We study a bilateral multi-issue bargaining procedure with complete information and endogenous unrestricted agenda, in which offers can be made in any subset of outstanding issues. We find necessary and sufficient conditions for this procedure to have a unique subgame perfect equilibrium agreement.multi-issue bargaining, complete information, agenda

Roberto Serrano

Younghwan In

English

Research Papers in Economics

 
 
 
 
 
Department of Economics 
Working Paper No. 0206 
http://www.fas.nus.edu.sg/ecs/pub/wp/wp0206.pdf 
 
 
Agenda Restrictions in Multi-Issue Bargaining (II): 
Unrestricted Agendas 
 
 
Younghwan In 
National University of Singapore 
Roberto Serrano 
Brown University 
August 2002 
 
 
Abstract:  We study a bilateral multi-issue bargaining procedure with complete information and 
endogenous unrestricted agenda, in which offers can be made in any subset of outstanding issues. We find 
necessary and sufficient conditions for t his procedure to have a unique subgame perfect equilibrium 
agreement. 
 
 
 
Key words:    multi-issue bargaining, complete information, agenda 
JEL Classification:  C72, C78, D74 
 
 
 
 
© 2002  Younghwan In  and  Roberto Serrano.  Younghwan In,  Department of Economics, National 
University of Singapore, 1 Arts Link, Singapore 117570, e -mail:  ecsyhi@nus.edu.sg, 
http://courses.nus.edu.sg/course/ecsyhi, tel: +65-6874-6261, fax: +65-6775-2646.  Roberto Serrano, 
Department of Economics, Brown University, Providence, RI 02912, U.S.A., e -mail: 
roberto_serrano@brown.edu, http://econ.brown.edu/faculty/serrano.    In gratefully acknowledges research 
support from Brown University through a Stephen R. Ehrlich fellowship, and Serrano from the National 
Science Foundation.  Views expressed herein are those of the authors and do not necessarily reflect the 
views of the Department of Economics, National University of Singapore 1 Introduction
This note attempts to shed light on the circumstances under which an unre-
stricted agenda multi-issue bargaining procedure, extension of Rubinstein’s
(1982) alternating oﬀers game, yields a unique subgame perfect equilibrium
(SPE) agreement. In and Serrano (2000) pointed out that restricting agen-
das may yield multiple SPE agreements, including those with arbitrarily long
delay. The procedure in In and Serrano (2000) exhibits a clear friction: bar-
gainers are forced to negotiate one issue at a time, the one chosen by the
proposer in each round.
We are thus led to examine a procedure where the issues can be bundled.
A basic observation in this respect is that if one studies a procedure where all
issues must be bundled in every oﬀer, Rubinstein’s uniqueness and eﬃciency
result extends (indeed, the same proof as in Osborne and Rubinstein (1990)
applies). Instead, we investigate a procedure suggested by Inderst (2000).
We refer to it as unrestricted agenda bargaining. In it, each proposer can
make an oﬀer on any subset of outstanding issues. We provide necessary
and suﬃcient conditions for this procedure to have a unique SPE, thereby
generalizing Inderst’s (2000) suﬃciency result to a considerably larger class
of utility functions (where separability across issues and concavity are not
assumed). The conditions we ﬁnd require the uniqueness of a stationary
equilibrium payoﬀ. In this sense, they are simple generalizations of those
found for the single-issue case, and we emphasize that they are compatible
with important classes of preferences ruled out in the earlier analysis of the
procedure.
2 Unrestricted Agenda Bargaining
Let L = f1;:::;lg be the set of issues. Bargaining takes place over potentially
inﬁnite discrete periods starting in period 0. For i = 1;2 and S µ L, the
procedure ΓS
i (±) induces an inﬁnite horizon game of perfect information and
it is deﬁned recursively.
² The ﬁrst move corresponds to player i, who makes a proposal: he
chooses an arbitrary subset of the pending issues T µ S;T 6= ; and
oﬀers the split (xT;1T ¡ xT), where 0T · xT · 1T.1
1We use the following vector notation for two arbitrary vectors x and y: x ¸ y means
1² Player j can then either accept or reject this entire proposal (he is not
allowed to accept the proposal only for a strict subset of T and reject
the rest).
– If the proposal is accepted, issues T are split accordingly and the
procedure Γ
SnT
j (±) is followed.
– If the proposal is rejected, the procedure ΓS
j (±) is followed with
probability ± (0 · ± < 1), whereas negotiations break down with
probability 1 ¡ ±. In the latter case, both players receive a zero
share from those issues on which there was no agreement.
The negotiations end either with the breakdown outcome, or when the
procedure Γ;
i(±) must be followed for i = 1;2.
We shall refer to the procedure ΓL
i (±) simply as Γi. Also, a subgame
ΓS
i (±) will be written simply as ΓS
i . We shall seek for the subgame perfect
equilibrium (SPE) agreements of the procedure Γi.
Both players are von Neumann-Morgenstern expected utility maximizers.
We make the following assumptions on the utility functions u1(x1;:::;xl) and
u2(1 ¡ x1;:::;1 ¡ xl):
² A0 (normalization):
ui(0L) = 0 for i = 1;2:
² A1 (continuity): ui(¢) is continuous on [0;1]l.2
² A2 (interior strong monotonicity):
ui(x) > ui(y) if x > y and x À 0L; and
ui(x) ¸ ui(y) if x > y and some element of x is zero:
We shall impose one more assumption that will turn out to be neces-
sary and suﬃcient for uniqueness of SPE in the procedure Γi. This extra
assumption plays the role of Lemma 3.2 in the proof of uniqueness of the
xk ¸ yk for all k; x > y means xk ¸ yk for all k and xk > yk for some k; and x À y means
xk > yk for all k. Also, xT = (xk)k2T, and we use 0T and 1T to denote vectors with jTj
zeros and ones, respectively.
2Lower case letters denote cardinality of sets, i.e., jLj = l, jSj = s, and so on.
2single-issue case found in Osborne and Rubinstein (1990). Before getting to
it, we introduce some pieces of notation.
Consider the game Γi, i = 1;2, and a subgame thereof ΓS
j (x¡S), j = 1;2,
prior to which the shares (xk;1 ¡ xk)k= 2S have been agreed. Moreover, sup-
pose that in this subgame both players have a non-degenerate set of feasible
utilities.3
We deﬁne the following amounts, the maximum and minimum utilities
that each player can get in this subgame:
u1(x¡S) = u1(0S;x¡S); ¯ u1(x¡S) = u1(1S;x¡S):
u2(x¡S) = u2(0S;1¡S ¡ x¡S); ¯ u2(x¡S) = u2(1S;1¡S ¡ x¡S):
The utility possibility set in this subgame is:
US(x¡S) = f(u1(xS;x¡S);u2(1S ¡ xS;1¡S ¡ x¡S)) : xS 2 [0;1]
sg:
Given the assumptions on utility functions, the Pareto frontier of the set
US(x¡S) is the graph of a continuous and strictly decreasing function fx¡S(u1)
deﬁned on [u1(x¡S); ¯ u1(x¡S)]. Let gx¡S(u2), deﬁned on [u2(x¡S); ¯ u2(x¡S)],
denote the inverse function of fx¡S(u1). Similarly, if S = L, the Pareto
frontier of the utility possibility set UL in the entire game is the graph of a
function f(u1), whose inverse is g(u2).
We now state our last assumption. It provides a necessary and suﬃcient
condition for uniqueness. We provide examples of speciﬁc utility functions
that imply the assumption in the sequel.
² A3. For every subgame ΓS
i (x¡S),
– (1) There exists a unique solution u¤
1(x¡S) to the equation
h(u1) = u1¡u1(x¡S)¡±[gx¡S((1¡±)u2(x¡S)+±fx¡S(u1))¡u1(x¡S)] = 0;
– (2) If h(z) ¸ 0, then z ¸ u¤
1(x¡S); and
– (3) If h(z) · 0, then z · u¤
1(x¡S).
3There are additional subgames of a trivial nature, where at least one of the players
has only one feasible utility level (if this player has accepted a zero share in some issue
and his utility is positive only when he receives positive amounts of all issues). But these
subgames are irrelevant for the analysis.
3Assumption A3 requires that the function h(¢), analogue to that used in
the proof of Lemma 3.2 in Osborne and Rubinstein (1990) as the basis of a
stationary construction, have a unique zero. This will turn out to be weaker
than additive separability and concavity, conditions used in Inderst (2000).
The next Lemma, an analogue of Lemma 3.2 in Osborne and Rubinstein
(1990), will be useful and sheds additional light on Assumption A3. The
simple proof is omitted.
Lemma 1 Under Assumption A3(1), in the subgame ΓS
i (x¡S) there is a
unique solution (u¤
1(x¡S);u¤
2(x¡S)) to the system of equations
u1 = (1 ¡ ±)u1(x¡S) + ±gx¡S(u2)
u2 = (1 ¡ ±)u2(x¡S) + ±fx¡S(u1):
If S = L and we consider the game Γi, the unique solution to the equa-
tions in Lemma 1 will be denoted (u¤
1;u¤
2). As in Rubinstein (1982), the
equations in Lemma 1 describe the indiﬀerence of player j = 1;2 when re-
sponding between accepting the proposal worth uj and rejecting it seeking
for a stationary continuation in negotiations behavior.
Proposition 1 Consider the unrestricted agenda multi-issue bargaining pro-
cedure Γ1(±) for any ± 2 [0;1). Under Assumptions A0-A3, there exists a
unique SPE payoﬀ, which is implemented immediately and is eﬃcient. It is
the payoﬀ pair (g(u¤
2);u¤
2).
Of course, in the game Γ2(±), the same proof also shows that the unique
SPE payoﬀ is (u¤
1;f(u¤
1)). With Lemma 1 in hand, the uniqueness part of the
proof is by now standard, based on the ingenious argument of Shaked and
Sutton (1984). Existence relies on an induction argument.
Proof of Proposition 1. First, we write down strategies that support the
desired payoﬀ. In doing this, we use induction on the number of issues. Since
when jTj = 1, the procedure ΓT
i reduces to Rubinstein’s (1982), we make the
following induction hypothesis: if jTj < l, the game ΓT
i (x¡T) has a unique
SPE payoﬀ when the set UT(x¡T) is non-degenerate. In it, player j receives
a payoﬀ u¤
j(x¡T) and player i its eﬃcient projection in the subgame. Notice
that this induction hypothesis holds for single-issue subgames: thanks to
Assumptions A0-A3, Rubinstein’s theorem applies.
Using the above induction hypothesis, SPE strategies for the game Γi
follow. In them, let S µ L.
4² (i) In subgames ΓS
i (x¡S) where the utility possibility set US(x¡S) is
non-degenerate, player i makes a proposal on all remaining issues. He
oﬀers to player j the payoﬀ u¤
j(x¡S) and asks for himself the eﬃcient
projection in the subgame of u¤
j(x¡S).
² (ii) In subgames ΓS
i (x¡S) where the utility possibility set US(x¡S) is
non-degenerate, player j accepts a proposal if and only if the utility
u that he would get by accepting (assuming he receives the SPE pay-
oﬀ speciﬁed in the induction hypothesis, for the issues still pending)
satisﬁes u ¸ u¤
j(x¡S).
² (iii) In degenerate subgames where at least one of the players receives
only zero utility, any SPE pair of strategies would do.
By the standard argument, which is based on the equations of Lemma 1
and involves the use of the one-time deviation property, it is easy to see that
these strategies constitute a SPE of the game Γi.
Next, we prove that the SPE payoﬀ is unique. Consider the games Γ1
and Γ2, subgames of each other. Recall that u1 = u2 = 0. Let M1 and m1
denote the supremum and the inﬁmum of u1, and M2 and m2 the supremum
and the inﬁmum of u2 in the SPE of Γ1. Let M0
1;m0
1;M0
2 and m0
2 denote the
similar magnitudes in Γ2.
First, M1 · g(±f(±M1)). This comes from combining M1 · g(±m0
2)
and m0
2 ¸ f(±M1): Therefore, by A3, part (3), M1 · z¤, where z¤ solves
z¤ = g(±f(±z¤)). Moreover, using the equations of Lemma 1, z¤ = g(u¤
2).
Second, m1 ¸ g(±f(±m1)). This is obtained from combining m1 ¸ g(±M0
2)
and M0
2 · f(±m1). Using A3, part (2), m1 ¸ z¤. And therefore, M1 = m1 =
g(u¤
2).
Finally, it is clear that M2 · f(m1) and m2 ¸ f(M1), but m1 = M1 =
g(u¤
2). Therefore, M2 = m2 = f(g(u¤
2)) = u¤
2.
Remark 1: Under Assumptions A0-A3, there might be multiple strategy
proﬁles supporting the unique SPE payoﬀ of Γi, but this can always be
supported by oﬀers involving all issues. To get uniqueness in the equilibrium
shares, additional conditions are needed (e.g., one should rule out cases like
identical linear preferences, which would yield a “thick” eﬃcient frontier in
the space of issues).
Remark 2: A3 is strictly weaker than concavity of the utility functions. It is
compatible with a convex frontier of the utility possibility set. For example,
5if u1(¢) = x1x2 ¢¢¢xl and u2(¢) = (1¡x1)(1¡x2)¢¢¢(1¡xl), the unique SPE
payoﬀ of Γ1(±) is ( 1
(1+±1=l)l; ±
(1+±1=l)l). This is supported by player 1 oﬀering
on all issues the same split: ( 1
1+±1=l; ±1=l
1+±1=l). Note how this Pareto frontier is
a convex function, the more convex the higher the number of issues.
Remark 3: Additive separability across issues has not been assumed either
in Lemma 1 or in Proposition 1. This, besides concavity, is the most striking
diﬀerence with respect to Inderst (2000). For an example, see Remark 2
again.
Remark 4: We now show that in the absence of A3 we will have multiple
SPE agreements. It suﬃces to develop an example to see that the multi-
plicity phenomenon happens as soon as the function h has multiple zeros.
To emphasize the connections with the single-issue analysis, we develop an
example that builds on it. For a single-issue, consider the following pair of
utility functions:
u1(x) =
( b1
a1x; x · a1
1¡b1
1¡a1(x ¡ 1) + 1; x > a1;
where 0 < b1 < a1 < 1, and
u2(1 ¡ x) =
( b2
a2(1 ¡ x); x · a2
1¡b2
1¡a2(¡x) + 1; x > a2;
where 0 < b2 · a2 < 1.
The equation for the Pareto frontier is:
² If a1 ¸ 1 ¡ a2,
u2 =
8
> > <
> > :
¡
a1(1¡b2)
(1¡a2)b1u1 + 1; u1 2 [0;
(1¡a2)b1
a1 ]
¡a1b2
a2b1u1 + b2
a2; u1 2 [
(1¡a2)b1
a1 ;b1]
¡
(1¡a1)b2
a2(1¡b1)u1 +
(1¡a1)b2
a2(1¡b1); u1 2 [b1;1]:
² If a1 · 1 ¡ a2,
u2 =
8
> > > <
> > > :
¡
a1(1¡b2)
(1¡a2)b1u1 + 1; u1 2 [0;b1]
¡
(1¡a1)(1¡b2)
(1¡a2)(1¡b1)u1 +
(1¡a1)(1¡b2)
(1¡a2)(1¡b1) ¡ 1¡b2
1¡a2 + 1; u1 2 [b1;
¡a2(1¡b1)
1¡a1 + 1]
¡
(1¡a1)b2
a2(1¡b1)u1 +
(1¡a1)b2
a2(1¡b1); u1 2 [
¡a2(1¡b1)
1¡a1 + 1;1]:
6Let f denote the function in the right-hand side of the above, and g
denote the inverse function of f. The Pareto frontier is a piecewise linear
curve with two kinks if a1 6= 1 ¡ a2, and with one kink if a1 = 1 ¡ a2.
For simplicity, assume a1 = 1 ¡ a2 and a2 = b2. Then,
g(u2) =
(
¡ 1¡b1
1¡a1u2 + 1; u2 2 [0;1 ¡ a1]
¡ b1
a1u2 + b1
a1; u2 2 [1 ¡ a1;1]
and
f(u1) =
(
¡a1
b1u1 + 1; u1 2 [0;b1]
¡1¡a1
1¡b1u1 + 1¡a1
1¡b1; u1 2 [b1;1]:
The two curves u1 = ±g(u2) and u2 = ±f(u1) have three distinct intersections
if a1 > 1
2 and b1 < 1
2, and if ±
1+± > max(1 ¡ a1;b1). They are
( b1±
a1(1+±); ±
1+±)
(
b1(1¡b1)±¡(1¡a1)b1±2
a1(1¡b1)¡(1¡a1)b1±2 ;
a1(1¡a1)±¡(1¡a1)b1±2
a1(1¡b1)¡(1¡a1)b1±2 )
( ±
1+±;
(1¡a1)±
(1¡b1)(1+±)):
For problems involving more than one issue, we can get a similar result, if
we replace x with x1x2 ¢¢¢xl for player 1’s utility, and (1¡x) with (1¡x1)(1¡
x2)¢¢¢(1¡xl) for player 2’s utility, or if we replace x with x1 +x2 +¢¢¢+xl
for player 1’s utility, and (1 ¡x) with (1 ¡x1)+(1 ¡x2)+ ¢¢¢ +(1 ¡xl) for
player 2’s utility. These are simple examples where the equation h(u1) = 0
in our assumption A3 has three solutions when S = L. Given the multiple
zeros of the function h, the construction of stationary SPE strategies around
each of them is standard and left to the reader.
References
In, Y. and R. Serrano (2000), “Agenda Restrictions in Multi-Issue Bar-
gaining,” working paper 2000-08, Department of Economics, Brown
University, forthcoming in Journal of Economic Behavior and Organi-
zation.
Inderst, R. (2000), “Multi-issue Bargaining with Endogenous Agenda,”
Games and Economic Behavior 30, 64-82.
Osborne, M. J. and A. Rubinstein (1990), Bargaining and Markets,
Academic Press, San Diego, CA.
7Rubinstein, A. (1982), “Perfect Equilibrium in a Bargaining Model,”
Econometrica 50, 97-109.
Shaked, A. and J. Sutton (1984), “Involuntary Unemployment as a
Perfect Equilibrium in a Bargaining Model,” Econometrica 52, 1351-
1364.
8

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