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464 ^{Chapter 15.} Repeated Games: General Results

Masin (1986, 1991),who establish also a result for a class of multiplayer games. ^A

reslt of Wen (1994) covers

^ll

multiplayer games.

The folk heorems for itely repeated games, Proposiions 460.1 and 461.1, are

due to Benoit and Krisn. (1985,1987).

Games n wich he players altenate moves, like he one in Exercise 459.3, are

sudied by Lanoff and Matsui (1997); Rubnsten and Wolnsky (1995) study a

closely related class of games.

he idea in Section 15.4 is due to Green and Porter (1984), who study a variant

of Comot's oigopoly game. he formulaion ^I use is taken rom Tuole (1988,

Section 6.7.1.1).

1

16.-1

16.2

16;3

14

Bargaining

Bargaining as ale:e�sive'gime ^'

465

Illustration: �ral

_� In a-:na rkt 477

Nash's axlomatiCiodel,"

'81

, Relation bew�en stit�

_g

k "and axiomatic models 489

Prerequisite; Chapter5an�

^Sections

4.1.3,6.1.1,

and

7.6

I ^{N MANY} siuaions, parties divide a "pie". A capitalist and he workers she hires

. diide he total revenue generated by he output produced; legislators divide

tax revenue among spendng programs valued by th�ir constituents; a buyer of

an object and a seller ivide the amont by wich he buyer's valuaion of he

object exceeds he seler's. n s chapter

^I

discuss two very dfferent models hat

are intended to capture "bargang" between he paries in such situations. One

model is an extensive game (see Chapter 5). The oher model takes an approach

not previously used in _{s book:}

t

considers he outcomes compaible wih a

ist of apparenly sensible properties. hough the models are Very diferent, he

outcomes hey isolate are closely related.

16.1

Bargaining as an extensive game

16: 1. 7 Extensions of the ultimatum game

One pont of departure for a theory of bargang is the "ultimatum game" studied

n

Section 6.1.1. n tis game, two players split a pie' of sze c that hey both value.

hroughout _{s section} I take

c ⁼

1. Player 1 proposes a division (XVX2) of he

pie, where

Xl + X2 =

_{1 and 0 5}

Xi

5 1 for i

⁼

I, 2. Player 2 eiher acceps

s division, n wich case she receives X2 and player 1 receives

^Xl,

or rejects it,

n wich case neiher player receives any pie. is game has a ique subgame

perfect equilibrium, in wich player 1 proposes the division (1,0), and player 2

accepts all ofers. The outcome of he equiibrium is hat player 1 receives all he

pie.

What accounts for tis one-sided outcome? Player 2 is powerless because her

only altenative to he acceptance of player l's proposal is rejection, wich yields

her no pie. Suppose, nstead, hat we give player 2 he option of makng a con­

terproposal after rejecting player l's proposal, wich player 1 may accept or reject.

hen we have the game illusrated n Figure 466.1, where

Y

means " accept" and N

means "reject".

465

466 ^{Chapter 16.} Bargaining

1

Figue

^{466.1 n} extension of the ultmatm game n which player 2, after ejecing player l's ofer, may make a contero. he top gray lngle presents he coninuum of possble poposals of player 1; he bottom gray tringle epesents he contnuum of possble conteproposals of player 2 r she jecs player 1's proposal

x.

he black lnes indicate thee of he iely mny termnal histories (namely

(x, y), (x,N,y, Y),

^nd

(x, N,y, N)).

n s game, player

¹

is powerless; her proposal at he start of he game is

irrelevant. Every subgame following player 2's rejecion of a proposal of player 1

is a variant of he ulimatum game n which player 2 moves irst. hus evey

suh subgame has a uique subgame perfect equilibrium, n wich player 2 offers

noting to player 1, and player 1 accepts all proposals. Using backward induction,

player 2' s opimal acion ater any offer (

xv

X2) of player ¹ with X2

^<

¹ is rejecion

(N). Hence in every subgame perfect equilibrium player 2 obtans all he pie.

n he extension of is game n which he players altenate ofers over many

peiods, a slar result holds: in every subgame perfect equbium, he player

who makes he ofer in the last period obtans all the pie. he feature of the model

responsible for is result is he players' indierence about he ng of an agree­

.

Real-fe bargaing takes ime, and ime

s

valuable, so we may reasonably

assume hat he players' preferences do not have this characterisic, but raher ex­

bit a bias toward early aeemns. he next section explores he consequences

of a paricular form of mpatience. ^.

16.1.2 A inite horizon game with altenating ofers and impatient players

Suppose hat he players alternate proposals, one per "period", and that each

player i regards he outcome n wich she receives all of he pie after t periods

of delay as equivalent to he outcome n wich she receives he raction _of of he

pie mmediately, where

° <

OJ

<

1 for i

⁼

1, 2. That is, suppose that each player i

"discounts" he fuure usng the constant discount factor OJ. (See Section ^14.2 for a

disussion of preferences wih discountng.)

wo-period dadline Cosider he game n wich two periods are pOSSible: if

player 2 rejects player l's niial proposal, player 2 may make a conterproposal

which, if rjected by player 1, ends he game wih a payof of

°

for eah player.

is game is llustrated n Figure 467.1.

'.'. !

(

16.1 Bargaining as an extensive game

467

1

Figur�

^{467.1 A} tw-peiod bargang game of altenang oers n whih each player j uses the actor 01 to dScont fuue payofs.

� ^{e maY} _. ^nd � e subgame perfect equilibria of tis game by usng backward in­

ducio _� , as m Se _� i

. o _� 16.1.1. he subgame starng after.a istoy in which player 2

has reJe _� ted an utial proposal of player 1 is simlr to an ultimatum game.

t

has a uque subgame pefect equilibrium, n wich player 2 proposes ( 0 , 1 _{) and}

player 1 accepts ll proposals.

s

equlibrium results in payoffs, viewed rom he

start of the game, of 0 for player 1 and � for player 2.

. Now consider the sUbgame following an itial proposal of player 1. If player 2

rejects he proposal, her payof

s

h as we have just found. hus she optimally

rejects any proposal hat gives her less han 02 and accepts any proposal hat ives

her more han 02; she is ndiffernt between acceptng and rjecing he proposal

(1- _{02, 02),}

. ^{Fna11 �} consider player l's niial proposal. Player 2 accepts any proposal

(�1,X2) Wlh X2

>

02· hus no such proposal is opmal: player 2 will accept

sihtly less, as long as he amont she ges exceeds 02, so hat player 1 can ncrease

he amount _� he re _� eives

. by oering player 2 less. No proposal hat gives player 2

� ^{ess n z}

⁸

opmal eIher: player 2 rejecs suh a proposal nd n he folow­

mg subgame proposes (0,1), which player 1 accepts, giving player 1 he payof

? hus he only p _: oposal of player 1 possible n a subgame perfect equilibrium

IS ( _: - 02,02). I clan ha

. t he game indeed has a subgame perfect equlibrium n

whih playr 1 makes is proposal, and n is equlibrium player 2 accepts he

proposal. f player _z were to rject it, player l's payof would ulimately be 0, so

hat player 1 could mcrease her payoff by raiSing the amount she niially ofers to

player 2 above 02, nducing player 2 to accept her proposal.

n conclusion, he game has a unique subgame perfect equilibrium in which

•

player l's initial proposlis (1 - _�,02)

•

player 2 accepts all proposals n wich she receives at least 02 and rejects all

proposals in wich she receives less than 02

•

player 2 proposes (0,1) after any history n which she rejects a proposal of

player 1

468 ^{Chapter 16.} 8argalnlng

• player 1 accepts all proposals of player ² at he end of the game (after a istory n which player 2 rjects pla�er l's openg proposal).

The outcome of

s

equilibrium is that player

1

proposes

(1- 02,02),

which player 2 accepts; player l's payof is 1-

02

and

pla

yer 2's s

02. s

inding s c�n­

sisent wih he

intuiion that he incenive to reach an ealy agreemnt embodied n he players' impaience leads to an outcome

n

whih player l's

. ^{payof is}_. ^pos­ iive. Player 2's "reat" to reject player l's

iial

p

r

opo

sal

s

credible

oly

f he

proposal gives player 2 less han

02,

because rejecion leads to a delay hat reduces r _{value of}

he

_{pie to}

z.

e ^EXERCISE

468.1 (wo-period bargaining wih constant cost of dela!)

^F

^�d h� su�­ game perfect equilibrium (equilibria?) of the variant of he g�e

m

. ^s

^s�cion

^m

wih

^playr i

's

^pay

off when she accepts e proposal

( Y 1, Y2) m

penod 21S

Yj

-^Ci, whre

⁰

< Cj < 1 (raher han

OtYi),

and her payoff to any teinal hi�t

oy

that nds

n

rejecion is ^-Cj (raher than

0),

for

i

⁼

1,

2. (Payofs

can

be negaive,

but a

pro

p

os

al

must still be a pair of nonegative numbers.)

Many-perod deadline

We may extend the game by allowing the pl.ayers to al�er­ nate proposals over many periods, raher than only two. For

^�ny

Jv�n �eadne, he game has a inite horizon, so we may use backward inducion to nd Its s�t of subgame perfect equilibia.

^As

for a wo-period deadne, the game has a ���e sub game perfect ebrium, in which player 2 immed�ately accepts h� lial proposal of player 1. This proposal d�pends o� he deadline.

Considr, for example, a hree-penod deadne.

• _By r analysis of he wo-peiod game, any subgame f?llowing a histoy in whih player l's openng proposal ^s rejected has a uque s�bgame per­ fect equilibium, in which player 2's proposal is

(0lt

1-

01),

whih player 1 accepts,

^r

es

l

n

g

in the pair of payofs

_(Of, 02 (

1^-

01»'

• The whole game has a unique sub game perfect equilibrium, n whih player 1 ofers player 2 he amont

02(1- 01)

at he start of

he

game (�g her infferent

between acceptance and

_rjecion).

Player

2 accepts is

^ofer,

generaing he pair

of payofs (1-

02(1- 01),62(1-01))'

• ^EXERCISE

468.2 (ree-peiod bargang

^ith

constant cost of

^delay � ^Find

^�e

subgame p

erfe

^c

t

equilibrium (equilibria?)

of the variant of the gme

^m

^Exer�e 468.1 n wich he game may last for ree periods, and ^{e cost to}

eah player

^I of . eah peiod of delay

^is

^Cj.

(Treat he cases c1 � C2

and Cl < C2 separately.)

16. 1.3 An Ininite horizon game with altenating ofers and impatient players

Deinition

^. n

appealing

version of he model assumes hat eah playe�, after r�­ jecing an ofer,

always

has

he

opporniy to make a counterofer. hat IS, there IS

no

deadne; the pl

^a

ye

^r

s may alternate offers indenitely. his game does not have a nite horizon: every ite sequence

(Xl, N, x2, N, ... )

n whih every ofer

Xl

" ,

., t

;

16.1 8argalnlng as an extensive game ₄₆₉

for

t

^{= I,} 2, ... is rejected ^s a possible ternal istory. Every oher termnal his­ toy is ite and takes he form

(x1, N,x2, N, ... ,xl, Y):

for some value of ^t, all pro­ posals hrough period

t

-

1

are rejected, ^d he proposal in period

t

is accepted. he game is called he

bargaining game of altnating ofers.

�

DEFINITION

469.1

(Bargaining game of altenating ofers)

The

bargaining game of

altenating ofers

s he follong extensive game wih perfect

infomation.

Players Two negotiators, say

1

and 2.

Terminal histories

Every

sequence of

he

form

(xl, N, x2, N, ... , Xl, Y)

for

t �

I,

and

every (ite) sequnce of he form

(xl, N, x2, N, ... ),

where each

xT

is a

division

of the

pie (a

par of

numbers

hat

sums to 1).

Player function

_P(0)

= 1 (player 1

makes he

irst

ofer),

and

I

2

t I

2

I

_{I

f

t

is evn

P(x ,N,x , N, ... ,x)

⁼

P(x , N,x , N, ... ,x , N)

⁼

2 ^f tis odd. Preferences _For

i

⁼ 1,2, player i's payoff to he termnal history

(xl, N,x2, N, ... ,

Xl, Y)

^s

of-1xf,

where

0

<

OJ

< I, and her payoff to every (ite) ternal history

(Xl, N, x2, N,

.^..

)

is ^O.

he irst wo periods of this game look

^e

he two-period game in Figure 467.1, except hat player l's rejection of an offer in the second period leads not to he end of he game (wih payofs

(0,0»,

but to a subgame n which the first move is a pro­ posal of player

1.

he sructure of ts subgame is the same as the sucture of he

whole

_game:

player

1 makes a proposal, which

player

₂

eiher accepts or rejects;

hen,

f player 2 rjecs he proposal, she makes a proposal, which

player 1

eiher accepts or rejecs; and so on.

ⁿ

fact, he sub game is

identical

to he whole game. hat is,

not

oly are he players, terinal histoies,

and

player uncion the same

in he subgame as hey are in he game, but so too are he playes' preferences.

he players'

payos

difer n

he game

and he subgame. For example, player 2's

acceptance of player

l's

ofer _{(Xl, X2)} n he irst peiod of he game generates he

payofs _(XI,X2), ile her acceptance of p

l

a

yer

l's

_offer

_(XI,X2)

_{in he}

rst period

of

he

subgame generates he payofs

( 0fxt,6fX2)' But

he playes'

prerences are

the same in he

_{game and}

the subgame: for any number k, eah

p

la

_yer

i

is

^indf­

ferent beween receiving

k

unis of payof

wih t

periods of delay

and

_receiving

_{6[ k}

its of payoff mmediatel.

Silarl, _all subgames staring ih a

proposal of player 1 (including he whole game) are idenical to

each oher. Further,

all subgames staring with

a proposal of

player 2 are identical to each ohe'. For his reason,

we say that he

structure

_{of he} game is

stationay.

Subgame pefect equilibrium Because the game does not have a ite horizon, we

canot u

se backward inducion to

nd its subgame

perfect equilibria.

^nstead,

I

argue hat he staionay sructure of he game suggests a certain form for an

equilibrium, and hen check that an equilibrium of such a form exists.

�

470 ^{Chapter 16.} Bargaining

A

player's srategy in he game is complicated.

A

strategy of player ^I, for example, speciies an offer n period

1;

a response (accept or reject) to every istory of ^e form (x,N,y), where x is an offer (of player

₁₎

n he irst period and

y

is an offer (of player

2) n e

second period; a conteroffer folowng every istory of e form

₍

x,

_{N, y, N);}

and so on. n particular, although each player faces he same subgame

r

she makes an ofer, she certaly is not restricted to makng he same ofer whenever t is her n to propose. Player l's oer at he sart of the game, for example, may differ rom her offer ater a history

(

^x, N,

y, N),

which

y

depend rarily on the values of x and

y.

However, he staionary srucue of he game s

t

reasonable to guess hat he game has a subgame perfect equilibrim in wich each player always makes he same proposal and always accepts the same set of proposa-that s, each player's strategy is

statonay.

The fact hat the strucre of he game s sry mplies neiher hat he game necessarily has an equilibrium n stationary srate­ gies, nor hat it does not have equlibia n srategies hat are not stationary. But stationary srategies denitely provide a reasonable staring pont n he searh for an eqibrium.

A

staionary strategy is specied by givng he offer he player alwas makes nd he ion she always uses to accept ofers. uition suggests hat n an eim eah player accepts ofers hat give her a fienly high payof and rejects

ll

oer ofers.

A

pair of staionary srategies in which each player uses suh a criteion for accping ofers takes he fom

• _player

₁

always proposes x* and accepts a proposal

y

if and oly if

_{Y1 :: Yi}

• _payer

₂

always proposes z* and accepts a proposal

w f

and

y f _W2 :: wi

for some proposals ^{w* , x* "}

y*

, and z*

.

Can we ind values of hese proposals such that e strategy pair is a subgame perfect eim? We found hat in a fnite horizon game every proposal ^s accepted n equilibrim.

A

reasonable guess ^s hat he same is true in he ite game, so hat ^x

_{i ::} wi

and

_{zi ::} y _i

.

f

eier of hese inequlities is srict, one of he players is wling to accept less han she is ofered, so hat the proposer can ncrease her payof by reducing her offer. hus for equiibrium we need

^xi

⁼

^wi

and

_zi

⁼

_Yi.

Under these conditions, he srategy pair we are considerng is one n wih

• _player

₁

always proposes x* and accepts a proposal

y

if and oly

f _{Yl :: Yi}

• _player

₂

always proposes y* and accepts a proposal x if and oly

f x2 :: xi.

Now consider a sbgame in wich he irst move is a response by player

2

to a proposal of player

_1. f

player

2

rejects player l's proposal, her srategy calls for her o propose

y*,

whih player

₁

accepts, yieldng player

2

he payof yz with one period of delay. us player

2

oply rejects any proposal x for

h _X2

<

_�yz'

accepts

y

proposal x for which

_X2

>

_62Y2'

and is ndiferent between acceping

.!

.1

q

j

16.1 Bargaining as an extensive game

nd rejecng a proposal x for

h

x

₂

=

_62Yi.

Hence we need x

_i

=

_62yi

471

(471.1)

for he strategy par to be a sbgame perfect eim. By a symetric argu­ ment for a subgame n whih the irst move is a response by player

₁

to a proposal of player

2

we need

y _i

⁼

_61Xi- (471.2)

We have x

_i

=

_{1 -}

_x

_i

_and

_{y z}

=

_{1 -Yi,}

so hese wo nequalities mply hat

1-02

x

_i

= _--

1-6162

*

_61(1-62)

Y1

⁼

_1-6162'

is argment shows hat f he sraegy pair we are consideng s a sbgame perfect ebrim, hen xi and

_Yi

^e given in hese wo equations.

n

fact, he strategy pair hus deined is indeed a subgame perfect equilibrim, as

I

show below. Furher, it is he only subgame perfect eim (so hat, n particular, he game has no subgame perfect equilibrim in h he players' srategies are not staionary). hat is, we have he following result.

•

PROPOSITION _471.3

(Subgame perfect eqlibrium of bargang game of ater­ naing ofers)

e argaining game of altenating oers has a unique subgame pefect

equilibrium, in. which

•

_plyer

1

always prposes x* and accps a proposal _{Y f} and only f Y1 :: Yi

•

_payer

2

always ps y* and acpts a prposal

^x

f and only _{f X2} ::

x

_i

,

whee

x· =

y*

⁼

( 1-62 62(1-61))

1-6162' 1-61�

(61(1-02) 1-01)

1-0102 '1-0102 ^.

he outcome of the equilbrim srategy pair is hat player

₁

proposes x* at he start of the game, and player

2

mmediay accepts is proposal.

I

have argued hat

f

a pair of staionry strategies

n

which every ofer is ac­ cepted s a subgme perfect ebrium, hen it takes he form given in he result. I now argue that his strategy pair is in fact a subgame perfect eqlibrium.

I

irst clam ithout proof the folOwing reslt, he argument for wich fol­ lows the nes of he argument for Proposition

439.2.

(For a statement of the one­ deviation proper, see page

38.)

•

PROPOSITION _471.4

(ne-deviaion property of sub game perfect eqlibria of bar­ gaining gme of alteg ofers) A

stratey proile in the bagaining game of alter­

nating oers s a subgame peect equilibrium f ad only f it satses the one-devation

property.

,

.. ^{472 Chapter 16.} Bargaining

e he strategy pir n Proposition

471.3

by

s·.

The

e s

wo ypes of sub game: one n which the rst move is an ofer, and one in which he irst move s a response to an ofer.

Frst consider a sub game n

h

he irst move s an offer. Suppose he ofer is made by player

I,

and ix player 2' s srategy to be

_si. f

player

1

uses he strategy si, her payof is

_xi. f

she deviates from si in he irst period of he subgame, she is worse of by he following arguments.

•

f

she ofers player 2 more than

_xi

in he rst period, then player 2 accepts her proposal, and her payof is less han

xi.

•

_f

she ofers player

2

less

n _xi

in he irst period, hen player

2

rejects her proposal and proposes

_{(yt, Yi).}

Player

1

accepts his proposal, obtaining he payof OlY", which is less han

_xi·

A symmeric argt shows hat player

2

canot proitably deviate in he irst period of a subgame hat starts wih her makng an ofer.

Now consider a subgame in which he irst move s a response to an offer. Sup­ pose hat he responder s player

1,

and ix player 2' s strategy to be

s2'

Denote by

(Yll Y2)

he ofer to whih player

1

s respondng. Player l's strategy

si

cals for her to accept he proposal

f

and only if

Y1 � _Yi.

If she rejects he proposal, she pro­ poses

_x·,

whih player 2 accepts, so that her payof is

_olxi,

wich is equal to

yj.

Thus no deviation in the rst period of the subgame increases player l's payoff. A symmetric argument shows hat player 2 cannot probly deviate n he irst period of a subgame in whih she responds to a proposal.

We conclude hat

s*

is a subgame perfect equm. The proof hat it is he

unique

subgame perfect eqibrium (so that, in plar, he game has no sub­ game perfect ebrium in which the players' strategies are not staionary) is a little ntricate, and

I

do not present it.

Props of suame perfect equilibrium

The equilibrium s· has some noteworthy properies.

Eiciency

Player

2

accepts player l's irst offer, so hat agreement is reahed

m­

mediatly; no resources are wasted in delay.

s

feaure of he equilibrium is nuiively appealing, given hat the players are perfectly nformed about eah oher's preferences. If he outcome were not reached imediately, ere wuld be an alteative outcome hat both players prefer; given their perfect formaion, one might epect he players to boh perceive and pursue his altenaive outcome. Neverheless, some variants of he model hat ain he players' perfect nformaion have subgame perfect eqilibria n which agreement is

not

reached immediately (he case

C1

⁼

C2

⁼

C

n Exercise

473.2

yields suh equlibria when ^c <

_i).

Efect of changes

n

patience

For a given value of 02, the value of

_xi,

the equi­ ibrium payoff of player

I,

ncreases as

01

ncreases to

1.

hat s, ng he

".

_;

,'.

i

16.1 Bagaining as an extensive game ₄₇₃

paience of player 2, player l's share increases as she becomes more paient. Further, as player ¹ becomes extremely patient (01 close to

I),

her share ap­ proaches

1.

Smmetrically, ixing he patience of player

I,

player 2's share increases to

1

as she becomes more patient.

First-mover advantage

If

01

⁼

02

⁼

0,

hen the only asmmetry n he game is hat player

1

moves irst. Player l's equiibrium payoff is

^(1- 0)/(1- 02)

⁼

1/ (1

⁺

6),

which exceeds

_!,

but approaches

_!

as

0

approaches

1. s

if the players are equally and only slightly impatint, player l's irst-mover advantage is smal and he e s almost symmeric.

I EXERCISE 473.1

(One-sided ofers) Consider he variant of he bargaing game of . altenatng ofers in which oly player

1

makes proposals: n every period, player

1

makes a proposal, which player

2

eiher accepts, ending he game, or rejects, lead­ ing to he next period, in which player

1

makes anoher proposal. Consider he strategy pail in wich player

1

always proposes

(Xl, 1- Xl)

and player

2

always accepts a prposal

(yi,Y2)

if and only

f Y2 � 1- Xl.

Find he value(s) of

Xl

for which tis srategy pair is a subgame perfect equilibrium.

(A

strategy pair is a sub­ game perfect eqibrium of

s

game

f

and only

f

it satisies the one-deviaion . p^roper.)

I EXERCISE 473.2

(Aaing ofer bargainng with constant cost of delay) Marx

(1973,65)

writes hat "Wages are deterned by the antagoistic struggle between capitalist and worker. Victory goes necessarily to the capitalist. he capitalist can e longer wihout he worker han can he worker wihout he capitaist." Per­ haps he has in ind he variant of the bng game of altenaing ofers n h eah player

i

loses

Cj

during eah period of delay (raher han discouning r payof), as in Exercises

468.1

and

468,2.

Show that if

C1

^<

C2,

hen is game has a subgame perfect ebrium in which player

1

always proposes

(1,0).

(n his case, n fact, he game has no o!er subgame perfect equilibrim.) Show also that

f CI

⁼

C2

⁼

c,

hn for every value of

Zl

with

C ^:; Zl ^:; 1

he game has a sbgame perfect eim n wih player

1

always proposes

(zl,l- Zl). (n

boh cases, a srategy pair is a subgame perfect equilbrium

f

and only

f

it satisies the one-deviaion property.)

16.1.4 Risk of breakdown

n

some siuations, a negoiator is motivated to reach agreement because she ks here is a chance, independent of her bhavior and hat of her adversat hat ne­ goiaions ll end prematurely. She may fear, for example, hat he pie hat is rly available will at some pont disappear because of the acions of

rd

parties, or hat her adversary will happen upon a more appealng venture and lose nterest in bargag wih her.

We may capture is idea in a variant of the bargang game of altenaing ofers in which ater any ofer is rejected, a move of chance ternates negotiaions