We find that bargaining over multiple issues as a bundle allows for credible cheap talk on the relative importance of the issues even when interests are directly opposed on each issue. This communication increases the chance of agreement by allowing the offerer to concede more on the issue of greater importance to the offeree. In contrast, if the issues are bargained over separately we find that the offeree will lie about which issue is of greater importance. The communication gains from bundling are in addition to the gains from bundling previously identified in the monopoly pricing literature. Applied to negotiations between an executive and a legislature, our results imply that bundling multiple issues in a single bill with no line-item veto increases communication and can benefit both the legislature and, in many cases, the executive as well.bundling; bargaining; cheap talk; veto

Archishman Chakraborty

Rick Harbaugh

English

Research Papers in Economics

Cheap talk comparisons in
multi-issue bargaining
Archishman Chakraborty∗†and Rick Harbaugh‡
Abstract
Bargaining over two issues as a bundle permits credible cheap talk about their relative
importance even when interests are directly opposed on each issue. The resulting communi-
cation gains can exceed the gains from bundling previously identiﬁed in the monopoly pricing
literature.
Key words: bundling, bargaining, cheap talk
JEL codes: D82, C78
∗Department of Economics and Finance, Baruch College, City University of New York, New York, NY 10010
†Corresponding author. Tel.: +1-646-312-3705; fax: +1-646-312-3451. E-mail address:
Archishman Chakraborty@baruch.cuny.edu
‡Department of Ecoomics, Claremont McKenna College, Claremont, CA 917111I n t r o d u c t i o n
In multi-issue bargaining both sides can beneﬁt by compromising on the issues they care least
about in exchange for a better deal on the issues they care most about. But in an asymmetric
information environment there is no assurance that the parties have either the incentive or the
credibility to communicate which issues to compromise on.
We analyze this problem in a “take it or leave it” bargaining game where an oﬀerer proposes
concessions on two issues to an oﬀeree after listening to messages sent by the oﬀeree. When
the issues are bargained over separately we ﬁnd that the oﬀeree will lie about which issue is
of greater importance so communication is not credible. This communication problem can be
solved by bundling the two issues together in a single oﬀer that must be accepted or rejected in
its entirety.
The communication in our model is non-veriﬁable “cheap talk.” The cheap-talk literature
shows that some signals can be credible if sender and receiver interests are partly aligned (Craw-
ford and Sobel, 1982), but in our model the two sides’ interests are directly opposed on each
issue. Considered separately, the oﬀeree has an incentive to lie about the importance of each
issue. But if both issues must be accepted or rejected together, a comparative statement about
which issue is better can be credible because it simultaneously reveals favorable information
about one issue and unfavorable information about the other issue.
A standard result of the monopoly pricing literature is that bundling multiple goods together
can increase a monopolist’s proﬁts because buyer valuations of a bundle are more predictable
than buyer valuations of individual goods (Stigler, 1963; Adams and Yellen, 1976; McAfee,
McMillan, and Whinston, 1989). This same logic clearly applies to our model of multi-issue
bargaining. The communication gains we identify are in addition to the standard beneﬁts of
bundling previously identiﬁed in the monopoly pricing literature.
2 The Model
We consider a game between two players, A and B, bargaining over two issues, 1 and 2. Player
B has private information vk ∈ [0,1] relevant to his value for each issue k. Let v =( v1,v 2). We
suppose that vk has continuous density f, and distribution F, i.i.d. across k ∈ {1,2}.
We model the potential for communication under two diﬀerent bargaining protocols. Under
the ﬁrst protocol, each oﬀer can be accepted or rejected separately. Under the second protocol,
the players either reach agreement on both issues or on none. We use the term “no bundling”
1to refer to the ﬁrst protocol and “bundling” to refer to the second.
For each bargaining protocol, communication is modelled as a cheap talk game. Player B
ﬁrst chooses a message m from some set M, possibly as a function of his private information v.
Upon hearing the message m,p l a y e rA will make a “take it or leave it” oﬀer xk ∈ [0,1] on each
issue k. The game ends when B accepts (agreement) or rejects (disagreement). Let x =( x1,x 2).
Notice that, for a ﬁxed protocol, the timing structure states that the oﬀer x has to be
optimal given player A’s inference about v upon hearing m. This distinguishes our cheap talk
model from a screening problem where player A ﬁrst commits t oam e n uo fo ﬀers and player B
chooses among them, with or without sending messages.
Given a realization v of player B’s private information and concessions x by player A,t h e
payoﬀs for each player from reaching agreement on issue k are equal to
UA,k =1− xk
for player A and
UB,k = g(vk,x k)
for player B.1 We assume that g is twice diﬀerentiable and strictly incr e a s i n gi ne a c ha r g u m e n t .
Payoﬀs are additive across issues and we denote by Ui = Ui,1 + Ui,2 the total payoﬀsf o ri ∈
{A,B}.
For each player there is a common outside opportunity equal to 0 for each issue k. We assume
that g(1,1) > 0 so that an agreement can be reached with positive probability, and g(1,0) ≤ 0
so that an agreement with no concession is worse than B’s outside option with probability 1.
Under these assumptions, there exists x ∈ [0,1) such that g(1,x)=0 . The oﬀer x corresponds
to the highest oﬀer that will give B ap a y o ﬀ less than his outside option with probability 1.
Let x =m a x {x ∈ [0,1]|g(0,x) ≤ 0}. When x<1, it corresponds to the lowest oﬀer that gives
B ap a y o ﬀ (weakly) more than his outside option regardless of v. Note that x < x. Note also
that, from the assumed properties of g,f o re a c hx ∈ [x,x] there exists a cutoﬀ value v(x) ∈ [0,1],
strictly decreasing and continuously diﬀerentiable in x, such that g(v(x),x)=0 .
We provide three diﬀerent examples of the function g below. In example 1 there is no
interaction between the realized uncertainty and the oﬀeree’s marginal beneﬁt of a concession
on the issue. It corresponds to the standard linear and additively separable utility function that
is usually considered in the literature on bundling by a multi—product monopolist. In example
1Our results depend only UA,k being strictly decreasing in x, not on its linearity.
22, player B’s utility is supermodular in the unknown parameter v and the concession x.I n
example 3, player B’s utility is submodular in v and x.
1. g(v,x)=v + x − u,w i t hu ∈ (1,2). Then x = u − 1, x =1a n dv(x)=u − x.
2. g(v,x)=vx− u,w i t hu ∈ (0,1). Then x = u, x =1a n dv(x)=u
x.
3. g(v,x)=l n ( v + x)+u with u ≥ 0. Then x =0 , x = e−u, and v(x)=e−u − x.
2.1 Babbling
Since we model communication as cheap talk, there is always a babbling equilibrium under either
bargaining protocol where A refuses to ascribe any meaning to B’s message m and makes her
oﬀers accordingly.
In a babbling equilibrium under no bundling, for each issue k ∈ {1,2}, player A chooses the
concession xk such that
xk ∈ arg max
x∈[0,1]
(1 − x)Pr[g(v,x) ≥ 0]. (1)
In contrast, in a babbling equilibrium under bundling, player A chooses the concessions x to
solve
x ∈ arg max
x∈[0,1]2(2 − x1 − x2)Pr[g(v1,x 1)+g(v2,x 2) ≥ 0]. (2)
We do not provide a general characterization of babbling equilibria in this note but turn now to
the existence of an informative equilibrium.
2.2 Rank-Revealing Equilibrium
We consider the possibility that B might credibly disclose his ordinal ranking of the diﬀerent
issues, i.e., whether v1 >v 2 or vice versa, without disclosing anything about the magnitude of
either v1 or v2. We call such an informative equilibrium a rank—revealing equilibrium (RRE).
We show below that, under a fairly general set of conditions, there does not exist a RRE unless
bundling is allowed. We assume without loss of generality that v1 ≥ v2 so that issue 1 is more
valuable to B than issue 2.
32.2.1 No Bundling
In a RRE (given a message from player B that v1 ≥ v2), for each k ∈ {1,2} player A chooses
xk to solve
xk ∈ argmax
x∈[0,1]
(1 − x)(1 − Fk(v(x))) (3)
where Fk(·) is the distribution of vk given v1 ≥ v2. Let fk(·) be the density associated with Fk(·).
Note that F1(v)={F(v)}2 and F2(v)=1− {1 − F(v)}2.
Since the objective function is continuous, problem (3) must have a solution xk. Moreover,
the solution must be interior, i.e., xk ∈ (x,x), k ∈ {1,2}. To see this note that, if xk ≤ x then
the oﬀer is rejected with probability 1 and A can do strictly better by making an oﬀer in (x,x).
Regarding the upper bound, when x = 1 then an oﬀer of xk = x earns A an expected payoﬀ of
0 so it is strictly dominated by an oﬀer in (x,x). And when x<1, an oﬀer of xk > x is strictly
dominated by the oﬀer x. Moreover, since v(x)=0w h e nx<1, the derivative of the objective
function at xk = x is −(1 − x)v0(x) < 0s ot h a ta no ﬀer slightly less than x dominates an oﬀer
of x.
At an interior solution we must have v(xk) ∈ (0,1) for all k ∈ {1,2}. Moreover, xk must
satisfy the ﬁrst—order necessary condition for an interior maximum:
−(1 − xk)v0(xk) −
1 − Fk(v(xk))
fk(v(xk))
=0 , (4)
and, furthermore, the left—hand side of (4) must be non—increasing in x at xk. Assume that (4)
has a unique solution.2 Then for all x ∈ [xk,x)w em u s th a v e
−(1 − x)v0(x) −
1 − Fk(v(x))
fk(v(x))
≤ 0.
This implies that x1 <x 2. For if not,
1 − F1(v(x1))
f1(v(x1))
= −(1 − x1)v0(x1) ≤
1 − F2(v(x1))
f2(v(x1))
.
Since v(x1) ∈ (0,1) and
1 − F1(v)
f1(v)
>
1 − F2(v)
f2(v)
for all v ∈ (0,1), we have a contradiction.
Since x1 <x 2 clearly B will lie for any v1 and v2 if the marginal value of a concession is
higher for higher v (supermodular g). The following shows that the problem is more general in
that for any g there are always some realizations of v1 and v2 such that B will lie.
2Suﬃcient conditions are that g is quasi—concave and
1−F(v)
f(v) is monotonically decreasing in v.
4Claim 1 If there is a unique solution to (4) then there is no rank—revealing equilibrium when
B can reject each oﬀer separately.
Proof. Suppose there is a rank-revealing equilibrium. Then, it follows from the arguments
above that A will choose concessions x1 and x2 such that x>x 2 >x 1 >xso that 1 >v (x1) >
v(x2) > 0. But if v1 and v2 are such that v(x1) >v 1 >v (x2) >v 2, then B has a strict incentive
to lie and claim issue 2 is more important. B will then accept the larger concession on issue
1 and reject the smaller concession on issue 2. This will give B ap a y o ﬀ of g(v1,x 2)w h i c hi s
strictly greater than 0, the payoﬀ from revealing the rank truthfully.
2.2.2 Bundling
In this case, given a message m from player B that v1 ≥ v2,p l a y e rA chooses x1 and x2 to solve
x ∈ arg max
x∈[0,1]2 (2 − x1 − x2)Pr[g(v1,x 1)+g(v2,x 2) ≥ 0|v1 ≥ v2]. (5)
For B to reveal the ranking truthfully it is suﬃcient that whenever v1 ≥ v2,
g(v1,x 1)+g(v2,x 2) ≥ g(v1,x 2)+g(v2,x 1). (6)
We consider two cases:
1. g is supermodular:
∂2g
∂v∂x > 0 for all v,x.
2. g is submodular:
∂2g
∂v∂x < 0 for all v,x.
Claim 2 If g is supermodular (respectively, submodular), then there exists a RRE with x1 >x 2
(resp., x1 <x 2), when B can only accept or reject the bundle.
Proof. Suppose g is supermodular. Then, for (6) to hold for all v1 >v 2 it is suﬃcient that
x1 >x 2.
In problem (5) A must choose x1 >x 2. For if x1 <x 2 then A can simply switch the
concessions, increasing the probability of agreement without increasing the total concessions in
the event of agreement. And if x1 = x2 = x then we must have x>x(otherwise agreement will
be reached with zero probability) and x<1 (otherwise agreement leaves A with no surplus).
But then there exists ε>0 such that an oﬀer x1 = x + ε and x2 = x − ε will increase the
probability of agreement without increasing the total concessions in the event of agreement.
5The symmetric argument applies to the case where g is submodular.
With bundling A wants to increase the probability of agreement by raising the total value
of the oﬀer to B for any given amount of concessions x1 + x2. B has an incentive to help A do
this by revealing the ranking. When g is supermodular, the value v and the concession x are
complements from B’s perspective and A concedes more on the more valuable issue to increase
the probability of acceptance. On the other hand, when g is submodular, the value v and the
concession x are substitutes from B’s perspective and A concedes more on the less valuable issue
to increase the probability of acceptance.
A is better oﬀ in an informative RRE with bundling compared to the babbling equilibrium
with bundling because she can enforce the same outcome in both cases and has more information
in a RRE. However, A’s expected payoﬀ in a RRE with bundling is not always higher than
her expected payoﬀ in the babbling equilibrium under no bundling. That is, the gain from
communication need not be greater than the ﬂexibility allowed by separate bargaining. Similar
remarks apply to B’s ex—ante expected payoﬀsf r o md i ﬀerent protocols.
2.3 Example
Consider the supermodular example where g(v,x)=vx−u. Assume u =1 /4a n dv is uniformly
distributed in [0,1]. We can think of the two issues as land in regions 1 and 2 along a common
border of two countries. Country A prefers to concede less land in each region, and prefers no
agreement to conceding too much. Country B prefers to receive a larger concession in each
region, and prefers no agreement to insuﬃcient concessions. Country B has private information
v1 and v2 about the marginal value of each unit of land conceded in each region.
Figure 1 shows the zones of acceptance under the three cases of no bundling, bundling
without communication, and bundling with communication. All three ﬁgures are drawn for the
case where v1 ≥ v2 so all of the probability mass is below the diagonal.
Without bundling Country B will not reveal which region is preferred, so each region is
treated identically by Country A.C o u n t r yA trades oﬀ the probability of acceptance, which is
increasing in the concession oﬀe r ,w i t ht h eg a i ni ft h eo ﬀer is accepted, which is decreasing in
the oﬀer. Solving problem (1), x = .5 for each region so the oﬀer on region k is accepted when
vk ≥ .5a ss e e ni nﬁgure 1(a).
With bundling but without communication, from (2) the optimal oﬀer is x = .564 on each
region and the bundle is accepted if v1 +v2 ≥ .887 as seen in ﬁgure 1(b). This increases payoﬀs
6Figure 1: Acceptance regions for bargaining protocols, v1 ≥ v2
for the same reason as in multi-product bundling models.3
Communication improves on the standard bundling outcome because the largest concession
is made for the preferred region. In the RRE, Country B makes the credible statement that each
unit of area in region 1 is more valued than in region 2. Solving problem (5), x1 = .917, x2 =0
and agreement is reached as long as v1 ≥ .525 regardless of v2. While the average concession
(x1+x2)/2=.459 is the lowest of the three cases, the payoﬀs for both countries are the highest.
As seen from the zones of acceptance in ﬁgure 1, the probability of agreement on both regions
rises from only 25% in (a) to about 60.7% in (b) and to about 72.4% in (c).
3C o n c l u s i o n
Two areas for further research are the existence of equilibria more informative than the rank
revealing equilibrium and the existence of a rank revealing equilibrium for larger numbers of
issues. The latter question is of particular interest since the monopoly pricing literature has
shown that the gains from bundling large numbers of products can be substantial (Armstrong,
1999; Bakos and Brynjolfsson, 1999). The rank order of issues becomes very informative as
the number of issues increases so the communication gains from bundling are also likely to be
signiﬁcant.
3Bundling does not always increase payoﬀs. For large u it is rare that both issues are desired enough to make
a bundle worthwhile so no bundling is preferred.
74 References
1. Adams, William J. and Janet L. Yellen (1976), “Commodity Bundling and the Burden of
Monopoly,” Quarterly Journal of Economics, 91, 475—498.
2. Armstrong, Mark (1999), “Price Discrimination by a Many-Product Firm,” Review of
Economic Studies, 66, 151—168.
3. Bakos, Yannis and Erik Brynjolfsson (1999), “Bundling Information Goods: Pricing, Prof-
its, and Eﬃciency,” Management Science, 46, 1613—1630.
4. Crawford, Vincent P. and Joel Sobel (1982), “Strategic Information Transmission,” Econo-
metrica, 6, 1431—1450.
5. McAfee, R. Preston, John McMillan, and Michael D. Whinston (1989), “Multiproduct
Monopoly, Commodity Bundling, and Correlation of Values,” Quarterly Journal of Eco-
nomics, 104, 371—383.
6. Stigler, George J. (1963), “United States v. Loew’s, Inc.: A Note on Block Booking,”
Supreme Court Review, 1963, 152—157.
8

Price Discrimination by a Many-Product Firm,”

United States v. Loew’s, Inc.: A Note on Block Booking,” Supreme Court Review,

Yannis and Erik Brynjolfsson

Credible Comparisons in Multi-Issue Bargaining

http://www.claremontmckenna.edu/rdschool/papers/2002-04.pdf

Credible Comparisons in Multi-Issue Bargaining

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