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Party voting discipline and the power of parties, legislators and states
539 Europ~iische Zeitschrift fiir Politische Okonomie] European Journal of Political Economy, 1/4 (1985) 539--561 © VVF, Munich
PARTY VOTING DISCIPLINE AND THE POWER OF PARTIES, LEGISLATORS AND STATES D. Marc Kilgour and Terrence J. Levesque*
1. Introduction
In many social decision procedures the actual decision-makers "represent" others who do not participate directly in the process but whose influence is posited to derive from the particular representatives they selected. Examples include parliamentarians or congressmen and the voters of their electoral districts, and members of university committees constituted along faculty lines and their faculty colleagues. Indeed, this structure lies at the heart of representative democracy. It is natural to ask how much influence constituencies have in this decision-making process, and how much influence belongs to the representatives themselves. Some answers have been provided in specific cases by the models and methods of n-person cooperative game theory. For example, the Banzhaf-Coleman and Shapley-Shubik indices of decision-making power have been calculated for American states under the Electoral College procedure for electing a president (Banzhaf, 1968; Mann and Shapley, 1962; Owen, 1975). We report elsewhere similar calculations for the Canadian provinces under the amending formula for the Canadian Constitution (Kilgour and Levesque, 1984). In each of these studies, an important and implicit assumption is that representatives are fully controlled by their constituencies, as if each representative were simply a mechanism for reporting the constituency Department of Mathematics and Department of Economics, respectively, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, Canada. The authors gratefully acknowledge the support of the Social Science and Humanities Research Council of Canada (Grant No. 410-84-0425) and thank the editor and an anonymous referee for suggestions which have improved the article.
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Party Voting Discipline and Power
decision. This is an empirically accurate characterization of the relationship between states and their electors under the Electoral College rule I . Its accuracy in the Canadian example, and in most others, depends on the extent to which electoral competition disciplines representatives to reflect constituency preferences accurately. It is intuitively clear that the more freedom the representatives have, the more p o w e r they have, and the less p o w e r remains with the constituencies. In many political decision-making situations legislators are members of parties which claim their allegiance in varying degrees. In other words, parties compete with constituencies for influence over legislators' voting behavior. A party's success in securing decision-making power thus depends on voting discipline - the ability to restrict coalition formation that crosses party lines (Luce and Rogow, 1956). In this essay we examine the sharing of decision-making p o w e r among the three major elements of a typical legislative setting, parties, legislators, and constituencies. We wish to ascertain, in particular, how the exercise of party voting discipline attenuates the p o w e r of constituencies. For example, we expect apriori that constituencies have no power when all legislators belong to a party and always adhere to its position. In contrast, when party discipline is negligible, a standard procedure (we use the BanzhafColeman power index) should measure constituency p o w e r accurately. In general, we believe that the p o w e r of one type of actor in a political decision-making setting can be assessed only with reference to the influence of all relevant actors, even when that influence is not part of the formal decision procedure. In other words, constitutions relate constituencies, legislators, and parties formally in terms of their roles in the political process, and a decision maker's p o w e r depends on his role -°, but it also depends on the conditions and conventions which govern the behavior of all components of the system. Our objective is to incorporate one such influence - the degree of party voting discipline - into the analysis of power in a legislature.
1
There have been a few instances in which electors did not cast their votes for their states' plurality presidential candidate, but this p h e n o m e n o n is rare.
'2 An example is the analysis of the distribution of formal power a m o n g the President, the Senate and the Congress of the United States (Shaplev and Shubik, 1954).
Party Voting Discipline and Power
541
2. Party Voting Discipline and Power We take as our starting point the proposition that parties in a legislature do not have total decision-making control themselves, but share it with the members of the legislature and the constituencies which elected them. For example, when the governing party depends on the support of a pivotal bloc of legislators it in some sense shares its power with that bloc even when voting discipline is strict. As voting discipline weakens some power flows to the constituencies insofar as their positions on issues are influential in determining their legislators' voting behavior. Our view, then, is that legislators have influence to the extent that they do not correlate their positions with either their parties' or their constituencies' preferences. We measure party voting discipline by the response of individual legislators to being "crosspressured", that is, to finding their party's position differs from their constituency's position on an issue. Party voting discipline is strict when legislators choose their party's position over their constituency's with certainty. Constituency control is strict when • all legislators always choose their constituency's position over their party's. Of course, there is a continuum of possibilities between these two cases. We are interested in assessing power not only at the extremes, but also through the range of intermediate cases. In accordance with the Banzhaf-Coleman power index, decision-making power is associated with an agent's opportunities to decide the outcome of an issue. Straffin (1978) refers to this view of power as answering a "question of effect on o u t c o m e " and Coleman (1971) calls it the " p o w e r to initiate". In both cases, the focus is on the ability of the agent to decide the issue. Analogously, our examination of party voting discipline concerns legislators' opportunities to decide issues, irrespective of tendencies of focal legislators under the relevant degree of party discipline. The constituency power that we define also has this property. However, both of our power definitions do take into account the effects of party affiliation and discipline on the behavior of nonfocal legislators. Straffin (1978) presents another approach that is natural here. He investigates the "question of group individual agreement", which asks how likely is an agent to agree with the group decision. His satisfaction index, a measure of the likelihood of groupindividual agreement, measures the value to the agent of partici-
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Party Voting Discipline and Power
pation in the decision-making procedure (rather than leaving it to others). Below we define a constituency's satisfaction and find, .just as Straffin did, that the value to a constituency of participation can be related simply to our index of the constituency's power. Several contributions in the literature describe extensions of the pivoting p o w e r concept that are related to our question. Kilgour (1974) discusses the effect on individual p o w e r of the refusal of some voters to participate in the same coalition (see also Brams (1975)). These "quarrelling players" bear some similarity to legislative members of different parties under rigid party discipline. Owen (1977, 1982) axiomatizes a p o w e r concept that includes the role of prior unions of voters. These unions can be thought of as formal models of legislatures divided strictly along party lines. Elsewhere, Owen (1975) examines the effects of political parties on p o w e r by modeling prior affinities among members to form coalitions. Another approach is taken by Straffin, Davis, and Brams (1982), w h o a t t e m p t to model explicitly the effects of ideology. Levesque (1984) formulates an empirical piw~ting p o w e r measure that accounts for restrictions on coalition formation associated with partisma identification. In all of this literature, specific restrictions on coalition formation determine a probability distribution over the potential legislative coalitions. Our model of decision p o w e r contrasts with those cited above (as applied to legislatures) in two respects. We attribute a definite role to constituencies as well as to parties and to legislators themselves. The earlier models postulate that parties or other prior arrangements place restrictions on legislators' decisions; our model focuses on the tension which arises when legislators' loyaltics are genuinely divided between party and constituency. Our model also yields natural measures of a constituency's p o w e r and satisfaction, whereas earlier models were unable to separate constituency effects and measures from the assessment of legislators' pivoting power.
3. The Model Legislature We now present our model of the legislative setting in preparation for our study of the p o w e r of parties, legislators, and constituencies. We assume that there are only two parties, A and B, and that each of the m members of the legislature represents one
Party Voting
543
Discipline and Power
o f the n constituencies. (A constituency may have more than one representative so that m / > n.) Every member of the legislature belongs to either party A or party B. Motions are passed in the legislature if they garner the support of a least a quota, q, of the legislators, where q > n/2. Finally, we assume that the legislative composition and the q u o t a are fixed. Our model legislature is restrictive on one dimension and general on two others. The assumption of a bipartisan legislature is empirically restrictive; we have given a generalization of the subsequent analysis to the multiparty case in Kilgour and Levesque (1985), where we consider the effects of various coalitions among the parties. While our assumption a b o u t the number of parties is specific, our characterization of constituencies and the legislative decision rule is quite general. Our analysis applies to multi-member as well as single-member constituencies, and to legislatures that operate with "special" majority decision rules, such as the twothirds majority required in the American Congress to overcome a Presidential veto. • To summarize and define notation formally, there are n/> 1 constituencies indexed by N =11, 2 . . . . , n I. Furthermore, each Constituency j e N is assigned rh,> 0 seats arid the two parties, A J • and B, hold mjA and mjB of these, respectwely. Of course m;A + mB = m.. The general form of our legislature is represented c~iaJ J Figure 1 : A General Legislative Composition Constituency 1
...
j
A
A
m1
B
m 1
mjA
...
n
...
m A n
Party
mB
B
...
m B n
grammatically in Figure 1. We also define the total party holdings mA = ~ m.A
m B = ~ m.B
j=l
j=l
J '
J "
Of course, the total n u m b e r of legislative seats is m = m A + m B .
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Party Voting Discipline and Power
Figure 2 shows a simple example legislature which we will use below for illustration. Figure 2: An Example ( q = 4) Constituency 1 2
3
A
2
1
1
B
0
1
1
Party
We assume that all propositions which arise in the legislature are binary, that is, there are only two relevant positions - that of party A and that of party B. We further assume that each constituency supports one of the two positions and we call a list of constituency preferences a constituency profile. In the example of Figure 2, the profile AAB indicates that constituencies 1 and 2 support party A's position, and constituency 3 supports party B's. Obsera,e that the party B legislator in constituency 2 is crosspressured; his voting decision will reflect the amount of party voting discipline exercised by party B. Constituency 2's party A member, in contrast, is not cross-pressured since both his party and constituency positions accord. The assumption that a constituency profile can exist merits some discussion since it includes the assumption that a definite constituency position on an issue exists. J u s t as we model each legislator as representing his entire constituency, so we model the constituency as having some majority preference on each issue. If a constituency tends to favor one side, say A, consistently, then most or all of its representatives will belong to party A, and those constituency profiles in which it supports A will be more c o m m o n than those in which it supports B. In this case, we do not allow a " m i n o r i t y " legislator to represent his "sub-constituency"; rather, we see him as cross-pressured in the sense defined above. In general, there are 2 n possible constituency profiles, since we exclude the possibility of constituency indifference on an issue. In deliberations on a large number of propositions, many different cleavages among the constituencies can be expected. That is, issues give rise to a distribution of constituency profiles. We assume that this distribution has the property that each con-
Party V o t i n g Discipline and P o w e r
545
stituency's position is (statistically) independent of the positions of all other constituencies 3 . Apart from this assumption, we do not restrict the distribution of profiles. We refer to the special distribution that makes all profiles equiprobable as the "a priori profile distribution". It is associated with a priori evaluations of power in our subsequent analysis. Similarly, an empirically-based distribution would correspond to an assessment of actual power. We turn now to the direct modeling of the actions of legislators in the presence of varying degrees of party discipline. We have already noted two extreme responses of legislators to being crosspressured - they always vote for the party or they always vote for the constituency. There is a range of intermediate responses that depend on the degree of party voting discipline. We model these in terms of a parameter p such that p e [0,1] where p = 1 corresponds to strict party discipline and p = 0 corresponds to strict constituency control. The parameter p represents the probability that a cross-pressured legislator votes with his party rather than with his constituency. We present in section 5 two specific models of legislators' actions consistent with this interpretation ofp. The parameter p describes legislators' responses only when they are cross-pressured. When the party position and the constituency position correspond we assume that the legislator votes for this c o m m o n position. In our example (Figure 2), party A's members in constituency 1 both vote for party A's position given the profile AAB - in total party A receives at least three and at most five votes under this profile.
4. Measures o f Power and Satisfaction In this section we present formal definitions of the measures of power and satisfaction for the model legislature described in Section 3. Equivalent definitions of our measures are also provided, along with some examination of their interrelationships. As discussed in Section 2, our first notion of the power of an actor (or group of actors) is ability to affect the outcome. In the c o n t e x t of our model legislature, this means the ability to deliver a victory to party A, or to party B, or to either one. Below we
3
It is possible to modify some of our analysis to cover more general cases. However, we chose to follow the present approach in the interests of simplicity.
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Party Voting Discipline and Power
connect this idea of being pivotal with a second concept of power that of getting one's own way. With each constituency we associate two kinds of p o w e r legislators' power and constituency power. Legislators' power is direct power - it measures the ability of a constituency's representatives, considered as a group, to affect the outcome. In contrast constituency power is indirect power - it measures the extent to which a constituency can affect the o u t c o m e by changing its position on an issue, thereby influencing its representatives' voting behavior. We begin with legislators' power. Note that this power is shared by all the representatives of the focal constituency, w i t h o u t regard to their pary affiliation. Legislators' Power for constituency j is the probability that there is a winning party which would not win if its support from j's representatives we are totally withdrawn. Legislators' power depends on the idea that the winning party "owes something" to the representatives of constituencyj if they were the difference between winning and not winning. In the calculation of how often constituency j's representatives are " o w e d something", no account is taken of constituency j's position, nor of the support which some of constituency j's representatives might automatically give because their constituency and party preferences agree. All that matters is w h e t h e r j's representatives, considered as a bloc, pivot for the winning party. Of course the preferences of constituencies other than j, and the party allegiance of their members, do make a difference in the calculation of the legislators' p o w e r for constituency j. We n o w provide a formal definition of legislators' power. For S c N, S ~ ~, let A(S) be the r a n d o m variable denoting the total n u m b e r of votes for party A from the constituencies in S, and define B(S) similarly 4. T h e n the legislators' power for constituency j is -
(1)
LP(j) = Pr 1A(N) ~> q I - PrIA(N - j ) ~> q l + PrIB(N)~ql
- PrIB(N - - j ) ~ q l
because, f o r e x a m p l e , the event I A ( N ) ~ ql means "A wins" and t A ( N - - i ) / > qt) means "A wins even with no support f r o m j " . For 4
For cooveoieoco we write
,lill
etc
547
Party Voting Discipline and Power
actual computation of LP(j), an expansion of (1) is usually more convenient: Ill
(2)
.
LP(j) = hZ=J1 [PrlA(N - j ) : q - h I PrIA0) ~> h l
+PrlB(N-j):q-h I PrlB(J)~>hl] We discuss briefly the behavior of LP(j) at extreme values of the party discipline parameter p. When p = 1, party discipline is exact, and for each constituency j, A(j) = m~ and B(j) = m~, independent of the profile of constituency positions. Neither party wins if m A < q and m B < q; in this case LP(j) = 0 by (1). Otherwise, one party wins for certain, and (1) leads to LP(j) =
ll0
ifmA--m'~< J otherwise
q~< m A o r m S - - m . S < q ~ < m S J
In the example of Figure 2, LP(1) = LP(2) = LP(3) = 1 when p = 1, since party A wins for certain, and depends on all its members to do so. Now suppose that p = 0, so that members' votes are completely controlled by their constituencies. Denote the event that constituency j supports party A by jA, and define j B similarly. Then
,3,
A(j) =
lm ifA 0
J if J.B
B(j) =
10if
J m.J if J.B
and it follows from (2) that (4)
LP(j) = P r l q -
mj ~ A ( N - - j ) < q l
PrljAI
In particular, under the a priori distribution of profiles, the legislature becomes a weighted voting game played by the n constituencies (constituency j has mj votes) with quota q, and LP(j)=Bj[q;ml, ...,
m n]
the (absolute) Banzhaf value of the weighted voting game. In the example of Figure 2, LP(1) = LP(2) = LP(3) = 1/2 when p = 0, under the a priori profile distribution. Examples of the calculation of LP(j) for intermediate values
548
Party Voting Discipline and Power
of p must await the introduction of specific models of legislator action in Section 5. We turn now to the second kind of power which is of interest, constituency power. As n o t e d above, a constituency's only opportunity to influence the legislature's decision is through its influence on its representatives. Gaining a constituency's support • can never harm a party, since, for example, at most mjA o f . ,j s representatives vote for A in the event jB, whereas at least m A vote for A in the event j A. Constituency power for constituency j is the probability that some party wins with the support of constituency j and would lose if constituency j were to reverse its position. Note that constituency power focuses on the actions of a constituency, and only indirectly on the actions of the representatives of that constituency, which are the focus of legislators' power. Constituency power takes no account of any tendency for the focal constituency to favor one party over the other, b u t it does depend on the information about other constituencies contained in the profile distribution. Indeed the definitions of legislators' and constituency power differ only in their focus on the representatives of a constituency, or on the constituency itself. We now provide a more formal definition of constituency power. F o r h = 1 , 2 , . . . , m j , l e t QA(h) = PrljA I PrIA(j)~> h IjA I P r l A ( j ) < ( h ] j B I Q~(h)=PrljBIPrIB(j))h
liB I P r I B ( j ) < h l j A I .
For example, QA(h) is the probability that constituency j supports party A and provides A with at least h votes, b u t would provide A with fewer than h votes if it were to reverse its position and support B. The interpretation of Q~(h) is similar. It follows that constituency power of constituency j is (5)
rail
CP(j) = E j
Pr A(N
-j)
= q - h Q (h)
h=l
+PrlB(N-j)=q-hlQ~(h)]
.
Comparison of (5) with (2) shows clearly the difference between legislators' and constituency power. It is convenient to use (5) to describe the behavior of CP(j) at
549
Party V o t i n g Discipline and P o w e r
p = 1. As noted above, A(j) = m Ai and B(j) = m~, without regard to whether ;A orjB occurs. In particular, for any h = 1, 2, . . . , mj, either P~'IA(j) ~>h I = 0 or PrlA(j ) < h~= 0 so that QA(h) = 0 for every h. Similarly Q~(h) = 0, ~o that CP'(j) = 0 when p = 1, by (3). This result corresponds to the idea, suggested above, that constituencies are powerless when party discipline is strict. When p = 0, (3) applies, so that for any h = 1, 2, . . . , mj, (6)
Pr IA(j) ~> hljAl= Pr~A(j) < hljB 1 = 1
and therefore QA(h)= Pr IjAI. Similarly, Q~ (h)= PrljB 1 . CP(j):Prlq-m
j~
PrljAI
+ Prlq - mj ~< B(N - j ) < q I PrljBI. Comparison with (4) now shows that, if p = 0, CP(j) = LP(j) for any profile distribution. In particular, under the a priori profile distribution CP(j)=Bj[q;m 1 . . . . . m n] when p = 0. The example of Figure 2 has, for j = 1, 2, and 3, CP(j) = 0 when p = 1 and CP(j)= 1/2 (under the a prioridistribution) when p -- 0. We now give a characterization of constituency power, applicable for any value of the party discipline parameter p but only under the a priori profile distribution. Under this distribution, Pr IjAI = PrljB l. Therefore 5 , PrljA and A(N) ~>q I - PrljB and A(N)>~q I = PrIA(N)>~qljA I PrljA I - P r I A ( N ) / > qljB I PrljB I =PrljA I [PrlA(N)>i qljA 1 - PrIA(N ) ~> qljBl]
rlA(N-j)--q-h I PrlA(j) > hljAI Ill.
= ZJ P r l A ( N - - j ) = q - - h l Q A ( h ) . h=l
5
The following calculation depends on the assumption that the positions of different constituencies are independent.
550
Party Voting Discipline and Power
A similar calculation can be performed when B wins (B(N) >~ q), and so, by (5), (7)
CP(j) = Pr IjA and A(N)>~ q l -
PrljB and A(N) >~ q I
+ PrljB and B(N)/> ql - PrljA and B(N)>t ql" We have found that under the a priori distribution, the constituency power of constituency j is equal to the difference between the probabilities that j supports a party that wins and that j opposes a party that wins. We now model a constituency's satisfaction with the legislative decision process. Our definition is adapted from the ideas presented in Straffin (1978). Satisfaction for constituencyj is the probability thatj supports a party that wins. Clearly satisfaction measures the frequency with which a constituency is in accord with the group decision. A formal definition of constituency j's satisfaction is (8)
S(j) = PrljA and A(N)~> q l + PrljB and B(N)/> ql"
An expansion for satisfaction as defined in (8) is now given. Observe that if the e v e n t s jA and A(N)/> q both occur, then precisely one of the following three statements must be true: either (i) A would win with no votes fromj's representatives; or (ii) A needs some support from j's representatives, but would receive sufficient votes even ifj were to support B; or (iii) A gets enough votes to win only ifj supports A. This observation, with (8), implies that 6 (9)
S(j) = PrljA I [PrIA(N - j ) ~> q I A
+
m2; j
PrIA(N-j)=q-hlPrIA(j)~>hljB
I]
h= 1
nl •
+ Ej PrlA(N-j)=q--h
IQJA.(h)
h=l
+PrljBl[Pr B(N-j)>~q
6
+
Here, and subsequently, we assume that a s u m m a t i o n equals zero when its lower limit exceeds it u pper limit.
Party
Voting
Discipline
m
551
and Power
B .
~J PrIB(N --j) = q - hlPrlB(j) ~> hljAl] h=l
~J P r I B ( N - j ) : q - h I Q]~(h) .
In. +
h = 1
Furthermore, comparison of (9) with (5) shows that (10)
S(j) - CP(j) = PrljAI [Pr 1A(N --j) ~ ql m
+ I~ j P r l A ( N - j ) = q - h l P r l A ( j ) = h l j B m.
I]
ta = 1
+ PrljBI [PrIB(N --j) ~> ql + g m .
Y/ PrlB(N -j)=q-
hi Prig(j) = hljAI] .
h= l
Of course, the right side of (10) is simply the probability thatj supports some party that would win even without j's support. Since this probability is non-negative, S(j)/> CP(j) follows. It is easy to verify, from (8) or (9), that when p = 1, PrljA 1 ifmA ~ q S(j) =
PrljB I 0
if m 1~ /> q otherwise.
When p = 0, use of (3) and (6) in (9) yields SO) = PrljAIPrIA( N --j) ) q -- mjl + PrljBIPr IB( N - j ) ~ > q - m j l . A simple relationship holds between constituency satisfaction S(j) and constituency power CP(j) under the a priori profile distribution. Observe first from (7) and (8) that S ( j ) - CP(j)= PrljB~ and A(N)>i ql, + PrljA and B(N) >~ q l"
552
Party Voting Disciplineand Power
(The difference between constituency satisfaction and power is the probability that the constituency opposes a winning party.) Further, consider (11)
S(j) -- I/2CP(j) = 1/2 PrlA(N) >f q I + 1/2Pr 1B(N) ~> q I
by (7) and (8). Following Straffin (1978), we identify the right side of (11) with the satisfaction of a dummy constituency which supports a party (according to the a priori profile i.e. Pr t D At = Pr ID B I = 1/,2 ) but which has no power to affect the outc~ome)of thd dedision process. Thus
1/2 CP(j) = S(D)= PrlDA and A(N)>1 q l
S(j)
+ PrIDB and B(N) ~> q I which leads directly to (12)
CP(j) = 2[S(j) - S(D)].
Under the a priori profile distribution, the constituency power measure CP(j) shares the property reported by Straffin (1978): j's power is proportional to the additional satisfaction j derives from its part in the decision process, relative to a dummy constituency. We end this section with a demonstration that LP(j) ~ CP(j). Conditions for equality arc also given below. First, subtract (5) from (2) to give (13)
LP(j) - CP(j) m
.
=h=l Zj P r l A ( N - - j ) = q - - h l ' [ P r l A ( J ) > ~ h l - - Q i ( h ) l m
+ h=l~J P r l B ( N - - j ) = q - h I
• [PrlB(j)~>h I --QB(h
)]
But, by the definition of Qi(h), PrlA(j ) ~> h I-- Q~}(h)= PrlA(j )/> h 1 --PrIA(j) ~> h andjA i erlA(j) < hljB I ~> PrIA(j)~> h I -PrIA(j)>~ h andjA I >~0 using Pr Ia(j) < hljBl~< 1 and the observation that the probabili-
553
l'arty Voting Discipline and Power
ty of a conjunction o f two events cannot exceed the probability of one of the events. This calculation and its analogue show that the bracketed quantities on the right side of (13) are non-negative, proving that LP(j)/> CP(j). We have already noted that for any profile distribution, LP(j) = CP(j) when p A 0. When p = 1, CP(j) = 0 whereas LP(j) = 1 if either m A - mj < q ~< m A o r m B - roB< q ~< m B , and LP(j) = 0 otherwise. These observations a m o u n t to necessary and sufficient conditions for the two kinds of power to be equal in this case. We have now proven that LP(j) 1> CP(j) for any value of p and any profile distribution, and we have reported the cases p = 0 and p = 1 in detail. It can also be shown that, for any value of p, LP(j) = CP(j) under the a priori profile distribution if and only if either (i) or (ii)
p = 0; p = 1 and b o t h m A - m . A J
or (iii) 0 < p <
1 and e i t h e r m .A--0 J
or P r l q -- m A < ~ A(N - - j ) < ql = 0, and either m.J~ = 0 orPrlq--m.Bj ~ O. However, an example in which LP(j) = CP(j) for all p e [0, 1] is not difficult to construct.
5. Models o f Legislators' Actions In this section, we make our power and satisfaction concepts more concrete by proposing two specific models for legislators' actions, and by applying these models to the example of Figure 2. As set out in Section 3, a model of legislators' actions describes the voting behavior of cross-pressured legislators. This behavior must depend on a parameter p e [0, 1] which corresponds to the probability a cross-pressured legislator supports party over constituency. There must be complete constituency control w h e n p = 0 and strict party discipline when p = 1. The two models presented here differ only in the degree of statistical dependence (within cross-pressured blocs) that they specify.
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Party Voting Discipline and Power
Independent Agent Model (Model A)
Under the independent agent model for legislators' actions (Model A), each cross-pressured legislator is viewed as an agent operating independently of all other cross-pressured legislators. Thus each cross-pressured legislator votes for his party with probability p, and the actions of different cross-pressured legislators are (statistically) independent. The formal version of Model A is not difficult to obtain. If constituency j supports party B and m~> 0, then the probability that party A receives h votes from j's representatives is (conditionally) . , m Aj , PrIA(j) = h I = ( h ~ l ph(1--p)m AJ - ,hh = 0 , 1 , . . the familiar binomial probability on parameters mix and p. Of course, Pr tB(j) = k !~ = Pr tA(j) = m j - k I~ for k• j= mB,'m~ + 1, ~-k ~ • .... mj. If m'j = 0, tfien Pr Ia(j ) = 01 =PrlB(j) = mj I = 1. In case constituency j supports p~rty A, thdprob~bilities are'similar. We now work out the a priori legislators' power, constituency power, and constituency satisfaction for the simple legislature shown in Figure 2. The calculation will be given in detail for constituency 1 arid summarized for constituency 2. (The situation of constituency 3 is identical to that of constituency 2.) We now assume the a priori profile distribution. We determine the a priori legislators' power associated with constituency 1 using (2): Observe that if 1A occurs, then A(1) = 2 a n d B(1)= 0. Since Pr I1AI = Prll BI = 1/2, it is easy to verify that (14) PrIA(1)~> 1l = 1--(1--p)2/2 PrIA(1) = 2I = (1 + pZ)/2 PrIB(1) ~> 11 = ( 1 - - p 2 ) / 2 Pr IB(1) = 21 = (1 - p)Z/2. Now (2) implies that LP(1)=PrIA(2,3)=3IPrIA(1)~> +PrIB(2,3):31PrlB(1)/>
1l 11
+ Pr 1B(2, 3)= 21Pr 1B(1) = 2I.
Party Voting Discipline and Power
555
It is easy to derive t h a t (15)
PrIA(2,3)=3I
=PrlB(2,3)=3I=(4p-4p~)/4
PrlA(2, 3)-- 21= PrIB(2, 3)= 21= (2 - 4p + 6p'-')/4 which, w h e n s u b s t i t u t e d , yields LP(1) = (1 - p + 6p '2 - 9p :~ + 5p4)/2 A similar calculation gives LP(2) = LP(3) = (1 -- p + 5p'-' - 8p :~ + 5p4)/2 Figure 3: .-! Priori Legislators' Power and C o n s t i t u e n c y Power in the Legislature o f Figure 2 u n d e r Model A
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PartyDiscipl|ne The graphs o f LP(1) and LP(2) are s h o w n ill Figure 3. Observe t h a t LP(1) = LP(2) = 1/2 at p = 0, LP(1) = LP(2) = 1 at p = 1, and LP(1) > LP(2) for 0 < p < 1. F u r t h e r m o r e LP(1) decreases f r o m p = 0 to p ~ . 1071 and increases t h e r e a f t e r - the m i n i m u m value o f LP(1) is ~ .4757. Similarly, LP(2) decreases f r o m p = 0 to p ~ .1435, increasing thereafter. Its m i n i m u m is ~ .4690. We n o w calculate a priori c o n s t i t u e n c y p o w e r in our e x a m p l e legislature. It is n o t hard to s h o w t h a t , u n d e r Model A,
556 (16)
Party Voting Discipline and Power
Q:](1) = Q~(2) = (1 - p)Z/2 Q'] (2) = Q~(I) = (1 - p 2 ) / 2
so that (5) and (16) yield C,(1) : Pr IA(2, 3 ) : 31 Q ] ( 1 ) + PrlB(2, 3 ) = 31 Q~ (1) + PrlA(2, 3 ) : 21 QA ( 2 ) + ' r l B ( 2 , 3 ) : 2 1 Q ~ l =(1-p+pZ
(2)
_p~)/2.
A similar calculation gives CP(2) = CP(3) = (1 - p + pZ _ 2p.~ + p4)/2. Note that CP(1) /> CP(2), with equality only at p = 0 and p = 1; both CP(1) and CP(2) are strictly decreasing functions for p e [0, 1]. The graphs of CP(1) and CP(2) are shown in Figure 3, which makes it easy to compare constituency power CP(j) with the corresponding legislators' power LP(j) in our example legislature. We now turn to calculation of a constituency's satisfaction, under the a priori profile distribution and Model A, in the example legislature of Figure 2. We use (12), which can be inverted easily to yield SO) = S(D) + CP0)/2 S(D) = [Pr IA(N) ~> ql + Pr 1B(N)~> q l ] / 2 . In the legislature of Figure 2, it is easy to show that S(D) = (2 -- 3p + 7p 2 -- 9p 3 + 5p4)/4 so that S(1) = (3 - 4p + 8p" - 10p ~ + 5p4)/4, S(2) = S(3) = (3 - 4p + 8p" - 1 l p ~ + 6p 4 )[4. As it t o b e expected from Figure 3, S(1) and S(2) are strictly decreasing functions of p, from S(1) = S(2) = 3/4 when p = 0 to S(1) = S(2) = 1/2 w h e n p = l, and S(1) > S(2) for 0 < p < 1.
Party Voting
Discipline
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and Power
Bloc Model (Model B) In the bloc model for legislators' actions (Model B) all crosspressured representatives of a constituency vote together, as a bloc. Note that all members of a particular bloc must belong to the same party. With probability p, the bloc supports this party with all its votes; otherwise the party receives no votes at all from this bloc. The actions of different blocs are assumed to be independent. It is simple to formalize Model B. If constituency j supports party B and m A j > 0, then, conditionally, PrIA(j)=mA. 1 = p ; J
PrIA(j)=OI=
1--p.
If m .A = O, then PrtA(j) = O l = 1. Of course PrIB(j ) : k I. PrIA(j) ~ " = m. J -- "k ~~ tot k" =~ m.,B m.B )+ 1, . . . , m j . When constituency j supports party A, the O e t m m o n s are similar. We n o w apply Model B to the example legislature of Figure 2, indicating how our earlier calculations for Model A must be modified. To calculate the a priori legislators' power for constituency 1, (14) must be replaced by
PrIA(1)~> I I = P r I A ( 1 ) = 2 1
=(l+p)/2
PrIB(1)~> 11 = P r I B ( 1 ) = 2 1
=(I--p)/2.
Since (15) remains valid 7 (2) implies that LP(1) = (1 + pZ)/2. An analogous calculation gives LP(2) = LP(3) = (1 - p + 2p2)/2. The graphs of LP(1) and LP(2) are shown in Figure 4. To calculate constituency l's a priori constituency power under Model B in the example of Figure 2, we revise our previous calculation under Model A. First, (16) must be replaced by
Q~ (1) = QA (Z) = ( 1 - p)/2 Q~ (1) = Q~ (2) = (1 - p)12. 7
T h i s p h e n o m e n o n o c c u r s h e r e b e c a u s e the b l o c s f r o m c o n s t i t u e n c i e s 2 a n d 3 h a v e only one member, and therefore Models A and B act identically on them.
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Party V o t i n g Discipline and P o w e r
Figure 4: A Priori Legislators' Power and Constituency Power in the Legislature of Figure 2 under Model B 7 0.9 0.8 0.7 0.6
LP(2)- LP(3) o. 0.4 0.3 0.2
_ ,
_ IP(2)." CP(3),
~
o.1 0
•
o
•
0.2
0.4
0.6
0.8
"
PoNy Discipline
Insertion in (5) and use of (15) gives CP(1)=(1-p+p2
p3)/2.
Similarly, CP(2) = CP(3) = (1 - p)/2. These functions are also shown in Figure 4. Finally, calculations of a priori constituency satisfaction similar to those for Model A give S(1)=(3-3p+3p
z -p~)/4
S(2) = S(3) = (3 - 3p + 2p 2)/4. A brief comparison of power u n d e r Models A and B is of interest. Since Model A and Model B agree at p = 0 and at p = 1, only values of p satisfying 0 < p < 1 will be reported. For constituency 1, legislators' power is always greater u n d e r Model B, and constituency power is always equal under the two models. For constituency 2, legislators' p o w e r is greater under Model A if
Party Voting Discipline and Power
559
p < .6, equal if p = .6, and less if p > .6. Constituency power for constituency 2 is always greater under Model A. A rough generalization, suggested by this and other examples, is that a bloc gets more power when its members act together, and less when other members aggregate into blocs. 6. Conclusions
Our purpose has been to describe a model of the distribution of decision-making power among the major elements of a typical legislative setting. Our analysis brings to light formally the role played by political parties in determining the balance of decisionmaking power between legislators and their constituencies. Parties determine the influence of legislators and constituencies by exerting coercive and persuasive control over representatives' voting behavior. Tile more power that parties exert, the less decision-making power constituencies have. The effect of increasing party discipline on legislators' power depends on whether the winning party relies crucially on their support or not, with pivotal legislators gaining power and non-pivotal legislators losing power. In the course of our analysis, we have related Straffin's two concepts of decision-making power based on "the question of effect" and "the question of group-individual agreement," finding in our own model the association that he reports. In addition we have uncovered an alternate definition of pivoting power in terms of the probabilities that a constituency supports or opposes a winning party. Our models also provide a potential framework for empirical analysis of the distribution of power in actual legislative settings. All that is required are the constituency seat distributions and an empirical referent for the party voting discipline parameter p. We have incorporated the effects of party discipline into the measurement of power in a legislature, and into the analysis of a representative democracy. Our models can no doubt be broadened, elaborated, and applied elsewhere, but we believe that one conclusion will remain firm: voting discipline concentrates power with parties - though parties may be always forced to share power with some legislative groups, other legislators, and all constituencies, are rendered powerless by extreme party discipline.
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Party Voting Discipline and Power
References Banzhaf, J.F. (1968). "One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College", Villanova Law Review, 1 3 , 3 0 4 - 3 3 2 . Brams, S.J. (1976). Paradoxes in Politics. New York: Free Press. Coleman, J.J. (1971). "Control of Collectivities and the Power of a Collectivity to Act", in B. Lieberman (ed.) Social Choice. New York: Gordon and Breach. Kilgour, D.M. (1974). "A Shapley Value of Cooperative Games with Quarrelling", in A Rapoport (ed.), Game Theory as a Theory of Conflict Resolution. Boston: D. Reidel. Kilgour, D.M. and T . J . l . e v e s q u e (1984). "The Canadian Constitutional Amending Formula: Bargaining in the Past and the Future", Public Choice, 4 4 , 4 5 7 - 4 8 0 . Kilgour, D.M. and T.J.l.evesque (1985). "Party Voting Discipline, Legislators' Power, and Constituency Representation in Multi-party Systems", in M.J. 1loller (ed.) The Logic of Multiparty Systems. Wiirzburg and Vienna: Physica-Verlag. 1.evesque, T.J. (1984). "Measuring S tate Power in Presidential Elections", Public Choice, 4 2 , 2 9 5 - 3 1 0 . l.uce, R.D. and A.A. Rogow (1956). "A Game-Theoretic Analysis of Congressional Power Distributions for a Stable Two-Party System", Behavioral Science, 1 , 8 3 - 8 5 . Mann, I. and L.S. Shapley (1962). "Values of Large Games, VI: Evaluating the Electoral College Exactly", Santa Monica: RAND Memorandum RM3158. Owen, G. (I 971). "Political Games," Naval Research Logistics Quarterly,
18,345-355. Owen, G. (1975). "Evaluation of a Presidential Election G a m e " , American Political Science Review, 6 9 , 9 4 7 - 9 5 3 . Owen, G. (1977). "Values of Games withA Priori Unions", in R. Henn and O. Moeschlin (eds.), Lecture Notes in Economics and Mathematical Systems. New York: Springer-Verlag. Owen, G. (1982). "Modification of the Banzhaf-Coleman Index for Games with A Priori Unions", in M.J. Holler (ed.), Power, Voting and Voting Power. Wfirzburg and Vienna: Physica-Verlag. Shapley, I..S. and M. Shubik (1954). "A Method for Evaluating the Distribution of Power in a Committee System", American Political Science Review, 4 8 , 7 8 7 - 7 9 2 . Straffin, P.D. (1978). "Probability Models for Power Indices", in P.C. Ordeshook (ed.), Game Theory and Political Science. New York: New York University Press. Straffin, P.D., M.D. Davis, and S.J. Brams (1982). "Power and Satisfaction in an Ideologically Divided Voting Body", in M.J. Holler (ed.), Power, Voting, and Voting Power. Wiirzburg and Vienna: Physica-Verlag.
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Summary We examine the distribution of decision-making power among the principal actors of a federal legislative system; parties, legislators, and constituencies. We are especially interested in how the balance of power between legislators and constituencies depends on the voting discipline exercised b y the parties. We derive power measures that are related to the BanzhafColeman index of decision-making power and illustrate with a simple example. Our power measures are novel because they are based on an explicit representation of party voting discipline as well as the structural properties o f the decision-making process, whereas previous analyses concentrate on structurally determined power alone.
PARTY VOTING DISCIPLINE AND THE POWER OF PARTIES, LEGISLATORS AND STATES D. Marc Kilgour and Terrence J. Levesque*
1. Introduction
In many social decision procedures the actual decision-makers "represent" others who do not participate directly in the process but whose influence is posited to derive from the particular representatives they selected. Examples include parliamentarians or congressmen and the voters of their electoral districts, and members of university committees constituted along faculty lines and their faculty colleagues. Indeed, this structure lies at the heart of representative democracy. It is natural to ask how much influence constituencies have in this decision-making process, and how much influence belongs to the representatives themselves. Some answers have been provided in specific cases by the models and methods of n-person cooperative game theory. For example, the Banzhaf-Coleman and Shapley-Shubik indices of decision-making power have been calculated for American states under the Electoral College procedure for electing a president (Banzhaf, 1968; Mann and Shapley, 1962; Owen, 1975). We report elsewhere similar calculations for the Canadian provinces under the amending formula for the Canadian Constitution (Kilgour and Levesque, 1984). In each of these studies, an important and implicit assumption is that representatives are fully controlled by their constituencies, as if each representative were simply a mechanism for reporting the constituency Department of Mathematics and Department of Economics, respectively, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, Canada. The authors gratefully acknowledge the support of the Social Science and Humanities Research Council of Canada (Grant No. 410-84-0425) and thank the editor and an anonymous referee for suggestions which have improved the article.
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Party Voting Discipline and Power
decision. This is an empirically accurate characterization of the relationship between states and their electors under the Electoral College rule I . Its accuracy in the Canadian example, and in most others, depends on the extent to which electoral competition disciplines representatives to reflect constituency preferences accurately. It is intuitively clear that the more freedom the representatives have, the more p o w e r they have, and the less p o w e r remains with the constituencies. In many political decision-making situations legislators are members of parties which claim their allegiance in varying degrees. In other words, parties compete with constituencies for influence over legislators' voting behavior. A party's success in securing decision-making power thus depends on voting discipline - the ability to restrict coalition formation that crosses party lines (Luce and Rogow, 1956). In this essay we examine the sharing of decision-making p o w e r among the three major elements of a typical legislative setting, parties, legislators, and constituencies. We wish to ascertain, in particular, how the exercise of party voting discipline attenuates the p o w e r of constituencies. For example, we expect apriori that constituencies have no power when all legislators belong to a party and always adhere to its position. In contrast, when party discipline is negligible, a standard procedure (we use the BanzhafColeman power index) should measure constituency p o w e r accurately. In general, we believe that the p o w e r of one type of actor in a political decision-making setting can be assessed only with reference to the influence of all relevant actors, even when that influence is not part of the formal decision procedure. In other words, constitutions relate constituencies, legislators, and parties formally in terms of their roles in the political process, and a decision maker's p o w e r depends on his role -°, but it also depends on the conditions and conventions which govern the behavior of all components of the system. Our objective is to incorporate one such influence - the degree of party voting discipline - into the analysis of power in a legislature.
1
There have been a few instances in which electors did not cast their votes for their states' plurality presidential candidate, but this p h e n o m e n o n is rare.
'2 An example is the analysis of the distribution of formal power a m o n g the President, the Senate and the Congress of the United States (Shaplev and Shubik, 1954).
Party Voting Discipline and Power
541
2. Party Voting Discipline and Power We take as our starting point the proposition that parties in a legislature do not have total decision-making control themselves, but share it with the members of the legislature and the constituencies which elected them. For example, when the governing party depends on the support of a pivotal bloc of legislators it in some sense shares its power with that bloc even when voting discipline is strict. As voting discipline weakens some power flows to the constituencies insofar as their positions on issues are influential in determining their legislators' voting behavior. Our view, then, is that legislators have influence to the extent that they do not correlate their positions with either their parties' or their constituencies' preferences. We measure party voting discipline by the response of individual legislators to being "crosspressured", that is, to finding their party's position differs from their constituency's position on an issue. Party voting discipline is strict when legislators choose their party's position over their constituency's with certainty. Constituency control is strict when • all legislators always choose their constituency's position over their party's. Of course, there is a continuum of possibilities between these two cases. We are interested in assessing power not only at the extremes, but also through the range of intermediate cases. In accordance with the Banzhaf-Coleman power index, decision-making power is associated with an agent's opportunities to decide the outcome of an issue. Straffin (1978) refers to this view of power as answering a "question of effect on o u t c o m e " and Coleman (1971) calls it the " p o w e r to initiate". In both cases, the focus is on the ability of the agent to decide the issue. Analogously, our examination of party voting discipline concerns legislators' opportunities to decide issues, irrespective of tendencies of focal legislators under the relevant degree of party discipline. The constituency power that we define also has this property. However, both of our power definitions do take into account the effects of party affiliation and discipline on the behavior of nonfocal legislators. Straffin (1978) presents another approach that is natural here. He investigates the "question of group individual agreement", which asks how likely is an agent to agree with the group decision. His satisfaction index, a measure of the likelihood of groupindividual agreement, measures the value to the agent of partici-
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Party Voting Discipline and Power
pation in the decision-making procedure (rather than leaving it to others). Below we define a constituency's satisfaction and find, .just as Straffin did, that the value to a constituency of participation can be related simply to our index of the constituency's power. Several contributions in the literature describe extensions of the pivoting p o w e r concept that are related to our question. Kilgour (1974) discusses the effect on individual p o w e r of the refusal of some voters to participate in the same coalition (see also Brams (1975)). These "quarrelling players" bear some similarity to legislative members of different parties under rigid party discipline. Owen (1977, 1982) axiomatizes a p o w e r concept that includes the role of prior unions of voters. These unions can be thought of as formal models of legislatures divided strictly along party lines. Elsewhere, Owen (1975) examines the effects of political parties on p o w e r by modeling prior affinities among members to form coalitions. Another approach is taken by Straffin, Davis, and Brams (1982), w h o a t t e m p t to model explicitly the effects of ideology. Levesque (1984) formulates an empirical piw~ting p o w e r measure that accounts for restrictions on coalition formation associated with partisma identification. In all of this literature, specific restrictions on coalition formation determine a probability distribution over the potential legislative coalitions. Our model of decision p o w e r contrasts with those cited above (as applied to legislatures) in two respects. We attribute a definite role to constituencies as well as to parties and to legislators themselves. The earlier models postulate that parties or other prior arrangements place restrictions on legislators' decisions; our model focuses on the tension which arises when legislators' loyaltics are genuinely divided between party and constituency. Our model also yields natural measures of a constituency's p o w e r and satisfaction, whereas earlier models were unable to separate constituency effects and measures from the assessment of legislators' pivoting power.
3. The Model Legislature We now present our model of the legislative setting in preparation for our study of the p o w e r of parties, legislators, and constituencies. We assume that there are only two parties, A and B, and that each of the m members of the legislature represents one
Party Voting
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Discipline and Power
o f the n constituencies. (A constituency may have more than one representative so that m / > n.) Every member of the legislature belongs to either party A or party B. Motions are passed in the legislature if they garner the support of a least a quota, q, of the legislators, where q > n/2. Finally, we assume that the legislative composition and the q u o t a are fixed. Our model legislature is restrictive on one dimension and general on two others. The assumption of a bipartisan legislature is empirically restrictive; we have given a generalization of the subsequent analysis to the multiparty case in Kilgour and Levesque (1985), where we consider the effects of various coalitions among the parties. While our assumption a b o u t the number of parties is specific, our characterization of constituencies and the legislative decision rule is quite general. Our analysis applies to multi-member as well as single-member constituencies, and to legislatures that operate with "special" majority decision rules, such as the twothirds majority required in the American Congress to overcome a Presidential veto. • To summarize and define notation formally, there are n/> 1 constituencies indexed by N =11, 2 . . . . , n I. Furthermore, each Constituency j e N is assigned rh,> 0 seats arid the two parties, A J • and B, hold mjA and mjB of these, respectwely. Of course m;A + mB = m.. The general form of our legislature is represented c~iaJ J Figure 1 : A General Legislative Composition Constituency 1
...
j
A
A
m1
B
m 1
mjA
...
n
...
m A n
Party
mB
B
...
m B n
grammatically in Figure 1. We also define the total party holdings mA = ~ m.A
m B = ~ m.B
j=l
j=l
J '
J "
Of course, the total n u m b e r of legislative seats is m = m A + m B .
544
Party Voting Discipline and Power
Figure 2 shows a simple example legislature which we will use below for illustration. Figure 2: An Example ( q = 4) Constituency 1 2
3
A
2
1
1
B
0
1
1
Party
We assume that all propositions which arise in the legislature are binary, that is, there are only two relevant positions - that of party A and that of party B. We further assume that each constituency supports one of the two positions and we call a list of constituency preferences a constituency profile. In the example of Figure 2, the profile AAB indicates that constituencies 1 and 2 support party A's position, and constituency 3 supports party B's. Obsera,e that the party B legislator in constituency 2 is crosspressured; his voting decision will reflect the amount of party voting discipline exercised by party B. Constituency 2's party A member, in contrast, is not cross-pressured since both his party and constituency positions accord. The assumption that a constituency profile can exist merits some discussion since it includes the assumption that a definite constituency position on an issue exists. J u s t as we model each legislator as representing his entire constituency, so we model the constituency as having some majority preference on each issue. If a constituency tends to favor one side, say A, consistently, then most or all of its representatives will belong to party A, and those constituency profiles in which it supports A will be more c o m m o n than those in which it supports B. In this case, we do not allow a " m i n o r i t y " legislator to represent his "sub-constituency"; rather, we see him as cross-pressured in the sense defined above. In general, there are 2 n possible constituency profiles, since we exclude the possibility of constituency indifference on an issue. In deliberations on a large number of propositions, many different cleavages among the constituencies can be expected. That is, issues give rise to a distribution of constituency profiles. We assume that this distribution has the property that each con-
Party V o t i n g Discipline and P o w e r
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stituency's position is (statistically) independent of the positions of all other constituencies 3 . Apart from this assumption, we do not restrict the distribution of profiles. We refer to the special distribution that makes all profiles equiprobable as the "a priori profile distribution". It is associated with a priori evaluations of power in our subsequent analysis. Similarly, an empirically-based distribution would correspond to an assessment of actual power. We turn now to the direct modeling of the actions of legislators in the presence of varying degrees of party discipline. We have already noted two extreme responses of legislators to being crosspressured - they always vote for the party or they always vote for the constituency. There is a range of intermediate responses that depend on the degree of party voting discipline. We model these in terms of a parameter p such that p e [0,1] where p = 1 corresponds to strict party discipline and p = 0 corresponds to strict constituency control. The parameter p represents the probability that a cross-pressured legislator votes with his party rather than with his constituency. We present in section 5 two specific models of legislators' actions consistent with this interpretation ofp. The parameter p describes legislators' responses only when they are cross-pressured. When the party position and the constituency position correspond we assume that the legislator votes for this c o m m o n position. In our example (Figure 2), party A's members in constituency 1 both vote for party A's position given the profile AAB - in total party A receives at least three and at most five votes under this profile.
4. Measures o f Power and Satisfaction In this section we present formal definitions of the measures of power and satisfaction for the model legislature described in Section 3. Equivalent definitions of our measures are also provided, along with some examination of their interrelationships. As discussed in Section 2, our first notion of the power of an actor (or group of actors) is ability to affect the outcome. In the c o n t e x t of our model legislature, this means the ability to deliver a victory to party A, or to party B, or to either one. Below we
3
It is possible to modify some of our analysis to cover more general cases. However, we chose to follow the present approach in the interests of simplicity.
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Party Voting Discipline and Power
connect this idea of being pivotal with a second concept of power that of getting one's own way. With each constituency we associate two kinds of p o w e r legislators' power and constituency power. Legislators' power is direct power - it measures the ability of a constituency's representatives, considered as a group, to affect the outcome. In contrast constituency power is indirect power - it measures the extent to which a constituency can affect the o u t c o m e by changing its position on an issue, thereby influencing its representatives' voting behavior. We begin with legislators' power. Note that this power is shared by all the representatives of the focal constituency, w i t h o u t regard to their pary affiliation. Legislators' Power for constituency j is the probability that there is a winning party which would not win if its support from j's representatives we are totally withdrawn. Legislators' power depends on the idea that the winning party "owes something" to the representatives of constituencyj if they were the difference between winning and not winning. In the calculation of how often constituency j's representatives are " o w e d something", no account is taken of constituency j's position, nor of the support which some of constituency j's representatives might automatically give because their constituency and party preferences agree. All that matters is w h e t h e r j's representatives, considered as a bloc, pivot for the winning party. Of course the preferences of constituencies other than j, and the party allegiance of their members, do make a difference in the calculation of the legislators' p o w e r for constituency j. We n o w provide a formal definition of legislators' power. For S c N, S ~ ~, let A(S) be the r a n d o m variable denoting the total n u m b e r of votes for party A from the constituencies in S, and define B(S) similarly 4. T h e n the legislators' power for constituency j is -
(1)
LP(j) = Pr 1A(N) ~> q I - PrIA(N - j ) ~> q l + PrIB(N)~ql
- PrIB(N - - j ) ~ q l
because, f o r e x a m p l e , the event I A ( N ) ~ ql means "A wins" and t A ( N - - i ) / > qt) means "A wins even with no support f r o m j " . For 4
For cooveoieoco we write
,lill
etc
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Party Voting Discipline and Power
actual computation of LP(j), an expansion of (1) is usually more convenient: Ill
(2)
.
LP(j) = hZ=J1 [PrlA(N - j ) : q - h I PrIA0) ~> h l
+PrlB(N-j):q-h I PrlB(J)~>hl] We discuss briefly the behavior of LP(j) at extreme values of the party discipline parameter p. When p = 1, party discipline is exact, and for each constituency j, A(j) = m~ and B(j) = m~, independent of the profile of constituency positions. Neither party wins if m A < q and m B < q; in this case LP(j) = 0 by (1). Otherwise, one party wins for certain, and (1) leads to LP(j) =
ll0
ifmA--m'~< J otherwise
q~< m A o r m S - - m . S < q ~ < m S J
In the example of Figure 2, LP(1) = LP(2) = LP(3) = 1 when p = 1, since party A wins for certain, and depends on all its members to do so. Now suppose that p = 0, so that members' votes are completely controlled by their constituencies. Denote the event that constituency j supports party A by jA, and define j B similarly. Then
,3,
A(j) =
lm ifA 0
J if J.B
B(j) =
10if
J m.J if J.B
and it follows from (2) that (4)
LP(j) = P r l q -
mj ~ A ( N - - j ) < q l
PrljAI
In particular, under the a priori distribution of profiles, the legislature becomes a weighted voting game played by the n constituencies (constituency j has mj votes) with quota q, and LP(j)=Bj[q;ml, ...,
m n]
the (absolute) Banzhaf value of the weighted voting game. In the example of Figure 2, LP(1) = LP(2) = LP(3) = 1/2 when p = 0, under the a priori profile distribution. Examples of the calculation of LP(j) for intermediate values
548
Party Voting Discipline and Power
of p must await the introduction of specific models of legislator action in Section 5. We turn now to the second kind of power which is of interest, constituency power. As n o t e d above, a constituency's only opportunity to influence the legislature's decision is through its influence on its representatives. Gaining a constituency's support • can never harm a party, since, for example, at most mjA o f . ,j s representatives vote for A in the event jB, whereas at least m A vote for A in the event j A. Constituency power for constituency j is the probability that some party wins with the support of constituency j and would lose if constituency j were to reverse its position. Note that constituency power focuses on the actions of a constituency, and only indirectly on the actions of the representatives of that constituency, which are the focus of legislators' power. Constituency power takes no account of any tendency for the focal constituency to favor one party over the other, b u t it does depend on the information about other constituencies contained in the profile distribution. Indeed the definitions of legislators' and constituency power differ only in their focus on the representatives of a constituency, or on the constituency itself. We now provide a more formal definition of constituency power. F o r h = 1 , 2 , . . . , m j , l e t QA(h) = PrljA I PrIA(j)~> h IjA I P r l A ( j ) < ( h ] j B I Q~(h)=PrljBIPrIB(j))h
liB I P r I B ( j ) < h l j A I .
For example, QA(h) is the probability that constituency j supports party A and provides A with at least h votes, b u t would provide A with fewer than h votes if it were to reverse its position and support B. The interpretation of Q~(h) is similar. It follows that constituency power of constituency j is (5)
rail
CP(j) = E j
Pr A(N
-j)
= q - h Q (h)
h=l
+PrlB(N-j)=q-hlQ~(h)]
.
Comparison of (5) with (2) shows clearly the difference between legislators' and constituency power. It is convenient to use (5) to describe the behavior of CP(j) at
549
Party V o t i n g Discipline and P o w e r
p = 1. As noted above, A(j) = m Ai and B(j) = m~, without regard to whether ;A orjB occurs. In particular, for any h = 1, 2, . . . , mj, either P~'IA(j) ~>h I = 0 or PrlA(j ) < h~= 0 so that QA(h) = 0 for every h. Similarly Q~(h) = 0, ~o that CP'(j) = 0 when p = 1, by (3). This result corresponds to the idea, suggested above, that constituencies are powerless when party discipline is strict. When p = 0, (3) applies, so that for any h = 1, 2, . . . , mj, (6)
Pr IA(j) ~> hljAl= Pr~A(j) < hljB 1 = 1
and therefore QA(h)= Pr IjAI. Similarly, Q~ (h)= PrljB 1 . CP(j):Prlq-m
j~
PrljAI
+ Prlq - mj ~< B(N - j ) < q I PrljBI. Comparison with (4) now shows that, if p = 0, CP(j) = LP(j) for any profile distribution. In particular, under the a priori profile distribution CP(j)=Bj[q;m 1 . . . . . m n] when p = 0. The example of Figure 2 has, for j = 1, 2, and 3, CP(j) = 0 when p = 1 and CP(j)= 1/2 (under the a prioridistribution) when p -- 0. We now give a characterization of constituency power, applicable for any value of the party discipline parameter p but only under the a priori profile distribution. Under this distribution, Pr IjAI = PrljB l. Therefore 5 , PrljA and A(N) ~>q I - PrljB and A(N)>~q I = PrIA(N)>~qljA I PrljA I - P r I A ( N ) / > qljB I PrljB I =PrljA I [PrlA(N)>i qljA 1 - PrIA(N ) ~> qljBl]
rlA(N-j)--q-h I PrlA(j) > hljAI Ill.
= ZJ P r l A ( N - - j ) = q - - h l Q A ( h ) . h=l
5
The following calculation depends on the assumption that the positions of different constituencies are independent.
550
Party Voting Discipline and Power
A similar calculation can be performed when B wins (B(N) >~ q), and so, by (5), (7)
CP(j) = Pr IjA and A(N)>~ q l -
PrljB and A(N) >~ q I
+ PrljB and B(N)/> ql - PrljA and B(N)>t ql" We have found that under the a priori distribution, the constituency power of constituency j is equal to the difference between the probabilities that j supports a party that wins and that j opposes a party that wins. We now model a constituency's satisfaction with the legislative decision process. Our definition is adapted from the ideas presented in Straffin (1978). Satisfaction for constituencyj is the probability thatj supports a party that wins. Clearly satisfaction measures the frequency with which a constituency is in accord with the group decision. A formal definition of constituency j's satisfaction is (8)
S(j) = PrljA and A(N)~> q l + PrljB and B(N)/> ql"
An expansion for satisfaction as defined in (8) is now given. Observe that if the e v e n t s jA and A(N)/> q both occur, then precisely one of the following three statements must be true: either (i) A would win with no votes fromj's representatives; or (ii) A needs some support from j's representatives, but would receive sufficient votes even ifj were to support B; or (iii) A gets enough votes to win only ifj supports A. This observation, with (8), implies that 6 (9)
S(j) = PrljA I [PrIA(N - j ) ~> q I A
+
m2; j
PrIA(N-j)=q-hlPrIA(j)~>hljB
I]
h= 1
nl •
+ Ej PrlA(N-j)=q--h
IQJA.(h)
h=l
+PrljBl[Pr B(N-j)>~q
6
+
Here, and subsequently, we assume that a s u m m a t i o n equals zero when its lower limit exceeds it u pper limit.
Party
Voting
Discipline
m
551
and Power
B .
~J PrIB(N --j) = q - hlPrlB(j) ~> hljAl] h=l
~J P r I B ( N - j ) : q - h I Q]~(h) .
In. +
h = 1
Furthermore, comparison of (9) with (5) shows that (10)
S(j) - CP(j) = PrljAI [Pr 1A(N --j) ~ ql m
+ I~ j P r l A ( N - j ) = q - h l P r l A ( j ) = h l j B m.
I]
ta = 1
+ PrljBI [PrIB(N --j) ~> ql + g m .
Y/ PrlB(N -j)=q-
hi Prig(j) = hljAI] .
h= l
Of course, the right side of (10) is simply the probability thatj supports some party that would win even without j's support. Since this probability is non-negative, S(j)/> CP(j) follows. It is easy to verify, from (8) or (9), that when p = 1, PrljA 1 ifmA ~ q S(j) =
PrljB I 0
if m 1~ /> q otherwise.
When p = 0, use of (3) and (6) in (9) yields SO) = PrljAIPrIA( N --j) ) q -- mjl + PrljBIPr IB( N - j ) ~ > q - m j l . A simple relationship holds between constituency satisfaction S(j) and constituency power CP(j) under the a priori profile distribution. Observe first from (7) and (8) that S ( j ) - CP(j)= PrljB~ and A(N)>i ql, + PrljA and B(N) >~ q l"
552
Party Voting Disciplineand Power
(The difference between constituency satisfaction and power is the probability that the constituency opposes a winning party.) Further, consider (11)
S(j) -- I/2CP(j) = 1/2 PrlA(N) >f q I + 1/2Pr 1B(N) ~> q I
by (7) and (8). Following Straffin (1978), we identify the right side of (11) with the satisfaction of a dummy constituency which supports a party (according to the a priori profile i.e. Pr t D At = Pr ID B I = 1/,2 ) but which has no power to affect the outc~ome)of thd dedision process. Thus
1/2 CP(j) = S(D)= PrlDA and A(N)>1 q l
S(j)
+ PrIDB and B(N) ~> q I which leads directly to (12)
CP(j) = 2[S(j) - S(D)].
Under the a priori profile distribution, the constituency power measure CP(j) shares the property reported by Straffin (1978): j's power is proportional to the additional satisfaction j derives from its part in the decision process, relative to a dummy constituency. We end this section with a demonstration that LP(j) ~ CP(j). Conditions for equality arc also given below. First, subtract (5) from (2) to give (13)
LP(j) - CP(j) m
.
=h=l Zj P r l A ( N - - j ) = q - - h l ' [ P r l A ( J ) > ~ h l - - Q i ( h ) l m
+ h=l~J P r l B ( N - - j ) = q - h I
• [PrlB(j)~>h I --QB(h
)]
But, by the definition of Qi(h), PrlA(j ) ~> h I-- Q~}(h)= PrlA(j )/> h 1 --PrIA(j) ~> h andjA i erlA(j) < hljB I ~> PrIA(j)~> h I -PrIA(j)>~ h andjA I >~0 using Pr Ia(j) < hljBl~< 1 and the observation that the probabili-
553
l'arty Voting Discipline and Power
ty of a conjunction o f two events cannot exceed the probability of one of the events. This calculation and its analogue show that the bracketed quantities on the right side of (13) are non-negative, proving that LP(j)/> CP(j). We have already noted that for any profile distribution, LP(j) = CP(j) when p A 0. When p = 1, CP(j) = 0 whereas LP(j) = 1 if either m A - mj < q ~< m A o r m B - roB< q ~< m B , and LP(j) = 0 otherwise. These observations a m o u n t to necessary and sufficient conditions for the two kinds of power to be equal in this case. We have now proven that LP(j) 1> CP(j) for any value of p and any profile distribution, and we have reported the cases p = 0 and p = 1 in detail. It can also be shown that, for any value of p, LP(j) = CP(j) under the a priori profile distribution if and only if either (i) or (ii)
p = 0; p = 1 and b o t h m A - m . A J
or (iii) 0 < p <
1 and e i t h e r m .A--0 J
or P r l q -- m A < ~ A(N - - j ) < ql = 0, and either m.J~ = 0 orPrlq--m.Bj ~
5. Models o f Legislators' Actions In this section, we make our power and satisfaction concepts more concrete by proposing two specific models for legislators' actions, and by applying these models to the example of Figure 2. As set out in Section 3, a model of legislators' actions describes the voting behavior of cross-pressured legislators. This behavior must depend on a parameter p e [0, 1] which corresponds to the probability a cross-pressured legislator supports party over constituency. There must be complete constituency control w h e n p = 0 and strict party discipline when p = 1. The two models presented here differ only in the degree of statistical dependence (within cross-pressured blocs) that they specify.
554
Party Voting Discipline and Power
Independent Agent Model (Model A)
Under the independent agent model for legislators' actions (Model A), each cross-pressured legislator is viewed as an agent operating independently of all other cross-pressured legislators. Thus each cross-pressured legislator votes for his party with probability p, and the actions of different cross-pressured legislators are (statistically) independent. The formal version of Model A is not difficult to obtain. If constituency j supports party B and m~> 0, then the probability that party A receives h votes from j's representatives is (conditionally) . , m Aj , PrIA(j) = h I = ( h ~ l ph(1--p)m AJ - ,hh = 0 , 1 , . . the familiar binomial probability on parameters mix and p. Of course, Pr tB(j) = k !~ = Pr tA(j) = m j - k I~ for k• j= mB,'m~ + 1, ~-k ~ • .... mj. If m'j = 0, tfien Pr Ia(j ) = 01 =PrlB(j) = mj I = 1. In case constituency j supports p~rty A, thdprob~bilities are'similar. We now work out the a priori legislators' power, constituency power, and constituency satisfaction for the simple legislature shown in Figure 2. The calculation will be given in detail for constituency 1 arid summarized for constituency 2. (The situation of constituency 3 is identical to that of constituency 2.) We now assume the a priori profile distribution. We determine the a priori legislators' power associated with constituency 1 using (2): Observe that if 1A occurs, then A(1) = 2 a n d B(1)= 0. Since Pr I1AI = Prll BI = 1/2, it is easy to verify that (14) PrIA(1)~> 1l = 1--(1--p)2/2 PrIA(1) = 2I = (1 + pZ)/2 PrIB(1) ~> 11 = ( 1 - - p 2 ) / 2 Pr IB(1) = 21 = (1 - p)Z/2. Now (2) implies that LP(1)=PrIA(2,3)=3IPrIA(1)~> +PrIB(2,3):31PrlB(1)/>
1l 11
+ Pr 1B(2, 3)= 21Pr 1B(1) = 2I.
Party Voting Discipline and Power
555
It is easy to derive t h a t (15)
PrIA(2,3)=3I
=PrlB(2,3)=3I=(4p-4p~)/4
PrlA(2, 3)-- 21= PrIB(2, 3)= 21= (2 - 4p + 6p'-')/4 which, w h e n s u b s t i t u t e d , yields LP(1) = (1 - p + 6p '2 - 9p :~ + 5p4)/2 A similar calculation gives LP(2) = LP(3) = (1 -- p + 5p'-' - 8p :~ + 5p4)/2 Figure 3: .-! Priori Legislators' Power and C o n s t i t u e n c y Power in the Legislature o f Figure 2 u n d e r Model A
1 0.
O.B 0.70.6-
~ O' L S ~P 0.3
(
3
)
O
.
4~(2) "
~
0.2
~
c~(2) = cP(3)
~
0.1 0
•
0
•
"
l
0.2
"
•
"
i
0.4
"
'
"
i
'
0.6
"
.
.
.
.
.
0.8
PartyDiscipl|ne The graphs o f LP(1) and LP(2) are s h o w n ill Figure 3. Observe t h a t LP(1) = LP(2) = 1/2 at p = 0, LP(1) = LP(2) = 1 at p = 1, and LP(1) > LP(2) for 0 < p < 1. F u r t h e r m o r e LP(1) decreases f r o m p = 0 to p ~ . 1071 and increases t h e r e a f t e r - the m i n i m u m value o f LP(1) is ~ .4757. Similarly, LP(2) decreases f r o m p = 0 to p ~ .1435, increasing thereafter. Its m i n i m u m is ~ .4690. We n o w calculate a priori c o n s t i t u e n c y p o w e r in our e x a m p l e legislature. It is n o t hard to s h o w t h a t , u n d e r Model A,
556 (16)
Party Voting Discipline and Power
Q:](1) = Q~(2) = (1 - p)Z/2 Q'] (2) = Q~(I) = (1 - p 2 ) / 2
so that (5) and (16) yield C,(1) : Pr IA(2, 3 ) : 31 Q ] ( 1 ) + PrlB(2, 3 ) = 31 Q~ (1) + PrlA(2, 3 ) : 21 QA ( 2 ) + ' r l B ( 2 , 3 ) : 2 1 Q ~ l =(1-p+pZ
(2)
_p~)/2.
A similar calculation gives CP(2) = CP(3) = (1 - p + pZ _ 2p.~ + p4)/2. Note that CP(1) /> CP(2), with equality only at p = 0 and p = 1; both CP(1) and CP(2) are strictly decreasing functions for p e [0, 1]. The graphs of CP(1) and CP(2) are shown in Figure 3, which makes it easy to compare constituency power CP(j) with the corresponding legislators' power LP(j) in our example legislature. We now turn to calculation of a constituency's satisfaction, under the a priori profile distribution and Model A, in the example legislature of Figure 2. We use (12), which can be inverted easily to yield SO) = S(D) + CP0)/2 S(D) = [Pr IA(N) ~> ql + Pr 1B(N)~> q l ] / 2 . In the legislature of Figure 2, it is easy to show that S(D) = (2 -- 3p + 7p 2 -- 9p 3 + 5p4)/4 so that S(1) = (3 - 4p + 8p" - 10p ~ + 5p4)/4, S(2) = S(3) = (3 - 4p + 8p" - 1 l p ~ + 6p 4 )[4. As it t o b e expected from Figure 3, S(1) and S(2) are strictly decreasing functions of p, from S(1) = S(2) = 3/4 when p = 0 to S(1) = S(2) = 1/2 w h e n p = l, and S(1) > S(2) for 0 < p < 1.
Party Voting
Discipline
557
and Power
Bloc Model (Model B) In the bloc model for legislators' actions (Model B) all crosspressured representatives of a constituency vote together, as a bloc. Note that all members of a particular bloc must belong to the same party. With probability p, the bloc supports this party with all its votes; otherwise the party receives no votes at all from this bloc. The actions of different blocs are assumed to be independent. It is simple to formalize Model B. If constituency j supports party B and m A j > 0, then, conditionally, PrIA(j)=mA. 1 = p ; J
PrIA(j)=OI=
1--p.
If m .A = O, then PrtA(j) = O l = 1. Of course PrIB(j ) : k I. PrIA(j) ~ " = m. J -- "k ~~ tot k" =~ m.,B m.B )+ 1, . . . , m j . When constituency j supports party A, the O e t m m o n s are similar. We n o w apply Model B to the example legislature of Figure 2, indicating how our earlier calculations for Model A must be modified. To calculate the a priori legislators' power for constituency 1, (14) must be replaced by
PrIA(1)~> I I = P r I A ( 1 ) = 2 1
=(l+p)/2
PrIB(1)~> 11 = P r I B ( 1 ) = 2 1
=(I--p)/2.
Since (15) remains valid 7 (2) implies that LP(1) = (1 + pZ)/2. An analogous calculation gives LP(2) = LP(3) = (1 - p + 2p2)/2. The graphs of LP(1) and LP(2) are shown in Figure 4. To calculate constituency l's a priori constituency power under Model B in the example of Figure 2, we revise our previous calculation under Model A. First, (16) must be replaced by
Q~ (1) = QA (Z) = ( 1 - p)/2 Q~ (1) = Q~ (2) = (1 - p)12. 7
T h i s p h e n o m e n o n o c c u r s h e r e b e c a u s e the b l o c s f r o m c o n s t i t u e n c i e s 2 a n d 3 h a v e only one member, and therefore Models A and B act identically on them.
558
Party V o t i n g Discipline and P o w e r
Figure 4: A Priori Legislators' Power and Constituency Power in the Legislature of Figure 2 under Model B 7 0.9 0.8 0.7 0.6
LP(2)- LP(3) o. 0.4 0.3 0.2
_ ,
_ IP(2)." CP(3),
~
o.1 0
•
o
•
0.2
0.4
0.6
0.8
"
PoNy Discipline
Insertion in (5) and use of (15) gives CP(1)=(1-p+p2
p3)/2.
Similarly, CP(2) = CP(3) = (1 - p)/2. These functions are also shown in Figure 4. Finally, calculations of a priori constituency satisfaction similar to those for Model A give S(1)=(3-3p+3p
z -p~)/4
S(2) = S(3) = (3 - 3p + 2p 2)/4. A brief comparison of power u n d e r Models A and B is of interest. Since Model A and Model B agree at p = 0 and at p = 1, only values of p satisfying 0 < p < 1 will be reported. For constituency 1, legislators' power is always greater u n d e r Model B, and constituency power is always equal under the two models. For constituency 2, legislators' p o w e r is greater under Model A if
Party Voting Discipline and Power
559
p < .6, equal if p = .6, and less if p > .6. Constituency power for constituency 2 is always greater under Model A. A rough generalization, suggested by this and other examples, is that a bloc gets more power when its members act together, and less when other members aggregate into blocs. 6. Conclusions
Our purpose has been to describe a model of the distribution of decision-making power among the major elements of a typical legislative setting. Our analysis brings to light formally the role played by political parties in determining the balance of decisionmaking power between legislators and their constituencies. Parties determine the influence of legislators and constituencies by exerting coercive and persuasive control over representatives' voting behavior. Tile more power that parties exert, the less decision-making power constituencies have. The effect of increasing party discipline on legislators' power depends on whether the winning party relies crucially on their support or not, with pivotal legislators gaining power and non-pivotal legislators losing power. In the course of our analysis, we have related Straffin's two concepts of decision-making power based on "the question of effect" and "the question of group-individual agreement," finding in our own model the association that he reports. In addition we have uncovered an alternate definition of pivoting power in terms of the probabilities that a constituency supports or opposes a winning party. Our models also provide a potential framework for empirical analysis of the distribution of power in actual legislative settings. All that is required are the constituency seat distributions and an empirical referent for the party voting discipline parameter p. We have incorporated the effects of party discipline into the measurement of power in a legislature, and into the analysis of a representative democracy. Our models can no doubt be broadened, elaborated, and applied elsewhere, but we believe that one conclusion will remain firm: voting discipline concentrates power with parties - though parties may be always forced to share power with some legislative groups, other legislators, and all constituencies, are rendered powerless by extreme party discipline.
560
Party Voting Discipline and Power
References Banzhaf, J.F. (1968). "One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College", Villanova Law Review, 1 3 , 3 0 4 - 3 3 2 . Brams, S.J. (1976). Paradoxes in Politics. New York: Free Press. Coleman, J.J. (1971). "Control of Collectivities and the Power of a Collectivity to Act", in B. Lieberman (ed.) Social Choice. New York: Gordon and Breach. Kilgour, D.M. (1974). "A Shapley Value of Cooperative Games with Quarrelling", in A Rapoport (ed.), Game Theory as a Theory of Conflict Resolution. Boston: D. Reidel. Kilgour, D.M. and T . J . l . e v e s q u e (1984). "The Canadian Constitutional Amending Formula: Bargaining in the Past and the Future", Public Choice, 4 4 , 4 5 7 - 4 8 0 . Kilgour, D.M. and T.J.l.evesque (1985). "Party Voting Discipline, Legislators' Power, and Constituency Representation in Multi-party Systems", in M.J. 1loller (ed.) The Logic of Multiparty Systems. Wiirzburg and Vienna: Physica-Verlag. 1.evesque, T.J. (1984). "Measuring S tate Power in Presidential Elections", Public Choice, 4 2 , 2 9 5 - 3 1 0 . l.uce, R.D. and A.A. Rogow (1956). "A Game-Theoretic Analysis of Congressional Power Distributions for a Stable Two-Party System", Behavioral Science, 1 , 8 3 - 8 5 . Mann, I. and L.S. Shapley (1962). "Values of Large Games, VI: Evaluating the Electoral College Exactly", Santa Monica: RAND Memorandum RM3158. Owen, G. (I 971). "Political Games," Naval Research Logistics Quarterly,
18,345-355. Owen, G. (1975). "Evaluation of a Presidential Election G a m e " , American Political Science Review, 6 9 , 9 4 7 - 9 5 3 . Owen, G. (1977). "Values of Games withA Priori Unions", in R. Henn and O. Moeschlin (eds.), Lecture Notes in Economics and Mathematical Systems. New York: Springer-Verlag. Owen, G. (1982). "Modification of the Banzhaf-Coleman Index for Games with A Priori Unions", in M.J. Holler (ed.), Power, Voting and Voting Power. Wfirzburg and Vienna: Physica-Verlag. Shapley, I..S. and M. Shubik (1954). "A Method for Evaluating the Distribution of Power in a Committee System", American Political Science Review, 4 8 , 7 8 7 - 7 9 2 . Straffin, P.D. (1978). "Probability Models for Power Indices", in P.C. Ordeshook (ed.), Game Theory and Political Science. New York: New York University Press. Straffin, P.D., M.D. Davis, and S.J. Brams (1982). "Power and Satisfaction in an Ideologically Divided Voting Body", in M.J. Holler (ed.), Power, Voting, and Voting Power. Wiirzburg and Vienna: Physica-Verlag.
Party Voting Discipline and Power
561
Summary We examine the distribution of decision-making power among the principal actors of a federal legislative system; parties, legislators, and constituencies. We are especially interested in how the balance of power between legislators and constituencies depends on the voting discipline exercised b y the parties. We derive power measures that are related to the BanzhafColeman index of decision-making power and illustrate with a simple example. Our power measures are novel because they are based on an explicit representation of party voting discipline as well as the structural properties o f the decision-making process, whereas previous analyses concentrate on structurally determined power alone.