This short paper demonstrates that the equilibrium payoffs of an alternating-offers bargaining game over a unit of surplus converge to equal division provided that the parties are allowed to bargain over all the surpluses generated by the "right" to be the first to make offers. The result obtained in the present paper may provide some "justification" for other division procedures such as the divide-and-choose or the moving-knife mechanisms.Bargaining theory, Alternating offers, First-mover advantage, Equal division.

Oz Shy

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 2 
  
Consistent Bargaining 
  
 
 
Oz Shy  
Federal Reserve Bank of Boston 
Abstract 
This short paper demonstrates that the equilibrium payoffs of an alternating-offers bargaining game over a unit of 
surplus converge to equal division provided that the parties are allowed to bargain over all the surpluses generated by 
the "right" to be the first to make offers. The result obtained in the present paper may provide some "justification" for 
other division procedures such as the divide-and-choose or the moving-knife mechanisms.
I thank Johannes Muenster, Helmut Bester, Lones Smith, as well as the participants at the WZB breakfast seminar for insightful discussions and 
comments. The views and opinions expressed in this paper are those of the author and do not necessarily represent the views of the Federal 
Reserve Bank of Boston or the Federal Reserve System. 
Citation: Oz Shy, (2010) ''Consistent Bargaining'', Economics Bulletin, Vol. 30 no.2 pp. 1425-1432. 
Submitted: Feb 27 2010.   Published: May 18, 2010. 
 
     1. Introduction
A division of one monetary unit, say a perfectly-divisible $1, in an alternating-oers mecha-
nism generally results in an unequal division. For example, if players' time discount factors
are not \too diverse," the player who is endowed with the privilege to be the rst to make
an oer ends up with more than 50% of the surplus.1 The natural question that arises
from this asymmetric result is why should the second player agree to this bargaining pro-
cedure? For this reason, the present paper also \allows" the bargaining parties to allocate
the surplus generated by the bargaining procedure itself which treats the bargaining parties
asymmetrically.
The above-mentioned asymmetry is of course known for a long time. One \remedy"
that has been proposed in the literature is to allow the parties to ip a fair coin at the
beginning of each bargaining period for the purpose of restoring equality, see Osborne and
Rubinstein (1990, p.53) and Muthoo (1999, p.192). There are two problems associated with
the application of this remedy:
(1) Because this lottery mechanism results in a more equal expected division, one should
raise the question why do we need the alternating-oers bargaining mechanism to begin
with? In fact, once a lottery becomes an integral part of the bargaining procedure, the
alternating-oers component can be completely dropped and the parties can simply use
a coin to allocate the entire surplus. And, even more importantly,
(2) The lottery \remedy" introduces a foreign element into the bargaining process, whereby
part of surplus is divided by a lottery and some other parts are divided by alternating
oers. This bargaining hybrid generates an inconsistency as dierent surpluses would
be divided according to dierent mechanisms.
The present paper shows that a near equal division is obtained if the bargaining parties
utilize one and only one bargaining procedure for the division of the entire surplus, including
the surpluses generated by the bargaining procedure itself, which assigns a higher surplus to
the player who is endowed with the right to be the rst to make an oer. The proximity to
equal division would depend on the number of subgames that the parties agree to bargain
upon, where each game divides a surplus generated by the privilege of being the rst to make
an oer in a subsequent game.
The extended bargaining procedure works as follows. Instead of having both parties
agreeing on dividing the surplus using only one alternating-oers mechanism, the parties
now also bargain over the dierence in payos generated by assigning the right to be the rst
to make an oer to one of the parties. Clearly, even the proposed pre-bargaining bargaining
game would result in an unequal division of surplus, which the parties may want to bargain
1As noted in Muthoo (1999, pp. 53{55), the rst-mover advantage vanishes as the time interval between
two consecutive oers becomes arbitrarily small. However, this interpretation implies that the bargaining
game is conducted between two computers for which the response time is indeed innetesimal.
1over in a second pre-bargaining game, and so on. It is shown that near-equal division is
obtained once we add up all the surpluses allocated to each party. The convergence to
equal division is extremely fast so one or two pre-bargaining games are sucient to obtain
a near-equal division of surplus.
Section 2 briey reviews the basic alternating-oers bargaining model. Section 3 extends
the bargaining process to include pre-bargaining bargaining stages where the parties bargain
over the surpluses generated by the right to be the rst to make oers during a series of
bargaining games that eventually lead to the standard nal bargaining procedure. Section 4
concludes with discussions.
2. The Alternating-oers Bargaining Procedure
This section describes the \basic" bargaining model which serves as the benchmark for the
present analysis. Two parties bargain over the division of a perfectly-divisible $1. In the each
even period t = 0;2;4;::: player 1 oers a division (xt
1;xt
2). In each odd period t = 1;3;5;:::
player 2 oers (yt
1;yt
2). The rst component is the payo to player 1 and the second is payo
to player 2. Since both parties prefer more to less, oers must satisfy xt
1+xt
2 = yt
1+yt
2 = $1.
Once being oered, the non-oering party can accept or reject the oer. The game ends
immediately with the proposed division of payos once an oer is accepted. If an oer is
rejected, the game advances one period and the other player makes an oer. Let 0 <  < 1
denote the players' common discount factor.2
Rubinstein (1982) proved that in the unique subgame perfect equilibrium (SPE) for this
game, the parties make the following oers.
x
t
1 = y
t0
2 =
1
1 + 
and x
t
2 = y
t0
1 =

1 + 
: (1)
where t = 0;2;4;::: and t0 = 1;3;5;:::. The equilibrium oers (1) are constructed such that
they are always accepted by the other party in a SPE. Therefore, player 2 accepts the oer
made by player 1 in t = 0 and the entire surplus is divided according to (x0
1;x0
2).
The important thing to realize is that the division rule (1) allocates dierent amounts
of the pie to the two bargaining parties, who are identical in all respects (having identical
preferences and the same time discount factor). Formally, we dene the surplus of the original
game generated by the right to be the rst to make an oer by
S0
def = x
0
1   x
0
2 =
1
1 + 
 

1 + 
=
1   
1 + 
: (2)
2As discussed in Section 4.1, the bargaining procedure described in this paper applies also to unequal
discount factors. However, in the general case, unequal division may be a consequence of the players' diverse
value of time which are not necessarily related to the rst-mover advantage (which is the subject of the
present investigation).
2Notice that S0 = 0 for  = 1 and increases to S0 = 1 as  declines to zero. That is, the
rst-mover advantage is small for highly patient players, and increases to the entire pie for
highly impatient players.
3. Consistent Division of All Surpluses
The surplus dened by (2) can be divided using a separate pre-bargaining procedure. In a
pre-bargaining game, player 1 oers player 2 part of the surplus generated by the advantage
of being the rst to make an oer in a subsequent bargaining game. If player 2 accepts,
the parties collect the payos as prescribed by division rule (1) in the form of transfers
between the players (to be added up to all transfers corresponding to the outcomes of all
pre-bargaining games). If player 2 rejects, the clock advances one period and it becomes
2's turn to demand from player 1 a compensation for having player 1 be the rst to make
an oer. If 1 rejects, the clock advances again and it becomes 1's turn to make a proposal,
and so on. Thus, the extended bargaining procedure developed in this paper is composed
of a (nite or innite) series of innite-horizon bargaining subgames. In each subgame, the
parties bargain over the surplus generated by the advantage of being the rst to make an
oer in a subsequent game.
Consider rst a single pre-bargaining game in which the parties bargain over the surplus
S0 dened by (2). By consistent bargaining I mean that the bargaining procedure used in the
nal game should also be used in the pre-bargaining game. More precisely, the division rule
(1) should also be the division rule for the surplus to be divided in all pre-bargaining games,
and not only for the nal game. Therefore, applying division rule (1) for the surplus given
in (2) implies that out of the nal share prescribed to player 1 in the nal game, player 1
has to \reimburse" player 2 an amount of S0=(1+) for the \privilege" to be the rst to be
making oers. Thus, sums of payos from one pre-bargaining game and the nal bargaining
game of player 1 and player 2 are

1
1 =
1
1 + 
 

1 + 

1   
1 + 
and 
1
2 =

1 + 
+

1 + 

1   
1 + 
: (3)
Note that 1
1 + 1
2 = 1 which makes (3) a feasible and ecient allocation.
The division (3) generates a surplus from the \right" to be the rst to make an oer how
to divide the surplus generated by the \right" to be the rst to make an oer in the nal
bargaining game. Formally, dene
S1
def = 
1
1   
1
2 =
(1   )2
(1 + )2 = (S0)
2 : (4)
Consider now a pre-pre-bargaining game which divides the surplus S1 according to the
rule prescribed by (1). Therefore, the bargaining outcome of this stage is that player 1
should reimburse player 2 an amount of S1=(1+ ) for the right to be the rst to oer in a
3subsequent pre-bargaining game. Hence, the total sums of payos collected by each player
are 2
1 = 1
1   (S0)2=(1 + ) and 2
2 = 1
2 + (S0)2=(1 + ). Hence,

2
1 =
1
1 + 
 

1 + 

1   
1 + 
 

1 + 

(1   )2
(1 + )2 and

2
2 =

1 + 
+

1 + 

1   
1 + 
+

1 + 

(1   )2
(1 + )2: (5)
Note that 2
1 + 2
2 = 1 which makes (5) also both feasible and an ecient allocation.
The payo allocation (5) generates a surplus from the sequence of bargainable \rights"
to have the right, and so on. Formally, dene
S2
def = 
2
1   
2
2 =
(1   )3
(1 + )3 = (S0)
3 : (6)
If the surplus S2 is to be divided in a pre-pre-pre-bargaining game according to rule (1),
player 1 must reimburse player 2 an amount of (S0)3=(1 + ) for the right to be the rst.
Thus, the total sums of payos are given by

3
1 =
1
1 + 
 

1 + 

1   
1 + 
 

1 + 

(1   )2
(1 + )2  

1 + 

(1   )3
(1 + )3 and

3
2 =

1 + 
+

1 + 

1   
1 + 
+

1 + 

(1   )2
(1 + )2 +

1 + 

(1   )3
(1 + )3: (7)
In view of (3), (5), and (7), the total prot of each player in a nite bargaining procedure
with n \pre-pre" bargaining subgames is

n
1 =
1
1 + 
 

1 + 

1   
1 + 

n 1 X
t=0

1   
1 + 
t
and 
n
2 = 1   
n
1: (8)
It can be easily veried that players' prots converge to 0:5 as the number of \pre-pre"
bargaining subgames n increases. Formally, n
1  ! 0:5 and n
2  ! 0:5 as n  ! 1.
Finally, as mentioned earlier, even a small number of \pre-pre" bargaining subgames is
sucient to have the parties agreeing on a near-equal division of the pie. To see this, Table 1
displays the payo to bargaining party 1 as a function of the players' common discount rate
 and the number of \pre-pre" bargaining games n.
Table 1 demonstrates the fast convergence to equal division. Even for highly impatient
bargaining parties ( = 0:2) the dierence in payos falls below 10% in only three \pre-
pre" bargaining rounds. For more patient bargaining parties, the dierence falls below 10%
already in the second pre-bargainig game. This means that only a small number of \pre-pre"
bargaining subgames are needed to get close to equal division.
4 0:2 0:4 0:6 0:8 0:9
0
1 0:8333 0:7143 0:6250 0:5556 0:5263
1
1 0:7222 0:5918 0:5313 0:5062 0:5014
2
1 0:6481 0:5394 0:5078 0:5007 0:5001
3
1 0:5988 0:5169 0:5020 0:5001 0:5000
4
1 0:5658 0:5072 0:5005 0:5000 0:5000
Table 1: Proximity to equal division in a small number of \pre-pre" bargaining games.
4. Discussions
Bargaining procedures must be agreed upon by all parties. Thus, the procedure itself should
also become an integral part of the negotiation. Therefore, any bargaining mechanism that
favors one party may be vetoed out by the other party. The major conclusion to be drawn
from this exercise is that unequal divisions tend to be realized only if the parties apply the
bargaining procedure over part of the surplus. This leaves some surplus to be allocated
outside the bargaining process without letting the parties bargain over it directly. Once this
surplus becomes an integral part of the bargaining process itself (consistent bargaining in
the language of this paper) the nal payos converge very fast to equal division.
4.1 Discount factors
The strength of the alternating-moves bargaining mechanism is the dependency of the bar-
gaining solution on the players' discount factors (in particular, when one player is more
patient than the other). In this respect, it should be pointed out that the procedure pro-
posed in this paper works perfectly well also for unequal discount factors, in which 1 6= 2.
The problem, however, is that under unequal discount factors, unequal division cannot be
attributed solely to the rst-mover advantage. That is, unequal division may be realized
because one player happens to be more patient than the other. For this reason, the analysis
in this paper is conned to equal discount factors. When 1 = 2 = , an unequal division
is a direct consequence of the rst-mover advantage (rather than the players' degree of im-
patience). The rst-mover advantage is the subject of the present paper in the sense that
the contribution here is to allow players to bargain on all surpluses including those that are
generated by rst-mover advantages.
54.2 Agenda and bargaining procedures
Perry and Reny (1993) analyze a continuous-time bargaining environment in which players
can choose when to make oers, but, each player must wait an exogenously-given amount
of time between making oers. Their model generally yields multiple equilibria of divi-
sions. Fershtman (1990) has already demonstrated the importance of choosing an agenda
in bargaining problems. The present paper makes the point that the choice of a bargaining
procedure (in our case the decision on which player moves rst) should be an integral part of
the bargaining procedure itself and not be decided by a dierent mechanism, such as a lot-
tery. In other words, bargaining over procedure is an essential component of the bargaining
agenda in every bargaining problem.
My argument for integrating the allocation of surplus generated by the specication of
the order of moves into the bargaining procedure is also motivated by general views on how
game theoretic models should be constructed in order explain \real-life" behavior. Quoting
from Nash (1953, p.129), \The negotiation process must be formalized and restricted, but
in such a way that each participant is still able to utilize all the essential strengths of
his position." Rubinstein (1991, p.919) discusses a view in which \...a game-theoretic model
should include only those factors which are perceived by the players to be relevant. Modeling
requires intuition, common sense, and empirical data in order to determine the relevant
factors entering into the players' strategic considerations and should thus be included in the
model."
Finally, the result obtained in this paper may provide some \justication" for other di-
vision procedures such as the divide-and-choose or the moving-knife mechanisms, see Brams
and Taylor (1996) for discussions, references, as well as extensions to more than two players.
As it turned out, economists for some reason don't make much use of these mechanisms. The
present paper demonstrates that the nal division of payos under these mechanisms can be
approximated by extending the basic alternating-oers bargaining procedure to consistently
apply to the division of all surpluses generated by the asymmetric treatment of players in
the alternating-oers bargaining procedure itself.
References
Brams, S and A. Taylor (1996) Fair Division: From Cake-cutting to Dispute Resolution,
Cambridge: Cambridge University Press.
Fershtman, C. (1990) \The Importance of Agenda in Bargaining" Games and Economic
Behavior 2, 224{238.
Muthoo, A. (1999) Bargaining Theory with Applications, Cambridge: Cambridge University
Press.
Nash, J. (1953) \Two-Person Cooperative Games" Econometrica 21, 128{140.
6Osborne, M and A. Rubinstein (1990) Bargaining and Markets, San Diego: Academic Press.
Perry, M and P. Reny (1993) \A Non-cooperative Bargaining Model with Strategically
Timed Oers" Journal of Economic Theory 59, 50{77.
Rubinstein, A. (1982) \Perfect Equilibrium in a Bargaining Model" Econometrica 50, 97{
109.
Rubinstein, A. (1991) \Comments on the Interpretation of Game Theory" Econometrica
59, 909{924.
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