Paper prepared for the 1998 Meetings of the American Political Science Association,Sept. 3-5, Boston, MA.

Proposal Powers, Veto Powers, and the

Design of Political Institutions

Nolan McCartyDepartment of Political Science

Columbia UniversityNew York, NY 10027

Version 1.0

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I. Introduction

The major preoccupation of rational choice theorists is the study of the mapping

between formal political powers and actual political outcomes. Using tools drawn from

economics, social choice, and game theory, rational choice theorists have attempted to

deduce propositions about the performance of many different types of political systems

and institutions. A number of recurring themes have emerged from the rational choice

enterprise. Two such themes are the roles of formal prerogative and sequence.

An important part of any rational choice model is an assignment of prerogatives,

formal powers, or more colloquially “who gets to do what.” In most rational choice

models of policy process, these questions boil down to “who may initiate policy change”

(proposal power) and “who may block policy changes” (veto power). In these models,

the relative influence of different political actors often depends on whether the actor

serves as an agenda setter or a veto player.

Sequence also plays an important role in rational choice models by adding the

question “when” to “who gets to do what.” When can a certain actor make a new

proposals? At what stage can new legislation be blocked? What is the time horizon of

the players? Using the tools of extensive form games and subgame perfection, the

importance of anticipated reactions and credibility have been established. The downside

is that many of these models are not robust when assumptions about sequence or time

horizons are changed.

Unfortunately, much of the theoretical work on political institutions has tended to

focus only on subsets of these questions. For example Baron and Ferejohn (1989b) have

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analyzed the effects of proposal power in a dynamic model, but only in a majority rule

setting with no veto players. Works such as Hammond and Miller (1987) and Tsebellis

(1994) have focused on the role of veto players, but have placed less attention on issues

of proposal power or sequence. Analyses based on the work of Romer and Rosenthal

(1978) have focused both on proposal power and veto power, but has dome so in

typically static contexts with very stylized sequences of actions.

In this paper, we attempt to synthesize these concerns into a single model that

incorporates very general allocations of proposal and veto powers while also accounting

while relaxing some of the assumptions about sequence and timing. The model we

propose is a generalization of the sequential choice model developed Baron and Ferejohn

(1989a). In this model a legislative assembly must decide how to allocate a fixed set of

resources across the legislative districts. Each member of the assembly has a set of

prerogatives which may include proposal powers, veto powers, and voting rights.1 In

each period, a member is chosen at random to make a proposal with a probability

determined by her proposal power. Each member then decides whether to support the

proposal. The veto rights of the opponents determine whether or not the proposal passed.

If the proposal passes, the game ends and resources are allocated according to the latest

proposal. In this model, expectations of the future will play an important role in the

process. Each players support or opposition to the current proposal will be based on

expectations about future proposals that are in turn are affected by the allocation of

parliamentary rights.

The paper proceeds as follows. In section II, we lay out the framework of the

basic model and make our conceptualization of proposal and veto powers much more

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precise. In section III and IV, we focus exclusively on veto power and draw a number of

prediction about when more veto power will be valuable. In section V, we compare these

results with the Shapley- Shubik procedure for measuring the distribution of power

within a legislature (Shapley and Shubik (1954)) . In section VI, we analyze the effects

of proposal power and compare these results to those on veto power in section VII.

II. The Basic Model

The model we employ is a generalization of the majority rule divide-the-dollar

game pioneered by Baron and Ferejohn (1989a). In each period a member of the

legislature is chosen via a random recognition rule to make a proposal as to the division

of the dollar. The proposal is then evaluated by the legislature who may reject it

according to some veto rule. If a veto of the proposal is sustained, the game continues to

the next period and a new proposal is made. Following any proposal that is not

successfully vetoed, the game ends and the dollar is divided according to the last

proposal.

We assume that the legislative body contains N members. Let the proposed

division by legislator O at time t and a history of play h be denoted as

xth th th Nthx x x= 1 2, , ,K< Awhere xith is the share going to district i under proposal xth.

Obviously, we require that xithi

N

=∑

1

≤ 1. In equilibrium this constraint will always bind.

Rather than arbitrarily limit the opportunities for legislative consideration of this

project, following Baron and Ferejohn (1989a), we assume that the legislature may

1 As we will see, voting rights are a special case of veto powers.

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consider the passage of the project for an infinite number of sessions.2 If the project does

not pass in a given session, it may be taken up again in the future. Each legislator

discounts future payoffs by δ so that legislator i’s payoffs are δ tithx where t is the time

period in which an agreement passes, O is the proposer at time t.3

The allocation decisions of the proposer are going to be determined in large part

by the allocation of parliamentary rights. In this paper, we focus the effects of the

recognition and veto rules. Let proposal powers be represented by a vector

p = p pN1, ,K; @ where pi is simply the probability that legislator i is chosen to make the

proposal in any given period. Obviously, we require that pii

N

=∑ =

1

1. We assume that p

remains the same over time so that the identity of the proposer is independently and

identically distributed. Thus, we differ from Baron and Ferejohn (1989b) who analyze

proposal power under the assumption that a committee has the right to make a proposal in

initial periods, but that the recognition probabilities are subsequently equal.

The notion of veto rights is a bit more subtle. Let k = k kN1, ,K; @ be the vector of

veto rights where ki is the number of votes required to overturn a veto by player i. Thus,

the larger ki, the more veto power held by legislator i. This formulation is quite general

and can be used to represent a number of legislative institutions.4 For example, suppose

that k Ni = +1 21 6 for all i. This is simple majority rule as a simple majority is sufficient

to offset player i’s opposition. If ki = N for some i and 0 for all other legislators,

2 The results will not be qualitatively affected by assuming that the process infinite.3 Deviating from the assumption that utilities are linear in the legislator’s shareprecludes any closed form solutions and does not substantively alter the comparativeanalysis of institutions.

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legislator i is a dictator. Finally, a qualified veto such as that of the U.S. president can be

represented by kN= 2

3 for the president and k N= +1 21 6 for the legislators.5

Following Baron and Ferejohn (1989a), we only consider stationary and

symmetric strategies. Stationary strategies are those that are time and history

independent.6 In this context, stationarity implies that proposals will only be a function

of the identity of the proposer as well as the basic parameters k, p, N, and δ, but not the

history of play. We also impose symmetry so that proposers treat all legislators of the

same type identically and all legislators of the same type play identical strategies.7 Since

the only ex ante differentiation of legislators is based on type, the stationarity and

symmetry assumptions suggest that strategies will depend only on the legislator’s “type”.

Given our setup, we can define a legislator’s type by the pair {pi, ki}. If {p i, ki} = {p j,

kj}, we can say that legislator i and j are the same type. Because of symmetry, we are

free to talk about the strategies of types rather than individuals. Therefore, let

4 This formulation is more general than other formulations that dichotomize thelegislature into veto players and non-veto players (e.g. Winter (1996)).5 Of course, the override requirement is concurrent 2/3 majorities. Concurrentmajorities require a slight modification to the model.6 Baron and Ferejohn (1989a) Baron and Ferejohn (1989a) show that any division can besupported in non-stationary equilibria using infinitely nested punishment strategies. Theintuition for this result is straightforward. Any proposer who deviates from theprescribed distribution of benefits will be “zeroed out” by the next proposer who wouldbe punished if she failed to do so. Because of the infinite number of periods, proposerswill always find it rational to carry out the punishment and so that the original proposerwill not deviate. Such strategies can only be implement if proposers can recall the entire(infinite) history of play so as to ascertain which proposers are to be zeroed out. Baronand Kalai (1993) argue that this is an attractive restriction of the strategy space due to thefact that it requires the fewest computations by the agents in the model. Further, withlegislative turnover, infinite recall of past histories is not a reasonable assumption.7 To be more precise, symmetry does not imply that the proposer proposes exactly thesame share to each member of a type class, but only that the mixed strategy probabilitydistribution be the same for all members of each class.

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m = m mT1, ,K< A where mτ is the number of legislators of type τ. Let pT = p pT1, ,K< A

be the recognition probabilities and kT = k kT1, ,K< A be the veto rules for each type

where superscripts denote types rather than individuals.

For each legislator, let vi be expected utility of the playing the game. Since we

will be interested in the ex ante payoffs to legislators of different legislators, vi will be the

main object of our analysis. However, these values play an important analytical role as

well. Because of the assumption of stationarity, the game begins anew each period.

Therefore, vi is also the expected utility to member i of defeating the current proposals.

For this reason we will refer to vi as the continuation value of legislator i. The

continuation value plays the same role as reservation utilities in spatial models of

legislative bargaining. However, the key difference is that continuation values are

endogenously derived through expectations of future play rather than given exogenously.

Given a continuation value, we can specify each legislator’s optimal strategy given a

proposal. If legislator i receives a share xi ≥ δvi, then she will not veto the proposal as the

share exceeds the discounted utility of continuing the game another period.8 Our

assumption of symmetry implies that if members i and j are type τ, then vi = vj = vτ.

The proposer’s strategy is a bit more complicated. Proposals may be formulated

to generate many different types of coalitions. To characterize these strategies, we need

some additional notation that focuses exclusively on veto types. Let ~ ~

, ,~

k = k k L1 K= B be

8 Implicit in this statement is that fact that we are assuming subgame perfection and thatplayers do not use weakly dominated strategies. Subgame perfection rules out non-credible threats to veto proposed shares that exceed the discounted continuation value.Elimination of weakly dominated strategies rules out bizarre equilibria where allmembers always vote in favor of every proposal because they are never pivotal.

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the set of distinct elements of k ordered from highest to lowest.9 Now, we can specify all

of the minimum winning coalition sizes.10 Clearly, if at least ~k1 receive their discounted

continuation values, the proposal will pass as all possible vetoes will be overridden. The

proposer can optimally create this coalition by choosing the ~k1-1 members with the

smallest continuation values and voting for the proposal herself. A second winning

coalition could be constructed with the votes of members with veto type ~k1 and

max ,~

&

~0 2

2

k ms

k ks

−%&'

()*>∑ other members. Thus, all

~k1 types accept and any veto by other

types will be overridden. If the proposer is type ~k1, she gives discounted continuation

values to the other legislators where members of her veto type class and to the

max ,~

&

~0 2

2

k ms

k ks

−%&'

()*>∑ lowest remaining continuation values. If the proposer is not type

~k1, she gives discounted continuation values all members of the

~k1 type and to the

max ,~

&

~0 12

2

k ms

k ks

− −%&'

()*>∑ lowest remaining continuation values. Both of these strategies

take into account that the proposer will support her own proposal.

Continuing inductively, we can see that all feasible minimum winning coalitions

can be constructed from the votes veto types greater than ~k jand from

max ,~

&

~0 k mj s

k ks j

−%&'

()*>∑ others. Each such coalition will be denoted as a

~k j- coalition. To

form a minimum cost winning coalition, the proposer will choose those from the

9 The ~ symbol is to distinguish these “veto types” from general types which are afunction of both proposal and veto power.10 Note that at this point we are not saying anything about the costs of differentcoalitions. A minimum cost coalition must be minimum winning, but not vice versa.

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remaining types with the lowest continuation values.11 In case of ties, we assume that the

proposer randomizes.12

Since all of these strategies correspond to different elements of ~k , we will use

~k

to represent the strategies of the proposer. Each proposer i will choose a strategy ~k j to

maximize her remaining share that is

z m v m v MIN k ms s

s k ki

j s

k ks j s j

τ τ τ

τ

δ δ= − − − − −���

���≠ ≥ ≥

∑ ∑1 12 7;

~ ~

~ if k k jτ > ~

[1]

and

z m v m v MIN k ms s

s k ki

j s

k ks j s j

τ τ τ

τ

δ δ= − − − − −���

���≠ ≥ ≥

∑ ∑1 1;

~ ~

~ if otherwise [2]

where MIN ri1 6 is a shorthand for the sum of the r smallest discounted continuation

values excluding i. These expressions account for the fact that the proposer will vote for

her own proposal. We will also allow for mixed strategies where the proposer

randomizes over various coalition sizes that we represent by σ ∈ Σ where Σ is the set of

probability distributions over ~k .

To complete the model, we turn to the computation of the continuation values.

Consider any proposal strategy profile σ = σ σ1, ,K T< A. Given στ, we can calculate the

probability that a type s is chosen by a type τ proposer. Let this be denoted as πτs .

11 We can again contrast our model with that of Winter (1996). Since he is notconcerned with overrides, his veto players are in all winning coalitions. Since there aremultiple levels of override in our model, we find that members with more veto power arein more, but certainly not all, winning coalitions.

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Further, we can compute the probability that a type τ is chosen as a coalition partner of

some other proposer as Πτ τ τττπ π= −

=∑m p ps s

ss

T

1

. Let Π = Π Π1, ,K T2 7.

Given the values of Πτ we can compute the continuation value of a legislator i as:

v p z vτ τ τ τ τδ= + Π [3]

Combining equations [1] or [2] with [3], we have a system of 2T equations with 2T

unknowns. Thus, given (Π1,…, ΠT), vτ and zτ are the solutions to this system which must

be full rank. A stationary, symmetric, subgame perfect Nash equilibrium requires the

following two conditions

i) vτ and zτ solve [1] ([2]) and [3] given Π*

ii) Πτ* maximizes zτ given Π ~*τ

Before turning to the analysis of veto and proposal powers, we state and (prove in

the appendix) some results that hold regardless of the distribution of legislative

prerogatives.

Lemma 1: In any stationary, symmetric, subgame perfect Nash equilibrium:

1) The bargaining with last only one round

2) vii

N

==∑ 1

1

12 This assumption can be rigorously justified as an essential part of a mixed strategyequilibrium. See Harrington (1990).

10

proof: See appendix

Lemma 1 is consistent with well-known efficiency results for bargaining models without

transaction costs or asymmetric information. Part 1) rules out any inefficiency due to

delay while part 2) suggests that aggregate welfare will be maximized at $1. While these

results are not terribly novel, they will often prove useful in the subsequent analysis.

III. The Effects of Veto Power

We will begin our analysis by looking exclusively at the effects of veto power

while holding proposal power constant at pi = 1/N for all i. The primary focus will be on

how the payoffs of different types of legislators are effected by veto power. Our first

result is that a legislator’s payoffs must be at least weakly increasing in her veto power.

Proposition 1: Suppose that pi = 1/N for all i. Then, in any symmetric, stationary,

subgame perfect Nash equilibrium, v vsτ ≥ if kτ < ks.

Proof: See appendix

The logic of proposition 1 is quite straightforward. Suppose type τ did get lower utility

than type s who has less veto power. Then each type τ would be an especially attractive

coalition partner relative to type s. First, type τ is less expensive because of her lower

continuation values. Secondly, it is more feasible to satisfy the override threshold of ks

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since the votes of type s are expensive. Thus, it must be the case that the probability of

being selected into a coalition is higher for τ implying that Π Πτ ≥ s . Since the

probability that a type s is selected is weakly lower, zs must be sufficiently higher than zτ

for vs > vτ. However, it can be shown that this is impossible or otherwise a proposer of τ

could defect to the strategy used by type s proposers.

It is possible however for vτ = vs even if kτ > ks. This can occur only when

Π Πτ = s . Thus, if both types equally attractive coalition partners a difference in veto

powers gives them no relative advantage over one another. As an example of this

possibility, suppose that there are three types 1, 2 and 3 where m1 = m2 = 1 and m3 = N-2.

Further, let k1 = N, k2 = N-1, and k3 = 2.13 Then, clearly all types have a dominant

strategy to build coalitions with type τ and s. The system of equations implied by

equations [1], [2], and [3] is

i) z v1 21= − δ iv) Nv z N v1 1 11= + −1 6δii) z v2 11= − δ v) Nv z N v2 2 21= + −1 6δiii) z v v3 1 21= − −δ δ vi) Nv z3 3=

Equations i) and ii) imply that z v z v1 1 2 2− = −δ δ . From iv) and v) this implies that

N N v N N v− = −δ δ1 2 or v1 = v2. Note that this calculation has only used up only two

equations so we can still plug ii) into iv) to get v vN N

1 2 1

2= =

− −δ1 6 . These results

13 Note that for type 3 to be distinct, we require N ≥ 4.

12

along with iii) and vi) imply vN N

3 1

2= −

− −δ

δ1 6 . To validate that the optimality of

excluding type 3 in all of the coalitions is a best response, we only need to check to see

that

Nv3 > v1 + v2 and (N-2)v3 + v1 > v1 + v2 which holds when N ≥−

−−

%&'()*max ,

2

1

3 2

1δδ

δ.

As this example shows, there is no guarantee that more veto power leads to

strictly higher payoffs. If N is sufficiently large, it is too costly to ignore either types 1

or 2 under any circumstances. The fact that type 1 has slightly more veto power is

irrelevant. As we will see in the more extensive analysis below, this irrelevance finding

holds in a wide range of cases -- not just in the (admittedly) special one here.

IV. The Case of Two Veto Types

We proceed now to a more elaborate example of the effects of veto powers. Let

there be two types with veto powers k1 and k2 where k k1 2> . Let the group sizes be m1

and m2 where m1 + m2 = N. Proposition 2 presents the ratio of the continuation values of

each type as a function of the veto powers, groups sizes and discount factors.

Proposition 2: The following are the ratiosv

v

1

2 for the symmetric, stationary Nash

equilibrium in the two type game:

If k1 ≤ m1, v

v

1

21= .

13

If k1 > m1, k2 ≥ m1, and k1 ≥ m2, then v

v

m k m

m

N k

N k

1

2

2 2 1

2

2

11=

− −−

−−

%&K'K

()K*K

min ,δ

δ2 71 6 .

If k1 > m1, k2 > m1, and k1 < m2, then v

v

m k m

m

m k k

m

1

2

2 2 1

2

1 1 2

11=

− −−

+ −%&K'K

()K*K

min ,δ

δ2 71 6 .

If k1 > m1, k2 < m1, and k1 ≥ m2, then v

v

N m

N k

1

2

1

1

1

1

1=−

− −−

%&'()*min ,

δ.

If k1 > m1, k2 < m1, and k1 ≥ m2, then v

v

k

m

1

2

1

1

1

1

1=−

−%&'()*min ,

δ.

The exact continuation values can be found by applying the result of Lemma 1 that

m v m v1 1 2 2 1+ = .

Proof: See appendix

In order to visualize the results of Proposition 2, Figure 1 contains results from

the case where m1 = m2 = 200 and δ = .9. Once again we find that under certain

conditions higher veto powers may be irrelevant. In the case where k k m2 1 1< < , both

types have the same continuation values. In this case it will always be cheaper to satisfy

the k1 override requirement than satisfy all m1 members of type 1. If v1 > v2, the type 1’s

would have a much lower probability of selection than type 2’s which would tend to

lower their continuation values and raise those of type 2 until v v1 2= . Not surprisingly,

we find that as the gap between k1 and k2 narrows, the ratio v

v

1

2 tends toward unity.

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Insert Figure 1 About Here

In order to see the effects of discount factors on veto power, Figure 2 plots the

continuation value ratios for different values of δ and k1 where N = 400, m1 = m2 = 200,

and k2 = 200. Clearly, patience benefits the members of type 1. Intuitively, veto power

is power to continue the bargaining in hopes of a better outcome in the future. If the

future is valueless, the veto is valueless and the ratio v

v

1

2 converges to 1.

Insert Figure 2 About Here

V. An Aside: Veto Power and Power Indices

One of the earliest methods for measuring the distribution of power within a

legislative body was proposed by Shapley and Shubik (1954). The Shapley-Shubik (SS)

index relates the probability that an individual’s support is pivotal to that member’s

power within the body. The SS index is the proportion of the permutations of the

legislators for which a given legislator converts a losing coalition to a winning coalition.

Formally, let S be the set of winning coalitions that are losing coalitions if member i is

removed. Then

SSs N s

Nis S

=− −

∈∑

12 7 2 7! !

!

where |s| is the number of members in coalition s.

15

To build some intuition about the relation between the SS index and our model,

consider the following example. Suppose that there are three legislators 1, 2, 3 where

k1 = k2 = 2 and k3 = 3. There are the following six permutations of these legislators:

1 2 3*

1 3* 2

2 1 3*

2 3* 1

3 1* 2

3 2* 1

Note that in all of the cases denoted by * the indicated legislator is pivotal because the

coalition of the members preceding it is a loser. Therefore, we can compute the SS

indices are {1/6, 1/6, 4/6} for 1,2, and 3 respectively.

While the SS index has received strong support and has well developed

behavioral foundations (see Roth (1977) ), our model suggests some of its liabilities.14

To see these, we will compare the SS indices from our example with the equilibrium

continuation values of our model. From Proposition 2, we get v v1 2

2 2

6 5= = −

−δδ

and

v3

2

6 5= −

−δδ

from our example. Thus, the continuation values only match the SS index at

δ = 6/7. Such a result might be less troubling if the SS index represented a limiting case

14 These liabilities, as we shall see, are not peculiar to the SS index but are true otherindices such as the Banzhaf. See Rapoport and Golan (1985) for a good review of theliterature on power indices and an application.

16

such δ → 0 or δ → 1, but δ = 6/7 is rather arbitrary. Not only does the SS index ignore

the time horizons of legislators, its accuracy depends on them.

A second problem with the SS index is that it assumes that coalitions are drawn

randomly. This assumption is more problematic than the assumption of randomization

over proposers. Because of random coalitions, the SS index for type τ will be strictly

greater than that of type s if kτ > ks. Proposition 3 which is stated and proven in the

appendix proves this claim for the case of two veto types. The logic of this result is that τ

will be a part of more possible coalitions which raises her pivot probability. However, as

we saw above, raising the override threshold may not increase the continuation values.

For example, in Proposition 2, if m1 > k1, then v1 = v2. The reason more veto power is

moot in this case is that the coalition with all members of type 1 never forms in

equilibrium. In general, SS indices may overstate the effect of veto powers.

VI. Asymmetric Proposal Power

Allocations of proposal power also may effect the distribution of power within a

legislative body. The proposer is advantaged in two ways. First, the proposer can

guarantee at least her discounted continuation value by making a proposal that would be

defeated. Secondly and most importantly, the proposer is the residual claimant on any

resources that remain once a majority coalition is formed. In fact, the proposer is the

only person who receives more than her continuation value as all other legislators either

receive 0 or δvi. So it would seem that the probability of proposing would have a

17

potentially large effect on a legislators payoffs. This logic turns out, however, to be only

partially correct.

Clearly, some proposal power is crucial. In our model, any legislator with a zero

recognition probability must receive a zero benefit share in equilibrium. This result,

summarized in Proposition 4, holds regardless of veto powers.

Proposition 4: If pi = 0, vi = 0.

The proof of this proposition is extremely straight forward. Equation [3] implies

that when pτ = 0, v vτ τ τδ= Π . Since δ and Πτ are less that equal 0, the only solution is

vτ = 0. Logically, if a legislator never receives a payoff higher than her average payoff,

her average payoff must be zero. It is interesting that no similar result exists for veto

powers. Even if kτ = 0, vτ may be positive since the votes of type τ may be used to

override the vetoes of other types.

For a more general examination of proposal powers, we focus on the effects of

proposal power for the case of ki = k. Like veto power, we establish that enhanced

proposal power increases continuation values, but only weakly.

Proposition 5: Suppose that ki is constant across i. Then, in any symmetric, stationary

Nash equilibrium, v vsτ ≥ if pτ > ps.

proof: See appendix

18

The logic of Proposition 5 is clear. If pτ > ps and vτ < vs, then type s would have

a lower probability of selection as both a proposer and as a coalition partner. However,

this is inconsistent with vτ < vs. Nevertheless, just as in the case of veto powers, it is

possible for pτ > ps and vτ = vs. If vτ > vs, the probability that τ is selected as a coalition

partner may be substantially lower than that of s. If pτ is only slightly larger than ps, the

negative effect will dominate driving vτ down to vs. The proposal power advantage must

therefore be substantial in order to overcome the effect of a lowered probability of

coalition participation. Proposition 6 presents the ratios of the continuation values

necessary conditions for proposal power to be valuable for the two type case.

Proposition 6: In a symmetric, stationary, subgame perfect Nash equilibrium for the

model with two types,

i) v

v

p

p

N m k m

N m

1

2

2

1

2 2

21

1= ⋅

− − −

− −

%&K'K

()K*K

max ,2 7 2 7

2 71 6δ

δ when m2 < k

= ⋅− −

%&'()*

max ,11

2

1

2

2

p

p

m

m kδ1 6 when m2 ≥ k

ii) v1 > v2 if and only if

p

p

N m k m

N m

1

2

2 2

2 1>

− − −− −

2 7 2 72 70 5

δδ

when m2 < k

p

p

m

m k

1

2

2

2 1>

− −δ0 5 when m2 ≥ k

19

iii) The exact continuation values can be obtained from i) and the result from lemma 1

that m v m v1 1 2 2 1+ = .

proof: See appendix

Figure 3 presents these ratios for 400 legislators and k = 200 as a function of m2.

Notice that the ratio of type 1’s proposals power to type 2’s must be substantially greater

than one for it to have any effect. Unlike veto power that tends to increase the number of

coalitions in which a member participates, proposal power tends to greatly reduce the

number. Only if the increased probability of proposing compensates is proposal power

valuable.

Proposition 6 also indicates, not unsurprisingly, that proposal power is more

important the more impatient the players are. Note that in the limit as δ → 0, the critical

value of p

p

1

2 goes to 1. In fact, in the limit continuation values are proportional to the

recognition probabilities.

VII. A Comparison of Veto and Proposal Powers

A comparison of the effects of veto power and proposal power is at best a slippery

exercise. After all, it’s hard to say where or not a .01 increase in the probability of

recognition is comparable to a 10 seat increase in an override threshold. Nevertheless,

we will attempt to undertake such a comparison. We will compare a legislature with

asymmetric proposal power with one with a asymmetric veto power. In each case, we

20

will limit ourselves to the two-type case. The primary question of interest is given an

asymmetric in veto power, how asymmetric must proposal power be to generate the same

legislative outcomes.15 This can be accomplished by choosing p

p

1

2 to equate the ratiov

v

1

2

from Propositions 2 and 6 for given values of k1 and k2. Since the relevant algebraic

solutions are messy and less than informative, Figure 4 presents these solutions for the

case of N = 400, k2 = 200, and δ = .9 as a function of k1 for different numbers of type 1’s

(and therefore type 2’s). Note that as k1 increases, p

p

1

2 must get substantially larger to

compensate. Also the fewer members of type 1, the more asymmetric proposal power

must be to compensate for a higher k1.

Insert Figure 4 Here

Acknowledging the difficulty of making precise comparisons, our results make a

much stronger case for veto powers over proposal powers than one of the most widely

applied analyses. The agenda setting model of Romer and Rosenthal (1978) has been

used to study policymaking in a large number of contexts. In the Romer-Rosenthal

model, a proposer chooses a policy alternative to an exogenous status quo. A veto player

chooses whether to accept the new policy or to maintain the status quo. The game ends

either with the proposal or the status quo as the outcome. The well-known result is that

the proposer does better than the vetoer. The only situation in which the vetoer can

15 An alternative approach to this question would be compare two groups within thesame legislature where one group is veto advantaged and the other is proposaladvantaged. We leave this extension to future work.

21

increase her utility beyond that of the status quo is when the proposer gets her most

preferred policy.

It is the dynamism of the sequential choice model that undermines the predictions

of the static Romer-Rosenthal model. As we pointed out before, veto powers are little

more than the option to continue the game in the chance that a member will become the

next proposer. In a static context, these powers are not as important. Proposal powers,

on the other hand, are less potent for two reasons. First, they make the proposer more

vulnerable to being omitted from future coalitions. Secondly, the option to continue the

game prevents the proposer from making credible ultimatums.

The effects of the future considerations can be seen by repeating the analysis of

Figure 4 for different levels of the discount factor.

Insert Figure 5 Here

Clearly, any advantage that veto power has over proposal power dissipates as the

future becomes more irrelevant. In the limit as δ → 0, proposal power clearly dominates

as v

v

p

p

1

2

1

2= for asymmetric proposal power while v

v

1

21= for any distribution of veto

power.

VIII. Conclusions

To argue that institutions may effect outcomes is no longer very controversial in

political science. The question now is to be able to make more precise and general

22

predictions about the relationship between certain procedures, rules, and outcomes. In

this paper, we have attempted to formulate a relatively general model to capture two

important institutional features: veto power and proposal power.

Of the predictions that our model, perhaps the most important are those about

when institutional prerogatives do not effect outcomes. For example, we find that veto

powers are ineffective when they are shared by many members and when members are

very impatient. In the same vein, we find that proposal powers are not important unless

they are especially asymmetrically distributed or when legislators are quite impatient.

Perhaps our model will provide a roadmap to explore which powers are likely to be

determinative in various institutional settings.

Another lesson of the model pertains to the question of how to design a legislative

institution. Rather than provide a theory that predicts whether designers will focus on

veto or proposal powers, our results suggest that any such theory is likely to be

underdetermined. Either veto power or proposal power can in principal be used to

engineer any set of (expected) outcomes in this model. The choice of institutional

arrangements is likely to be determined by considerations outside the purely distributive

setup we have employed.

23

Bibliography

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Political Science Review 89: 1181-1206.

Baron, David P. , and John A. Ferejohn. 1989b. “The Power to Propose.” In Models of

Strategic Choice in Politics, edited by Peter C. Ordeshook. Ann Arbor: University

of Michigan Press.

Baron, David P. , and Ehud Kalai. 1993. “The Simplest Equilibrium of a Majority Rule

Division Game.” Journal of Economic Theory 61: 290-301.

Hammond, Thomas H., and Gary J. Miller. 1987. “The Core of The Constitution.”

American Political Science Review 81: 1155-1174.

Harrington, Joseph E. 1990. “The Power of a Proposal Maker in a Model of Endogenous

Agenda Formation.” Public Choice 64(1): 1-20.

Rapoport, Amnon, and Esther Golan. 1985. “Assessment of Political Power in the Israeli

Knesset.” American Political Science Review 79: 673-692.

Romer, Thomas, and Howard Rosenthal. 1978. “Political Resource Allocation,

Controlled Agendas, and the Status Quo.” Public Choice 33(1): 27-44.

Roth, Alvin E. 1977. “The Shapley Value as a von Neumann-Morgenstern Utility.”

Econometrica 45: 657-664.

Shapley, Lloyd S., and Martin Shubik. 1954. “A Method for Evaluating the Distribution

of Power in a Committee System.” The American Political Science Review 48(3):

787-792.

Tsebellis, George. 1994. “Decision-Making in Political Systems: Veto Players in

Presidentialism, Parliamentarism, Multicameralism, and Multipartism.” British

Journal of Political Science 25: 289-325.

Winter, Eyal. 1996. “Voting and Vetoing.” American Political Science Review 90(4):

813-823.

Appendix

Lemma 1: In any stationary, symmetric, subgame perfect Nash equilibrium:

1) The bargaining with last only one round

2) vii

N

==∑ 1

1

.

proof: First we wish to show that vii

N

=∑ ≤

1

1. Because of discounting, the maximum sum

of continuation values must come when agreement is reached immediately. When there

is immediate agreement, vi = E(xi) where E is the expectations operator over the

probability distribution of proposals. Feasibility requires and optimality requires

xii

N

=∑ =

1

1 and the laws of the expectations operator therefore imply E xii

N

1 6=∑ =

1

1. Thus, if

we have immediate agreement, vii

N

=∑ =

1

1 otherwise vii

N

=∑ <

1

1.

To show that we in fact do have immediate agreement, we need to show that

z vi i≥ δ . The most undesirable case for any proposer is ki = N for all i so if we can

establish the claim for this case we are done. In this case, z vj ii j

= −≠∑1 δ which may be

re-written as z v vj j ii

N

− = −=∑δ δ1

1

which is positive since vii

N

=∑ ≤

1

1 and δ < 1. Therefore,

given immediate agreement vii

N

=∑ =

1

1.

♦

The following lemma is useful in several of the propositions proven below.

Lemma 2: In any stationary, symmetric Nash equilibrium,

i) If πτs > 0, then z v z vs sτ τδ δ− ≥ − .

ii) If πτs = 0 and πs

s > 0, then z z vs sτ δ≥ −

iii) If πτs = 0 and πs

s = 0, then zτ ≥ zs.

iv) If πτs = 0 and πτ

τ = 0, zs ≥ zτ.

proof:

i) Suppose that z v v zs sτ τδ δ+ − < . Then the type τ could raise her payoff choosing

the same coalition as a type s proposer. Since πτs > 0, one of type s’s best

responses must include a coalition with at least one type τ. Therefore, τ may

mimic s with a coalition which requires one additional type s and includes one

fewer type τ. This defect will necessarily pay since z v v zs sτ τδ δ+ − < .

Therefore, we must have z v z vs sτ τδ δ− ≥ − .

ii) Suppose that z v zs sτ δ+ < . Then τ can mimic s at the cost of an additional type s.

iii) In this case, τ can exactly mimic s.

iv) In this case, s can exactly mimic τ with perhaps one fewer type s.

♦

Proposition 1: Suppose that pi = 1/N for all i. Then, in any symmetric, stationary,

subgame perfect Nash equilibrium, v vsτ ≥ if kτ < ks.

proof: Suppose not i.e. vτ < vs and kτ > ks. In this case, there is not optimal to include a

type s in a coalition at the expense of a type τ. Type τ is cheaper and is required for more

coalitions than s. Therefore, π πτr r

s≥ in for all types so that Π Πτ ≥ s as well.

Since pi = 1/N for all i, recall that Nv z N vτ τ τ τδ= + Π and Nv z N vs s s s= + δ Π .

Consider two cases. First, suppose π πτs s

s= = 0. From Lemma 2(iii), this implies that

z zsτ ≥ so that N N v N vs s− ≥ −δ δΠτ τΠ which is inconsistent with vτ < vs and

Π Πτ ≥ s . Now suppose that πτs > 0. Then using the results of Lemma 2(i) and doing

some algebra, we find that N N v N N vs s− − ≥ − −δ δ δ δτ τΠ Π which is inconsistent

with vτ < vs and Π Πτ ≥ s .

♦

Proposition 2: The following are the ratiosv

v

1

2 for the symmetric, stationary, subgame

perfect Nash equilibrium in the two type game:

If k1 ≤ m1, v

v

1

21= .

If k1 > m1, k2 ≥ m1, and k1 ≥ m2, then v

v

m k m

m

N k

N k

1

2

2 2 1

2

2

11=

− −−

−−

%&K'K

()K*K

min ,δ

δ2 71 6 .

If k1 > m1, k2 > m1, and k1 < m2, then v

v

m k m

m

m k k

m

1

2

2 2 1

2

1 1 2

11=

− −−

+ −%&K'K

()K*K

min ,δ

δ2 71 6 .

If k1 > m1, k2 < m1, and k1 ≥ m2, then v

v

N m

N k

1

2

1

1

1

1

1=−

− −−

%&'()*min ,

δ.

If k1 > m1, k2 < m1, and k1 ≥ m2, then v

v

k

m

1

2

1

1

1

1

1=−

−%&'()*min ,

δ.

proof:

k1 < m1: There are two cases m2 ≥ k2 and m2 < k2. Consider the case of m2 ≥ k2. Given

any values of v1 and v2, it is cheaper to build a k2 coalition. Suppose that v1 > v2, then

type 1’s would not be included in any coalitions since there are at least k2 members of

type 2. Therefore, Lemma 2 iv) implies that z z2 1≥ . The continuation values for each

type solve vz

N1

1

= and vz

N

k

Nv2

2 221= + − δ . Together these imply that

Nv N k v1 2 2≤ − −δ δ which cannot hold if v1 < v2. The only remaining possibility is

v v1 2= . The case of m2 < k2 is similar.

k1 > m1, k2 > m1, and k1 > m2: There are two possible types of equilibria in this case.

There is a pure strategy equilibrium where both types form k2 coalitions and a mixed

strategy equilibrium where type 2 proposers mix between k1 and k2 coalitions. In the

pure strategy equilibrium, both types make offers to both types since m1 < k2. Lemma 2.i

suggests that z v z v1 1 2 2− = −δ δ which implies that N N v N N v− − = − −δ δ δ δΠ Π1 1 2 2 .

Since all proposers give to all type 1’s NΠ1 = N-1. Since type 2’s are used to complete k2

coalitions, type 1 proposers will select them with probability k m

m

2 1

2

− and type 2’s select

them with probability k m

m

2 1

2

1

1

− −−

. Thus, Nm

mk m k mΠ2

1

22 1 2 1 1= − + − −2 7 . Substituting

these above and doing a little algebra yields v m v m k m1 2 2 2 2 11− = − −δ δ1 6 2 7 .

In the mixed strategy equilibrium, a type 2 proposer must be indifferent between a

k1 coalition which costs δ δk m v m v1 2 1 2 21− + −2 7 2 7 and a k2 coalition which costs

δ δm v k m v1 1 2 1 21+ − −2 7 . Indifference requires that N k v N k v− = −1 1 2 22 7 2 7 . If

m k m

m

N k

N k

2 2 1

2

2

11

− −−

> −−

δδ

2 71 6 , then the pure strategy equilibrium does not exist and if

m k m

m

N k

N k

2 2 1

2

2

11

− −−

< −−

δδ

2 71 6 the mixed strategy does not exist. Therefore,

v

v

m k m

m

N k

N k

1

2

2 2 1

2

2

11=

− −−

−−

%&K'K

()K*K

min ,δ

δ2 71 6 .

The other cases can be proven in a similar fashion.

♦

Proposition 3: In the two type model if k1 > k2, SS1 > SS2.

proof:

Let there be two types where k1 > k2. There are two distinct ways that a type 2 can be

pivotal:

1. Appear in the k2-th position following all m1 members of type 1. In this case, the

vetoes of other type 2s will be overridden. The number of permutations of this form

can be given as k N m

k m

2 1

2 1

1 1

1

− − −

− −2 7 2 7

2 7! !

!.

2. Appear in the k1-th position following j < m1 members of type 1. In this case all

vetoes will be overridden. The number of permutations of this form is

k N kN m

k j

m

j1 1

1

1

1

11

1− −

− −− −

���

������

���2 7 2 7! ! .

Therefore, the Shapley-Shubik Index for Type 2 is

k N m

k m N

k N k

N

N m

k j

m

jj

m2 1

2 1

1 1 1

1

1

0

11 1

1

1 1

1

1− − −

− −+

− − − −− −

���

������

���=

−

∑2 7 2 72 7

2 7 2 7! !

! !

! !

!

For a type 1 to be pivotal, the possibilities are

1. Appear as the m1-th member of type one in any position greater than or equal k2 and

less than k1. The number of permutations satisfying this requirement is

i N m

i mi k

k − −

−=

−

∑1 1

1

1

2

1 1 6 2 72 7

! !

!.

2. Appear in the k1-th position following j < m1 members of type 1. As above this is

given by k N kN m

k j

m

j1 1

1

1

1

11

1− −

− −− −

���

������

���2 7 2 7! ! .

Therefore,

SSi N m

i m N

k N k

N

N m

k j

m

ji k

k

j

m

1

1

1

1 1 1 1

1

1

0

11 1 1

12

1 1

=− −

−+

− − − −− −

���

������

���=

−

=

−

∑ ∑1 6 2 7

2 72 7 2 7! !

! !

! !

!.

To prove the claim, note that SS SSi N m

i m N

k N m

k m Ni k

k

1 2

1

1

1 2 1

2 1

1 1 1

2

1

− =− −

−

�!

"$## −

− − −

−=

−

∑1 6 2 7

2 72 7 2 7

2 7! !

! !

! !

! !

which must be positive for any k1 > k2 since the first term of the summand is greater than

k N m

k m

2 1

2 1

1 1

1

− − −

− −2 7 2 7

2 7! !

!.

Proof of Proposition 5: Suppose that ki is constant across i. Then, in any symmetric,

stationary, subgame perfect Nash equilibrium, v vsτ ≥ if pτ > ps.

proof: The proof is nearly identical to that of Proposition 1 except that 1/N is replaced

with pτ and ps.

♦

Proposition 6: In a symmetric, stationary, subgame perfect Nash equilibrium for the

model with two types,

i) v

v

p

p

N m k m

N m

1

2

2

1

2 2

21

1= ⋅

− − −

− −

%&K'K

()K*K

max ,2 7 2 7

2 71 6δ

δ when m2 < k

= ⋅− −

%&'()*

max ,11

2

1

2

2

p

p

m

m kδ1 6 when m2 ≥ k

ii) v1 > v2 if and only if

p

p

N m k m

N m

1

2

2 2

2 1>

− − −− −

2 7 2 72 70 5

δδ

when m2 < k

p

p

m

m k

1

2

2

2 1>

− −δ0 5 when m2 ≥ k

Proof: We will consider only the case of m2 < k. The other case is very similar. From

Proposition 5, we know that if p1 > p2, v1 ≥ v2. Suppose that v1 > v2. Then all proposers

will choose type 2’s first. Since m2 < k, both proposers will have to choose some of both

types. Therefore, from Lemma 2.i, z v z v1 1 2 2− = −δ δ which implies

1 111

12

2

2− − = − −δΠ δ δΠ δv

p

v

p. Since all type 2’s are always chosen, Π2 21= − p .

Type 1’s are chosen with probabilities k m

m

− −−

2

1

1

1 and

k m

m

− 2

1 by types 1 and 2

respectively. Therefore, Π1 1 2 2 22

11= − − + −

p k m p mk m

m2 7 . Plugging these in and doing

algebra, we v

v

p

p

N m k m

N m

1

2

2

1

2 2

2 1= ⋅

− − −

− −2 7 2 7

2 71 6δ

δ which must be greater than 1 if v1 > v2.

Otherwise, we must have v1 = v2. The rest of the proposition follows trivially.

♦

Figure 1

20.080.0

140.200.

260.320.

380.

100

180

260

340

1

3

5

7

9

11

Veto Power of Group 1

Ratio

Veto Power of Group 2

Continuation Value Ratios

Figure 2

Effect of Discounting on Veto Power

1

3

5

7

9

11

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

Override Threshold for Type 1

Rat

io v

1/v2

Delta = .9 Delta = .7 Delta = .5 Delta = .3

Figure 3

Critical Proposal Power Ratios

1

3

5

7

9

11

10 30 50 70 90 110

130

150

170

190

210

230

250

270

290

310

330

350

370

390

Number of Type 2 Members

Rat

io p

1/p

2

delta = .9 delta = .7 delta = .5

Figure 4

Proposal Power Needed to Balance Veto Power

0

20

40

60

80

100

120

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

Override Threshold for Type 1

Rat

io p

1/p

2

m1 = 100 m1 = 200 m1= 300

Figure 5

Proposal Power Needed to Balance Veto Power

1

3

5

7

9

11

13

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

Override Threshold for Type 1

Rat

io p

1/p

2

Delta = .7 Delta = .5 Delta = .3