Arbitrations when players have different information-香川大学学術情報リポジトリ

Kagawa U Vol 70， No 1， June1997，55-95

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T E E L -A i h u Q U O

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Abstract

Two arbitration procedures， double-o任erarbitration (DOA) and combined arbitra

-tion (CA)， have been proposed to improve thec1assical final-o任erarbitration (FOA) procedure An important result of DOA and CA is that the offers of disputants criss -cross if the disputants share a common estimate about arbitrator's notion of a fair sett1ement This paper considers these three arbitration procedures under the assump -tion that disputants have different information about arbitrator's notion of a fair settle -ment In this analysis， we define that the 0任ersconverge if they crisscross or they are

c10se in some probability sense， and show that the revised DOA is the only procedure under which the probability that offers converge is always positive

Key words: Combined arbitration (CA)， Double-o任er arbitration (DOA)， Final-o妊er

arbitration (FOA)， Game， Nash equilibrium 1 Introduction

There are several arbitration procedures used to settle a dispute between two disputants. Among them， conventional arbitration (COA) and final-offer arbitration (FOA) are extensively studied Under COA， the arbitrator hears the arguments of both disputants， analyzes the demands of disputants and selects a settlement which hejshe thinks fair.. In this case it is common for the arbitrator to“split the difference" and select the

本Departmentof Applied Mathematics and Physics， Facul

ty of Engineering Kyoto University， Kyoto 606-01Japan

-56- Kagawa University Economic Review 56

midpoint of the two demands as a compromise“ However， this type of

compromise tends to encourage disputants to make extreme demands， and

it is unlikely that two disputants can reach an agreement on theirO Wll..

This is a fatal weakness of COA if we take a view that a good arbitration

procedure should have a power to make the two offers to come close To

remedy this， Stevens (1966) introduced arbitration procedureFOA Under

FOA， each disputant is required to make a final0任erthat expresses his/her

demand for settlement Ifthe offers of two disputants do not crisscross，

then the arbitrator chooses one of them， which is closer to his notion of a fair settlement， as the final settlement Since the disputant with an

extreme 0任eris1ikely to lose， this procedure induces disputants to make

compromises; therefore two offers are expected to come closer than those

of COA Several papers however show that FOA is not strong enough to

make two 0任erscrisscross， and some improvements have been proposed，

see Brams (1990) or Zeng， N akamura and Ibaraki (1996) for surveys. The

most e丘'ectiveimprovements are the double-o丘町 arbitration (DOA) of

Zeng， Nakamura and Ibaraki (1996)， and the combined arbitration(CA) of

Brams and Merrill (1986) DOA asks each disputant to make two 0丘ers

respectively， while CA combines COA and FOA

To model a dispute， we consider a transaction between sellers and

buyerb;s hopes a high price and b hopes a low price If they cannot reach

an agreement on the price， they ask an arbitrator to give a settlement. It

is natural to suppose that after investigation， the arbitrator has a notionZa

of a fair settlement Although none of s and b has completely knowledge

ofz.α， it is assumed that each disputant has an estimate (i.e..， probabi1ity

distribution) ofZa Many papers， including Brams and Merrill (1986) and

Zeng， N akamura and Ibaraki (1996)， assume that two disputants share a

57 Arbitrations When Players Have Di任erentInformation -57-ー

that disputants have di妊erentestimates because of their di任erentinforma

-tion， see Samuelson (1991)， Guo and Chen (1995) Indeed， Samuelson

(1991) further argues that the common estimate obviates the need for

arbitration

The purpose of this paper is to reveal how the above three procedures

behave when two disputants hold di妊erentestimates， which are common

knowledge to al1participants. In this situation， Guo and Chen (1995) gave

a discussion on FOA. In order to further compare FOA， DOA and CA， we

relax the meaning of convergence from crisscross to“being close

ぺ

where

two offers are judged with certain probability to be convergent if the

distancet between the0任ersis small The idea is originated from reallife

Ifthe offers are su伍ciently close， the difference between disputants

becomes unimportant and the arbitrator can judge that the disputants have

reached an agreement. The resulting procedures are called the revised

FOA， CA and DOA respectively We then show that the convergence

probability under the revised DOA is always positive， whereas the conver

-gence probability under either of the revised FOA and the revised CA may

become zero in some cases

This paper is organized as follows In Section 2， we describe our

mathematical model and give some basic assumptions.. The probabi1ity is

introduced there because different arbitrators may take different views of

closeness， and therefore the meaning of“close" is described as a random

variable.. The revised FOA， DOA and CA are them analyzed in Sections 3，

4 and 5 respectively.. In Section 6， we compare all the results and conclude

that DOA is most desirable in the sense of having positive convergence

-58 Kagawa University Economic Reviω 58

2 Basic Assumptions

In the revised FOA， DOA and CA， we consider that two 0妊erscon

-verge if they crisscross or are close， where closeness is described as a

random variable; ie， the distancet E [0，∞)between two 0ぽersis thought

to be close with probabilityh(t)， where

h(t)=l， ift=O

Oぐh(t)<l，iftε(0， C) (1)

h(t)= 0， ift ミ C，

hold for a positive number C In other words， the offerXs of s and the

offerXb ofb are always considered to be convergent ifXs ::;;: Xb， as these two disputants can reach an agreement(XS+Xb)/Z on their own Furthermore， even ifXs

>

Xb， the arbitrator judges that the two 0妊ersare convergent with probabilityh(t)， if their distancet = Xs -Xb is smaller than C The disputant s (resp..b)does not have a complete knowledge ofZa but has its information in the form of probability density functionん(・)(resp， ん(・))over (一∞，∞ト Asmentioned in Section 1， this paper considers the situation in which ん(・)宇fb(・) We denote the distribution function offs (respん)by 1九(resp“Fb)with median ms (respド mb) Also we assume throughout the paper that all these functionsh(t)，ん

and fbare common knowledge to all participants. As we analyze the

arbitration procedures from game-theoretic view point， we shall refer

disputants as players in what follows.. The Nash equilibrium is a funda

-mental concept in noncooperative games.. This paper only considers pure

Nash equilibrium， and therefore we may omit the word “pure" from now

on.. For a game in which strategies of two players s and b are of one

-dimension， such asXs and Xb for FOA or CA， we say that a pair of

59 Arbitrations When Players Have Different Information

-59-response to X6and simultaneously X6 is local1y a best response to xi':Le..，

れ(χs，xt)ζ 日(x;i，X6) and れいに Xb)二三 Vb(X，"sxt) (2)

hold for any xsENs(ば)and xbENb(X6)， where Ns(x6) (resp Nb(Xt)) is an

open neighborhood of xi'(resp. x6，) and 日(Xs，Xb) (resp九(.Xs，Xb)) is the expected payo任tos (resp句 b)when offers are(xs， Xb) Therefore， ifVs and 九 aredi貨erentiable，a local1y N ash equilibrium(X:， xt) must satisfy the fol1owing first-order conditions : 川 (Z;xt

L

=

0 and oV;b(ZJ xt

L=O

(3) In addition to the above first-order conditions， the fol1owing second-order conditions are sufficient to ensure that (xi， xt) is a localN ash equilibrium :

♂れ(xixt

L<O

and

2

.

2

九

校

，

xt

L

<

O

.

.

OXs OX，i

If(2)holds for al1Xs and Xb， then0任ers(x:，ば)are said to form a global

N ash equilibrium If the strategies of players are of two-dimension， such as (xs， Ys) and (Xb，日)forDOA， a pair of strategies (x;， y，'iX6， y6) form a localN ash equilibrium if Vs(Xs， Ys， X6， Y6)三二日(X;'yi， X6， y6) (4) Vb(xi， y，'iXb， Yb)三三 Vb(xi， ぱ，X6， yt)

hold for any (xs， ys)εN

s

C

x，'iy;) and (Xb， Yb)ENb(X6， Y6)， where Ns(x:，

y;) (resp. Nb(X6， y6)) is an open neighborhood of (.xi，ば)(resp. (X6，ぽ))， and Vs(xs， Ys， Xb， Yb) (resp.. Vb(xs， Ys， Xb， Yb)) is the expected payoぽtos (respb)when 0任ersare (xs， Ys， Xb， Yb). Therefore， if Vs and Vb are di任erentiable，then a local1y N ash equilibrium (xi， yi， X6， Y6) must sat -isfy the following first-order conditions: 。九(x;，y;， X6， Y6) _ 1¥ OVb(X:， y;， X6， Y6) = 0， (5) aXs -v， OXb

60-ー Kagawa Universify Eヒ0招omicReview 60 3Vs(xi， yi， X6， Y6) _" OVb(xi. yi. X6. Y6) = 0.. (6) OYs v， OYb In addition to the above first-order conditions， the following second-order conditions are su伍cientto ensure that(x:， yi， X6， yt) is a local N ash equilibrium:

。

2Vlx;.yi. X6. yt) み

1 3 b

<O，

ど五位

Ly;.

X6. Y6) O2九(X;.yt. X6. Y6) ax.~ Òy~

一(川悦

Y6.X6. Y6

)y

₎

_刈

OXsOYs

。

2Vb(X;. ペ 努 '?L

<

O

.

oX，i

芝五位

Ly;

，X6， Y6);]2Vb(X;， y;， X6， Y6) oxg oyg

-(川悦

y

:

，xt，

バ

L

y

OXbOYb (7) (8) (9)

。

。

If (4) holds for all(Xs， Ys)and (Xb， Yb)， then0妊ers(x:， y:， X6， yt) form a global N ash equilibrium As noted before， we consider in this paper the case in which fsand ん are different However， for mathematical convenience， we consider the following special case in which probability density functionsfsand fbare related in some specific manner

Assumption 2.1 (i) The medians ms and mb ofん andfbsatisfyms> mb

(ie， player s estimatesZa higher than playerb)

(ii) Two density functionsfsand fbsatisfyf~(mo) >0 and ん(mo)>0， where

mo = (ms+mb)/2

(iii) Both fsand ん havethe same shape， and one is obtained from the

other by a shift of distancems-mb; i..e..，ん(X)=ん(x+mS-mb).

61 Arbitrations When Players Have Di妊erentInformation -61

their medians Furthermore， der村ativesf;(mo) and f~(mo) exist

Although the estimates fsand fbsatisfying the above assumption is

very severe， it is a natural generalization of the case ofん=ん， and is

believed to reveal an essential behavior of arbitration procedures when ん

and fb do not coincide

The following lemma follows from the assumption on fsand fb

Lemma 2.1 Ifん andん satisfyAssumption 2.，1. then equation

与詐

L=

殻

ト

竹 H M l ( has a unique solution mo

Proザ ByAssumption 2..1(iii) and (iv)， we have

fb(mO)=ん(2mb-mO)=ん(2mb-mO+mS-mb)=ん(m仏 ( 12) and

F

b

(

m

h

f

f

b

(

x ) 改

=

i

:

o

ys ( χ + m ヲ-mb)dx

=

1

β

1

:

0

了

〉

〉

o

弘

V

f

ん以山仇s的((ο?

=1

ρf

ツ〉

ysμω(

_い

ωx

刈

x)d枕

x

戸=1ト一イ

4

引

F

九引~(川

ω(

附附削)

ω

By Assumption 2..1 (ii)， mo is a solution of Equation_(l1)“ In the following， we show the uniqueness. LetX be a solution of(11). Then ん(x)>0，ん(x) >0.. If X

<

mo then _ん(x)>ん(x)and Fs(x)+Fb(x)<F_わ，no)

₊

Fb(mO)= L 1-Fs(x)_~ Fb(x) T 恥h恥ere伽ff

与与与料よ<毛i

_{ん ( 叫 ん(}惇_χ昇件₎

L

Summing叩

…

nOωot伽加hee目rχ件 牛 刊m附ois a山 tion

of(11)

口

Remark 2.1 Assumption 2..1evidently implies that F

s

C

x)ζ _{Fb(X) for all x，}

and F.(x)

<

Fb(X) if F

_s

_C

x)

_<

1 and Fb(X)

_>

0.. Furthermore， according to(13)，

it holds that

-62 Kagawa University Economic Review 62

We then describe the assumptions we make on the definition(1)of

convergence

Assumption 2.2 (i)The parameter C of (1) is not too large; specifically，

C<与志~+祭去す 2勺弘行L= 告訴

(ii) Functionh(t) of(1)is convex and twice di妊erentiableover interval [0，

∞)， thereforeh"(t)ミ

o

fortε[0，∞)

Assumption 2..2 (i)is introduced for the convenience of comparing

DOA and FOA. Our purpose is to show that the probability of convergence

under DOA is positive even if C is small， but the same property does not hold for FOA or CA， therefore this assumption is acceptable. Assumption 2..2 (ii) implies thath'(t)is continuous over [0，∞)， and therefore h(t)=万(t)= 0 fortE[C，∞)， h(t)>0 and h'(t)く

o

fortε[0， C) ( 15) (16)

There are many functionsh(・)satisfying Assumption 2..2(ii) Example 4..1

in Section 4..4contains such an h(・)which is cubic in interval [0， C].

Lemma 2.2 Under Assumption 22， the following relation holds :

h'(t) ___ h(t)

当 主 斗 >_{fs(mo) "} 0(1 ，---

b

&

:

'

M

¥

¥

fortE[O， C)

2(1-Fs(mo))

Proof:Sinc怠 h(・)is convex and h(C)

=

0， we have

-h'(t)>

与

for但 [0，C)

Therefore the conclusion follows from Assumption 2..2 (i).

3 FOA When Players Have Different Information

口

Under the original FOA of Stevens (1966)， players s and b make offeres

Xs and Xb， respectively. If the offers crisscross， i..e..xs三三Xb， then the set

63 Arbitrations When Players HaveDi任erentInformation -63ー which is closer to his/herZa， as the settlement. Therefore the payoffs to s and b are U:OA(Xs，ね ) = 九 九 円 主)+xs(1-Fs(

今生))叫

and utOA(Xs

，

Xb)

=

mb -Xb

九(弓吋

-Xs(1

刊 号

X

，

b

)

)

respectively In this game， Guo and Chen (1995)obtained the following conclusion. .

Lemma 3.1 (Guo and Chen， 1955)1 Offers

mIWL

誠

一

務

ト 一 j 日 一 以 十 一 m m 一一一一 A A A V ハ υ F S F b x x f t i l l -E く 1 l I B --、

。

7) are the unique2 _{localN ash equilibrium offers under FOA}

In what follows， FOA is revised in the way as explained in Section 2 That is， in theIevised FOA， even if two offers Xs and Xbdo not crisscross

(Le.， Xs

>

Xb)， they are considered to be convergent with probabilityh(χs

-Xb}. Therefore， the expected payo任sto s and b under the revised FOA

are v:OA(Xs， Xb)

=

(

ヰ

Xb-ms )h(xs-Xb)十U:OA(Xs，Xb)(1-h(XS-Xb)) and

v

t

げ 伊PP倒叩0A respectively Based on this， we show that(17)is also the uniqueN ash 1 In the original theorem of Guo and Chen(1995)， the following conditions forfs(・) andん(・) are necessary:2/;(問。)>ー(1-Fs(mo))j;(mo)， 2R{mo)> Fb(mo)/，f(mo) Note that these conditions hold by Assumption 2..1 2 In Guo and Chen(1995)， there is no argument about uniqueness， but the uniqueness is evident

-64- Kagawa University Economzc Reviω 64 equilibrium for the revised FOA

Theorern 3.1 Under the revised FOA， (17) is the unique local N ash equilib -nu口1

品00/:By Assumption 2..2(i)， h(X[OA-x;tOA)= 0 holds， and therefore

v:ω(Xs， Xb)= u{倒(Xs，ぬ)and vtOA(xs， Xb)= utOA(xs， Xb)

hold in an open neighborhood of(x{ω，x;tOA).. Hence Lemma 3.1 shows that (17)is alsοa local Nash equilibrium of the revised FOA To prove the uniqueness， it is sufficient to show that no equilibrium0百ers(ん，.Xb) satisfyX s <Xb

+

C. For this， we assume contrarily that there exists such equilibrium0妊ers(ム，れυ) Then (3) leads to

。

U[OA(ん，.Xb) すh(れ れ ) + コ;，cs， ;，cbl(1 -h(んーん))

+(与主

-ms-U{OA(

れれ加

(18) and

。

utOA(Xs，れ) ーすh(XS-Xb)+ 払'"s， A bl (1-h(ムーん))

+(斗ム

-mb十utOA(X s， X b) )h'(ん -Xb)= 0 側 Summing側 and(19)yields

(

似

似

(Xs，ん

L+o

U.[OA(定品川、_)(1-h( れ一れ)) OXs OXb )

。

。

+(れ+れ-mS-mb-

U

:

倒(れ，Xb)+utω(ム，

x

.

b))h'(れ_-ib)= O. F or convenience， let us denote.Xs

=

.X

+

u and

x

.

b

=

X -u.. Since.xs

<

.X b

+

C was assumed， we conclude 0

<

u

<

C/2(note that.Xs>.xb evidently holds) In this notation， (20)is equivalent to (u(ん(X-)ーん(X-))十1-Fs(X-)-Fb(X-))(1-h(2u)) +2u(Fs(X-)

+

Fb(X-)-l)グ(2u)

=

O. (21) Clearly

x

= mo satisfies2(1). If

x

>

mo， then1-Fs(X-) -Fb(X-)

<

0 by (13)and

65 Arbitrations When Players Have Different Information 65←

ん(x)ーん(x)孟

o

by Assumption 2.1. Since 0< U < C!2， we have 1-h(2u)

> 0 and h'(2u)

<

0 by (16)， which imply that the left hand side of(21)is negative川 Similarly，if

.

x

<

mo， then the left hand side of(2)1is positive，

Therefore

x

= mo is the only solution of(21) New by subtracting(19) from

(18) and by using(12)， we have

h(2u)+2( -uん(mo)+I-Fs(mo))(I-h(2u))= 0，. (22)

However， since u<C!2<(I- Fs(mo))!.ん(mo)，we have -u/s(mo)+1

-Fs(mo) >0， and therefore the left hand side of(22)is positive by (16) This

is a contradiction and hence the theorem holds

口

Corollary3.1 Under the revised FOA， two 0妊ersin the unique local N ash

equilibrium(1 do n7) ot converge.

Proof : Immediate from Theorem 3，.1and Assumption 2，.2 口

4 DOA When Players Have Different Information

This section first revises DOA of Zeng， N akamura and Ibaraki(1996)in

the way as explained in Section 2， Then we derive a N ash equilibrium for

this revised DOA， and show that the probability that two double-o紅白s

converge in the equilibrium is always positive for any arbitrarily smallC.

4.1 Revised DOA

In the original DOA， playerss and b make double-offers (.xs， y，) and

(

χb， Yb) respectively， where Xs and Xb are cal1ed the primary0妊ers，y

，

and

Yb are called the secondary0任ers. The revised DOA proceeds as follows

(also see the fiowchart of Figure1). If the arbitrator judges that Xs and Xb are close， then the settlement is (xs+xb)!2; if not， but the secondary0妊ers

Ys and Yb are close， then the settlement is (Ys+Yb)!2 If neither are close，

then the arbitrator evaluates the offers in the same way as in the original

DOA and determines the settlement. That is， the arbitrator evaluates the

66 Kagawa University Economic Review C

s

C

Xs， yslza)

=

αIYs-Xsl +(1一α)(Ys-Za) Cb(Xb， Yblza)=αIYb-Xbl十(1一α)(Za-Yb) The arbitrator chooses the disputanti satisfying C

ベ

Xi，Yilza)

<

C;(x;， y;lza) 66 (23) as the winner， and determines the final settlement to be equal to the primary 0任erXi， where i = s orb， and

i

is i's opponent.. IfCs(xs， YslZa)

= Cb(χb， YbIZa)， then a lottery determines the winner， which is equivalent

~

67 Arbitrations When Players Have Different Information 67-ー

(in the sense of the expected value) to say that(xs十Xb)/2 is the settlement

Here the parameter αE(O， 1) is a constant to be determined and announced

in advance by the arbitratoL

Lemma 4.1 Let (xi， yi， X6， y;) be the0紅白sof s and b， which form a

local N ash equilibrium under the revised DOA Then x.

i

ミ

Y

i

ミ

Y

6

さよxt，

and

x

:

>.x6

Proof: Similar to the proofs of Lemmas 4..1and 4..2of Zeng， Nakamura

and Ibaraki(1996)

口

Since this paper is concerned with N ash equilibria， we suppose that o任ers(xs， Ys， Xb， Yb)always satisfy Xs二"::Ys主主 Yb主主Xband Xs >Xb

。

。

In addition to Assumptions 2J and 2.，2.the following assumption on α is necessary in this section

Assumption 4.1 The parameter αin (23)is set to 0くαく1/2

Under Assumption 4.，1. the original DOA of Zeng， Nakamura and

Ibaraki (1996) induces a crisscross of secondary0釘ersiffs(・

)=.M

・) In the following part of this section， we derive a N ash equilibrium of the revised DOA. We can see that in the equilibrium， the probability that two secondary0任ersconverge is positive for arbitrarily smallC. Finding out a N ash equilibrium consists of three steps， which are separately de -scribed in the following subsections. At first， Subsection 4 2 analyzes the first-order conditions (5) and (6)， and Subsection 4..3 checks the second-order conditions(7)~(lO) In Subsection 44， we further show that the derived local N ash equilibrium is also globa.l 4.2 First-order conditions This subsection derives some equations from the first-order conditions (5)~(6)リ Theseequations are necessary conditions for a N ash equilibrium We then show that the equations have a unique solution.

l

i

-68 Kagawa Universi~y Economic Revieω 68

Let

A = A(xs， Ys， Yb， Yb)=

i

1-2α) (Ys+ Yb)+α(約 十Xb)

2(1-α)

。

5)

Ifo妊ersdo not converge， then s wins if and only ifZa> A(xs， Ys， Xb， Yb)“

Note that

。

'A oA α oA oA 1-2α

百

7

五

7

一夜

1-a

了'万

J

一万

J

一夜仁五了

If two primary 0丘町sare not close， which happens with probability 1

h(XS-Xb)， then， by Lemma 4J， the expected payo妊sto players s and b

become as follows : 印 刷(Xs，Ys， Xb， Yb)=

(ヰ~-

ms )h(ys -Yb)

+

(.Xs-ms)(l-h(Ys-Yb)) 十(xb-Xs)Fs(A(xs，Ys， Xb， Yb))(l-h(Ys-Yb))， (26) UfOA(xs， Ys， Xb， Yb)= ( mb-

斗斗

(Ys-Yb) +(mb-xs)(l-h(Ys-Yb)) 十(Xs-xb)Fb(A(xs， Ys， Xb， Yb))(l-h(Ys-Yb)) Since the probability that two primary0任ersare close ish(XS-Xb)， the total expected payo妊sto s and b are given by 印刷(.Xs，Ys， Xb， Yb)=

(

斗

!:.E...-ms)h(ゐ-Xb) + UpOA(xs， Ys， Xb， Yb)(l-h(xs-xb)) and げOA(XS，九 九 九)= (mb

ーヰ斗

(XS-Xb) 十UfOA(xs，Ys， Xb， Yb)(1-h(xs -Xb))， respectively.. For the convenience of analysis， we define an auxiliary game， called

69 Arbitrations When Players Have Different Information

-69-u

;

r

OA， respectively. For comparison， the original game of the revised DOA is called V -game， in which two playerss and b have payo妊functions

VSDOAand vtOArespectively Restating (5) and (6)， the following four equations are necessary condi“ tions for(xs， Ys， Xb， Yb)= (x:， Y:， xt， . yt)to form a local N ash equilib -rium of the U-game:

。

UpOA_

r

，.. T:" ( 11*¥ I 1_..* _.*¥.L I /l*¥ a

1

ー す 一 =_:s

l

l

-

Fs(A *)+(xt -x:)fs(A*

切らけ

L 'Z~l- a) J (1-h(y: -yt)) = 0，

。

U!?OA_

r

T:' { 11*¥ ， {..* __*¥ 1 ( 11*¥ a

l

す 一 =

l

-

Fb(A

*)+(バ -xt)f川)茄~a)

J

以'Xb L ' Z~l -a;J (1-h(y: -yt))= 0， aupOA 1 ホ ホ * 句 円

寸y

，

すh(Yi-yt)+(xt-x:)ん(A*)

百ゴ了

(l-h(ぱ-yt))

(幻) (28) (xt -x:)民(A*)h'(y: -yt)

+

(

ヰ

丘

-x:)h'(y:-yt) = 0， (29)

。

U!?OA_ 1 U..* ..*¥， {.* *¥ 1 ( 11* 旬 日 寸Yb

ーが(バ

-yt)+(ぬ -Xt)fb(A*)

茄ゴ了

(l-h(Y:-yt))

+

(

χ:

-X6)Fb(A *)h'(

バーぱ)十円

Yt-xi )h'(Y:-ぽ) = 0，

。

。

where A本 =A(x:， y:， xt， yt)

Lemma 4.2 In any local Nash equilibrium(x:， y:， xt， yt)of the U-game，

the secondary0任erssatisfyyi

>

yt

Proof: Assume that the lemma does not hold. By Lemma 4.1， this means thaty:= Ytholds in a N ash equilibrium， which impliesh(yi -yt)= 1

Then， by (29) and (30)，

-70ー Kagawa University Economi( Review 70

and

3UbDOA(

い

X6，Y6

」す

+(CXi-X6)九(A*)+ yi

1

Y6 xi

)

h

'

(0)

SinceFb(A *)ミFs(A

つ

byRemark 2，.1， we have ei出er

(xi-x6)九(A*)+

ぷ子主

-x，iζ

。

。

1

)

or

(Xi-X6)九(A*)+

ぷヂ

ι

-xi>O (32)

。

U/OA(xi，yi， X6， Y6)

If (3D holds， then V V S ¥-"8，，'_oY

:

:

_s

s

， A b， Y b I

>

0 ; if (3~ holds then

/ V， OUbD似_{(xi，Yi， X6， y6)} < 0， Since these two derivatives should be zero in a local N ash equilibrium， (xi， Yi， X6， y6) is not a local N ash equilibrium， which concludes the proof， By Lemma 4，.2，的)and (28)are equivalent to 1-Fs(A *)+(X6-xi)ん

(rh27=O

-Fb(A*)十(xi-x6)fb(A

っす持てで=

_2(1-α)

口

(33)

。

。

Ifん(A

つ=

0 thenFs(A *)= 1 by (33) Hence ，M A

つ=

0 and Fb(A *)= 1 by

Assumption 2，.1(i) and (iii) Therefore the left hand side of (34)becomes-1，

implying that (34)is false， which shows thatfs(A*)>0 Similarly， we con

-cludef~(A *)>0 Equations (33) and (34)then imply that (1-Fs(A *))/IsCA

つ

=

Fb(A*)lfb(A勺 From Lemma 2，1.， we haveAホ=mo Let

x

= (xi

+

X6) /2 Then the unique solution of (33) and (34)is

[

X

J

=

Z

耕 平

X6

x-

与務

:

)

0

)

う

三

Note that， by Assumptions 2，.2(i) and 41，，. (35)

71 Arbitrations When Players Have Di任erentInformation -71-J-xt=

血

7L

与志タ心

2

与

訪

7

2

L

>

C

。

。

Therefore

x

i

and X6are not close.. Furthermore， by (25)，

A* = mn

一息二

2α)(

バ

+

yt)+2αf

一 川u- 2(1-α) (37)

According to倒 and附， equations (29) and (30) are equivalent to

0=th(hJ)

一平

(l-Fs(附 ))(1-h(y;-Y6)) (10""，1. ，¥1-α1 -Fs( mo)，~ ~ ¥ +( (2Fs(附)一1 ) 7

官

r

:

OJ十7一 山

;-d)

(38) and 0=ーが(ば-Y6)+

弓句

(mo)(l-h(y; -yt)) (10""，1...¥ ，¥1一α1-}九(mo) 一 一 ¥ +( (2Fb(mo)-1)一一ームマ午会午2L

+

y-.x)h'(yi-yt) (39) \α Ts~mo) /

respectively， wherey

₌

(ぱ十Y6)/2.. Summing equations (38) and (39)， and considering

ω

， we obtain

(タ-x)グ(yi-yt)= 0“

If h'(yi-yt)= 0， thenyi-Y6 主主 C by Assumption 2..2 (ii)

and (39) imply that

Fわno)

=

1 >0

=

Fb(mO)，

側 Therefore (38)

which contradicts(14). Hence (40) implies y

₌

.

x

， Furthermorey

=

.

x

=

mo by (37)， Then (35) can be rewritten as Let

(…噌平

X6=

均一背

:

)

0

)

平

yi= mo十u本 ，Y6 = mo-u*， 制)

ω

)

Then u *

_>

0 holds from Lemma 42，.. and both (38) and (39) are equivalent to saying thatu* is a solution of

-72 Kagaωa Unzversity Economic Revieω 72

1-2

2

h(2u) 一万立(1-Fs(mo))(1-h(2u))

十(2Fs(mo)

内旦勺閥均

(2u)= 0 (43)

Summarizing the above analysis， we obtain the following lemma

Lemma 4.3 Under Assumptions 2J， 2..2and 4.，1.any local N ash equilib

-rium of the U…game (xi， yi， X6， yt)satisfies(41)and (42)， where u * is a

positive solution of(43)..

The next lemma says that equation(43)always has a unique solution

Lemma 4.4 Equation(43)has a unique solution， which is located in interval

(0， C/2).

Proof: Denote the left hand sider of帥byL(u) Then L(u) is a continu -ous function ofu over[0，∞).. Sinceh'(O)<0， h(O)= 1 and F山?勾)<1/2， it is easy to check thatL(O) >0ド Onthe other hand， since h(u)=万(u)= 0

for alluE[C，∞) (see(15))and α< 1/2by Assumption 4.，1.it follows that

唱口

L(u) = 一万立(l-Fs(mo))<O，foruE[C/2，∞). (44)

Hence L(u) = 0 has a solution， which is located in interval (0， C/2)

Moreover， from Assumption 2..2(ii)，

2(1-2α)

L'(u) = グ(2u)+~こ坐L(1-Fs( mo))グ(2u) α <0 2(1一α)(2Fs(mo)-1)(1-Fs(mo)) h"(2u)

α

ん(mo) foruE(O， C/2) This inequality shows thatL(u) is strictly decreasing in interval (0， C/2)， and hence by (44)， the solution ofL(u) = 0 is unique

口

CoroIlary 4.1 Equations(27)~側 have a unique solution(x

ム

バ

，

d

，

yt)i Furthermore the probability thatyi and Y6are considered to be close is always positive for arbitrarily small

c

.

.

73 Arbitrations When Players Have Di任erentInformation 一万一

Proof: By (35) andω)， from Lemmas 4.3 and 44，. equations(2)7，，-，(30) have a

unique solution(xi， yi， X6， Y6). Furthermore， Lemma 4.4says thatyi -Y6くC，therefore the probability thatyiand Y6are considered to be close is always positive for arbitrarily smal1C

口

The solutionu * of帥;)evidently depends on parameter α W e further prove that thisu * = u *(α) is strictly increasing as a function of α Lemma 4.5 As a function of α， the solutionu* of (43) is strict1y increasing :

(

u

て

っ

α)>0

Proof: Viewing u * as a function of αand computing the derivative of (43) with respect toα， we obtain: ( _1(1一α)(1一九(mo))(1-肌 川h"仰"

て

ηr吋位(2μu ん(mo)) 十(α+2(1一 勾)(1一Fs(附 )))h'(2U*))(川 (α) 一 (1-FsCmo))(1-2Fs(mo))( -h'(2u*)) ー ん(mo) h(2u*) -2(1-Fs(mo))(l-h(2u *))_.!!3.す4 山 1-2α

=一羽百戸

(2u引 下 玄(1-Fs(附 ))(1-h(2u*)) h(2u勺 -2(1-Fs(附))(1-h(2u *))_.!!3.す4

=一司誌了

h(2f)-tz(l-A(附

)

)

(

1

-h

(

μ 収 0，

where the second equality is from倒“ Theabove relations imply出at (u

て

っ

α)

>

0 by Assumption 2..2 (ii)

口

Remark 4.1 Sinceu*(α)E[O， C) is bounded， the above lemma implies that limαーou*(α)exists. Let the limit be uo* Then from ω.) we have

j山no)

グ(2uo*)= _1-2F1 ~~';"l:M

¥

(1-h(2uo*))

s(mo)

-74ー Kagawa UniversiかEconomiιReview 74

>0

Remark 4.2 Since

h

(・

is strictly decreasing by Assumption 2..2(ii)，

Lemma 4..5 says that the probability that two secondary0妊ersare close

becomes higher when a takes a smaller value.. However， as implied by

Remark 4.，1.the probability does not converge to 1 even ifa converges to

O

For the V -game， we have the following conclusion

Lemma 4.6 The strategy(

，

x

:

，

y

，

:

，

X6

，

yt)given by臼])and (42)also satisfies

the first-order conditions of a local N ash equilibrium of the V -game..

Proof : The first-order conditions， which are necessary for a local Nash

equilibrium of the V -game are : OVpOA 1 71 .. ¥ ， ( ¥ 一 一 一 =_{OXs 2 ¥ 2} ~h(XS-Xb)+( 主主主-ms-

U

P

OA )h'(XS-Xb) "1/'$ 1"...I，':;i } o[印 刷 +寸子一(l-h(Xs-xb)) uんS = 0，

tZ

竺 =

ι

h(xs -Xb) -( mb

一主学

ι-U(jOA )h'(xs-Xb) 【ノIんb ム ¥ ム / oUfOA

+1:

トー(1-h(XS-Xb)) u.んb = 0， oVf似 _o[印 刷 ~ー_{= (l-h(xs-xb))} 寸 Lー =0，

a

'ys ¥..1. "，¥..I¥;S .A.l_{0/ /} ò，Y~ oV，fOA

打

コ

D似 一~.~ = (1-h(XS-Xb)) U~.~ ー₌₀ OYb _¥.l_nV.s_.Abjj OYb 的) By (36) and (15)

，

h(x': -xt)= h'(x': -X6)= 0 holds.. Since(x

，

:

，

y

，

:

，

X6

，

yt) sa tisfies all_(却側， it also satisfies the above four equalities..

_口

4.3 Second-order conditions To check whether0妊ers(x':

，

y

，

:

，

X6

，

Y6)of(4])and (42)really form a

-75-Arbitrations When Players Have Di任erentInformation 75 ofUPOAwith derivative partial second-order The conditions -order respect toXsis given by

P

UpOA(xi， yi， X6， y6) àx~

=

(

-

似

附

)+(x6-x:)f;(m

匂三分布匂(日

(ν)) α(2!.;(mo)

+(1-

Fs(mo))j;(mo))_(1-h(2u*))

_<

0“ 2(1-α)ん(mo) Furthermore， l i s

-。

2UpOA(X;， y;， X6， y6) aXsavs = ( -fs(mo)十 ( 日 ) が(mo

初出

(l-h(ν )

)記念

and 32UpOA(X;， y;， X6， Y6) - ( 守 口 ¥ = (1 +(x; -X6)ん(m作~: )h'(2u*) +(mo-x;+(x;-x6)Fs(m

ω

h"(2u*)

+

ω

-x;)

バ

(mo

炉多

(1-h(2u*)) Therefore the Hessian determinant becomes

。

2UPω(x;，y;， X6， Y6) a2UP似(x;，y;， X6， y6) àx~ àv~ 、_¥ l I / x

一

y

-b

* J h_{一 批} x

一

A 一 D L U

一

ぺ O 一 ， f t t t

、 、

2f}(mo)

+

(1一九(mo))

五

(mo) = 有 ら ( トh(2u*))

{

-(1+

血子

L(l-Fs(m

川

h'(2u*) 1-α(1-2Fs(mo))(1-Fs(mo)) ，-，'/rL.*¥ i 十 一 一 - _ん(mo) r(2d)}

_，

_.

_{\~... I J} 一 点 川 信 手 ( 1 一 町))2 (46) Similarly， for playerb， the second-order partial der討ativeof

u

t

OAwith

76 Kagawa University Eωnomic Review -76 α(2R(m日)-Fb(mo)

五

(mo))_(1-h(2u*))<0 2(1一α)ん(mo) respect toXbis

32ueOA(x;， y;， Xb， Yb)一

ax~ Furthermore， the Hessian determinant is 32ueOA(x;， y;， Xb， Yb) o2ueω(x;， y;， Xb， y;) IX~ òy~

(

問

削

(x;，y;， Xb， Yb)V OXbOYb ) 2j~(mo) -Fb(mo)umo)

=茄じ了

(I-h(2u*)) 九(mo)

{

-

(

1

+

皇 子 弘

(mo)

万

)

(2u*)+ u h 叫 ノ 、 山 u m 一 位 / べ -v n h f 一 一

寸

仇

ロ

伊

川 m a u 川合一日

ι

心川町 i J m α 一 げ 1 一 一 α

↓

(47) We are interested in the situation that both (46) and 仰 arepositive， as it ensures that(xi，ぱ，Xb， Yb)is a 1oca1 Nash equi1ibrium is positive if and on1y if (46) is positive By (12) and (13)，仰) We are happy to conclude :

口

The Hessian determinant (46) is positive Proof:See Appendix A. This estab1ishes the following conclusions. Lemma 4.7 0丘町s(xi， yi， Xb， y;)of (41)and (42) form a 1oca1 N ash equi1ib

-Offers(xi， yi， Xb， y;)of附 and(42) form a 1oca1 Nash equi

-librium of the V -game (ie.， the revised DOA). rium of the U-game Theorem 4.1 Lemma 4.8 仰) Therefore， by Lemma Proof:SinceX i -Xb

>

C by (36)， VpOA(Xs， Ys， Xb， y;)= UfOA(xs， Ys， Xb， y;) VeOA(xi， yi， Xb， Yb)= UeOA(xi， yi， Xb， Yb) ho1d in an open neighborhood of(x;i， y;， Xb， y;)

77 Arbitrations When Players Have Different Information - 77-ー 4..8， (x;， yi， X6， Y6) is also a local N ash equilibrium of the V -game

口

Remark 4.3 According to(36)， the expected payoffs to s and b in the local N ash equilibriumχ(

i

，

y

;

， X6， y6)are V，fOA(xi， yi， X6， Y6)

=

UpOA(X;， y;， X6， Y6) = mo-ms+(ト 2Fs(mo))

勺間斗平日

(ν))

>

mo-ms and VfOA(xi， y;， Xb， y6)

=

UfOA(xi， yi， X6， y6) mb一 向 十 ( 机(mo)-4

掛 平

(1-h(2u*))

>

mb-mO， respectively.. 4.4 Global N ash equilibrium The purpose of this subsection is to show that (x;， y;， X6， y6) of(41) and (42)is actually a global N ash equilibrium of the revisedDOA. For this， it is necessary and su伍cientto check that V，fOA(XS， Ys， X6， y6)~ VfOA(xi，ぱ， X6， y6)for all Xs and Ys and VfOA(xi， yi， Xb， Yb)三二 VfOA(xi， yi， X6， Y6) for all Xb and Yb First， let us consider the U-game instead of the V -game.. Recall that Lemma 4..4states that (xi，ぱ)is the unique solution satisfying all the first order conditions of a maximal point of UfOA(xs， Ys， X6， y6)， and it is

actually a maximal point by Lemma 4..8.. In addition， the following lemma

shows that an infinitely large offer is not an optimal choice for players，

therefore (xi，ぱ)is a global maximum point of UpOA(ぬ， Ys， X6， Y6) ..

Lemma 4.9 0妊ers(xi，

y

;

， X6， y6)of(41)and (42)form a global Nash equi

-78- Kagawa UniversiかEconomiじReview 78

Proof: See Appendix B

口

To show that (xi， Yi， X6， Y6)is actually a global Nash equilibrium of

the V -game， the following lemma is useful

Lemma 4.10 For the Nash equilibrium (x':， vi， X6， yt) of the U-game

following two inequalities hold : 3UP似 (Xs，Ys， X6， y6) 3 r s > O ifzs<xt+C，

。

U

;

;

ω(x':

，

y

，

:

，

Xb. Yb) 一一-

e

z

AL<O if h>zf-Cl (49) Proof: See Appendix B

口

The U-game is different from the V -game when xs-XbくC The

following conclusion shows that(x:，

v

:

， X6，ぽ)is also a global N ash

equilibrium of the V -game

Theorem 4.2 0任ers(xi， vi， X6， y6) of臼1)and (42) form a global N ash

equilibrium of the revised DOA

Proof: If Xs-X6 主主 C， then倒 holds， Therefore Lemma 4.9 tells us that

(.xi，ぱ)is a best reply to (X6， Y6).. If Xs-X6くC，then by (49)， we have

()UpOA/()XS>O. If (xs， Ys)is a best reply to (X6， Y6)， then

UpOA(Xs， Ys， X6， yt)とよ UpOA(X:， ぱ，X6， y6)>mo-ms，

where the second inequality is from Remark 4..3 Also， from xi -X6 > C，

we have Xs< x.:， and hence (xs十x6)/2-ms<(xi十x6)/2-ms= mo-ms

Therefore， 当主主-ms-UpOA(xs， Ys， X6， Y6)<0 Furthermore， since h'(xs-x6)三二OandO ζ h(xs-xt) < 1 hold， we have ()VP倒 / 一 一 、 一τケー(Xs，Vs， X6， y'b)>O V，^'S by the first part of倒 Thiscontradicts the assumption that (xs， Ys) is a best reply to (X6， y6) The conclusion for b can be shown analogously.

79 Arbitrations When Players Have Di任巴rentInformation - 7

9-口

From this theorem， we can derive the following corollary that states an

important advantage of using the revised DOA..

Corollary4.2 Under the revised DOA， the probabi1ity that the Nash

equilibrium(xi，ぱ，X6， Y6) ofωand 仰)is considered to be convergent is

always positive.

Proof : The result directly follows from Corollary 4..1and (1)6

口

Comparing this with Corollary3..，1 the revised DOA has a stronger

power of inducing convergence than the revised FOA.

Usually， two disputants go to arbitration because their estimates are

di任erent The difference can be measured by the di任erenceof medians m.s

-mb under Assumption 2..1 The following corollary shows that the dis

-tance between two secondary offers may be shorter thanms-mか

Corollary4.3 If Cζ ms -mb (which holds in Example 4..1below)， then the secondary0丘町sin the N ash equilibrium(xi，バ， xt，ぽ)of仙 and(42) satisfyYi -Y6

<

C 三二mS-mb.. ExamPle4.1 Let F

s

C

x) and Fb(X) be the triangular distributions over intervals [0， 2] and [-1， ，]1 respectively That is， (t for0 三二O 三二1 (t十1for-1 ::;;:t 三三O ん(t)

=

~2-t for1 三二 t ::;;:2 and ん(t)

=

~l-t for0 三二 t ::;;:1

l

O

otherwise

l

O

otherwise We set parameter

α=

1/3. Then mo

=

1/2and xi

=

4， X6=ー3.. If iftE[O， 1] otherwise， then Assumption 2..2holds Equation(43)then becomes 21 す

h

(

2

u

)

-

t

(

1

-

h

(

2

u

)

)

ーすグ

(

2

u

)

= 0， or

-80- Kagawa University Economic Review 80

1l(1-2u)3+63(1-2U)2-7 = 0

The solution of this equation is u * ~

o

.

338， and hence

y

i

~ 0838， Y6~

-0..162 The distance Yi-Y6 = 2u 本~0..676is judged to be close with prob

-abilityh(2u*)~0..034 >0. By Theorem 4，.2.0丘町s(xi， Yi， X6， y;)form a

globa1 Nash equilibrium of the revised DOA game

口

In the above examp1e， the probabi1ityh(2u*) is positive but is very

small. There are main1y two reasons for this.. First， ms-mo = 1 is too

1arge; ie， the two estimates are too far from each other.. Ifms.-mo→0

(thereforeFs(mo)→1/2)， then it is easy to see that the solutionu* of(43)

converges toO. That is， two secondary 0妊ersbecome very close and

convergence probabilityh(2u*) becomes very close to L This observation

is consistent with the resu1t of the origina1 DOA of Zeng， Nakamura and

Ibaraki(1996)， which says that two secondary 0任erscoincides ifん(・)=

fo(・)

The second reason is that parameter va1ue α= 1/3 is 1arge， U nder the

revised DOA， two secondary 0丘町sbecome closer ifαtakes a smaller va1ue

(see Lemma 45)， and therefore the convergence probability becomes higher

However， we point out that， under the assumption ofんキん， the conver

-gence probi1ity does not converge to1 even ifαconverges to0 (see Remark

42) Intuitive1y， if a p1ayer (for examp1e s)compromises in the secondary

o妊er，the arbitrator may judge that two secondary offers converge there

-fore (ys+円)/2is the fina1 settlemenL Note that (Ys+Yo)/2くYs，if Ys and

.yo do not crisscross，. In other words， each p1ayer cannot ensure that the fina1 resu1t is better than his/her secondary 0任ereven he/she compromises， hence a p1ayer suffers a heavy 10ss for giving compromise， In contrast， under the origina1 DOA， two secondary offers will not be judged as conver -gent un1ess they crisscross. Therefore each p1ayer can ensure that the fina1 resu1t will not be worse than his/her secoddary offer if he/she compromise..

81 Arbitrations When Players Have Di任erentInformation 81

As a result， different from the original DOA， under the revised DOA， two secondary0任ersdo not crisscross even ifαis small. Besides， we mention

that two primary offers are farther ifαtakes a small value because of (41) In practice， we suggest that the arbitrator should make his/her prefer -ence of fairness more clear， so that two disputants can make closer esti -mates.. As another means to increase the convergence probability， we suggest to use a small αin arbitrator's criterion functions(23) However， two primary 0任ersof the (revised) DOA may be far from each other in this

way. Therefore， if the secondary0任ersdo not converge， then the arbitra

-tor has to choose an offer， which may be very extreme as the settlement To alleviate this problem， as sugggested in Brams and Merrill(1986)， we can give the disputants a final chance to negotiate a settlement once again. Since the primary offers are extreme， a great pressure is given to the disputants so that they bargain seriously in this additional final stage

5 CA When Players Have Different Information

Procedure CA can also be revised in the way as described in Section2. The ftowchart of the revised CA is shown in Figure 2.

For mathematical convenience， this section further makes the follow -ing assumption， which is stronger than Assumption 2..1(ii)

Assumption 5.1 Two estimates

f

s

and fbare close in the sence that

.i~S[lsCx) 一伽0)

]ぬ =L70[ん(x)一 伽0)

]ゐ寸

(50) Assumption 5..1implies Assumption 2..1(ii)， because otherwise the left hand side of(50)is1/2， which is larger than the right hand side

Ifplayerss and b make offersXs and Xb satisfyingXs

>

Xb， and the two

o任ersare considered not close in the sense of(1)， then the expected payoff

i l k -i i 82 Kagaωa Universiか EconomicR， evieω 82 UfA(XS， Xb)

=

l

;

b

x

ん(χ)dx

吋ア

fs(x)枕

+

X

s

メ

L

J

s

(

X

)

改

fx

ん(x)dx，-ms

=ひん(χ)此ーひん(叫んbFs(弓~

)

-

xbF，(Xb)叫 仏 ) 寸 sFs(

守主

)-ms Fig 2: Flowchart of the revised CA

83 Arbitrations When Players Have Different Information 83-= -

L"sxdFsCx)+XbFs(斗~

)-XbFs(Xb)

+

xsFs(xs) -xsFs(

当主)

=fR(z)いbFs(ヰ吋 -xsFs(斗~)，

ω

where the third equality holds becauseんissymmetric. Similarly， when xs

and Xb do not converge， the expected payo妊tob is

UtA(九

xh-f

削 減 -XbFb(

当主)+れれ(守主)

Lemma 5.1 (Brams and Merrill (1986))Iff is symmetric and strictly

unimodal， then UfA(XS， Xb) <0

μ

宇

ι>

ms = 0

if 斗~=ms

v ↓ 小 > 0 if 一万一~く ms Prooj: See Appendix C. (52)

口

Since the offers Xs and Xb are thought to be close with probability h(xs -Xb) and not be close with probability 1-h(XS-Xb)， the total expected payoff to s is VSCA(Xs， Xb)

=

(

斗

Xb-ms )h(xs -Xb)叫 A(XS，xb)(l一 恥 -Xb))， and the total expected payoff tob is VbCA(九 ぬ )= (mb

一斗

Xb)h(xs-Xb)

+

UtA(xs， Xb)(1-h(XS-Xb)) Lemma 5.2 Any nonnegative solution of the following equation is located in the interval (ms-mb，∞) and there is at least one solution in this inter -val:

84 Kagawa University Economic Review

[

生

p

i

c

;

u

F

s

(

χ

)

枕 十

M

川 )]h'(2U)+

ヤ

(2u) +[Fs(mo+u)-Fs(mo)-ufs(mo)](1-h(2u))= 0 Proof:See Appendix C Theorem 5.1 Under the revised CA， (xY= m山 CA

X~A

mパ CA) 84 (53) 口 (54) form a local Nash equilibrium if and only ifUCAsatisfies(53)and uCAE(ms

mb，∞) holds

Proof:We first note thatUCAミ

o

holds similarly to Lemma 4..1 The proof

of Theorem 5..1consists of two parts， which consider the first-order and the second-order conditions separately (i) Firsl-order conditions. The partial derivatives of (3) are

。

VSCA_ ( ¥ 寸 ア=("'S寺坐- ms-UiA(xs， Xb) )h'(XS-Xb) where and where 見ノんg ¥ '" ノ oUEA

+

τ

_二

_一

(1-h(xs， Xb))

十戸

(XS-Xb)，

安こ =ι[lsCx)-fs(斗~)]仇

OVbCA (Xム附 ¥ 寸 : 十 =( .As ~ .Ab -mb-UfA(xs， Xb) )h'(XS-Xb) U，..{，b ¥ L， コUfA + U~.b (1-h(xs， Xb))

一甘い

S-Xb)， uんb ム

努土=バプ[fb(弓~)ーん(X)

此

]

If(54)is a local N ash equilibrium， then o VsCA !OXS = 0， which is equivalent to (53)by the above expressions. Similarly， for player， b， OVbCA

!

O

.

Xb= 0

85 Arbitrations When Players Have Di妊erentInformation 85ー (which is the second part of (3)) is equivalent to

[血手立に

;uFb(x)ゐ十2uFb(mo)]h'(2u)-

~

h(初) +[Fb(mo-u)-Fb(mO)

+

μ

九(mo)](l-h(2u))

=

0 (55) We then show that(55)is also equivalent to(53).. lndeed， by Assumption 2..1 (i)， we have F

s

C

x)= 1-Fb(ms+mb-x)，

which is more general than(13). Hence

(56)

L

f

;

u

九(x)此 =2M

L

f

;

u

N

m

s

十mb-x)改 = 2u

ー

に

u九(x)仇 which implies

生ヂーに

M M + 2 u九(mo) 戸民η，t ) 白 u (

=門戸三

-

L

f

J

7

F

s

(

χ

)

ぬ十2uFs(mo)] Furthermore， by (56)again， we have Fb(mo-u)-Fb(mo) = -[Fs(mo+u)-Fs(mo)] (58) Relations~日) and (58)then show that倒 isequivalent to(53).. (ii) Second -order conditions“ By calculation， we have íJ2 V~ω 一(， " 0USCA(Xs. Xb)¥ 寸二~ = (1-2 uus ~~s ， .Abj )h'(XS-Xb) VA<S ¥ ミノ-^'S

+(守主-

m

s

-

UfA(.xs，ぬ))hll(XS-Xb) 02_打ω 十 寸 才(l-h(Xs-xb)) (59) First we consider the case ofXfA ::;:2ms-mo Sinceん(X)二とん(mo)holds for allxE[mo， 2ms-mo，]we have

o

UfA(xfA.X~A) _

r

x

主A

r

-86- Kagawa UniversiかEconomκReview 86

g

L

:

m

-

m

o

[

ん(χ)ーん(mo)]ド

2

f

[

ん(x) ん(附)]dx

寸

，

側

where the last inequality is from 側 Onthe other hand， ifX

f

A

>

2 ms -mo，

then/.(x)<ん(mo)for allxE(2ms-mo， xs] Therefore it is easy to see that 側stillholds As h(・)is decreasing， the first term of the right hand side of 側 isnegative for any Xs Also asんisstrictly concave， we conclude that

r

UfA(

ダ

，

xY

L

= f

s

C

X;A)_/.( XfA

芋判

xf

与坐

fdXfA

主判

¥ 2 J 告 ¥ 2 J くん(mo+uω)ーん(mo)<O，

where the last inequality comes from UCA

>

ms -mb Therefore the third

term of the right hand side of(59)is nonpositive.. Finally， UfA(xs， Xb)二:0:0

follows from 倒， and by Assumption 2..2(ii)， the second term of the right

hand side of側 isalso nonpositive It is evident that all three items in仰)

can not be equal to 0 simultaneously.. Therefore，

。

2VlA(xfA，XfA)

ペ フ LL<O

(j

X

s

Similarly， by applying the above argument toXband /~， we have

r

VbCA(xfA， XfA)

ペ ヌ >0

(jXI;

Therefore(54)is a local N ash equilibrium ifu is a solution of(53)

“ 口

CorolIary 5.1 If Cζ2( ms -mb) (an in Example 5..1below)， then have XfA

-XfA

>

C， and hence the0任ersare not considered to be close“

The above conclusion shows that if C is small， then the probability that

two offers converge is 0れ Comparingthis with Corollary 4，.2. we conclude

that the revised DOA is better than the revised CA in the sense of inducing

convergence

CorolIary 5.2 In the equilibrium(5)4of the revised CA， the distanceXfA

_XfA between the0任ersis always larger than 2(ms -mb)リ

87 Arbitrations When Players Have Di任erentInformation -87-ー

gap ms-mb between the two estimates of the disputants“ Comparing with

Corol1ary 4..3， from another viewpoint， this reveals that the revised DOA is

better than the revised CA in the sense of inducing convergence.

ExamPle5..1 Consider the revised CA in the same situation as in Example

4.，1.where functionsfs(・)， f~( ・)and h(・)satisfy Assumptions 2.，1.2..2 and

5..，1 and C = 1

<

2(ms -mb) Equation(53)is then equivalent to 7 1/3 ¥2 u 8 す け-u) す υ which has a unique solutionUCA = (2+花)/2:::::::L707in tinterval(ms-mb， ∞) Therefore xf>l之 2..207and .xtA ::::::: -L207. SinceX;A -xt A _{;:::::3414> 1} = C， these two offersXfAand xtA are impossibly considered to be close Comparing this result with that of Example 4J， we observe that

yi<X;Aくxiandx6く χtAく

Y

6

These relations provide an intuitive explanation why the revised DOA has

a larger power of making the offers ofs.and b close than that of the revised

CA The revised DOA provides an opportunity for each player to separate

the estimate from the demand so that two secondary0任ersbecome close.

6 Comparison Of FOA， CA And DOA

Arbitration procedure DOA of Zeng， Nakamura and lbaraki(1996)，

and CA of Brams and Merri1l(1986)both exhibit the convergence of offers

when two players share the same estimate about arbitrator's fair settlement

Zα(Zeng and Shishido(1996)develope the convergence result of DOA to

more general cases) ln this sense， these two procedures are better than

FOA since FOA does not have such convergence property. ln order to

examine the behavior of arbitration procedures in more general setting，

this paper considered three procedures FOA， DOA and CA under the

-88- Kagaωa UniversiかEwnomzιReview 88

Furthermore， this paper adopts a revised definition of the convergence

which is based on the probabilistic estimateh(t)of being c1ose， where h(t)

>

0 if the distance between two 0妊ersis less than C and 0 otherwise

For the revised FOA， DOA and CA， we first derived optimum strat

-egies that form N ash equilibria of the corresponding games. As results，

Corollary3.1 concludes that in the unique local N ash equilibrium of the

revised FOA， the distance of two offers are larger than C， and therefore

the offers are impossibly judged to be c10se Contrary to this， Corollary4..2

confirms an important property of the revised DOA， that the distance

between the secondary offers in the Nash equi1ibrium is always smaller than

C， and therefore the probability that the secondary offers are judged to be

c10se is always positive. Under the revised CA， Corollary 5J shows that

the distance between the two 0丘ersin the Nash equilibrium is larger than

C if Cζ 2(ms-mb) (ie， the probability that two 0任ersare judged to be

convergent is zero) Also， Corollary5..2says that the distance between the

two 0丘町sof CA is larger than ms -mb. As ms -mb intuitively measures

the gap between two estimates， it means that the revised CA cannot

surmount the gap， whereas Corollary4..3tells us that the revised DOA

surmounts the gap ms -mb successfully“

In conc1usion， in the sense of achieving larger convergence probability，

the revised DOA is better than either of the revised FOA and the revised

CA

Acknowledgements

This work was partially supported by the Scientific Grant in Aid by the

Ministry of Education， Science and Culture ofJ apan， which is gratefully

89 Arbitrations When Players Have Different 1nformation 89 References

S.J Brams， NegotiationGames A}ψlying Game Theoη to Bargazning and Arbitration， Routledge， London， 1990

S. J Brams and S Merrill UI， Binding versus final-offer arbitration: a combination is best， Management Scienα32 (1986) 1346-1355

W. -G. Guo and T. Chen， Final-o任erarbitration with incomplete information， Control and Deιision 10 (1995) 40-44 (1n Chinese)

W. F. Samuelson， Final-ofter arbitration under incomplete information， Management Scie珂ce37 (1991) 1234-1247

C M. Stevens， 1s compulsory arbitration compatible with bargaining?， lndustrial Rela -tions 5 (1966) 38-52

D ωZ Zeng， Game-theor巴ticstudies on arbitration procedures and fair settlement， Ph

0..Dissertation， Kyoto University， Japan， 1996

D -Z Zeng， S. Nakamura and T 1baraki， Double-o丘町arbitration，MathematiwlSocial Sczences 31(1996) 147-170

D -Z. Zeng， S. Shishido， Th巴existenceof equilibrium in systemmetric arbitration games

FOA and DOA， Kagawa University Ewnomic Review 69 (1996)， 317-332

Appendix A: Proof of Lemma 4.7

In order to prove Lemma 4.，7.we first prove two more lemmas Recall that Lemma 4..3 affirms that u*

=

(Yi-Y6)/2 is the unique solution of (43).. W e view this solution as a function ofα， and derive the following conc1u

-SlOn

Lemma A.l For the solutionu* of (43)， 1-2α

h(2u*)<

τ

士

J

holds for all0 <α< 1/2

Proof: Since u* is a solution of附， we have that from Lemma 2..2，

唱 A

ーすh(2u*)+-z旦(1-Fs(附 ))(1-h(2u*))

一段〉ー Kagawa University EwnomI(Review

=

(1-2Fs

川 平

(l-Fs(附))(

-殺す)

>(ト2F.(mo))

王

示

h(2u*)， which can be rewritten as 1-2α 2.la h(2u *)[1-2(1 α)Fs(mo)]く 一 一 一(1一日(mo))(l-h(2u*)) α Therefore， 1-h(2u*)¥1-2(1-α)F.(mo) ¥ G > -h(2u*) "2(1-2α)(l-Fs(mo)) / 1-2α' 90

where the last inequality is from Fs(mo)

<

1/2" Inequality (61)follows direct

-ly from the above relations

口

Lemma A.2 For xE(O， 1/2]， the solutionu* of (43) satisfies the following

relation:

αe正予(1一 α)十4α

ー

が

(2u*)+

守主

(1-2α)2(1-h(2u*)) >0 側

Proof: Denote the left hand side of (62) by L(x)引 Then

L(

す

)

=

勾

(2a-l)h(ν )

+

2(1-2a)2(1-h(ν)) = 2(1-2α)(α-1)h(2u*)+2(1-2α)2

>

-2(1-2α)(1-2α)+2(1-2α)2 = 0， where the last inequality is from (6，)1 Since L'(x)=

寸当

fh(ν )一3日 げ(1-h(2u*))叫 it holds thatL(x)注 L(1/2)>OforxE(0，1/2]

口

Proof of Lemma ，.47: SinceFs( mo)ε(0， 1/2)， x = Fs(mo) satisfies側，

which can be rewritten as

一 一 一 .

2(1-α)¥2(l-Fs(m0)) 1-2Fs(mo) 1一α/

91 Arbitrations When Players Hav巴Di妊erentInformation

>信表

(1-h(2u*)) From附， we obtain

珂壬五了苛坐ゐ了一時弓間山

'(2u*)

ー

ヌI

土

石

了

2(1-Fs(m日))

#ヰヂ(-

~

h(2u*) 1-2Fs(mo) (1-α)2 ¥

十弓匂一五

(mo))(l-仰木)))

>

伝

説

;(1-h(2f))

Furthermore， by Assumption 2..2 (ii)， we have

1-α(1-2Fs(mo))(1一九(mo)) h"(2u*)ミ0， α ん(mo) and hence the Hessian determinant倒islar ger than -91ー 制)

有ら了

(1

一

h(2u*))山

0

)

(

一知

L

)

{

l

+

1

l

i

ヂ(1-

Fs(mo))} 一斥(mo)

安市多

(1-h(2u*))2

一時勺関心

(2u

*

)

)

一庁仇(例吋附 >川淵2//斥;

3

汽μ (切m附0心)川(仕1ト一→h(

ω

川

ν

初

*

つ

切

刈

)η)

ベ

(

茄

乞

了

口

凡

以

2

(

ど

2

;

みゐムん)刀₎

」手苛守弓椋斗均叫的万

K

仰(ω2川u*川水つ) 一イfβ爪仇刷f瓦

μ

仇(仰m附例削州0)

省

茄

乏

帯

勃

券

弥

(1ト 一 山 ) 2

-92ー Kagawa UniversiかEconomたReview 92

>0，

where the last inequality is from (63)

口

Appendix B: Proofs of Lemmas 4.9and 4.10

Prooj of Lemma 4日9:We only show the case for player5， since the case forb can be analogously treated By Assumption 2.，1. Fs(t)= 1 for allt 主 Therefore， by (お，)and Assumption 4.，1.there exists a su伍ciently 1品(mo) large number T (without loss of generality， we letT ミYb

+

C) such that Fs(A(xs， Ys， Xb， y;))= 1 for all (xs， Ys) with ( 11xs， Ys) 11二三 T， where 11・1i1s

the Euclidean norm For all those (xs， Ys)， we have

UpOA(九 九 Xb，y

ト(当主主一

ms)h(ysイ )

+(x6 -m' s)(l-h(ys-y;)) (64)

from

。

位

On the other hand， by Assumption 2..1(iv)， UpOA(xs， Ys， Xb， y;)

is a continuous function of (xs， Ys)， and therefore has a maximum point(x~，

yg)in the closed regionR = {(xs， Vs)lll(χs， ys)11ζ T+ 1} Note that， ifys

ミ バ +C， then the right hand side of制 iSXb-ms<mo-ms<仏(x:，

v

:

，

Xb， y6')， where the second inequality is from Remark 4..3 Therefore， in

the case of ysミ ぱ +C， double-o妊er(xs， Ys) can not be a best response to

(Xb， Y6)， Le引，(xs， Ys) キ (x~， v~).. Hence V~<Y6 十 C 孟 T Since the right

hand side of(64)does not depend on Xs and (64)holds for all (xs， Ys) such that

TζII(xs， ys)11ζ T+1， we can suppose that(x~， y~) is in {(xs， ys)lll(xs，日)11

三

二 T} (otherwise， we can choose another maximum point in the region)，

therefore it is an interior point of R. Then (x~， y~) is a stationary point of

UpOA(XS， Ys， x:，バ，)， because UPOA(xs， Ys， Xb， y;) is di妊erentiable by

Assumptions 2.1(iv) and 2..2(ii) However， (x:， v:)is the unique station

-ary point by Lemma 4ムtherefore(x2，ぽ)= (x:， y

，

n

which concludes the

proof

口

93 Arbitrations When Players Have Di丘erentInformation

inequa1ity can be shown similarly.. First， we show that

(1一α)(χ6'+C)<(1-2α)yi+

似;

In fact， since α< 1/2， we obtain from Assumption 2..2(i) that

C<2

勺抑止<士号抑止

By Lemma 4.，2.yi -mo

>

0 holds， and hence

C_l

一一一一一一一一一一一 <u<~(v

勺間企/ハ

l

二

ヲ

ー

ー Therefore，

附

-

与

訪

3

E

L平

+

ω

市

y

;

which is equivalent to紡，).. Now， Ys三二 X fs.ollows from (24) Therefore if xs< x6 + ' ，C then A(x，s.Ys， x，'6y;)= 皇二包)(y8+Y6' )+ α(~~土 x6') 2(1-α) 三二 (1一α)χ8+(1-2α)y'6

+

αx:6' 2(1一α) < 込 二α)(X6+C)+(1-2α)Y'6十 仰lt 2(1一α) <i1-2α)y:+αχ;十(1-2α)y'6+αd 2(1-α) =A(

χ

;

，ぱ，X6， Y6')(= mo)， 93 (65)

where the third inequality is from航) BecauseFs(t)and fs{t) are nonde

-creasing over(一∞，mo]by Assumption 2..1(iv)， we have

。

A

1-Fs(A(xs， Y，s.x，'6y;))+(x6'-x8)ん(A(x，s.Ys， x，'6y;))τァ

巴//vs

aA

>

1-Fs(mo)+(x'6-x:)fs(mo)

五

J=O Therefore the first relation of (49) follows from (27)

Appendix C: Proofs of Lemmas 5.1 and 5.2

口

-94 Kagawa University Eω抑 制icRevieω

show that UiA(xs， Xb)ぐOwhenlxs-msl>lxb-mbl. From(5，)Iwehave

裂

と

=ι[

ん(χ)

→門主

)

J

d

x

Since fsis decreasing over [ms，∞)， we have oUSCA / ¥(:~ ~_L:~C~~ XS+ Xb ー す よ ー <0 if Xs satisfies~ーす一一 >ms ， αXs ~ and hence for this xs， USCA(Xs， Xb)< !im USCA(t， Xb)= 0， r↓乙m S -X'b which concludes the proof Proof of Lemma 5..2:Denote the left hand side of (53) by L( u) 94

口

(i) We first show that L(u)

=

0 has no solution in interval [0， ms-mb] Let " ^今 /'mo+u L(u) =

竺

ラ

工

"s

ー

ム

ー

Flx)の+2uF例。)，

and uE[O， mS-mb] Then L~(u) = - Fs(mo+u)-Fs(mo-u)+2Fs(mo)

and Li'(u)=ーん(mo+u)+ん(mo-u) Since mo<ms， Li'(u)三二 Oholds by

Assumption 2..1 (iv). Hence L~(u) 三二 L~(O)

=

0， which implies that Ll(U)

ζ Ll(O)= (mb-ms)/2<0引 Furthermore，for uE(O， mS-mb]，

九(附+u)-Fs(mhMmhfh(x)ーん(附)]政>0 側

Therefore 1

心

i)本

o

holds for all uE[O， mS-mb] as shown in the following..

If h(2u)>0， then L(u)>(1/2)h(2u)>0; otherwise (Le， h(2u)= 0)， we

have u >0 and hence L(u) >0 by (66)

(ii) We then show that (53) has a solution in interval (ms-mb，∞)

For uミ C/2，we have 1i(2u)

=

h(2u)

=

0， and therefore

L(u) = Fs(mo十u)-Fs(mo)-uf~(mo)< 1-

u

.

Mmo)

Recall that Assumption 5.1implies Assumption 2..1(ii); henceん(mo)>O

口 ( ∞ ‘ q ut -S ut) I B A1 8 l uf uf u O f ln l o s B S問 ( E S ) p

問

、

n on u qu O : : l s f ( n )7 ‘ o n Z Z p UB (A n - 9 6 -UO!~EUllO JUI f U d ld i j! G < l AEH Sl < l A El d U < l l j M . SUO!lE l~!qlV 9 6