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Rhetoric in Legislative Bargaining with Asymmetric

Information1

Ying Chen

Arizona State University

yingchen@asu.edu

H¨ ulya Eraslan

Johns Hopkins University

eraslan@jhu.edu

June 22, 2010

1We thank Ming Li and Jack Stecher for helpful comments and stimulating conversa-

tions, along with seminar participants at the Carnegie Mellon University, University of British

Columbia and Quebec Workshop on Political Economy. Any errors are our own.

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Abstract

In this paper we analyze a legislative bargaining game in which parties privately in-

formed about their preferences bargain over an ideological and a distributive decision.

Communication takes place before a proposal is offered and majority rule voting deter-

mines the outcome. When the private information pertains to the ideological intensities

but the ideological positions are publicly known, it may not be possible to have informa-

tive communication from the legislator who is ideologically distant from the proposer,

but the more moderate legislator can communicate whether he would “compromise” or

“fight” on ideology. If instead the private information pertains to the ideological po-

sitions, then all parties may convey whether they will “cooperate,” “compromise,” or

“fight” on ideology. When the uncertainty is about ideological intensity, the proposer

is always better off making proposals for the two dimensions together despite separable

preferences, but when the uncertainty is about ideological positions, bundling can result

in informational loss which hurts the proposer.

JEL classification: C78, D72, D82, D83

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1 Introduction

Legislative policy-making typically involves speeches and demands by the legislators

that shape the proposals made by the leadership. For example, in the recent health care

overhaul, one version of the Senate bill included $100 million in Medicaid funding for

Nebraska as well as restrictions on abortion coverage in exchange for the vote of Nebraska

Senator Ben Nelson. As another example, consider the threat by seven members of the

Senate Budget Committee to withhold their support for critical legislation to raise the

debt ceiling unless a commission to recommend cuts to Medicare and Social Security is

approved.1Would these senators indeed let the United States default on its debt, or

was their demand just a bluff? More generally, what are the patterns of demands in

legislative policy-making? How much information do they convey? Do they influence

the nature of the proposed bills? Which coalitions form and what kind of policies are

chosen under the ultimately accepted bills?

To answer these questions, it is necessary to have a legislative bargaining model in

which legislators make demands before the proposal of bills. One approach is to assume

that the role of demands is to serve as a commitment device, that is, the legislators refuse

any offer that does not meet their demands.2While this approach offers interesting

insights into some of the questions raised above, it relies on the strong assumption

that legislators commit to their demands.3In this paper, we offer a different approach

that allows legislators to make speeches but do not necessarily commit to them when

casting their votes. The premise of our approach is that only individual legislators know

which bills they prefer to the status quo. So even if the legislators do not necessarily

undertake what they say, their demands can be meaningful rhetoric in conveying private

information and dispelling some uncertainty in the bargaining process.

We model rhetoric as cheap-talk messages as in Matthews (1989). In our framework

1http://thehill.com/homenews/senate/67293-sens-squeeze-speaker-over-commission

2This is the approach taken by Morelli (1999) in a complete information framework. He does not

explicitly model proposal making and voting stage. As such, the commitment assumption is implicit.

3Politicians often carry out empty threats, for example, http://thehill.com/homenews/news/14312-

gopsays-it-can-call-reids-bluffs.

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(1) three legislators bargain over an ideological and a distributive decision; (2) bargainers

other than the proposer are privately informed about their preferences; (3) communica-

tion takes place before a proposal is offered; (4) majority rule voting determines whether

the proposal is implemented. By introducing communication into legislative bargaining,

our goal is take a step towards answering fundamental questions of political economy,

“who gets what, when and how” (Lasswell, 1958), together with fundamental questions

of communication theory, “who says what to whom in what channel with what effect”

(Lasswell, 1948).

We begin by analyzing the case in which the legislators’ positions on a unidimensional

ideological spectrum are publicly known, but their trade-offs between the ideological

dimension and the distributive dimension are private information. So the proposer of

a bill (also referred to as the chair) is unsure how much private benefit he has to offer

to a legislator to gain support on a policy decision. When no equilibrium coalition is

a surplus coalition (i.e., at most one legislator other than the proposer gets positive

private benefit), we obtain two main findings: (1) the rhetoric of the legislator who is

ideologically more distant from the proposer is not informative in equilibrium; (2) it is

possible for the more moderate legislator to have meaningful rhetoric.

To establish these results, we first explore the legislators’ expected payoffs in different

coalitions. Suppose one legislator is offered positive private benefit while the other is

offered none (call this a minimum winning coalition). Then the legislator who is excluded

strictly prefers the status quo and will vote against the proposal whereas the legislator

who is included becomes pivotal and hence can guarantee a payoff at least as high

as the status quo. Alternatively, suppose no legislator is offered private benefit (call

this a minority coalition). Then the chair’s optimal proposal is the one that makes

the moderate legislator just willing to accept. Hence in a minority coalition the more

moderate legislator gets a payoff equal to the status quo but the more distant legislator

is made worse off than the status quo. It follows that the more distant legislator would

like to maximize his chance to be included in a coalition, thereby undermining the

credibility of his rhetoric. As to the more moderate legislator, it is possible for him to

have (at most) two equilibrium messages signaling his ideological intensity. When he

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puts a relatively high weight on the ideological dimension, he sends the “fight” message,

and the chair responds with a minority coalition that excludes both legislators as their

demands indicate that there is no room for making a deal. When he puts a relatively

low weight on the ideological dimension, he sends the “compromise” message and the

chair responds by offering some private benefits in exchange for moving the policy closer

to his own ideal. The threshold type is indifferent between sending the “fight” and the

“compromise” messages because either way he gets a payoff equal to the status quo, and

a single-crossing property of the utility function guarantees that other types’ incentive

constraints are satisfied as well. It is impossible for even the moderate legislator to

convey more precise information about his ideological intensity. In particular, once the

chair believes that the moderate legislator puts a relatively low weight on ideology and

hence includes him in a minimum winning coalition, the legislator now has the incentive

to exaggerate his ideological intensity and demand a better deal from the chair, but

this undermines the credibility of his demands. Somewhat ironically, the proposal of a

minority coalition induced by the “fight” message always passes in equilibrium, but the

minimum winning coalition induced by the “compromise” message may fail to pass.

Next, we consider the case in which the legislators’ ideological intensities are known,

but their ideological positions are uncertain. The setup is related to Matthews (1989)

which models presidential veto threats as cheap talk in a bilateral bargaining game over

a unidimensional policy and assumes that the president’s position is his private infor-

mation. Our model differs from Matthews (1989) by having multiple senders and a

distributive dimension in addition to an ideological dimension. In this case, we find that

equilibrium demands from either legislator may convey limited information about their

preferences. In particular, legislators can signal whether they will “cooperate,” “com-

promise” or “fight.” If either legislator makes a cooperative speech, the chair responds

by proposing his ideal policy and a minority coalition in which he extract all the surplus

. If both legislators make tough demands by sending the “fight” message, the chair gives

up on the ideological issue and again does not give out any private benefits. Otherwise,

he proposes a compromise policy, which depends on whether one or both legislators sig-

nal willingness to compromise. Again, only the minimum winning coalition induced by

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a “compromise” message may fail to pass in equilibrium whereas the minority coalitions

induced by the “cooperate” or “fight” message always get passed in equilibrium.

Since the legislators in our model bargain over both an ideological dimension and a

distributive dimension, a natural question arises as to whether it is better to bundle the

two issues together in one bill or negotiate over them separately. An obvious advantage

of bundling the two issues together is that the chair can exploit difference in the other

legislators’ trade offs between the two dimensions and use private benefits as an instru-

ment to make deals with them on policy changes that he wants to implement. Indeed,

when the uncertainty is about the ideological intensities of the legislators, we find that it

always benefits the chair to bundle the two dimensions together. But bundling may also

result in informational loss when the uncertainty is about ideological positions: once side

payments become a possibility, it might be too tempting for a legislator to declare that

his position is not especially close to the chair’s in the hope that the chair will respond

with a more attractive deal. This incentive to distort one’s demand may result in less

information transmitted in equilibrium, hurting the legislators in the end. (By contrast,

if the uncertainty is about ideological intensity, then rhetoric does not matter if the two

dimensions are separated and hence bundling never results in information loss.) If we

interpret bundling as the possibility of using pork barrel spending to gain support on

policy reform, our finding points out a potential harm of pork barrel spending that, to

our knowledge, was not pointed out before.

Before turning to the description of our model, we briefly discuss the related litera-

ture. Starting with the seminal work of Baron and Ferejohn (1989), legislative bargaining

models have become a staple of political economy and have been used in numerous ap-

plications. The literature is too large to list comprehensively here. The papers most

closely related to ours are Austen-Smith and Banks (1988), Banks and Duggan (2000),

Jackson and Moselle (2002), and Diermeier and Merlo (2004), which include an ideolog-

ical dimension and a distributive dimension. All these papers (and others that build on

Baron and Ferejohn) take the form of sequential offers but do not incorporate demands.

A smaller strand of literature, notably Morelli (1999), instead model legislative process

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as a sequential demand game where the legislators commit to their demands.4With the

exceptions of Tsai (2009), Tsai and Yang (2009a, b), who do not model demands, all of

these papers assume complete information.

Existing cheap talk literature has largely progressed in parallel to the bargaining

literature. Exceptions are Farrell and Gibbons (1989), Matthews (1989), and Matthews

and Postlewaite (1989). Of these Matthews (1989) is the most closely related, but

as discussed earlier, there are a number of important differences between his model

and ours. Our paper is also related to cheap talk games with multiple senders (see,

for example, Gilligan and Krehbiel (1989), Austen-Smith (1993), Krishna and Morgan

(2001), Battaglini (2002) and Ambrus and Takahashi (2008)). Our framework differs

from these papers because it has voting over the proposal made by the receiver and also

incorporates a distributive dimension.

In the next section we describe our model. In Section 3, we analyze the bargaining

game when the legislators’ ideological intensities are uncertain. In Section 4, we analyze

the bargaining game when the legislators’ ideological positions are uncertain. We discuss

extensions and generalizations in Section 5.

2 Model

Three legislators play a three-stage game to collectively decide on an outcome that con-

sists of an ideological component and a distributive component, for example, setting the

level of environmental regulation and dividing government spending across states. We

assume that legislator 0 is the chair (proposer) of the legislature in charge of formulating

a proposal. Denote an outcome by a vector z = (y;x) where y is an ideological decision

and x = (x0,x1,x2) is a distributive decision. The set of feasible ideological decisions is

Y ⊂ R, and the set of distributions is denoted by X. For x ∈ X, xidenotes the cake

share of legislator i. Suppose c(≥ 0) is the size of the cake for division and a proposal

(y;x) satisfies?2

to the other legislators, so xi≥ 0 for i = 1,2. The status quo allocation is denoted by

4See also Vidal-Puga (2004), Montero and Vidal-Puga (2007), Breitmoser (2009).

i=0xi= c. We also assume that the chair cannot give negative share

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s = (˜ y; ˜ x). We assume that ˜ y ∈ Y and normalize ˜ x to be (0,0,0). The set of possible

outcomes is thus Y × X where X = {(x0,x1,x2) :?2

The payoff of each legislator i depends on the ideological decision and his own cake

share. We assume that the legislators’ preferences are separable over the two dimensions.

i=0xi= c,x1≥ 0,x2≥ 0} ∪ ˜ x.

Specifically, legislator i has a quasi-linear von Neumann-Morgenstern utility function

ui(z,θi, ˆ yi) = xi+ θiv (y, ˆ yi),

where z = (y;x) specifies the outcome, ˆ yidenotes the ideal policy of legislator i and

θi∈ [0,∞) is a parameter that measures the intensity of legislator i’s preference over the

ideological dimension relative to the distributive dimension.5When θi= 0, legislator

i cares about only the distributive dimension. In the other extreme, when θi → ∞,

legislator i cares about only the ideological dimension.

We make the following assumptions on the function v (·,·): (1) v (·,·) is continuous;

(2) v (·, ˆ yi) is single-peaked at ˆ yi; (3) v (·,·) satisfies the single-crossing property, i.e., if

type ˆ yiis indifferent between two policies y?and y and y?> y, then the higher types

prefer y?and the lower types prefer y. Formally, if v (y?, ˆ yi) = v (y, ˆ yi) and y?> y, then

(ˆ y?

i− ˆ yi)(v (y?, ˆ y?

We assume that the ideological intensity θiand the ideal policy ˆ yiof legislator i ?= 0

are observed only by legislator i and let ti= (θi, ˆ yi) denote the type of legislator i. All

i)) − v (y, ˆ y?

i)) > 0.

other legislators believe that tiis a random variable, independent of all other tjfor j ?= i,

with a distribution function Fiwhose support is Ti= [θi,θi]×[yi,yi] ⊂ R+×R, possibly

with yi= yior θi= θi. For simplicity, we assume that the preference of the chair is

commonly known, that is, the chair’s ideal policy ˆ y0and ideological intensity θ0∈ (0,∞)

are observed by all legislators. Without loss of generality, we assume ˆ y0< ˜ y so that the

chair would like to lower the status quo policy.

The bargaining game consists of three stages. In the first stage, each legislator i ?= 0

observes his type ti and simultaneously sends a message mi ∈ Mi to the chair. We

5The model can be easily extended to allow for a more general vi(y) in the place of v (y, ˆ yi). For

expositional convenience, we use the simpler form and let ˆ yi parameterizes legislator i’s ideological

preference.

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assume, without loss of generality, that Mi= Tifor i = 1,2. In the second stage, the

chair makes a proposal in Y ×X. In the last stage, the legislators vote on the proposal.

Without loss of generality we assume that the chair always votes for the proposal. The

voting rule is majority rule, so if at least one of legislators 1 and 2 votes for the proposal,

then it is accepted. Otherwise, status quo s = (˜ y; ˜ x) prevails.

A strategy for legislator i ?= 0 consists of a message rule in the first stage and

an acceptance rule in the third stage. A message rule mi : Ti → Mi for legislator

i specifies what message he sends as a function of his type. An acceptance rule for

legislator i is a function γi(.,ti) : Y × X → {0,1} that specifies how he votes when his

type is ti: he votes for a proposal z = (y;x) if γi(z,ti) = 1, and he votes against it if

γi(z,ti) = 0. A strategy set for legislator i consists of the measurable pairs of functions

(mi,γi) satisfying these properties. The chair’s strategy set consists of all proposal rules

π : Y × X × M1× M2→ [0,1] where π(z,m) is the probability that he offers z = (y;x)

when the message profile is m = (m1,m2).

Following Matthews (1989), we define the equilibria directly in terms of the derived

properties of perfect Bayesian equilibria.6An equilibrium is a strategy profile (m,γ,π)

such that the following conditions hold for all i ?= 0, ti ∈ Ti, y ∈ Y,x ∈ X and

m ∈ M1× M2:

(E2) if π(z,m) > 0 then u0(z,t0) ≥ u0(s,t0).

If in addition?

z ∈ arg max

(E1) γi(z,ti) =

1 if ui(z,ti) ≥ u(s,ti),

if ui(z,ti) < u(s,ti);0

{mi(ti)=mi}dFi(ti) > 0 for all i ?= 0, then

z?∈Y ×Xu0(z?,t0)β(z?|m) + u0(s,t0)(1 − β(z?|m)),

where

β(z|m) = 1 − (1 − β1(z|m1))(1 − β2(z|m2))

6The formal definition of perfect Bayesain equilibrium requires only that the optimality conditions

hold for almost all types and pairs of messages. This would not change any of our results.

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is the conditional probability that z is accepted and βi(z|mi) is the conditional proba-

bility that legislator i ?= 0 votes to accept the proposal:

?

βi(z|mi) =

{mi(ti)=mi}γi(z,ti)dFi(ti)

?

{mi(ti)=mi}dFi(ti)

.

Otherwise, π(z,m) = π(z,m?) for all z where m?is any arbitrary message profile such

that such that?

(E3) if mi(ti) = mi, then mi∈ argmaxm?

{mi(ti)=m?

i}dFi(ti) > 0 for all i ?= 0.

iV (m?

i) where

V (m?

i) = E [βj(z|mj)ui(z,ti) + (1 − βj(z|mj))max{ui(z,ti),ui(s,ti)}]

where j ?= i and the expectation is taken over z using π(.,m?

mj(tj), and over tjusing Fj.

i,mj), over mjusing

Condition (E1) is subgame perfection: it requires the legislators to accept proposals

that they prefer to the status quo.7Condition (E2) requires that the equilibrium pro-

posals maximize the payoff of the chair and the belief be consistent with Bayes’ rule.

Condition (E3) requires that the legislators demand only their most preferred proposals

among the ones that are possible in equilibrium (in the sense that there is some demand

that generates it), taking into account the acceptance rule of the other legislator.

Say that a proposal z is induced by a message profile m if π (z,m) > 0. A proposal z

is an equilibrium proposal if it is induced in equilibrium with positive probability. Given

an equilibrium strategy profile (m,γ,π), a proposal z is induced by message miif there

?

proposal a minimum winning coalition if either x1> 0 or x2> 0 but not both, i.e., only

one legislator (other than the chair) is given positive private benefit, a minority coalition

exists a message mj with

{mj(tj)=mj}dFj(tj) > 0 such that π (z,mi,mj) > 0. Call a

if x1= 0 and x2= 0, and a surplus coalition if both x1> 0 and x2> 0. Also, say that

a proposal (y;x) includes legislator i if xi> 0 and excludes legislator i if xi= 0.

7We assume that a legislator accepts a proposal whenever he is indifferent. This assumption simplifies

the exposition but is not necessary. It is easy to show that this assumption must be satisfied in any

equilibrium.

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3 Uncertain ideological intensity

We start by analyzing the case in which legislator i’s (i = 1,2) ideological position, ˆ yi,

is commonly known,8but his ideological intensity, θi, is his private information. Given

this restriction, we redefine legislator i’s type to be θiand assume that the distribution

of θihas full support on [θi,¯θi] for i = 1,2.

If v (ˆ y0, ˆ yi) ≥ v (˜ y, ˆ yi) for either i = 1 or i = 2, i.e., legislator i prefers the chair’s ideal

to the status quo policy, then the chair’s problem is trivial: he proposes his ideal and

leaves all the private benefit to himself, forming a minority coalition. To avoid triviality,

assume v (ˆ y0, ˆ yi) < v (˜ y, ˆ yi) for i = 1,2 for the remainder of the section. Also, without

loss of generality, assume that ˆ y0≤ ˆ y1≤ ˆ y2, so legislator 1 has an ideal closer to the

chair’s than legislator 2 has.

Say legislator i’s message rule is informative if and only if there exist two messages

miand m?

ithat are sent by legislator i with positive probability and induce different

distributions of proposals. To see whether an informative equilibrium exists, we first

establish a few lemmas on properties of equilibrium proposals.

Lemma 1. Suppose an equilibrium proposal (y;x) includes i but excludes j, i.e., xi> 0

and xj = 0, then legislator j strictly prefers the status quo to (y;x) and rejects the

proposal, and legislator i is pivotal.

Proof. We first show that legislator j prefers the status quo to (y;x). Suppose to the

contrary that some type of legislator j prefers (y;x) to the status quo.Note that

legislator j’s preference over (y;x) and (˜ y; ˜ x) is independent of θjas both give him zero

private benefit. So if any type of legislator j prefers (y;x) to (˜ y; ˜ x), he will accept (y;x)

with probability 1. But then the chair can make an alternative proposal with xi= 0

(not giving i any private benefit either) and still have it accepted by legislator j. This

alternative proposal gives the chair a strictly higher payoff, a contradiction. So legislator

j rejects the equilibrium proposal (y;x) if j is not included and it follows that legislator

i is pivotal.

8So the support of ˆ yiis degenerate: yi= ¯ yi.

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The next lemma shows that if a minimum winning coalition is formed with legislator

i when he sends message m, then for the highest type who sends m, the proposal cannot

be strictly better than the status quo,9but since legislator i is pivotal, he can guarantee

the status quo payoff.

Lemma 2. Suppose legislator i’s message m induces a proposal (y;x) with xi> 0,xj= 0.

Let θ∗

i= sup{θi: mi(θi) = m}. Then type θ∗

proposal (y;x) and type θ∗

imust weakly prefer the status quo to the

i’s payoff when inducing (y;x) is equal to his status quo payoff.

Proof. For xi> 0, if xi+ θiv (y, ˆ yi) ≥ θiv (˜ y, ˆ yi), then either (i) v (˜ y, ˆ yi) − v (y, ˆ yi) > 0

and therefore xi+ θ?

and xi+ θ?

proposal (y;x) to the status quo, then any lower type θ?

iv (y, ˆ yi) > θ?

iv (˜ y, ˆ yi) for 0 ≤ θ?

i< θi. So if type θiof legislator i prefers a

i< θior (ii) v (˜ y, ˆ yi) − v (y, ˆ yi) ≤ 0

iv (y, ˆ yi) > θ?

iv (˜ y, ˆ yi) for 0 ≤ θ?

imust prefer (y;x) to the status

quo as well. Suppose the chair proposes (y;x) in response to m that has xi> 0, xj= 0

and makes type θ∗

istrictly better off than the status quo. Then there exists a proposal

(y;x?) with x?

i< xi, x?

j= 0 that still makes type θ∗

istrictly better off than the status quo

and hence is accepted by legislator i with probability 1. But the chair strictly prefers

(y;x?) to (y,x), a contradiction. So type θ∗

imust weakly prefer the status quo (˜ y; ˜ x) to

the proposal (y;x). Since legislator i is pivotal when he is included and legislator j is

excluded and he can always reject the proposal, type θ∗

igets a payoff equal to the status

quo payoff.

The next lemma establishes some properties of minority coalitions.

Lemma 3. Suppose the chair proposes a minority coalition in equilibrium. If ˆ y1≥ ˜ y,

then the ideological outcome is ˜ y. If ˆ y1< ˜ y, the proposed y satisfies y < ˜ y, v (y, ˆ y1) =

v (˜ y, ˆ y1) and is accepted by legislator 1, but legislator 2 prefers the status quo to the

proposal (strictly if ˆ y1< ˆ y2).

Proof. In a minority coalition, x1= 0,x2= 0. Since ˆ y1≤ ˆ y2, it follows that if ˆ y1≥ ˜ y,

then both legislators will vote against any y < ˜ y. Hence the resulting ideological outcome

9More precisely, the highest type who sends m may not exist, so we define θ∗

bound in the lemma. Although the message sent by type θ∗

ias the lowest upper

imay not be m, by continuity his equilibrium

payoff must be the same as what he gets if he sends m.

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is ˜ y. If y1< ˜ y, then legislator 1 accepts any y that satisfies v (y, ˆ y1) ≥ v (˜ y, ˆ y1). Since

v (ˆ y0, ˆ y1) ≤ v (˜ y, ˆ y1) and v (·,·) is single-peaked, it is optimal for the chair to propose y

that satisfies v (y, ˆ y1) = v (˜ y, ˆ y1). Since v (·,·) satisfies the single-crossing property and

ˆ y1≤ ˆ y2, legislator 2 prefers the status quo to the proposal and strictly so if ˆ y1< ˆ y2.

In what follows, we derive results regarding the legislators’ equilibrium message rules

under the assumption that no equilibrium coalition is a surplus coalition. This is satisfied

under reasonable conditions on the type distributions. For example, it is satisfied if the

density of θiis weakly increasing (e.g., uniform distribution), as is typically assumed in

related cheap-talk models.10Moreover, if only one legislator has an unknown type, then

no equilibrium coalition is a surplus coalition (see discussion on page 18).

The first proposition shows that legislator 2 cannot convey meaningful information

in equilibrium if his ideal point is not the same as legislator 1’s and legislator 1 wants

to lower the status quo as the chair does.11

Proposition 1. If ˆ y1< ˜ y and ˆ y1< ˆ y2, then legislator 2’s message rule is not informative

in equilibrium.

Proof. Suppose to the contrary that there exist two messages m?and m??that are sent

with positive probability in equilibrium by legislator 2 and induce different distributions

of proposals. As Lemma 3 shows, if a proposal excludes both legislators, then legislator

2’s payoff is strictly lower than the status quo. Let θ?

2= sup{θ2: m2(θ2) = m?} and

θ??

2= sup{θ2: m2(θ2) = m??}. As Lemma 1 shows, if a coalition includes legislator 1

but excludes 2, then legislator 2’s payoff is strictly lower than the status quo. Moreover,

since legislator 1 is pivotal in this case, the proposal does not depend on legislator 2’s

message. If a coalition includes legislator 2 but excludes 1, then legislator 2 is pivotal

and as Lemma 2 shows, the payoff of type θ?

2(θ??

2) is the same as his status quo payoff.

10In a supplementary appendix not intended for publication (also available on our web pages), we

provide sufficient conditions for no equilibrium coalition to be a surplus coalition.

11If ˆ y1≥ ˜ y, i.e., both legislators’ ideals are higher than the status quo, then in a minority coalition

y = ˜ y and both legislators 1 and 2’s payoffs are equal to the status quo. In this case, for certain

parameters we can construct an equilibrium in which legislator 2’s messages induce different proposals.

It has the same properties as the informative message rule described in Proposition 3.

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So, the probability of legislator 2 being included in a coalition must be the same for

messages m?and m??because otherwise either type θ?

2or type θ??

2would have an incentive

to deviate to the message that induces a higher probability of inclusion. Note also that

conditional on legislator 2 being included in a coalition, the proposal does not depend on

legislator 1’s message. Since the inclusion probabilities are the same for m?and m??, to

prevent deviation messages m?and m??must induce the same proposals when legislator

2 is included. But this implies that m?and m??induce the same distribution of proposals

for legislator 2, a contradiction.

Now that we have established that legislator 2, whose ideology is furthest from the

chair’s, cannot convey useful information in equilibrium, we would like to see whether

the message of legislator 1, whose position is closer to the chair’s, can be informative.

We will call legislator 1 the moderate legislator.

Suppose legislator 1’s message rule takes the following form:12send message mkif

θ1∈ (θk

“size” of legislator 1’s message rule. If K = 1, then legislator 1’s speech is uninformative,

1,θk+1

1

] (k = 1,...,K) where θi= θ1

i< θ2

i< ... < θK+1

i

=¯θi. We refer to K as the

but if K > 1, then legislator 1’s speech conveys information about his preference.

Lemma 4. If mkinduces a proposal that includes legislator 1 who accepts it with positive

probability, then any proposal induced by mlwhere l < k must include legislator 1.

12This is the form of equilibrium message rules in both the classic model of Crawford and Sobel (1982)

and the related model of Matthews (1989). In our model, it is without loss of generality to consider

a message rule of this form if the chair plays a pure strategy. To see this, first note that there can be

at most one equilibrium proposal that does not include legislator 1 because such a proposal depends

only on 2’s message, but as already shown, 2 sends the same message in equilibrium. If such a proposal

is induced in an equilibrium, then there exists a type θ1such that only types higher than θ1induces

it. Next, consider two messages that are sent in equilibrium by legislator 1: m and m?. Suppose m

induces z, m?induces z?and both z and z?include legislator 1. Without loss of generality, assume that

x1> x?

z to z?and any type θ1> θ∗

m?, it follows that an equilibrium message rule has the partition form. An argument similar to that

1. Let θ∗

1be the type who is indifferent between z and z?. Then any type θ1< θ∗

1strictly prefers z?to z. Since this holds for any pair of messages m and

1strictly prefers

in the proof of Proposition 3 shows that there can be at most one equilibrium proposal that includes

legislator 1.

12

Page 15

Proof. Denote by zkthe proposal induced by mkthat includes legislator 1 and is accepted

with positive probability. Since zkis a best response for the chair, it must give the chair

an expected payoff at least as high as any proposal that excludes legislator 1. Also, since

zkis accepted with positive probability, type θk

1must strictly prefer zkto the status quo

and therefore any type θ1< θk

with l < k, there exists a proposal zl=?yl;xl?such that yl= ykand 0 < xl

is accepted with probability 1 by legislator 1. Moreover, zlgives the chair an expected

payoff strictly higher than any proposal that excludes legislator 1. It follows that any

1strictly prefers zkto the status quo. Hence, for any ml

1< xk

1and zl

proposal induced by mlmust include legislator 1.

The next proposition shows that in an informative equilibrium, the ideologically more

distant legislator is never included in a coalition.

Proposition 2. If legislator 2 is included in a proposal with positive probability in equi-

librium, then legislator 1’s message rule is not informative.

Proof. Suppose to the contrary that K ≥ 2. Let mkbe the highest message sent by

legislator 1 that induces a proposal, denoted by zk, that includes legislator 2 and excludes

legislator 1.

Suppose mk= m1. Then by Lemma 4, any proposal induced by m2must exclude

legislator 1. Since mkis the highest message that induces a proposal that includes 2, it

follows that no proposal induced by m2includes 2. Hence the proposal induced by m2

excludes both legislators, and by Lemma 3, legislator 1’s payoff is equal to his status quo

payoff. Since type θk+1

1

’s payoff is lower than the status quo payoff from the proposal

zk, type θk+1

1

(and types in the neighborhood below it) has an incentive to deviate and

send m2, a contradiction.

Suppose mk> m1. We next show that any proposal induced by mkdoes not include

legislator 1. Suppose not. Then by Lemma 4, any proposal induced by message ml

with l < k includes legislator 1 and hence gives legislator 1 a payoff at least as high as

his status quo payoff. Since type θk+1

1

’s payoff is equal to his status quo payoff when

legislator 1 is included and strictly lower than his status quo payoff when legislator 2

is included, it follows that type θk+1

1

(and types in the neighborhood below it) has an

13

Page 16

incentive to deviate and send ml, a contradiction. So if a message mkof legislator 1

induces a proposal that includes legislator 2, then any proposal induced by mkmust

exclude legislator 1. Now consider a message mn?= mksent by legislator 1. Recall

that legislator 1’s payoff is strictly lower than his status quo payoff when legislator 2 is

included and weakly higher than the status quo payoff when legislator 2 is excluded. So,

to prevent profitable deviations, any proposal induced by mnmust also exclude legislator

1 and the probability that legislator 2 is included must be the same for messages mk

and mn. Since a proposal does not depend on legislator 1’s message if legislator 2 is

included, it follows that messages mkand mninduce the same distribution of proposals,

a contradiction.

It follows from Proposition 2 that in an informative equilibrium, the ideologically

more distant legislator is always excluded in a coalition. The next proposition describes

what messages are sent and what proposals are induced in an informative equilibrium.

Let y∗

1= min{y : v (y, ˆ y1) = v (˜ y, ˆ y1)}, i.e., the lowest y that gives legislator 1 the status

quo payoff.

Proposition 3. If legislator 1’s message rule is informative in equilibrium, then it must

have size two: message m1induces a proposal (y,x) with y < y∗

1, x1 > 0 and x2 =

0, which is accepted with positive probability by legislator 1; message m2induces the

proposal (y∗

1;c,0,0), which is accepted by legislator 1 with probability 1. In an informative

equilibrium, legislator 1’s payoff is higher than the status quo and legislator 2’s payoff is

lower than the status quo.

Proof. We first show that if a message mkinduces a proposal zkthat includes legislator

1 and is accepted with positive probability, then mk= m1, i.e., the lowest message.

Suppose not, then mk−1exists and by Lemma 4, any proposal induced by message mk−1

must include legislator 1. Among all proposals induced by mk−1, let zk−1denote the

proposal that is accepted with the highest probability. Let θ?

1denote the type who is

,θk

indifferent between zk−1and the status quo. Note that θ?

1∈ (θk−1

1

1]. Let θ??

1denote the

type who is indifferent between zkand the status quo. Note that θ??

1∈ (θk

1,θk+1

1

]. Next,

we show that type θ?

1strictly prefers zkto zk−1. Since type θ?

1is indifferent between

14

Page 17

zk−1and the status quo, we have xk−1

xk−1

1

= θ?

1

the status quo, we have xk

1

+ θ?

1v?yk−1, ˆ y1

?= θ?

??. So

1v (˜ y, ˆ y1), which implies that

?v (˜ y, ˆ y1) − v?yk−1, ˆ y1

?zk,θ?

??. Similarly, since type θ??

1

?zk−1,θ?

= (θ??

1is indifferent between zkand

1= θ??

?− u1

?v (˜ y, ˆ y1) − v?yk, ˆ y1

1, ˆ y1

u1

1, ˆ y1

?= xk

1+ θ?

1v?yk, ˆ y1

?− θ?

1v (˜ y, ˆ y1)

1− θ?

1)?v (˜ y, ˆ y1) − v?yk, ˆ y1

?zk,θ?

??.

?zk−1,θ?

Since θ??

1> θ?

1and v (˜ y, ˆ y1)−v?yk, ˆ y1

?> 0, we have u1

1, ˆ y1

?−u1

1, ˆ y1

?>

0, i.e., type θ?

1strictly prefers zkto zk−1. So the expected payoff that type θ?

sending mk−1is equal to the status quo and the expected payoff he gets by sending mk

1gets by

is strictly higher than the status quo. Hence type θ?

1(and types in the neighborhood

immediately below it) has an incentive to deviate, a contradiction. So if a message mk

induces a proposal zkthat includes legislator 1 and is accepted with positive probability,

then mk= m1. The argument also implies that there can be at most one message above

m1and the proposal it induces does not include legislator 1. Since in an informative

equilibrium, legislator 2 is always excluded, the proposal induced by m2excludes both

legislators. Since the chair’s optimal proposal when both legislators are excluded is

(y∗

1;c,0,0), it follows that m2induces (y∗

1;c,0,0). Since legislator 1 is pivotal in all

equilibrium coalitions, his payoff is higher than the status quo payoff. Legislator 2 is

always excluded and his payoff is therefore lower than the status quo payoff.

Proposition 3 says that legislator 1 may be able to convey limited information about

his ideological intensity. When legislator 1 is intensely ideological, he sends a “fight”

message, signaling that the chair will not be able to “buy” his vote on an ideological

compromise through private benefits, and the chair responds with a proposal of minority

coalition and a policy that makes legislator 1 just willing to accept. When legislator 1 is

not intensely ideological, he sends a “compromise” message and the chair responds with

a proposal of a minimum winning coalition that includes legislator 1 and a policy closer

to the chair’s ideal.

Proposition 3 shows that in an informative equilibrium, the chair forms a coalition

only with the legislator whose ideological position is closer to his own. It is worth noting

that in an uninformative equilibrium, it is possible that the more distant legislator is

15

Page 18

included in a coalition. This happens if his position is not too extreme relative to the

other legislator and the chair believes that he puts much less weight on ideology than

the other legislator and hence it is less costly to gain his support.

To illustrate what an equilibrium looks like when legislator 1’s message is informative,

we provide the following example.

Example 1. Suppose ˜ y = 0, ˆ y0 = −1, ˆ y1 = −1

legislator i’s utility function is xi− θi(y − ˆ yi)2, θ0 = 1, and θ1,θ2 are both uniformly

distributed on [1

5, ˆ y2 =

1

2, c = 1. Also, assume that

4,4].

Consider the following strategy for legislator 1: send m1if θ1 ∈ [1

θ1∈ (θ2

for any θ2.

4,θ2

1] and m2if

1,4]; and the following strategy for legislator 2: always send the same message

To find the chair’s best response, note that if the chair wants to make a proposal

that legislator 1 accepts when θ1 ≤ θ∗

type θ∗

1, then he would propose (y,x) so as to leave

1indifferent between the status quo and the proposal (y,x). If x1> 0, we have

x1− θ∗

1(y − ˆ y1)2= −θ∗

1(˜ y − ˆ y1)2. So x1= θ∗

1(y − ˜ y)(y + ˜ y − 2ˆ y1) and the chair solves

max

y∈[y,¯ y](c − θ∗

1(y − ˜ y)(y + ˜ y − 2ˆ y1)) − θ0(y − ˆ y0)2

subject to

θ∗

1(y − ˜ y)(y + ˜ y − 2ˆ y1) ≥ 0.

θ0ˆ y0+θ∗

θ0+θ∗

If an interior solution exits, then y =

1ˆ y1

1

. Since the chair can always propose

y = 2ˆ y1− ˜ y, x1= 0 and have it accepted by legislator 1 with probability 1, we must

have y = min{θ0ˆ y0+θ∗

Given the strategy of legislator 1, when the chair receives message m1, he infers that

1ˆ y1

θ0+θ∗

1

,2ˆ y1− ˜ y}.

θ1∈ [1

˜ y−2ˆ y1) will be accepted with probability 1. Suppose θ2

13There are many other values of the threshold θ2

4,θ2

1]. Using the calculation above, the proposal y =

θ0ˆ y0+θ2

θ0+θ2

1ˆ y1

1

, x1= θ2

1(y − ˜ y)(y +

1= 2.13Plugging in the numbers,

1that satisfy the equilibrium conditions. We pick

θ2

1= 2 just as an example. What is important is that the chair’s optimal proposal when receiving m2

involves a minority coalition and his optimal proposal when receiving m1involves a minimum winning

coalition with legislator 1.

16

Page 19

we find that if the chair wants to make a proposal that is accepted by all θ1∈ [1

then the optimal proposal is y = −7

is the chair’s optimal proposal for θ1∈ [1

that is not accepted with probability 1 by legislator 1 is suboptimal and it is also optimal

4,θ2

1],

15and x1=

4,2] and θ2∈ [1

14

225. Calculation shows that indeed this

4,4]. In particular, any proposal

to exclude legislator 2.14

When the chair receives message m2from legislator 1, he infers that θ1 ∈ (θ2

Calculation shows that when θ2

1,4].

1= 2, it is optimal to propose y = ˜ y − 2ˆ y1= −2

5and

x1= 0,x2= 0, and legislator 1 accepts the proposal with probability 1.

We next check that legislator 1’s incentive constraints are satisfied. Let z1denote

the proposal induced by m1and z2denote the proposal induced by m2, i.e., z1=

?−7

x2

15;211

225,14

225,0?and z2=?−2

5;1,0,0?. Type θ2

1is indifferent between z1and z2because

?< y2?= −2

he gets the status quo payoff either way. Since y1?= −7

types above θ2

155

?and x1

1

?=

14

225

?>

1(= 0), types below θ2

1(who puts relatively low weight on ideology) prefer z1to z2and

1(who put relatively high weight on ideology) strictly prefer z2to z1.

In this equilibrium, when legislator 1 puts relatively low weight on ideology and high

weight on private benefit, he sends message m1, which can be interpreted as signaling

willingness to form a coalition with the chair. The chair responds to m1with a proposal

that includes legislator 1 in the coalition and moves the policy towards the chair’s ideal.

When legislator 1 puts relatively high weight on ideology and low weight on private

benefit, he sends message m2. The chair responds with a proposal of a minority coalition

because it is too costly to form a coalition with either legislator: legislator 1 is too

intensely ideological and legislator 2 has an ideological position that is too far away.

Next, we use our results to shed light on a number of interesting questions.

Seniority and uncertainty: What happens if only one of the legislators has private

information on his preference? This applies if one legislator is a senior member of

the legislature whose preference has been revealed from past experience and the other

legislator is a junior member whose ideological intensity remains uncertain. To see what

14In this example, the minimum winning coalition that the chair proposes in response to m1is

accepted with probability 1. This is a special feature of the example and does not hold in general, i.e.,

it may happen that the minimum winning coalition fails to pass with positive probability.

17

Page 20

happens in this case, first note that no surplus coalition is ever proposed in equilibrium

because there is no uncertainty about whether the senior legislator will vote for any

given proposal. In particular, if it is optimal for the chair to include the senior member

in a coalition, it must be optimal for him to propose a policy that ensures the senior

member’s vote and also leave the junior member out of the coalition.

Is it possible for the junior member’s speech to be informative in equilibrium? As this

is a special case of the preceding general analysis (in particular, if θi=¯θi, then there

is no uncertainty about legislator i’s preference), we can apply the results to obtain

the following observations. Whether the junior legislator’s speech can be informative

depends on the relative positions of the legislators on the ideological spectrum. If the

junior member has an ideological position closer to the chair’s, then it is possible for his

messages to be informative. The message rule and the resulting outcomes are similar

to those described in Proposition 3. By contrast, if the senior member is the one whose

ideological position is closer to the chair’s, then it is impossible for the junior member’s

message to be informative in equilibrium, as shown in Proposition 1.

Benefits of bundling the two dimensions: Since the legislators bargain over

both an ideological dimension and a distributive dimension, a natural question to ask

is whether the proposer is better off bundling the two dimensions together or negoti-

ating them separately. When the uncertainty is about the trade-off between the two

dimensions, the answer is unambiguous: the chair (weakly) prefers to bundle the two

dimensions together. To see why, note that if the two dimensions are bargained over

separately, then the legislators’ private information is irrelevant since it is about how

they trade off one dimension for the other, not about their preferences on each dimen-

sion. The resulting outcome is (y∗

1;c,0,0), i.e., the chair gets all the private benefit and

the policy he proposes is the one that make the closest legislator just willing to accept.

When the two issues are bundled together, the proposal (y∗

1;c,0,0) is still feasible and

will pass with probability 1. It immediately follows that bundling can never make the

chair worse off. In fact, there are two potential advantages from bundling: (1) Useful

information may be revealed in equilibrium, as seen in Proposition 3. (2) Given the

information he has, the chair can use private benefit as an instrument to make better

18

Page 21

proposals that exploit the difference in how the players trade off the two dimensions.

This result that legislators prefer to make proposals for the two dimensions together

despite separable preferences is analogous to the finding in Jackson and Moselle (2002),

although their model of legislative bargaining has no asymmetric information or com-

munication. But as we will see in the next section, this result is sensitive to the nature

of the uncertainty. When legislators’ ideological positions, rather than intensities, are

uncertain, it may be better for the chair to separates the two dimensions.

4 Uncertain ideological position

Suppose θiis commonly known,15but legislator i (i ?= 0) is privately informed about his

ideological position ˆ yi. As such, we redefine legislator i’s type to be his ideal point, i.e.

ti= ˆ yi, and assume that the distribution of tihas full support on [yi, ¯ yi] for i = 1,2.

As before we analyze equilibria in which the message rule for legislator i (i ?= 0) takes

the following form: send message mk

iif and only if ti∈ (τk

= ¯ yi. So Kiis the size of legislator i’s message rule. What

i,τk+1

i

] (k = 1,...,Ki) where

yi= τ1

i< τ2

i< ... < τKi+1

i

follows is a number of useful lemmas about equilibrium proposals.

Lemma 5. Any equilibrium proposal (y;x) satisfies y ≤ ˜ y. If an equilibrium proposal

(y;x) satisfies y = ˜ y, then x must be (c,0,0) and is accepted with probability 1.

Proof. Since the proposal (˜ y;c,0,0) is accepted with probability 1 and ˆ y0< ˜ y, the chair

will never make a proposal with y > ˜ y. Hence any equilibrium proposal z = (y;x)

satisfies y ≤ ˜ y. Suppose an equilibrium proposal z = (y;x) satisfies y = ˜ y, but x

?= (c,0,0). Then x0 < c and there exists a legislator i(?= 0) with xi > 0 and hence

will vote yes on z. But since the status quo is (˜ y;0,0,0), the chair can instead propose

z?= (˜ y;x?) with 0 < x?

i< xi. Legislator i will still vote for the proposal, but the chair

is better off with z?, a contradiction.

15So the support of θiis degenerate: θi= θi.

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Page 22

Lemma 6. Suppose a proposal (y;x) with y < ˜ y is induced by?mk1

with positive probability by legislator i in equilibrium. Then, type τki

prefers (y;x) to the status quo.

1,mk2

2

?and accepted

i of legislator i strictly

Proof. Since v (y,ti) is single-peaked and satisfies the single-crossing property in (y,ti)

and uiis quasi-linear, it follows that if type tiweakly prefers a proposal z?= (y?;x?) to

z = (y;x) and y?> y, then type t?

i> timust strictly prefer z?to z. Assume, without loss

of generality, that (y;x) is accepted by legislator 1 with positive probability. Suppose

type τk1

1

weakly prefers (˜ y; ˜ x) to (y;x). Since y < ˜ y, this implies that if t1> τk1

1, then

type t1must strictly prefer (˜ y; ˜ x) to (y;x), contradicting the assumption that (y;x) is

accepted by legislator 1 with positive probability. Hence, type τk1

1 strictly prefers (y;x)

to the status quo. The same argument applies if (y;x) is accepted by legislator 2 with

positive probability.

Lemma 7. Suppose a proposal (y;x) with y < ˜ y is induced by mk

i(k ≥ 2) and accepted

with positive probability by legislator i in equilibrium. Then any equilibrium proposal

induced by mk−1

i

must be (ˆ y0;c,0,0).

Proof. Lemma 6 implies that type τk

iof legislator i strictly prefer (y;x) to the status

quo. In equilibrium type τk

imust be indifferent between sending mk

iand mk−1

i

and

therefore indifferent between the distributions of outcomes induced by the messages.

Next, we show that all the equilibrium proposals induced by mk−1

i

must be (ˆ y0;c,0,0).

Suppose not. Then at least one proposal induced by mk−1

i

is not (ˆ y0;c,0,0). Among

these proposals find the one that gives type τk

ithe highest payoff, and denote it by

(y?;x?). Consider the following two cases. (i) Suppose type τk

iprefers (ˆ y0;c,0,0) to

(y?;x?). Then it follows that type τk

istrictly prefers (ˆ y0;c,0,0) to (˜ y; ˜ x). Because ˆ y0< ˜ y,

uiis quasi-linear and v (y,ti) satisfies the single-crossing property in (y,ti), it follows

that for any type ti< τk

i, legislator i strictly prefers (ˆ y0;c,0,0) to (˜ y; ˜ x) and therefore

will vote for it. Since (ˆ y0;c,0,0) gives the chair the highest possible payoff, his optimal

response to mk−1

i

must be (ˆ y0;c,0,0), a contradiction. (ii) Suppose type τk

to (ˆ y0;c,0,0). Then (y?;x?) is the proposal among those induced by mk−1

iprefers (y?;x?)

that type τk

i

i

likes best. It follows that type τk

i(and all the lower types) strictly prefers (y?;x?) to

20

Page 23

(˜ y; ˜ x). This implies that instead of proposing (y?;x?), the chair can propose a policy

lower than y?or shares lower than x?to the other legislators and still get the proposal

accepted with probability 1, but this alternative proposal makes the chair strictly better

off, a contradiction.

Let˜ki= max{k : mk

positive probability by legislator i}. Lemma 7 implies that˜ki≤ 2 since any equilibrium

message mk

iinduces a proposal other than (˜ y;c,0,0) and it is accepted with

iwith k <˜kimust induce (ˆ y0;c,0,0).

Lemma 8. Suppose an equilibrium message mk

ieither induces (˜ y;c,0,0) or induces a

proposal that is rejected by legislator i. Then k = Ki.

Proof. Suppose to the contrary that k < Ki. Suppose (y;x) is a proposal induced by

mk+1

i

and accepted by legislator i with positive probability. If y < ˜ y, then by Lemma 7,

mk

imust induce (ˆ y0;c,0,0). Since mk

from Lemma 5 that x = (c,0,0). Hence a proposal induced by mk+1

iinduces (˜ y;c,0,0), we must have y = ˜ y. It follows

i

must either be

(˜ y;c,0,0) or rejected by legislator i. Note that when a proposal is rejected by legislator

i, then it depends only on legislator j’s message. Hence messages mk

iand mk+1

i

induce

the same distribution of proposals, a contradiction.

It follows from Lemma 8 that there can be at most one equilibrium message mk

k >˜ki. Since˜ki≤ 2, the maximum number of equilibrium messages for legislator i is 3,

i.e., Ki≤ 3. To summarize, when the legislators’ ideological positions are uncertain, it

is possible for their demands to affect the proposals and outcomes in equilibrium, but

iwith

the information conveyed is still limited.

Proposition 4. An equilibrium message rule has at most size three, i.e., Ki≤ 3 (i =

1,2). For a size-three message rule mi(·), m1

induces a proposal (˜ y;c,0,0) or induces a proposal that legislator i rejects; compromise

iinduces the proposal (ˆ y0;c,0,0); m3

ieither

proposals with y ∈ (ˆ y0, ˜ y) are induced only by m2

i.

It is useful to interpret m1

ias the “cooperate” message, m2

ias the “compromise”

message and m3

ias the “fight” message. Legislator i sends message m1

ionly when

21

Page 24

his ideology is sufficiently aligned with the chair’s and in particular, he prefers the

chair’s ideal to the status quo policy. By sending the “cooperate” message, he signals

his willingness to vote for the chair’s most preferred policy even without getting any

private benefit. With the assurance of this cooperative ally (one such legislator is enough

under the majority rule), the chair proposes his ideal policy without giving out any

private benefit to the other legislators. By contrast, when legislator i has an ideological

position that is distant from the chair’s, he send message m3

ito signal a tough stance on

policy change. If both legislators send the “fight” message, the chair realizes that both

legislators’ ideals are too far from his own and the best proposal he can put forward

is to keep the status quo policy unchanged and give out no private benefit. When a

legislator has an ideological position that is somewhat aligned with the chair’s, he sends

a “compromise” message and the chair responds with a policy that is in between the

status quo and his own ideal unless the other legislator indicates willingness to cooperate.

To illustrate what an equilibrium with meaningful rhetoric looks like, we provide the

following example.

Example 2. Suppose ˜ y = 0, ˆ y0= −1, c = 1, legislator i’s utility function is xi−(y − ti)2

and ti(i = 1,2) is uniformly distributed on [−2,2] for i = 1,2.16

Consider the following strategy profile. The message rule for legislator i = 1,2 has

size three: specifically, mi(ti) = m1

iif t ∈ [τ1

i= −2 < τ2

ifor either i = 1 or 2, propose (ˆ y0;1,0,0); if mi = m3

both i = 1,2, propose (˜ y;1,0,0); if m1= m2

i,τ2

i], mi(ti) = m2

iif t ∈ (τ2

i,τ3

i] and

mi(ti) = m3

iif (τ3

i,τ4

i] where τ1

i< τ3

i< τ4

i= 2. The chair’s strategy is

the following: if mi = m1

ifor

1and m2= m3

2, propose (y?;x?

0,1 − x?

0,0); if

m1= m3

(y??;x??

1and m2= m2

2, propose (y?;x?

0,0,1 − x?

2and propose (y??;x??

0); if mi= m2

ifor both i = 1,2, propose

0,1 − x??

By using the indifference condition of type τ2

0,0) with probability1

0,0,1 − x??

0) with probability1

2.

iand type τ3

iand conditions for the

chair’s proposals to be optimal conditional on the messages received, we find that τ2

i≈

−0.80, τ3

16For expositional simplicity, we assume that legislators 1 and 2 are ex ante identical in this example,

i≈ 0.54, y?≈ −0.23,x?

0≈ 0.7,y??≈ −0.45,x??

0≈ 0.87.

but equilibrium message rules of size-three may still exist even when the legislators are not ex ante

identical.

22

Page 25

In this equilibrium, if at least one of the legislators signals his willingness to cooperate

by sending message m1

i, then the chair proposes his ideal policy ˆ y0and keeps the whole

cake to himself. Because legislator i’s ideal is in [τ1

i,τ2

i] = [−2,−0.80], he is willing to

go along with the chair’s ideal policy even without any transfer of private benefit. This

proposal of a minimum winning coalition passes with probability 1.

If both legislators act tough and send m3

i, then the chair proposes the status quo

policy ˜ y = 0 and still keeps the whole cake to himself. Since both legislators’ ideal

policies are high (ti∈ (0.54,2])), it is too costly (i.e., the cake shares needed in exchange

for their votes are too large) for it to be optimal for the chair to try to change the

status quo policy. Legislators 1 and 2 are indifferent between voting for and against the

proposal. In equilibrium they vote yes and this proposal of a minority coalition passes

with probability 1.

If legislator i signals willingness to compromise by sending m2

iwhile legislator j sends

the tough message m3

j, then the chair tries to gain the vote from legislator i while giving

up on legislator j. He proposes a compromise policy (y?≈ −0.23) and forms a minimum

winning coalition with legislator i (x?

i≈ 0.3) to legislator i and zero share to legislator

j. The proposal is rejected by legislator j, but is accepted by legislator i since any

ti∈ (τ2

Perhaps the most interesting case is when both legislators signal willingness to com-

i,τ3

i] strictly prefers it to the status quo.

promise by sending m2

i. In the equilibrium we constructed, it is equally costly (in

expectation) for the chair to win the vote of either legislator. So he randomizes with

equal probability between two proposals that involve the same policy (y??≈ −0.45) and

the same cake share (x??

0≈ 0.87) for himself, but differ with respect to which legislator

he chooses to form a coalition with. Compared with the case in which only one legislator

signals willingness to compromise while the other shows a tough stand, here the com-

promise policy is even closer to the chair’s ideal and the cake share that the chair keeps

for himself is also larger. Intuitively, when both legislators signal willingness to compro-

mise, they create competition between themselves. Since the chair needs only one vote

to have a proposal passed, his optimal proposal involves less ideological compromise and

less distributive concession. It is interesting to observe that in equilibrium the legislator

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who is excluded from the coalition still votes for the proposal if he finds it more attrac-

tive than the status quo. In particular, suppose legislator i gets a positive share while j

does not. Then legislator i votes for the proposal if ui(y??,1 − x??

ti∈ [−0.80,−0.075] and legislator j votes for the proposal if uj(y??,0,tj) ≥ ui(˜ y,0,tj),

i.e., if tj ∈ [−0.80,−0.224]. Although the probability that legislator j votes for the

proposal is lower compared to legislator i, both vote for the proposal with positive

0,ti) ≥ ui(˜ y,0,ti), i.e., if

probability. Moreover, this proposal is rejected with positive probability in equilibrium.

To compare the informative size-three equilibrium with the uninformative babbling

equilibrium (which always exists), note that in the babbling equilibrium, what the chair

should propose becomes a simple optimization problem. Calculation shows that in this

example the chair’s optimal proposal is (0;1,0,0), i.e., he gets the whole cake and

implements the status quo policy. So the chair’s equilibrium payoff in the babbling

equilibrium is 1 − (0 + 1)2= 0. Since the chair always benefits from more information

revelation, his expected payoff is higher in the size-three equilibrium than in the babbling

equilibrium. Calculation confirms that his expected payoff in the size-three equilibrium is

0.56, higher than that in the babbling equilibrium. Interestingly, the other legislators also

have higher expected payoffs in the size-three equilibrium (−2.662) than they do in the

babbling equilibrium (−2.666). So they also benefit from more information transmission.

Seniority and uncertainty: One special case of our analysis is when only one

legislator’s preference is uncertain, perhaps because the other legislator is a senior mem-

ber with known preference, as discussed in section 3. If the senior member prefers the

chair’s ideal to the status quo policy, then the chair can propose (ˆ y0;c,0,0) and have

it passed. In this case, the junior legislator’s message has no effect on equilibrium out-

come. The interesting case is when the senior member prefers the status quo to the

chair’s ideal. The message rule of the junior legislator still has at most size three. To

illustrate with an example, suppose legislator 2 is the senior member with a known ide-

ological position ˆ y2= −0.2 but otherwise keep the parametric assumptions in Example

2. We find that legislator 1 has an equilibrium message rule of size three with the cutoffs

τ1

1= −2,τ2

message with the proposal (ˆ y0;1,0,0), the “compromise” message with the proposal

1= −0.87,τ3

1= −0.3 and τ4

1= 2. The chair responds to the “cooperate”

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(y?;1 − x?

(y??;1 − x??

when the junior member sends the “fight” message, the chair does not respond with the

1,x?

1,0) where y?= −0.65, x?

2,0,x??

1= 0.0325 and “fight” message with the proposal

2) where y??= −0.6 and x??

2= 0.12. A few things are worth noting. First,

minority coalition of (˜ y;c,0,0) because it is better to form a coalition with the senior

legislator. The proposal will be accepted by the senior legislator, but rejected by the

junior one. This observation holds in general as long as the senior member’s preference

is sufficiently close to the chair’s so that it is better for the chair to form a coalition

with him than to implement (˜ y;c,0,0). Second, the senior legislator is included in a

coalition only when the junior legislator sends the “fight” message. To see this, note

that if the chair were to respond to the “compromise” message with a coalition that

includes the senior legislator, then the “fight” message and the “compromise” message

induce the same proposal (y??;1 − x??

when the junior legislator sends the “compromise” message in equilibrium, the chair

2,0,x??

2), effectively becoming the same message. So

always proposes a minimum winning coalition that includes the junior legislator.

Disadvantages of bundling the two dimensions: As we have seen in section

3, bundling the ideological and distributive dimensions together affords the legislators

the flexibility of trading private benefits for policy compromises and is always better

for the chair when the uncertainty is about the legislators’ ideological intensities. As

will be shown in the discussion that follows, however, when the uncertainty is about

the legislators’ ideological positions, bundling has two potential disadvantages. The first

disadvantage is that by combining the two dimensions together, the chair risks losing

the cake if negotiation breaks down whereas no such risk exists if the legislators bargain

over the distributive dimension separately.

This disadvantage can be easily illustrated using Example 2. Suppose the legislators

negotiate over the two dimensions separately. Then bargaining over the distributive

dimension becomes a simple ultimatum game and the chair keeps the whole cake to

himself. As to the ideological dimension, it is straightforward to show that the most-

informative equilibrium is the following17: both legislators 1 and 2 play the message

17Matthews (1989) considers a bargaining game between two legislators over ideology and shows that

an equilibrium has at most “size two.” Although there are three legislators in our game, we can modify

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rule such that mi(ti) = m1

iif t ∈ [τ1

i,τ2

i], mi(ti) = m2

iif t ∈ (τ2

i,τ3

i] where τ1

i= −2 <

τ2

i< τ3

and he proposes a compromise policy y if both legislators send m2

i= 2; the chair proposes his ideal ˆ y0if at least one of the legislators send m1

i,

i. The indifference

condition of type τ2

iimplies that τ2

i≈ −0.81. If both legislators send m2

i, the chair

responds with the proposal y ≈ −0.62, and the legislator i votes for the proposal if

and only if ti≤

payoff on the policy dimension is −0.36. Since he keeps the whole cake, his payoff on the

distributive dimension is 1. So the chair’s total expected payoff is 1−0.36 = 0.64, higher

than his expected payoff (= 0.56) in the size-three equilibrium if the two dimensions are

y+˜ y

2

≈ −0.31. Calculation shows that the chair’s expected equilibrium

bundled together in negotiation. The reduction of payoff from bundling comes from the

loss of the cake when the legislators fail to reach an agreement, which happens when

both legislators send the “compromise” message but their ideological positions are too

far from the chair’s for them to find the resulting proposal attractive enough. Although

failure to reach an agreement also happens even if the legislators negotiate the ideological

dimension separately, the distributive dimension is shielded from such failure.

Another, perhaps less obvious, disadvantage of bundling is the information loss that

may result from bargaining the two dimensions together (or, as another interpretation,

the chair’s lack of commitment of not using private benefit in exchange for votes on

ideological decisions). This matters even if there is no risk of “losing the cake.” To

illustrate, suppose c = 0, so the break-down of agreement does not result in the dissipa-

tion of private benefits. In this case, we can interpret bundling of the two dimensions

together as (the possibility of) using side payments to gain support on an ideological

decision and separation of the two dimensions as the unavailability of side payments.

As shown in the previous paragraph, if no side payments are allowed, then there exists

an informative equilibrium in which the legislators send m1

iif tiis below τ2

iand send

m2

iif ti is above τ2

i. When side payments are allowed, however, this is no longer an

equilibrium strategy if the chair puts a relatively low weight on the distributive dimen-

the argument provided in Matthews (1989) to show that each legislator’s message rule has at most

size two when they bargain over just the ideological dimension. Because it does not provide much new

insight, we omit the details of the modified argument here.

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sion. For example, suppose θ0= 9, but keep all the other parametric assumptions in

Example 2 unchanged. If the legislators were to follow the message rule with the cutoff

τ2

i= −0.81, then, upon receiving m2

respond by proposing y equal to his ideal point ˆ y0and a side payment (= 3.8415) to one

ifrom both legislators, the chair would optimally

of the legislators. But this undermines the legislator i’ incentive to send m1

ibecause it

induces the chair’s ideal without the possibility of getting any side payment whereas the

other message m2

iinduces the chair’s ideal with a side payment as well. Indeed, in this

example there exists no informative equilibrium when the chair can use side payments in

his proposals. Because of this informational loss, the chair’s equilibrium payoff is lower

when side payments are allowed.18

5 Concluding remarks

In this paper, we develop a new model of legislative bargaining that incorporates pri-

vate information about preferences and allows speech making before a bill is proposed.

Although the model is simple, our analysis generates interesting predictions about what

speeches can be credible even without commitment and how they influence proposals

and legislative outcomes. We are also able to provide new insight into when legislators

should make proposals for different issues together and when they should make proposals

separately.

We believe that both private information and communication are essential elements of

the legislative decision making process. Our paper has taken a first step in understanding

their roles in the workings of a legislature. There are many more issues to explore and

many ways to generalize and extend our model and what follows is a brief discussion of

some of them.

Our motivation for incorporating private information into legislative bargaining is

that individual legislators know their preferences better than others. Another possible

18Harstad (2007) shows in another bargaining game that side payments may be harmful because

they increase conflict of interest and incentive to signal, resulting in more delay. The reason for side

payments to be harmful is different here.

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source of private information is that some legislators may have better information (per-

haps acquired through specialized committee work or from staff advisors) regarding the

consequences of certain policies, which is relevant for all legislators. Although the role

of this kind of “common value” private information in debates and legislative decision

making has been studied in the literature (e.g. Austen-Smith (1990)), it is only in the

context of one-dimensional spatial policy making. It would be interesting to explore it

further when there is trade off between ideology and distribution of private benefits.

In our model the chair does not have private information about his preference, con-

sistent with the observation that bill proposers are typically established members with

known positions. But sometimes legislators can be uncertain about what exactly the

legislative leaders’ goals are, in particular, how much compromise the leaders are willing

to make to accommodate their demands in exchange for their votes. In this case, apart

from speeches, the proposal that the chair puts on the table may also reveal some of his

private information. This kind of signaling effect becomes particularly relevant when the

legislators have interdependent preferences or when the proposal is not an ultimatum

but can be modified if agreement fails.

We have focused on a specific extensive form in which the legislators send messages

simultaneously. It would be interesting to explore whether and how some of our results

change if the legislators send messages sequentially. In that case, the design of the

optimal order of demands (from the perspective of the proposer as well as the legislature)

itself is an interesting question. Another design question with respect to communication

protocol is whether the messages should be public or private. Although this distinction

does not matter for our model because we assume simultaneous speeches and one round

of bargaining, it would matter if either there were multiple rounds of bargaining or the

preferences were interdependent.

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