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464

Chapter 15. Repeated Games: General Results

A
Masin (1986,1991),who establish also a result for a class of multiplayer games.
reslt of Wen (1994) covers llmultiplayer 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

Bargaining

16.-1 Bargaining as ale:e�sive'gime 465
16.2 Illustration: �ral� In a-:na rkt 477
16;3 Nash's axlomatiCiodel," '81
14 , Relation bew�en stit�g k "and axiomatic models 489
Prerequisite; Chapter5an� Sections 4.1.3,6.1.1, and 7.6
'

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.

I

16.1
16: 1. 7

Bargaining as an extensive game
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

Chapter 16. Bargaining

466

16.1

1

Figue

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

(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

467

1

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

Bargaining as an extensive game

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.

!

(

�e maY.nd �e subgame perfect equilibria of tis game by usng backward in­
ducio�, as m
Se�io� 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 llproposals. 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 hat 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

Chapter 16. 8argalnlng

468


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).

16.1

"

Every sequence of he form (xl, N,x2, N, ... ,Xl, Y) for t � I, and
2
every (ite) sequnce of he form (xl,N,x , N, ...), where each xT is a
division of the pie (a par of numbers hat sums to 1).

Player function

Preferences



The whole game has a unique subgame 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 perfect 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 players may alternate offers indenitely. his game does not have
2
a nite horizon: every ite sequence (Xl, N,x , N,...) n whih every ofer Xl
.

=

1 (player 1 makes he irst ofer), and

For i

=

=

., t
;

P(x I, N,x2 , N, ... ,xI , N)

=

{I

f t is evn
2 f tis odd.

2

1,2,player i's payoff to he termnal history (xl, N,x , N,... ,
< I, and her payoff to every (ite) ternal

1
Xl, Y) s of- xf, where 0 < OJ
2
history (Xl, N,x ,N, . . .) is O.

�ny

-

P(0)

2
P(xI,N,x , N,...,x)t

Many-perod deadline We may extend the game by allowing the pl.ayers to al�er­

(Of,

Two negotiators,say 1 and 2.

Terminal histories

=

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
02 (1 01»'
accepts,reslng in the pair of payofs

DEFINITION 469.1 (Bargaining game of altenating ofers) The bargaining game of
altenating ofers s he follong extensive game wih perfect infomation.
Players

whre 0 < Cj < 1 (raher han OtYi), and her payoff to any teinal hi�toy that
nds n rejecion is -Cj (raher than 0), for i 1, 2. (Payofs can be negaive,but a
proposal must still be a pair of nonegative numbers.)



=



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
. 21S Yj - Ci,
wih playr i's payoff when she accepts e proposal ( Y 1, Y2) m penod

Jv�n �eadne,
nate proposals over many periods,raher than only two. For
nd Its s�t of
to
inducion
he game has a inite horizon,so we may use backward
has a ���e
game
the
deadne,
subgame perfect equilibia. As for a wo-period
h� lial
accepts
ly
immed�ate
2
player
which
subgame perfect ebrium, in
deadline.
he
o�
pends
d
proposal

proposal of player 1. This
Considr,for example,a hree-penod deadne.

469

for t I, 2,... is rejected s a possible ternal istory. Every oher termnal his­
1
2
toy is ite and takes he form (x , N,x , 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.

,

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 player 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
.
. he
iive. Player 2's "reat" to reject player l's iial p roposal s credible oly f
02,
han
because
rejecion
leads
to
a
delay
hat
reduces
proposal gives player 2 less
r value of he pie to z.

8argalnlng as an extensive game

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 subgame 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 (X ,X2) n he irst peiod of he game generates he
l
payofs (XI,X2), ile her acceptance of player l's offer (XI,X2) in he rst period
of he subgame generates he payofs ( 0fxt,6f X2)' But he playes' prerences are
the same in he game and the subgame: for any number k, eah player 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 use 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.



16.1

Chapter 1 6. Bargaining

470

A player's srategy in he game is complicated. A strategy of player

471

Bargaining as an extensive game

nd rejecng a proposal x for h x2

I, for

=

62Yi. Hence we need

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

for he strategy par to be a sbgame perfect eim. By a symetric argu­

subgame r she makes an ofer, she certaly is not restricted to makng

ment for a subgame n whih the irst move is a response by player 1 to a proposal
of player 2 we need

the game, for example, may differ rom her offer ater a history ( x, N, y, N), which

We have xi

xi

he same ofer whenever t is her n to propose. Player l's oer at he sart of

y depend rarily on the values of x and y.

However, he staionary srucue of he game s t reasonable to guess hat

=

1-xi and yz

=

he game has a subgame perfect equilibrim in wich each player always makes

*

Y1

player's strategy is statonay. The fact hat the strucre of he game s sry

perfect ebrim, hen xi and Yi

gies, nor hat it does not have equlibia n srategies hat are not stationary. But
an eqibrium.



rejects ll oer ofers. A pair of staionary srategies in which each player uses

* *
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 xi :: 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
=

.!





player 2 always proposes y* and accepts a proposal x if and oly f x2

.1
j

=

y*

=

( 1-62 62(1-61))
1-6162' 1-61�
(61(1-02) 1-01) .
1-0102 '1-0102

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.

::Yi

I irst clam ithout proof the folOwing reslt, he argument for wich fol­

::xi.

1. f player 2 rejects player l's proposal, her srategy calls for
1 accepts, yieldng player 2 he payof yz with one
period of delay. us player 2 oplyrejects any proposal x for h X2 < �yz
'
accepts y proposal x for which X2 > 62Y2 and is ndiferent between acceping
'
her o propose y*, whih player

·
x

he outcome of the equilbrim srategy pair is hat player 1 proposes x* at he

q

lows the nes of he argument for Proposition 439.2. (For a statement of the one­
deviation proper, see page 38.)

Now consider a sbgame in wich he irst move is a response by player 2 to

a proposal of player

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 :: xi,

whee

=

player 1 always proposes x* and accepts a proposal y if and oly f Yl

PROPOSITION 471.3 (Subgame perfect eqlibrium of bargang game of ater­


n wih


given in hese wo equations.

argaining game of altenating oers has a unique subgame pefect
equilibrium, in.which

"

wi
ncrease her payof by reducing her offer. hus for equiibrium we need xi
one
is
considerng
are
we
pair
and zi
Yi. Under these conditions, he srategy

e

naing ofers) e

suh a criteion for accping ofers takes he fom

2 always proposes z* and accepts a proposal w f and y f W2 :: wi

=

1-02
1-6162
61(1-62)
1-6162'
--

srategies are not staionary). hat is, we have he following result.

eim eah player accepts ofers hat give her a fienly high payof and

payer

=

particular, he game has no subgame perfect equilibrim in h he players'

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



(471.2)

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

stationary srategies denitely provide a reasonable staring pont n he searh for

::Yi

61Xi-

(471.1)

is argment shows hat f he sraegy pair we are consideng s a sbgame

mplies neiher hat he game necessarily has an equilibrium n stationary srate­

player 1 always proposes x* and accepts a proposal y if and oly if Y1

=

62yi

1-Yi, so hese wo nequalities mply hat
xi

he same proposal and always accepts the same set of proposa-that s, each



yi

=



PROPOSITION 471.4 (ne-deviaion property of subgame 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.

,

..

16.1 Bagaining as an extensive game

Chapter 16. Bargaining

472

e he strategy pir n Proposition 471.3 by s·. The e s wo ypes of
subgame: 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 subgame n h he irst move s an offer. Suppose he ofer
is made by player I, and ix player 2's srategy to be
f player 1 uses he strategy

si, her payof is

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

si.
xi. f she deviates from si in he irst period of he subgame, she



02 0, hen the only asmmetry n he game is
2
hat player 1 moves irst. Player l's equiibrium payoff is (1- 0)/(1- 0 )
1/( 1 + 6),

=

s

if

. 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

xi·

period of a subgame hat starts wih her makng an ofer.

!, but approaches ! as 0 approaches 1.

I EXERCISE 473.1 (One-sided ofers) Consider he variant of he bargaing game of

Player 1 accepts his proposal, obtaining he

2 canot proitably

which exceeds

=

advantage is smal and he e s almost symmeric.

f she ofers player 2 less n xi in he irst period, hen player 2 rejects her

A symmeric argt shows hat player

=

the players are equally and only slightly impatint, player l's irst-mover

her proposal, and her payof is less han xi.
proposal and proposes (yt,Yi).
payof OlY", which is less han

Smmetrically, ixing he patience of player I, player 2's share

First-mover advantage If 01

xi in he rst period, then player 2 accepts

f she ofers player 2 more than

1.

increases to 1 as she becomes more patient.

;

". ,'.

is worse of by he following arguments.


473

ing to he next period, in which player

1 makes anoher proposal. Consider he
(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

deviate in he irst

strategy pail in wich player

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

1

always proposes

which tis srategy pair is a subgame perfect equilibrium. (A strategy pair is a sub­

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.

I EXERCISE 473.2 (Aaing ofer bargainng with constant cost of delay) Marx

A symmetric argument shows hat player 2 cannot probly deviate n he irst

capitalist and worker. Victory goes necessarily to the capitalist. he capitalist can

game perfect eqibrium of s game f and only f it satisies the one-deviaion

.

Thus no deviation in the rst period of the subgame increases player l's payoff.
period of a subgame in whih she responds to a proposal.
We conclude hat

unique

s* is a subgame perfect equm.

(1973,65) writes hat "Wages are deterned by the antagoistic struggle between

e longer wihout he worker han can he worker wihout he capitaist." Per­

The proof hat it is he

haps he has in ind he variant of the bng game of altenaing ofers n

subgame perfect eqibrium (so that, in plar, he game has no sub­

Cj during eah period of delay (raher han discouning
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
h eah player i loses

game perfect ebrium in which the players' strategies are not staionary) is a

r payof), as in Exercises

little ntricate, and I do not present it.

Props of suame perfect equilibrium
properies.

proper.)

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

The equilibrium s· has some noteworthy

that f

CI

=

C2

=

c, hn for every

value of

Zl with C :; Zl :; 1

sbgame perfect eim n wih player 1 always proposes

he game has a

(zl,l- Zl). (n

boh cases, a srategy pair is a subgame perfect equilbrium f and only f it satisies

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

the one-deviaion property.)

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
yields suh equlibria when

c

<

i).

=

C2

=

C n Exercise 473.2

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

16.1.4

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

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