Bargaining Power in Repeated Negotiations with
Outside Options
Christian Groh
Department of Economics
University of Mannheim
L 13, 15, D - 68131 Mannheim, Germany
Tel.: ++49 181 3421
email: [email protected]
December 20, 1999
Abstract
For repeated ultimatum bargaining, the Folk Theorem of repeated game theory suggests that there are many equilibria, among them equilibria in which the
second mover, say, the buyer, appropriates all the surplus. This paper shows that
the presence of outside options in form of alternative trading partners eliminates
some of these equilibria by reducing the cope for extracting surplus in repeated
bargaining. The structure of optimal equilibria is characterized.
Keywords: noncooperative bargaining, repeated games, outside options
JEL Classication: C78, C73, C72
This paper was partly written while the author was visiting University College London whose
warm hospitality is gratefully acknowledged. I am especially grateful to Tilman Borgers and Martin
Hellwig for their encouragements and generous advice. I have also beneted from the comments of
Malte Cherdron, Tony Curzon-Price, Philippe Jehiel and Ennio Stacchetti. All remaining errors are
mine.
1 Introduction
In this and the companion paper (Groh [10]) I investigate market power and collusion
in situations involving bilateral bargaining. Market power and collusion are usually
studied in models with homogenous markets, i.e. the Cournot model or the Betrand
model. In contrast, my transactions are based on bilateral bargaining with "markets"
providing outside options in the form of new trading partners.
I consider repeated negotiatons for repeated one-shot transactions. The analysis is
complicated by the Folk Theorem of repeated game theory which suggests that even
in the absence of outside options there are many equilibria, in particular equilibria in
which the buyer can extract all the surplus from an agreement. Starting from this
observation the present paper analyses how the ability to extract surplus is aected by
the presence of other market participants, in particular the ability to trade one partner
o against the other. In the companion paper I discuss the scope for collusion in this
setting.
I introduce the basic model in section 2. The game is constructed so that in each
one-shot transaction market power rests with the seller; the seller makes a take-it-orleave-it oer to the buyer, who can accept or reject this oer. In a repeated setting this
ultimatum structure of the stage game does not preclude the buyer from extracting
all the surplus from the bargining so the eect of the Folk Theorem is most dramatic.
The key of the argument is the observation that threats to refuse any slightly unsatisfactory oer - even if this would be advantageous - are credible if acceptance induces
a reputation for softness and implies worst equilibria in the rest of the relation. This
is shown in section 3.
In section 4 I then show that the presence of outside options in form of alternative
trading partners reduces the buyer's power to extract surplus because it reduces the
credibility of refusing advantageous oers. The point is that the outside option provides
for more benign equilibria than the worst equilibrium in the continuation of the current
relationship.
The argument presumes that the alternative trading partners have no information
2
about one's trading history so they cannot be integrated into a system of punishment
strategies. This suggests that information or communication about private bargaining
histories may serve as a tool for collusive behavior among the buyers. This is the
subject of the companion paper.
The present paper spells out the above argument and shows to what extent buyers
can extract surplus even though alternative trading partners provide outside options. I
employ ideas from the literature on mechanisms that enforce cooperation in a Prisoner's
Dilemma without information ow and show that this is possible if equilibria involve
decreasing price sequences. Such equilibria have the property that the outside option
is not better than the continuation of the partnership with the current partner. This
makes more severe punishment strategies available. In section 4 I characterize such
equilibria and determine the maximal surplus which buyers can extract.
1
In section 5, I study a more symmetric setting in which the proposer in each match
is selected randomly and the agent who responds to an oer can terminate the ongoing
partnership. Then neither buyers nor sellers can extract more surplus than in the
one-shot-transaction; a Folk Theorem does not hold. I conclude with a discussion in
section 6.
The paper relates to several aspects on the literature of one-shot and repeated bargaining. Most of the literature states that outside options are useful or at least not
harmful and focusses on when the presence of an outside option makes a player strictly
better of. The outside option principle says that outside options have an positive eect
on a player's equilibrium utility if and only if a player's outside option payo is strictly
greater than the equilibrium payo in the situation without outside option. Moreover,
it matters when a player can take the outside option. The analysis here shows that the
presence of the outside option may actually be harmful because it reduces a player's
ability to commit himself to threats.
2
Bilateral repeated bargaining situations with an innite horizon have been studied
by Muthoo [14]. He discusses repeated Rubinstein bargaining games between one
See Ghosh and Ray [9] and Datta [5].
See Shaked and Sutton [20], Sutton [21] and, for repeated Rubinstein bargaining games, Muthoo
[15].
1
2
3
seller and one buyer and shows that stationary strategies yield a unique subgame
perfect equilibrium. With non-stationary strategies, a Folk Theorem obtains. My
model without outside options is a special case of his model.
Rubinstein and Wolinsky [17], [18] and Gale [8] study one-shot bargaining between
many agents in a market. In these models, agents are matched randomly each
period and cannot choose a new trading partner. The main point of these papers is to
establish a game theoretic foundation for competitive equilibria; the answers are mixed.
For example, Rubinstein and Wolinsky [18] derive a Folk Theorem-like result in which
buyers can extract all the surplus from bargaining. This results depends crucially on an
outside option in the case of discounting; I discuss the exact relation of these ndings
to my work in section 6.
3
The literature on sequential bargaining under incomplete information examines market power in bargaining situations as well. Gul, Sonnenschein and Wilson [11] show for
the gap-case (gains from trade are bounded away from zero) that the seller cannot extract any surplus from the bargaining as the length between two consecutive price oers
from the seller tends to zero. Moreover, buyers' equilibrium strategies have to satisfy
a Markov property. In the no-gap case (there may exist no gains from trade) the seller
cannot extract any surplus if the Markov property holds. Ausubel and Deneckere [3]
demonstrate for the no-gap case that without the Markov property many equilibria are
possible. The set of seller payos expands from zero to static monopoly prots. In the
same context, Hart and Tirole [13] and Schmidt [19] develop a nite horizon-imperfect
information version of my basic game between a single buyer and a single seller. They
show that when the horizon is long the seller charges a price according to the lowest
valuation of the buyer. These approaches do not consider outside options and market
interaction and cannot analyse the role of the market as an incentive system.
Finally, Bush and When [4], Fernandez and Glazer [6] or Haller and Holden [12]
link the structure of one-shot bargaining game with the structure of a repeated game:
players can receive positive payos while disagreeing in the bargaining process. In
these models, as in my model, there exist multiple equilibria and delay is possible.
3
Osborne and Rubinstein, [16], ch.6-9, provide an excellent discussion of this literature.
4
2 The Model
There are two nite sets of agents, buyers and sellers. The set of sellers is denoted by
S , with s 2 S ; s = 1; :::; N , the set of buyers is denoted by B, with b 2 B; b = 1; :::; N .
Time is discrete and indexed by t = 1; 2; :::. At each point in time, each seller receives
an endowment of one indivisible unit of some good. The good is homogenous, i.e. all
sellers' endowments are perfect substitutes. The good is also perishable. It cannot be
stored from period to period. Sellers' valuation of their endowment is zero. Sellers
derive utility from money. A seller who sells his unit in some period at price p receives
utility p in that period. Sellers maximise the sums of discounted present values of their
period utilities. Sellers' discount factor is S 2 (0; 1).
Buyers have no endowment of the good, but they have money. In each period, each
buyer has exactly one unit of money which is nonstorable. Buyers are potentially
interested in buying at most one unit of the good. Every buyer's valuation of the good
is equal to 1. A buyer who pays a price p for the good therefore receives in that period
utility 1 , p. Buyers maximise the sums of discounted present values of their period
utilities. Buyers' discount factor is b 2 (0; 1):
I now explain the strategic game which buyers and sellers are playing. At time t = 1,
all agents are matched. Each buyer is assigned to one seller, and no seller is matched
to more than one buyer. If a match between two agents b and s has formed, I denote
the pair as (bs). In the relationship (bs), seller s makes a take-it-or-leave-it oer to
buyer b by proposing a price p(t) 2 R to b. The buyer may accept this price, (Y ), or
reject the oer, (N ). Denote this action as rb(t) 2 fY; N g. Then, the buyer can decide
whether to stay with his partner or not. When b stays, (bs) play the same bargaining
game again. Otherwise buyer b and seller s return to the matching pool. Denote this
action as b(t) 2 fstay; poolg. I shall specify the domains and ranges of all strategies
(being functions) more detailed below.
In period t = 2, n new buyers and n new sellers are added to the matching pool.
Then, all agents in the matching pool are matched. These are agents who are fresh
in the pool and all agents whose relation was split up by a buyer's exit decision in
5
period 1. Then the same bargaining game as before is played: each seller proposes a
price to the buyer with whom he is matched. The buyer can accept or reject this price.
Then the buyer can decide whether to continue with the current trading partner, or to
separate. This continues indenitely.
In each period in which he has a partner each agent only observes the price proposed,
the response to this price, and the decision of the buyer whether or not to separate.
Moreover, agents do not observe what happens in matches in which they are not
involved. Agents do not observe the identities of their trading partners. Note also
that any seller who starts a new relation does not know if his new trading partner is
new in the matching pool or if he has traded before. It is not possible for a seller to
reconstruct any new trading partner's private bargaining history. This is possible since
there is a ow of new agents in the matching pool in each period.
Let b(t) denote the buyer with whom a given seller is matched in period t and dene
s(t) similarly. A seller's private bargaining history is Hs (t) =
fps(1); rb (1); b (1); :::; ps(t , 1); rb t, (t , 1); b t, (t , 1)g
and a buyer's private bargaining history is Hb(t) =
fps (1); rb(1); b(1); :::; ps(t , 1); rb (t , 1); b(t , 1)g.
(1)
(1)
(
1)
(
1)
(1)
Buyer b's acceptance decision is given by
rb (t) : Hb (t) R ,! fY; N g
and the outside option strategy for b is given by
b (t) : Hb (t) R fY; N g ,! fpool; stayg:
A buyer's strategy set is given by
b = frb(t) : Hb (t) R ! fY; N gg fb(t) : Hb (t) R fY; N g ! fpool; staygg:
The formal description of a strategy for the seller is given by
ps (t) : Hs (t) ,! R:
Denote the set of strategies for a seller as
s = fps (t) : Hs (t) ,! Rg
6
As usual in the study of extensive games I want to use an equilibrium concept that
is stronger than Nash equilibrium and embodies sequential rationality. Given that the
game has no subgames and that sellers' strategies are continuous variables the usual
denitions of subgame perfect equilibrium or sequential equilibrium do not apply. Even
so the notion that no player wants to deviate from the equilibrium strategy at any point
in time given that all other agents stick to their equilibrium straegy is well dened.
This is because the assumption about information in histories makes it possible to treat
behavior in any one period as if the pair played a genuine subgame. I do not give a
formal denition of this notion of sequential rationality.
I impose a restriction and consider equilibria which are characterized by (i) a sequence
of target prices fpE ( )g1
which the buyers would like to sustain and (ii) a sequence
of punishment prices fpD ( )g1
indicating the continuation following acceptance of
prices higher than the target price.
=1
=1
Given the two sequences fpE ( )g1 ; fpD ( )g1 the associated strategies are specied as followed.
=1
=1
in the rst period of any relation, the seller proposes pE (1) and the buyer accepts
any p pE (1) and rejects any p > pE (1).
regardless of the history, the buyer never leaves
if in periods 0 < the buyer has rejected any oer p > pE ( 0) or if no such oer
has been made, the seller proposes pE ( ) and the buyer accepts any p pE ( )
and rejects any p > pE ( ) for all > 0.
if in periods 0 < the buyer has accepted any oer p > pE ( 0), then the seller
proposes pD ( ), which the buyer accepts, for all > 0.
D
1
The two sequences fpE ( )g1
; fp ( )g are referred to as an equilibrium if for
any and any history up to it is sequentially rational for any player to follow his
prescribes strategy given that all other players follow their strategy.
=1
=1
7
3 Bilateral Monoploy
Suppose there is only one buyer and one seller. The buyer does not have any outside
option.
Proposition 1. Let pE 2 [0; 1]. Then, for B 1=2 there is an equilibrium such that
in each period the seller proposes the price pE and the buyer accepts this price.
Proof. Fix a price pE 2 [0; 1]: The proposed equilibrium strategies are:
buyers agree to all p pE and reject all p > pE ,
sellers propose pE .
in all periods in which the buyer has rejected all price oers p > pE , and otherwise:
buyers agree to all prices;
sellers propose pD = 1.
It is straightforward to show that the constructed optimal penal code sustains the
equilibrium for all b 1=2. Suppose the seller proposes a price p > pE . The buyer
follows his equilibrium strategy if
0 + (1 , pE ) 1 ,b (1 , p) + 0 1 ,b b
b
(1)
for all p. Rearranging terms yields
b 1,p
(1 , pE ) + (1 , p) :
(2)
Note that the right hand side of this inequality is decreasing and continuous in p. Since
the inequality has to hold for all p > pE , I set, from continuity, p = pE and get
1
2
b :
It is straightforward to check that this implies that all other equilibrium conditions are
satised, too.
8
Proposition 1 says that the buyer can obtain any desired surplus from the bargaining
in a subgame perfect equilibrium. The buyer supports any such equilibrium by building
up a reputation for being a tough bargaining partner. Suppose the buyer whishes to
sustain a price of zero. Any soft behavior, as the acceptance of a higher price, leads
to the harshest punishment possible (see Abreu [1], [2]). The buyer cannot avoid the
punishment and is labelled as a weak bargainer from now on. Anticipating this, the
buyer has an incentive to accept only prices equal to zero and to stay tough. The seller
recognizes the buyer as a tough bargaining partner and does not demand a price higher
than zero.
The game between one seller and one buyer without outside option is a special case of
the game considered by Muthoo [14]. He studies repeated Rubinstein games. His stage
game has an innite horizon and a new transaction takes place only after an agreement.
There are two discount factors: one captures the value of future bargaining situations
(as b in my model) and one captures the cost of delay in the current bargaining
situation. One nding of his paper is that the Folk Theorem holds even for small
discount factors b as long as players are patient enough within each single transaction.
Then, gains from accepting a deviant oer are close to zero and losses are strictly
positive.
4 Disadvantageous Outside Options
Suppose now that there are many buyers and sellers in the market and that buyers
have the outside option available. I introduce the following denition for stationary
price oers from the seller.
Denition 1. An equilibrium price sequence pE (t) is a a stationary equilibrium
sequence if pE (t) = pE (t + 1) := pE for all t = 1; 2:::.
Proposition 2. If equilibrium price sequences are stationary, the only equilibrium has
the buyer extracting no surplus. The only equilibrium price sequence is given by pE (t) =
1 for all t and for all matched pairs.
9
Proof. Suppose there is an equilibrium with stationary price sequences pE 2 [0; 1)
such that buyers extract positive surplus from the bargaining. Suppose that the seller
proposes a price p > pE . In the equilibrium it has to be the case that the buyer rejects
such a price, otherwise the seller would have a protable deviation. The price pE can be
sustained only if it can be enforced with the optimal punishment available. But, due to
the buyer's outside option, the strongest punishment for a deviant buyer is pE again.
For the buyer who deviates can always use his outside option and quit his partner,
avoiding any harsher punishment. The new seller cannot reconstruct the deviant's
bargaining history and cannot "punnish" appropriately. Then, accepting p > pE is
protable. But then, sellers propose p = 1 in all periods. This argument holds for all
pE 2 [0; 1):
Proposition 2 asserts that stationary strategies cause a complete loss of buyers' bargaining power if an outside option is available. The market provides buyers with no
incentives to stay tough in bargaining at all. With an outside option buyers can always
leave their partner if punishment is prescribed from the next period on. The new trading partner does not know the deviant buyer's trading history and allows the buyer
to obtain the good for the low price. But then there is no incentive for the buyer to
build up a reputation for being tough in his rst partnership; the buyer gets weak and
accepts higher prices. Sellers know about the buyers' dilemma and propose a price of
one right away.
Most of the bargaining literature states that outside options are useful or at least not
harmful and focusses on when the presence of an outside option makes a player strictly
better of. The outside option principle says that outside options have an positive eect
on a player's equilibrium utility if and only if a player's outside option payo is strictly
greater than the equilibrium payo in the situation without outside option. Moreover,
it matters when a player can take the outside option. The analysis here shows that the
presence of the outside option may actually be harmful because it reduces a player's
4
See Shaked and Sutton [20], Sutton [21] and, for repeated Rubinstein bargaining games, Muthoo
[15].
4
10
ability to commit himself to threats.
If I consider more general price sequences the buyer can receive positive utility. I
introduce a further denition.
Denition 2. An equilibrium price sequence fpE (t)g1t is a decreasing equilibrium
price sequence if pE (t + 1) pE (t) for all t and pE (t + 1) < pE (t) for at least one t.
=1
Proposition 3. Assume that equilibrium price sequences are not restricted to be stationary. If b 2 (1=2; 1) there is an equilibrium which generates the following decreasing
sequence of prices: pE (1) = 2 and pE (t) = 0 for all t = 2; 3; :::. This equilibrium yields
strictly positive utilities for all buyers.
Proof. I propose the following equilibrium strategies:
Let pE (1) = 1 and pE (t) = 0 for all t = 2; 3; ::
For any match (bs): s proposes pE (t), b accepts all p(t) pE (t) and rejects all
p(t) > pE (t). Moreover, b never leaves s.
If b accepts any p(t) > pE (t), restart the price sequence, that is, pD (t + 1) =
pE (1) = 1 and pD (t + k ) = 0 for all k = 2; 3; :::
Again, the only interesting subgame is where s proposed p(t) > pE (t) for some t 2.
The incentive constraint for b reads
b
0 + 1 , (1 , p(t)) + b ,1 + 1 , b
b
b
(3)
for all p(t); t 2. Letting p(t) = 0 and rearranging terms yields
b 1
2
(4)
It is straightforward to check that all other equilibrium conditions are satised as
well and that buyers' equilibrium utilities are strictly positive. In particular, it is not
protable for a buyer to leave his current trading partner and try to escape from his
punishment.
11
Proposition 3 states that a decreasing price sequence can yield a strictly positive
utility level for a buyer. The market provides buyers with some incentives if there
are small frictions to build up a new reputation for being tough. A high initial price
keeps a buyer from leaving his current partner since any deviation triggers a restart
of the decreasing price sequence with the same trading partner. If the buyer takes
his outside option the new partnerhsip starts with the high rst period price as well.
Taking the outside option then is not a protable deviation. Hence, in a situation with
outside option it helps buyers to build up a reputation slowly. The gradual process of
building a reputation increases incentives to stay tough and stick to the equilibrium in
expectation of low prices in the future. Any soft bargaining leads immediately to the
loss of the reputation and the buyer has to built up a reputation for being tough from
anew.
There are other (decreasing) price sequences that yield buyers positive utilities for b
suciently large. I picked a particular decreasing sequence which will be crucial in the
next section.
Next, I look for optimal equilibria from the perspective of the buyer. An equilibrium
is optimal if it yields the highest payos for the buyer for a given discount factor b.
I will not follow the Folk-Theorem approach which gives results for discount factors
"suciently large". Rather, I suppose that agents are not absolutely patient. Denote
an optimal price sequence by fp(t)g1t and the equilibrium utility from fp(t)g1t as
U (p (t)) U .
5
6
=1
=1
I look for prices p(1); p(2); ::: that maximise the buyer's objective function and that
are sustainable as an equilibrium. In this equilibrium, trade always occurs and buyers
never leave their sellers. Moreover, the buyer always rejects any prices higher than the
ones specied by the equilibrium price sequence. If a buyer accepts a higher price the
equilibrium price sequence gets restarted. By this construction I impose the most severe
punishment on a defector. The most severe punishment is the restart of the sequence
since the buyer can always leave the seller and start a new partnership. Hence, any
see Fudenberg and Tirole [7], ch. 5, for a discussion of various versions of the Folk-Theorem for
perfect and imperfect information.
6see Abreu [1], [2] for a seminal analysis in this spirit.
5
12
punishment that the buyer's trading partner imposes cannot be harsher than a restart
of the equilibrium price sequence and any optimal equilibrium must be supported by
the most severe punishment.
The incentive constraints for the buyers that must be satised by any optimal equilibrium price sequence p(t) are given by
1
X
(1 , p( ))b ,t , b
=t+1
"
#
1
X
bt,1(1 , p (t))
t=1
(1 , p(t))
(5)
for all t and for all p(t).
Equation (5) compares, at each point in time, the trade-o between a one-period gain
from a deviation from the prescribed equilibrium, 1 , p(t), where p(t) is any proposal
from the seller, and the discounted future losses which are given by the left hand side
of equation (5). The rst term on the left hand side is the expected discounted payo
from following the equilibrium strategy whereas the second term on the left-hand side
ressembles the restart from the price sequence p(t). Incentive compatibility requires
that discounted future losses exceed the one-period deviation gain at each point in time.
Let E (t; b) be the set of all price sequences that satisfy the incentive constraints.
An optimal price sequence is a solution to problem (P ):
(P )
1
X
max
t,1 (1 , p(t))
p(t)2E (t;b) t=1 b
(6)
In words: if I look for a price sequence that maximises buyers' expected discounted
utility subject to p(t) satisfying the incentive constraints, I must end up with p(t)
itself. Since p(t) depends on b, I write p (t; b) and denote buyers' utility from an
optimal equilibrium sequence as U (p (t; b)).
Proposition 4. For all b 2 (1=2; 1), there exists a decreasing price sequence fp(t)g1t
=1
which is an optimal price sequence. This simple decreasing price sequence
(i) yields
U =
2b , 1 1 :
1 , b b
13
(ii) has a positive rst period prices p (1) and has prices p(2) = p(3) = ::: = 0,
where
1
p (1) =
b
Proof. See Appendix.
5 Random Proposer
I now study a more symmetric situation. With probability 1/2, agents nd themselves
in the situation we studied so far (without information leakage ): sellers propose pi(t),
buyers say yes or no and have then the possibility to leave their partners. With
probability 1/2 however, agents nd themselves in the reverse situation: buyers propose
pi (t), sellers accept or reject and then sellers have then the possibility to leave their
partners.
I show that the only equilibrium of this game is the stage game equilibrium. The
intuition for this would be that any equilibrium that supports market power on one side
of the market leads immediately to a termination of any partnership from an agents on
the other side of the market. The reason for this is that, due to imperfect information
about the trading history of other matches, increasing or decreasing price sequences
are needed. But then, there is always an incentive for some agents to leave his partner
and go back to the pool since this allows me to start a new relationship with a high
(low) price from anew.
Let q(t) be the average of the prices pi (t) and pj (t) at which trade occurs in each of
the two cases.
Proposition 5. The only equilibrium of this game has q(t) = 1=2.
Proof. We concentrate on the buyers. The argument for the sellers is just the same.
Suppose q(t) = q(t + 1) 6= 1=2 for all t were an equilibrium. As in Proposition 1, this
cannot be the case: buyers would always accept higher prices since they cannot be
punished appropriately.
14
Suppose now that q(t) 6= q(t). We know that q(t) cannot be increasing for all t if we
want to have Ui > 1=2. In particular, q(t) is has to be decreasing for t 2 1; t for some
t 2. But then, in period 2, sellers can always deviate protably and terminate the
relationship.
Proposition 5 illustrates the eects of outside options in repeated bargaining in the
most striking way. With only one buyer and one seller - and no outside options multiple equilibria obtain in the situation with random proposer. If multiple parties
and outside options for all agents are introduced the equilibrium price sequence is
unique.
6 Discussion
My ndings demonstrate that the role of the market as an incentive system is not clear
at all. In my model, the market lowers incentives to be tough in negotiations rather
than to enhance them.
The outside option leads to lower equilibrium utilities for the buyers as in the situation
without outside option. I mentioned already that this stands in contrast to most of
the literature on outside options. Shaked and Sutton [20] nd that an outside option
increases a rm's payo in negotiations with a worker. My model shows the opposite:
if the rm can re the worker and hire a new worker its payo is lower than in the
situation if it cannot re the worker. The result provides a rational for long-term
partnerships: if rms were able to commit themselfes to stay forever with the same
worker they would force themselfes to be tough bargaining partners in the rst place.
Rubinstein and Wolinsky [18] derive a Folk Theorem-like result in which buyers can
extract all the surplus from bargaining. Their result holds for preferences without
discounting. For preferences with discounting almost all prices between 1/2 and 1 can
be supported as equilibrium prices if the seller has an outside option and if the discount
factor is large enough. If the seller has no outside option, the only subgame perfect
equilibrium price is 1/2 as the discount factor tends to 1. Hence, the outside option
15
enhances the sellers' payos which stands in sharp contrast to the prediction in my
model.
The order of moves determines the role of outside options as well. In my model,
the choice of the exit option is not costly, that is, the buyer can nd a new partner
immediately. The buyer can avoid punishment without any further cost. If the buyer
can leave his partner (i) only after any proposal from the seller or (ii) before any
proposal of the seller, exit implies that the buyer cannot trade for one period. If the
buyer cannot trade for one period it would be possible to implement an equilibrium
that sustains a price of zero, for b large enough.
A further point concerns my informational assumptions. Suppose that sellers in the
pool could have (costly) access to information about any buyer's trading history. The
analysis shows that the sellers which are currently trading are not interested in such
an information ow at all. For if this information were available a buyer's deviation
could be detected by his new trading partner and could be punished. But then buyers
could extract all the surplus from the bargaining. Hence the restricted availability of
information for "outsider" helps the "insiders". It does not hurt the "outsiders" : rst,
in equilibrium they are not actively involved, second, by my matching assumptions,
every "outsider" gets an "insider" sooner or later.
Since buyers cannot extract all the surplus through individual bargaining power alone
one could ask if buyers can use the presence of many buyers in the market and form
a collusive agreement. Moreover, one wonders why buyers cannot choose to switch to
matched sellers inducing competition among them. In Groh [10] I show how competition and information leakage serve as a tool for implicit collusion among buyers. The
collusive agreement has the same avour as the equilibria that we know from the literature on collusion in innitely repeated oligopoly games. With collusion, buyers can
improve upon the maximal payo derived in this paper for any xed discount factor.
16
Appendix
Suppose, I wish to implement a price sequence with p(1) > 0; p (2) = p(3) = ::: = 0.
Then, in equilibrium, the following incentive constraints have to be satised for the
buyers after each proposal from a seller.
I have, for t = 1,
0 + 1 ,b (1 , p(t)) + bU b
for all p(t) p (1):
(7)
for all p(t) 0:
(8)
For all t = 2; 3; :::,
0 + 1 ,b (1 , p(t)) + bU b
To satisfy the requirement of sequential rationality within each relationship , I have
to nd the incentive constraints which put the hardest restrictions on U for any given
b. The other constraints will be satised as well. It is easy to see that he crucial
constraint is the one from period 2 onwards. Sequential rationality requires that I set
p(1) = p (1) and p(t) = 0 for all t = 2; 3; :::.
This yields, from equation (8),
U which I let hold with equality.
2b , 1 1 ;
1 , b b
(9)
I show next that the proposes price sequence is optimal indeed. Let pA (t) denote any
arbitrary price sequence with 0 pA (t) 1 for all t. Let U A (pA(t)) be the utility from
an arbitrary sequence and let U A (t) be the continuation utility from period t on for all
t.
Lemma 1.
U (p (t)) U A (pA (t))
for all sequences pA (t).
Proof. Suppose that there exists an alternative price sequence fpA (t)gt=11 such that
U (p (t)) < U A (pA (t)) and that pA (t) 6= 0 for at least one t 2 and pA (1) < p (1).
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Fix the equilibrium with the sequence fp(t)g1t . In particular, choose p(1) such
that the incentive constraint for the equilibrium in period 1 holds with equality:
=1
0 + 1 ,b = (1 , p(1)) + bU :
(10)
b
Suppose that there exists a sequence fpA (t)g1t with pA (t) 6= 0 for at least one t 2
and pA (1) < p (1) such that U A (pA (t)) > U (p(t). For fpA (t)g1t to be an equilibrium,
I require in period t = 1,
=1
=1
0 + bU A (2) (1 , pA (1)) + bU A :
(11)
Suppose that this constraint is satised. Note that, by assumption, U A > U and
pA (1) < p (1). Then, the LHS of (11) is < than the LHS of (10) and that the RHS of
(11) is larger than the RHS of (10). But then, given b, if fp(t)g1t is an equilibrium,
(11) cannot hold. This in turn implies that fpA (t)g1t cannot be an equilibrium,
yielding a contradiction.
=1
=1
The derivation of p(1) = 1=b follows immediately from solving the equation
(1 , p (1)) +
2b , 1 1 :
=
1 , b
1 , b b
b
References
[1] D. Abreu. Extremal Equilibria of Oligopolistic Supergames. Journal of Economic
Theory, (39):191{225, 1986.
[2] D. Abreu. On the Theory of innitely discounted repeated Games with discounting. Econometrica, (56):383{396, 1988.
[3] L. Ausubel and R. Deneckere. Reputation in Bargaining and Durable Goods
Monopoly. Econometrica, (3):511{531, 1989.
18
[4] L. Busch and Q. Wen. Perfect Equilibria in a Negotiation Model. Econometrica,
(3):545{565, 1995.
[5] S. Datta. Building trust. Mimeo, 1993.
[6] R. Fernandez and J. Glazer. Striking for a Bargain between Two Completely
Informed Agents. American Economic Review, (81):240{252, 1991.
[7] D. Fudenberg and J. Tirole. Game Theory. MIT Press, Cambridge, Mass., 1991.
[8] D. Gale. Limit Theorems for Markets with Sequential Bargaining. Journal of
Economic Theory, (43):20{54, 1987.
[9] P. Ghosh and D. Ray. Cooperation in Community Interaction without Interaction
Flows. Review of Economic Studies, (63):491{519, 1996.
[10] C. Groh. Market Power and Collusion in Markets with Bilateral Negotiations.
Working Paper, University of Mannheim, 1999.
[11] F. Gul, H. Sonnenschein, and R. Wilson. Foundations of Dynamic Monopoly and
the Coase Conjecture. Journal of Economic Theory, (39):155{190, 1986.
[12] H. Haller and S. Holden. A Letter to the Editor on Wage Bargaining. Journal of
Economic Theory, (52):232{236, 1990.
[13] O. Hart and J. Tirole. Contract Renegotiation and Coasian Dynamics. Review of
Economic Studies, (55):509{540, 1988.
[14] A. Muthoo. Bargaining in a long-term relationship with endogenous termination.
Journal of Economic Theory, (66):590{598, 1995.
[15] A. Muthoo. Bargaining Theory with Applications. Cambridge University Press,
Cambride, UK, 1999.
[16] M. Osborne and A. Rubinstein. Bargaining and Markets. Academic Press, San
Diego, New York, 1990.
[17] A. Rubinstein and A. Wolinsky. Equilibrium in a Market with Sequential Bargaining. Econometrica, (53):1133{1150, 1985.
19
[18] A. Rubinstein and A. Wolinsky. Decentralized Trading, Strategic Behavior and
the Walrasian Outcome. Review of Economics Studies, (57):63{78, 1990.
[19] K. Schmidt. Journal of Economic Theory, 1993.
[20] A. Shaked and J. Sutton. Involuntary Employment as a Perfect Equilibrium in a
Bargaining Model. Econometrica, (52):1351{1364, 1984.
[21] J. Sutton. Non-Cooperative Bargaining Theory: An Introduction. Review of
Economic Studies, (53):709{724, 1986.
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