M wh , , h

. A , h S Ab I

k h. A h h , Tk,

S, A, E wh M h

. Bh

b b , , b . S

Ab I h h w wk

h , wh Tk b I h b -

b h - h h

w h h .3 S

h Tk w w

h h .T h h b w,

n k . A s h

Zwh h h h

h . I h . S h h h wh h h , h (.., bh wh h

k k b). W ,.., j j , h wh h bh h

wh w h h h h .Th hh , w, . I

h h h

h h- b b -

h h. M, bh h hh

b h .4 I S V, w

h h .

3O , Sh (1985), h f bw ,

Rb Lb (2008) R (2007).4F , Tk h Ib qk h b h (hh h w ), whh wh q h h hh . Fh, h f k, wh , whh wh .


5/31


Th b h q: w h h -

h h wh h b

h? Th b z

wh wk b h q

h ( Bb Jk 2004). Sh P . O , hw, hhh h wh h

h b b; .., wh j

h w h h .

Rb (1999) Bb, Mh, Sh (2001) b w j . Rb (1999) b whn, wh b j ,

- , b h

{1,2, ,k} k w. Bb, Mh, Sh(2001) b whh b h b w .5 L(2009), wh h

, h . Th h A

Rb (2000, 2006) Lzz P (2004), h - b Bb Jk(2004), b wk.

Tw h Chw (1994) G Jh (2005).Chw wh b

h b z h h

bw w h, b. I Chw , hw, h h-

z b.6 G Jh wh

. Th hw h h

h b h , h h qb b -

wh h . I , P

b , b q qb

k wh 1. W

hz ( q) , whh G Jh ( , wh , w h h -

b). F, b h ,wh G Jh, h 1, h

(.., h b h ).F, wk h -

b h.7A bw h h

h, b , w h

h j ( j) h b h h w

5Bb, Sh, Zh (1991) b q, h b w -

b b ( h h q) . Th h h b , wh h, h , h w q q q .

6Th k bw Chw b h A.7O , , .., M (1997); R Vh (1997, 1999); S

W (1998); Kh R (2003). O b h, Ek . (1999) Sh (2002).


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h wh h j. Th wh h -

- b b h . I ,

h wk h b b (R Vh 1997,1999), wh w h h w.8

Th h z w. S I h . S II . I S III, w

h (-) Mk qb h bh h q bw

h qb h hz S II. S IV

wh b , wh S V

h h E 2. S VI .

A A ; , ,

h A.

I. Environment

Th . T , b ( 1).Th saes whh w b . Thh h , |X| h b X, | | | | h b - , . S bh

, h b w h -

k (.., z , h -j, h wh w ). Ahh

b , b b w- (b b ) h h h k, b .

Th b s0. Th b h - h h b N . F 1, h s . A X alii, w h b . Eh s hz b ({wi(s)}i,s). H, h s , wi(s) () i . P s b b h wiialiis s, s, (b ) b . Th w

z , h wh j j- , w. F , s j q

k, s b h j; s - i , s h i.

9 S s h , h k .10

8Oh wk Bk W (2000), wh h b; Jh Sh (2001), wh hw h h q j j b h h ; A, A, E (2005),wh h b EU ; B B (2003), wh h h EU .

9

P z b h s , h wh k ,hw k , . Th b h - . W w hw h h h qb , j s .

10O w h wh , s s, h b h w .


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Thh h , w h w .

ASSUMPTION 1 (W C):Fr ay sae s ,s saisfes

()I X, Y, X Y, a Xs he Ys.

()IX,Ys, he X Y.

P () h X w s, h h z h w h. P () h b hw j w h . Is= , s ex-eusly sable. N hz whh

h b .

W h w b . Fx,y , w w

(1) x y i : wi(x) = wi(y).

I h w x ypay-equivale, equivale. M

h b z. F z , z b

(2) y z x {i : wi(y) wi(x)} z.

I,y z x h h h w (

z) wh h b wk y x. N h - b z. F, b . I ,wi (y) wi(x) h i wh y h h

x. Whh h h w h

wh. S, h z - k h b h q. I

h h . Th, h z z whh h wk y x w

h w , whh . Wh h b

, q (2) h, w hy weakly preerrexz. I h

h , h h h y x, h h w b xy h

h w , h b

y h x. R z b

(3) y z x {i : wi(y) > wi(x)} z.

I(3) h, w hy srily preerrexz.11

11R q ; xy y z, hx z. I , h b zz b . Nh, x,z , w hxz x, wh z , w hxz x. F A 1 w h h x,y,z ,y z xxz y, y z xxz y.


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Th j w

.

ASSUMPTION 2 (P): Pays {wi(s)}i, s saisy he llwi

prperies:

() Fr ay sequee saess1,s2, ,ski,

sj+1 sj sj r all 1 j k 1 s1 sk sk.

() Fr ay sequee saess,s1, ,ski S wihsjs sr1 j kasj sl 1 j < l k,

sj+1 s sj 1 j k 1 s1 s sk.

mrever, i rx,y,s Swe havexs say s x, hey s s.

A 2 j (b wk h ). P () h y xx, z y y,

xz zh , h h h , w h . P () h y s x, xs z, z s y( h x,y, z -q). A h, h y s x,z s y,xs z.

12 I q

h b .13

Ahh A 1 2 , h

b hz . M -

, h ( Th 4 h A). I A 1 2, w b q b h w()q.

ASSUMPTION 3 (Cb): Fr x,y,s suh ha xs s, y s s, ax y, eiher y s x r xs y.

A 3 h w xy s( s), h q, hxy s-b. Th - b q qb.

II. Axiomatic Characterization

B h - , w -

b hz b . Th h w

12Nh A 2 b h h. E 6 7 h A h h wh h 2() 2() .

13T w q h s,x,y z,y s x,xs z y s z. I, h whz = s.


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. F, h h k h h

k h

- . S, h h -

h hz h qb h

h . I , , Th 2, w k h hz.

Th k h b hz hwih su-

fiely rwar-lki behavir, a iiviual shul wish rasi a

sae ha will uliaely lea aher sae ha ives her lwer uiliy. Th b

h b h hz (axiaially) sable saes.M , hz

h h , : b s h s0. W h w h .

AIOM 1 (Db): I x,yare suh ha y = (x), he eiher y =x ry xx.

AIOM 2 (Sb): I x,yare suh ha y = (x), he y = (y).

AIOM 3 (R): I x,y,zare suh ha z xx, z = (z), a z xy, hey (x).

A h h wh w h b. A 1

q h h h x h y h w h h . A 2 -

h b b; b,

b h b . A 3 h -

b q h h b z bhxy b w-

x, h h ky x.14 N h wh h , h h .

W h h h bw b .

DEFINITION 1 (A Sb S):Le { : : saisfesA 13}. A sae sis (axiomatically) stable i(s) = s r se .the se sable saes (fxe pis) r appi is = {s : (s) = s}a he se all sable saes is = {s : (s) = s r se }.

Th h bh h b h w

Th 2, whh bh h q bw qb h

14A 2() h y xx z xy, h z xx. Th, A 1 , h hqz xx b A 3.


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( S III bw) b . A Th 1 A A.15

THEOREM 1 (A Chz Sb S):Suppse Assupis 1

a2 hl. the

() the se is epy. tha is, here exiss a appi saisyi Axis 13.

()Ay a be reursively srue as llws. orer he saes as{1, ,| |}suh ha r ay 1 j < l | |, ljj. Le(1) = 1.Fr eahk= 2, , | |, efe

(4) k = {s {1, ,k1} : s k ka(s) = s}.

Th

k k = (5) (k) = { . s k: z kwih z k s k (I here exis re ha es k suh haz k wihz ks, pik ay

hese; his rresps uliple uis).

(iii) the sable ses ay w appis1,2 iie; i.e., 1 = 2 = .

(iv)I, i aii, Assupi 3 hls, he r ay w appis1a2i, 1(s) 2(s) r all s .

Th 1 hz h h A 13. I, A 2() h h 1 h h h h s wh s 11. Tk 1

b, w h { 1, , | |} j 2 h h. Th, w h k ,k= 2, , | |, h b h k(h , s h h (s) = s s kk). Wh h k , h b h k( k) b b w . I h ,w h (k) = k. Wh k , h h b h(k) = s h s. I ( h --), h hz hhh h

b hz h I: k b b

15Th h b wk . P () A 2 b b bh w : Fr ay sequee saes s,s1, ,sk i Swih sjs sr1 j k, sj+1s sj r all1 j k 1 wul iply s1s sk.


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h b w k h sable (.., b h h k= ). Th h .

P () Th 1 hw h h b h h . F w 1 2 , b

h 1(s0) 2(s0) s0, b h h h . Th , h h b , q b h w .16 F, 4 hw h

wh A 3 h, b b

-q. I h w, s1= 1(s0) s2= (s0), h s1 s2 h h w , b h h

h .

W h h Th 1 wh h h,

wh w-k, b h

b b ( h ; .., t t). Th h h b h .

III. Noncooperative Foundations of Dynamically Stable States

W w b h - h

S I hz h Mk qb (MPE) h . Th h q bw h MPE h h

Th 1.W h . A

h , i z

(6) Ui() = (1 )=

ui( ) ,

wh (0, 1) . W

ASSUMPTION 4 (P):the sae pays i (6) are ive by

wi(s) s = s1(7) ui() = { . wi s s1F h i x, w h

wi < wi(x) .

A 4 :

whh h , h b w h h w

16I h A, w h w h, N-M b Chw . U A 1 2, bh wh .


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h wh h . G z wi(s) > 0, A4 ; .., wi= 0 i. S w h 1, h h .17 I , ( ) b s h, wi(s) h h-

. Sb, h h - MPE.18

W : () q - h ; () . V q b bw; h q whh w .19 W

h q ,{s}s, prl. L Ksb b h s . Th, s

s : {1, ,Ks}

h s . Th, h s q , h q - (h Ks hh h q s). Is(k), h -wh w k h . A, s(k), h whh

. Th, h - h bh -

h w b - .

W k h w {s}s:

ASSUMPTION 5 (P): Fr eah s , e (r bh) he llwi wiis hls:

() Fr ay saez \{s}, here exissk : 1 k Ks suh has(k) =z.

() Fr ay playeri here exissk : 1 k Ks suh has(k) = i.

Th h h qs b h

h h q s w b - - k b h . W h s h s; h h h w

.

17M , _

= i,x|wi(x) wi|, whh h z . Th

_

, h 0< 1 h h Th 2 h > 0. Th , whh h A, h h h b wh .

18E 4 5 h A h h q(7), (-) qb . Wh h b

, h b b. A (.., h h ) wh w wh h .

19Th q w MPE wh h h q qb. A, E, S (2009) qb, Mk b h qb, whh qb bh wh wh q .


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I h b, = 0, s0 (h h - h ). Sbq ( 1), h w:

Periodb wh s1 h h . Fork= 1, ,Ks1, h kh Pk, w. Is1(k),

h Pk,= s1(k). Is1(k), h s1(k) h Pk, . IfPk, s1, h h q bwPk, s1(w w hw

h h q h h qb ). Eh yes( Pk,) ( s1). L Yk, h whyes. IYk, s1, h Pk, ; hw (Yk,s1), j. IPk,= s1, h w h - h h Pk, j.

IfPk, , h s= Pk,k , h . IPk, j h bPk,= s1 k< Ks1,h h 2 wh k b 1; k= Ks1, h s= s1, h .

Attheendoftheperiod,eachplayerreceivesstagepayoffui().

A MPE h h b qb

(SPE) wh - . H - h s b b, h - , w , -

h ( h A ). AMk h , s s, wh s= s1

h s1. I whw, w h MPE qb hb. N, w

b .

DEFINITION 2 (D Sb S): Sae s is a dynamically stablestate i here exis a iiial sae s0, a se prls {s}s, a mPE sraey

prfle , a t< suh ha al he equilibriu pah we have s= sr all

t.

P , s b h t hs= s

t. O bj () whh b h b b

hz h h s0, () bh hq bw b hz

h .

W whh h hh 0(0, 1) A A. Th h

z h w h.

THEOREM 2 (Chz D Sb S): Suppse haAssupis 1,2,4, a5 hl. the here exiss 0(0, 1) suh ha r all > 0,he llwi is rue.


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() Fr ay here exiss a se prls {s}sa a pure-sraeymPE he ae suh ha r ay s0, s

=(s0) r ay 1; hais, he ae reahes (s0) aer e peri a says i his sae hereaer.therere, r eah s0, s =(s0) is a yaially sable sae.

() mrever, r ay se prls {s}shere exiss a pure-sraey mPE.Ay suh mPE has he prpery ha here exiss suh ha r ayiiial sae s0, s

=(s0) r all 1. therere, all yaially sablesaes are axiaially sable.

() I, i aii, Assupi 3 hls, he he mPE is esseially uique: ray se prls {s}s, ay pure-sraey mPE, ay iiial sae s0,a ay , s0

(s0).

P () () Th 2 h h b h b b hz Th 1 -; h A 13 h -MPE h MPE h . A h h hz b

q (5) b b .Th q h Th 1 2 . H b

h-h (), h w b h h h w h w h qb h. Wh

, hw ( > 0), h b h h h h . Cq, w - w h () b Th 1.

Th Th 2 h , b h . F

, w j h - z s. W MPE h

s wh b h . W h

hw h b, h j.C, MPE, w hw h s w

(s), h A 2(i) hw h qb h h -q (s). F,w h (s) A 13.

A b E 1 h I, h bw -

b b w h .

S, P b wh h h b-

w. Th h h whh P

w .

THEOREM 3 (P E): Suppse ha r every w saes x a yhere is a sae z suh ha {wi(z)}i = {wi(y)}i az x, a saeis exeusly sable (i.e.,s r eah s ). the, every (axiaiallyr yaially) sable sae is Pare efie. oherwise, sable saes ay bePare iefie.


15/31


Th h wh h h h h -

-k h wh

w (whh h b h h xz h h y wh w h ),

P b.

IV. Ordered States and Agents

Th 1 2 hz

b h s0 hA 1 2 . Wh h

hk, A 2 b wh . I h

, w hw h wh h whh - -k-

( h w {s}s ), A 2 . Th b h-w h w .

I b , h h , h wxy b k. Wh h , w k, wh

, b b . S, . G h h h , w

.20 A h w

h .

DEFINITION 3 (S-C S-Pk):give , ,a{wi(s)}i, s, he single-crossing condition hls i r ay i,j a x,y suh ha i wi(x) iplies wj(y) > wj(x) a wj(y) < wj(x)iplies wi(y) < wi(x).

give a{wi(s)}i, s, preerees are single-peaked i r ay i here exiss a saexi suh ha r ayy,z , y < z xi rxi z >y iplieswi(y) wi(z).

W z h h

(.., h j wh h b).

DEFINITION 4 (Q-M V): give a{s}s, player i is aquasi-median voter(i sae s) i r ay X Ws suh ha X= {j : a j b}

r se a,b we have i X.

D h q- s bms. L 1 h Th 4 hw h, h A 1 , h .

20Rh (1990) A-Sh Bk (1999) h , - .A G S (1996) hw, h q - h b wk.


16/31


DEFINITION 5 (M M V P):give a , hese wii aliis {s}shas monotonic median voter property i r eah

x,y saisyi x


17/31


W h h b h w b h . F , hh -

h b ,

b. S h k q, w

h b

wi(s) = vi() + V(Z) ,

wh V(Z) , vi() h :

vi() vj() wh i > j.

Th , , h h q ,

{ i}i=1 , (wk) . I , , wh

h q vi() = ( i)2.

S -()j Zh h h ,, h h h bw mZ mZ, wh, b,

mZ h q- . F, h (,Z), whZ b mZ mZ.

Th h hh . G

(., ) w h h- b, b h hh (., ) h h.Th h h h h wh -

h h h b h Zh w h h h . D h, h h

b h hz h

qb.

W bh h A 1 2 , h h

qb h b hz b Th 1

2. T h h , w h wi(s)wi(s) i ss, whh h A 6 h. Th, w (s0) h , bh b, h

s0.

PROPOSITION 1:

() Fr ay eree (super) ajriy,Assupis 1 a2 are saisfe ahus theres 1 a2 apply i his evire. I pariular, here exiss

0< 1 suh ha r ay isu ar > 0, a equilibriu exiss.

()AssueV(Z) be(srily) ireasi(wheeverZZ,ZZ ipliesV(Z) < V(Z )). the r ay iiial saes0, (s0)=s = (Z,)wih Z -aii a leas e he exree players, 1 r.


18/31


T h , w h {s1, ,s| |} h s wk. W h bh h

h, Th 4.21 Th P 1 hw

h wh V , b h

h b h . I, h h b h h h h w h -

. Th, h w b w h

h .

Th h h w b b

. Th h q. I wh w, w h

V(Z) vi() -k i .

PROPOSITION 2:defe A V( ) iV( \{i}) a Ai V( ) V({i}).

() Suppse v1( 1) v1( m) < A a v( ) v( m) < A. the ray iiial saes,Z((s)) = .

() Suppse v1( 1) v1( m) >A1 a v( ) v( m) >A. thereexiss k suh ha i he iiial saes0saisfes|Z(s0) | k, he(i)whe

Z0ilues he ile player(r a leas e he w ile players i is eve),Z((s0)) = , a(ii)wheZ0ilues e he exree players,

Z((s0)) .

()I > 1_

, i.e., he rule is uaiiy, he r ay iiial saes0,Z((s0))= .

Th h hw h

( h w ), h .M , h hw h wh h

, w wh

wh h .

Th h h h , h h

b b h q b h . I, h w w

h h h b

.22 Th h q whh h h

w h

h w -

. Wh h b b , w b h

b w -

b k w h b

21N, hw, h h h h h b ; w k h .

22Th h Th 3, whh hw h wh b h wh h w w , qb P . (R h wh P h ).


19/31


h b h . (O hh w h w

h .)F, w h wh h

h ()j . I , w s = (,Z,), wh, wh [,1] . Th, , h w .

PROPOSITION 3:

()I his evire, Assupis 1 a2 are saisfe a hus theres 1a2 apply.

() Suppse ha ais > 1_ . the r ay equilibriu a r aysaes0,Z((s0)) = .

Th h b h

whh h j -

h. A, wh b , -

. I, h k ( ) , h. W h, hw, h h h

h h h w b. I

, h h I

w h A, h-w h w b b (wh) j h wh k, h b h w b b

q.23 A h , ,

h w b h h

Tk h b h h .

VI. Conclusion

A k , h

h h , b -, j , b h -

bh, h h h

k, b, b h b

h . Th h -

z h h.

W wk h b. W

bh hz b

hw h h () b b .Th hz hhh h s b h

b k w ( s) b . Th h b

23S A, E, S (2008) .


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b P ; h h hh

, b b.

O b h . Sb

, h , h.

Oh, h h , b , hh h h . A b , -

h b h h h b

h b w h , -k

b h b h h .

Appendix

PROOF OF THEOREM 1:

Part 1: W , b , q {1, ,| |}hh

(A1) 1 j < l | | , h l j j.

A 2(i) h , hz h h x,xz z. A h , w b 1.Nw, w h j j k 1, wh k | |. A h h \{ 1, ,k1}, w h h

k (A1) h k.Th , b , : . F k= 1, (k) = k. S w h (j)

j k 1 wh 2 k | |. D h k (4).Th h b whh h b whh - (s) = s kwh k. Ik , h w (k) = k. Ik , h k (k) =zkh h

(A2) s k z sk

( A 2(ii) k, w h h zkh h s kz, h s kz, sk). P 2 k | |, wb q (5).

T h , w h (5) A 13.Th hw A 1 2. I , b , h

(k) = k( h h h ), (k) k.I h ,(k) kk ((k)) = (k) b (4). T hk A 3, - h kh y h hy kk,y = (z), y k(k).Th y kk, b wh (A1), hy { 1, ,k1}, hyk. B h y k(k) (A2). Th h h y , h A 3 .

Part 2: Th q h w: , q

{ 1, ,| |}wh h (A1), (k) (5) k, h


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A 13. S h (k) b (5) k= 1.Th (1) 1, (1) = l l > 1. I h , A 1,b l11 b (A1). Nw, k> 1 b h k whh (k) b (5). S, b , h A 13 h. Th k

(4) w , h k= k. Ik= (k) b (5), h (k) k. Th, A 1 2 (k) kk((k)) = (k). S k= , w h h (k) = l l > k, b h (k) kk (A1). Th h h A 1 A 2 ( bh). Ik, h l= (k). Il > k,h A 1 . Il = k, h A 3 ( h, k

z k b h z kk, z k(k) (z) =z). I l < k, hA 1 A 2 (k)k. Th, (k) b (5), h y kh hy k(k). B h A 3,

y k(k),y

kk, (y) =y . W h b b

.

Part 3: S, b , h 12. Th k:1 k | |h h j1j2 j < k, b h k1 k 2, k 1 k2. Wh , h k1 k 2. Th 2 h 2(k) = l l < k. A A 1 2 2, w b lkk 2(l) = l; h hl2. S, b hh, j1j2 j < k, w h l1.Th, lkk, lk1(k)(b 1(k) = k), 1(l) = l, b h

A 3 1.

Part 4: S A 3 h. S, b , h

s, 1(s) 2(s). P 3 h Th h 1(s) = s 2(s)= s; 1(s) 2(s), w h h 1(s) s2(s). A 1 h 1(s) s s, 2(s) s s, A 3 h h 1(s) s2(s) 2(s)s1(s). Wh , h h h . Th

y = 2(s) h z = 1(s) h hz s y,z s s, 2(z) =z(h hb 1(s) = s b A 2, h 2(s) = s b 3 h Th). Thw A 3 2 s h 2(s) y, .

PROOF OF THEOREM 2:

Part 1: A h w :

(A3) i x,y ,

wi(x) < wi(y) wi(x) < (1 | |)wi + | |wi(y).

T 2, w w h w :

(A4) i x,y,z ,

wi(x) < wi(y) 1 _

(wi(z) (1 ) wi) + wi(x) < wi(y).


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I , h b (A3) (A4). Th, h 0(0, 1) h h > 0, (A3) (A4) h.

Pk s0 S. W MPE h h h h 1, s= (s1). F i s,q ,

(1 )wi(s) s = q(A5) Vi(s,q) = { }(1 ) wi s q

wi((q)) (q) = q+ { }. (1 ) wi+ 2wi((q)) (q) q

I h qb w bw, Vi(s,q) w b h i h s h q. I h , w

.

F h s , k Ks | | 1. Tk s( ) h h A 5 h, (s) s, h s(Ks) = (s). C * w:Eh i Pk(yes) :

() h k= Ks(w h ), PKs = (s) Vi(s,(s)) >Vi(s,s);

() Vi(s,Pk) > Vi(s,(s)).

I , s(k) k, h h Pkb.Th * Mk. W w hw h MPE h .F, w hw h h *, h (s) (s) s

(s) = s. S h (s) s, h A 1 h

Xs {i : wi((s)) > wi(s)} s.

Nw, (A3) > 0 h i Xs, w h

Vi(s,(s)) = (1 ) wi + wi((s)) > (1 )wi(s)

+ (1 ) wi + 2wi((s)) = Vi(s,s) .

Cq, (s) s, h *, h (s) Ks h.

L w hw h h Xs s Pk h h Vi(s,Pk) >Vi(s,(s)) i Xs; .., h wh Vi(s,Pk) > Vi(s,(s)) w s. T b , h h

Xs Pk. Th, Pk s ((s)) = (s), w w h h i Xs, wi((Pk)) > (1 ) wi + wi((Pk)) Vi(s,Pk)

> Vi(s,(s)) (1 ) wi + wi((s)),


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h, b (A3),

wi((Pk)) > wi((s)) i Xs.S,Xs s (Pk) s(s), whh, h (s) s s, (Pk) s sb A 2(ii). B (Pk) s(s), (Pk) s s, ((Pk)) = (Pk) - A 3. Th, h wh Vi(s,Pk) > Vi(s,(s)) w s. Th h *, (s) = s, (s) s, h b h , h PKs = (s) .

S, w h *, - q b (A5). I q s , h h q, wh , h h i

(1 )wi(s). I h , h ( (q)) * (q) q, h h h qb h.H, h b (A5).

Th, w hw h h b * . F -, h h b h h k . F

, h Mk, w b

h h h (wk) , h wh * b. I, (s) s, h h , h i Vi(s,(s)) h Vi(s,s) j. I h , i Vi(s,Pk) Pk

Vi(s,(s)) j (b (s) w b (s) s (s) = s). Th, h -b * h (A1). Th, * b s , h * MPE h .

Part 2: W h MPE , h h MPE h h . W A 13. Tk q { 1, ,| |} (A1). Th, w h b Th 1. F, w (1) = 1. I l 2 w h l= , h (l) = l;

hw, Zl lb

Zl = {z l : s l : s z s l z}.

Th Zl, w A 2(b) l. Ch - Zl (l) w. L Yl b h l h h j Yl, l(j) l(j) Zl, l h z Zl: wi(z) > wi(l), wh i = l(j). B A 5, Yl ; kl

* b h Yl. Il(kl*) , h (l) = l(kl*), whl(kl

*) , h (l) b z Zl h h wi(z) > wi(l) i = l(j ). P b l, w .

W w qb h . I h qb, - i h s q , Vi(s,q), b (A5); , h i Vi(s,s). G


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h , h b w () wh b Vi(s,q). W MPE h b bkw .

Case (i):(s) s. G s, h ks* b h . Iks* h , h Ksw ks*+ 1 w h w. S h h , h h s. C h 1, w h SPE,s b s = (s) j hw. B b Ys, s b b i. P bkw h - , w h h i h wi((s)) wi(s), h h s, whh h s = (s) h b . B (s) h h h Ksh h h s. W h

ks*+ 1, SPE h b ks*+ 1 wh s .

C w ks*. B h , (s) b . A h , h h h

, s(ks*) = i , h i b (s): i , hs . H, b h ks*, h SPE wh (s) .

I ks* 1, w wh bkw . A ks* 1, h h (s) b , w h h h (s)

j h ( ks*

). I h h - i, h , w k . P bk-w, w h SPE h h s h (s) s.

Case (ii):(s) = s. Tk h , h s s . F i s, wi((s )) > wi(s) h. S(s) = s, hw, h w . Cq,w h h h w b -

. Cq, h - , b h,

h b . W bkw wh .

N h bh , w h b Mk b h h q b wh . H

s , w Mk . B h h -, h b (A5), h- h b -h (hw, w b SPE s). Th hw h MPE.

O bh h h MPE . Tk

{s( )}s - MPE . F s, h qh h qb h w- ( q = s - j) (s) = q. F, h : h :(s) s h > 1, (s) s(wh 2(s) ((s)), .). Th b bh b . S h h h (s) = s, b


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(s) s. D bJs {1, ,Ks}h s wh - Pk h qb h . B , h Js (s). Tw b.

Case (i): k Js, +1(Pk)(Pk) . Th h - k Js. IPk , h i wi, Pk j, i (1 )wi(s) + wi> wi. B Pk b MPE, h .

Case (ii): k Js, +1(Pk) =

(Pk) . D h h kbJsJs; , h Js Js. L kb h Js; h+1(Pk) =

(Pk) | | 1. C h kJsh k. A h k, Pk, wi h i, wh j (1 | |) wi+

| |wi(| |(Pk)) > wi. Th, Pk

b MPE, whh bh h

.

Th h (s) = | |1(s) | | 1.D (s) = | | 1(s), , wh h h 0(s) s,

(A6) (s) = { {0} : (s) = (s)}.

E, 0 (s) | | 1, (s) = 0 (s) = (s) = s.

M,

(A7) ((s)) = ((s)) = ((s)) = (s)

s, w h . F,

(1 )wi(s) (s) = s(A8)

_

Vi(s) = { } + wi((s)),(1 ) wi (s) swhh h qb i h qb (s) , , h b h ,

_

Vi,

(1 )wi(s) s = q(A9)

_

Vi(s,q) = { }+ _Vi(q) .(1 ) wi s qC,

_

Vi(s,q) h i s q , qb ( ) w. W w ; h w w h (s) A 1 2, h h (s) = (s)(whh s= (s0) 1), h A 3.

Pr ha i prpsals Pkj a Pkl, j < l, are prpse a aepe i sae s,he (Pkj) (Pkl)a (Pkj) (Pkl). W h wh(s) s, h, (s) 1. F h s k h Jh


26/31


h h kJ, h Pk . LJ= {k1, ,k|J|}, wh kj< klj < l(w s ); hJ. I qb, Pk1 , (s) = Pk1 (Pk1) = (s). S h Pkl 1 l |J| h qb, h 1 l < |J|,

_

Vi(s,Pkl) _

Vi(s,Pkl+1) w

s. F h ,

(A10) (1 (Pkl)+1) wi + (Pkl)+1wi((Pkl))

(1 (Pkl+1)+1) wi + (Pkl+1)+1wi((Pkl+1)),

h, (A3), wi((Pkl)) wi((Pkl+1)); h (Pkl) s(Pkl+1). W h h

_

Vi(s,Pk|J|) _

Vi(s,s) w s, h ,

(A11) (1 (Pk|J|)+1)wi + (Pk|J|)+1wi((Pk|J|))

(1 )wi(s) + ((1 (s))wi + (s)wi((s)))

> (1 (s)+1) wi + (s)+1wi((s)).

F (A3), w wi((Pk|J|)) wi((s)) = wi(Pk1); h, (Pk|J|) s(Pk1).

A 2(ii) w h (Pkj) (Pkl) 1 jwi(s). W h h h s h h (s) s, (s) s s, h A 1 h.

Pr hasaisfes Axi 2 hw ((s)) = (s) (A7).Pr ha(s) = (s). I(s) = s, h (s) = s = (s) h

. L h (s) s, h (s) k ; .., h (s)= (s)(, q, (A6)(s) = 1 wh (s) s).C w .

Case (i): (s) = Pkj j : 1 j |J|. I h , (Pkj) = 0 A 2 h. B w h (Pkl) wk l,h, ((s)) = (Pk1) = 0, h (s) = 1.

Case (ii):(s) = Pkj h j. Th h (Pk1) 1 (s) (s). S h k, h Pk= (s) ( h qb h). Th , h i w _

Vi(s,Pk) = (1 ) wi+ wi((s)), j, h w

_

Vi(s,x) (1 )wi(s) + (1 ) wi + 2wi((s))


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xh h (x) = (s). A wh wi((s)) > wi(s) , (A3), h

_

Vi(s,Pk) >_

Vi(s,x). S (s) s s(A 1), Pk= (s) wb .

B A 5, h w b

k, h w b -. I h , k J, b h (s) = Pkj h j, . Ih , i h h wi((s)) > wi(s) h - k, h h Pk= (s) qb, w b , w . P Pk= (s), hw, w _

Vi(s,Pk) wh k h qb w _

Vi(s,x). F i,_

Vi(s,Pk) >_

Vi(s,x) w , h h h b . Th h qb, whh h (s) = (s) s .

Pr hasaisfes Axi 3. S h A 3 h. Th h h s,z h h (z) =z,z s s(whh z s),

z s(s)(whh (z) (s)). A b, h k, h Pk=z ( qb h). I , h i w

_

Vi(s,z) = (1 ) wi+ wi(z), j, h w

_

Vi(s,x) (1 )wi(s) + (1 ) wi + 2wi((s))

xh h (x) = (s). Nw, (A4) h_

Vi(s,z) >_

Vi(s,x) wh- wi(z) > wi((s)); .., w s. Th, Pk= zw b .

S (z) (s), b hz h qb h.B A 5, h b h b h - k. Wh wh wi(z) > wi((s)) b h -,z b h. Th h qb,

h bh h A 3. Th h 2 h Th.

Part 3: Th w Th 1 2 h

Th.

PROOF OF THEOREM 3:S, b , h b s P .

Th h x, wi(x) > wi(s) i . B hh, hy h h y s wi(y) = wi(x) > wi(s) i . Tk - . h A 13. C w . I(y) =y, h (s) = s y s s w y s(s), A 3 ( h z h h(y) =y, y s s, y s z, h z (s)). I (y)y, h A 1 wi((y)) > wi(y) > wi(s) w y, whh w s, h (y) s s (y) s(s). A 2 h ((y)) = (y).A, w h A 3.

PROOF OF THEOREM 4:

Th , h A, hz q-

. R hms h q- s.


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LEMMA 1: give , , pay uis {wi(s)}i, s, a wiialiis {s}ssaisyi Assupi 1, he llwi are rue.

() Fr eahs, he sems is epy.

()I he sile-rssi prpery i defii 3 hls, he r ay saesx,y,z ,

xz yi a ly i r alli mz, wi(x) >wi(y), a

xz yi a ly i r alli mz, wi(x) >wi(y).

()I i eia ver ii i defii 5 hls, he here is aereasi sequee{s}s players suh hasmsr alls .

PROOF OF THEOREM 4:

Part 1: W wh A 2(i). S h h s1, ,slh h sk+1sksk 1 k l 1 s1slsl. Tk q {s}s. R h s w s, h, x z,xz z, h wx(z) > wz(z). Nw hs w wh s h ;.., s= {X : s X}. D h bw b, w h h xz z, hxz

z. Cq, h w s1, ,slh h sk+1sksk 1 k l 1 s1slsl, h w h sk+1 sk sk

1 k l 1 s1 sl sl; h, . Nw k h h (whh b ). Wh , h s2 h w ( s2 s1 s2 s3); h s2 s1 s2 s3.S s3 s2 s2 s2 s1 s1, w h ws2(s3) > ws2(s2) ws1(s2) > ws1(s1).B s2 s3 s2 s1, h, ws2(s3) ws2(s2) > 0 ws1(s3) ws1(s2) > 0. Cb h wh ws1(s2) > ws1(s1), w h ws1(s3) >ws1(s1). B h s3 s1 s1, s1 h s1. Th h s2 b k h , h h {sk}k=1

l h h.

T A 2(ii), k s s ms. S h s1, ,slh h sk+1s sk 1 k l 1, s1s sl, sj sk1 j < k l. Wh , h slz h - s s1, ,sl. Th ws(sl) ws(s1), A 6 ws(sl) > ws(s1). B h, b L 1, s1s sl, h h - . F, x,y h h xs s y s x, h i ms w h wi(y) > wi(x) > wi(s), whh, , y s s. Th hwh A 2(ii) h h 1.24

24Th b Th 4.6 A-Sh Bk (1999).


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Part 2:25 L = ss; h , w , - A 1. L * b b x*y {i : wi(x) >wi(y)} . S -k, Th 4.1 A-Sh Bk (1999) h * , h . C,

A 2(i) w b *; A 6, w A 2(ii). H, h . F, Th 4.1 A-Sh Bk (1999) h h s -, xs s y s xy s s.

Prs Prpsiis i Sei V

PROOF OF PROPOSITION 1:

Part 1:S > 1_2

h , A 1 . E s1, , s(wh = | |) h h sk wk k(h wh h b b). Wh h , h- D 3. I, sk< sl i < j, h

(wj(sl) wj(sk)) (wi(sl) wi(sk))

= (vj(sl) vj(sk)) (vi(sl) vi(sk)) 0,

b sk sl v h . Nw q q- ; h w

h h. F sk, k sk h hsk sk < sk+1 sk < , sk = hw. Th sk q h sk, wk , q- skbh b . W w 1 Th 4 hw h A 2 Th 1 Th 2 .

Part 2: S h s wh 1, Z(s) b. SZ(s) = [a,b]

Z = [a 1,b + 1]. Th mZ mZ mZ mZ, h s = (s,Z ) b . B h V(Z), sP - s. Tk h A 13 x= (s ). ThA 1 h wi((s )) wi(s ) > wi(s) w s, h, b L 1, i ms . Th, h h i ms, h w s. S ((s )) = (s ), w A 3. Th h s b, h 1 h b Z(s) b s. H, s0, h wb .

25W hk h 2 Th 4.


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Part 1: N h xwhZ(x) = b, wh (x)xw A 1. I, m x m

-k, q- mw b w y why > x, mw b w y < x. Nw , b , h s h hZ(s), (s) = s. C h w. C (): (s) < m. Tkx= ( m, ); b hh, w1(x) > w1(s), h

w1(x) = v1( m) + V( ) > v1( 1) + V(s) v1(s) + V(s) = wi(s) .

S (x) > (s), h wi(x) > wi(s) i . B w h xb, h (s) = s A 3. C (): m s m. Tk

x= (s, ) h wi(x) > wi(s) i . S w h(x) =x, w A 3. C (): s > m.Th (). I , (s) = s .

Part 2: L k= 1. L s b h hZ(s) kZ(s) h . D h Z(s) b i k x= ( i, ), whh b . Th (x) =x . I(s) x, h A 3 , wi(x) > wi(y) y x. Th h (s) =x.

I s h 1 Z(s) k, h h s = (

1,{1}) s = ( ,{}). S h (s) =x= (, ); h m m, v1() v1( m) v() v1( m) b -k. Th h ,

w1(x) = v1() + V( ) v1( m) + V( ) < v1( 1) + V({1}) = w1(s) ,

h A 1 . S, h ,w(x) < w1(s), A 1 . Th h (s) x.

Part 3: Tk s; , b , h (s) =xh h

Z(x). B A 1, wi(x) > wi(s) i . Cy = (x,); h - hy b (x). A hw , (y) =y, wi(y) > wi(x) > wi(s) i . B h (s) =x A 3, .


Part 1:Th w h 1 P 1.

Part 2: S, b , h s, (s) =xh hZ(x) h A 1 3. B A 1,wi(x) > wi(s) w s. Cy = (,x, ) h h >

1_

; hy -b. B (y) =y, wi(y) > wi(x) i . Th wi(y) > wi(s) w x. B h (s) =x A 3, .


31/31


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