Bicameralism and majoritarian equilibrium

Public Choice "/4: 169-179, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

Bicameralism and majoritarian equilibrium*

GEOFFREY BRENNAN Research School of Social Sciences, Australian National University, Canberra, ACT 2601

ALANHAMLIH

Department of Economics, University of Southampton, Southampton S09 5NH

Received 12 October 1990; accepted 21 January 1991

Abstract. Recent papers have established that bicameralism can support a non-empty core in majority voting games in two dimensional policy spaces. We generalise this result to the n- dimensional case, and provide a discussion of multi-cameralism. Bicameralism generates a core of potentially stable equilibria by institutionalising opposition between mutually oriented median voters, this provides a clear link with the standard median voter model and with more traditional analyses of bicameralism.

1. Introduction

Recent papers by Hammond and Miller (1987) and Miller and Hammond (1990) have investigated bicameral decision making and the interaction between bicameralism and other aspects of the constitution - particularly committee structures and the operation of an executive veto. They conclude, inter alia, that bicameralism supports a non-empty core provided that the Pareto sets of the two houses of the bicameral legislature are sufficiently dissimilar. While the analysis of these papers offers considerable insight into mechanisms involved in constitutional government, and hence into the principles of constitutional design, one major theoretical limitation of the analysis is the restric- tion to a two dimensional policy space - so that bicameralism is offered as a potential remedy for the revolving majority problem only when there are exact- ly two issues to be decided. Indeed Hammond and Miller repeat, with apparent support, a conjecture of Cox's that adding more issues or policy dimensions may undermine the stabilizing influence of bicameralism; and that n- cameralism (a legislature of n houses) may be required for stability (in the rele-

* An earlier version of this paper (Brennan and Hamlin, 1990), written in ignorance of the work of Hammond and Miller (1987, 1990), benefited from comments at the Public Choice Society meetings, Tucson; the European Public Choice Society meetings, Meersburg, the Center for Study of Public Choice, and the Universities of Chicago and Oxford. Hamlin is grateful for the support of visiting fellowships at ANU and All Souls College, Oxford.

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vant sense of a non-empty core) in the n-dimensional case (Hammond and Miller, 1987:1156-1157). This conjecture may be based on the intuition that if uni-cameralism supports a non-empty core in the case of a single issue (the standard median voter theorem), and bicameralism supports a non empty core in the case of two issues, then, in general, we might require the number of issues (the dimensionality of the policy space) and the number of legislative chambers required to generate a non empty core, to be the same.

The main purpose of this brief paper is to examine and refute both parts of this conjecture. We shall argue that bicameralism can support stable voting equilibria in an n-dimensional policy space on conditions that are natural generalizations of those studied by Hammond and Miller (1987, 1990) in the two dimensional case. We shall also investigate the generalization to n- cameralism, and will argue that the impact of increasing the number of effective chambers in the legislature is to increase the size, and the dimensionality, of the core.

While we operate within the same spatial voting framework as Hammond and Miller (1987, 1990), our approach is slightly different. It is clear that bicameralism is neither necessary nor sufficient for stability, and so we adopt a constructive approach to show that, for any set of legislators, there exist bicameral partitions that support stable equilibria. Typically there are many such partitions and, for any given partition, multiple stable equilibria. To illustrate this approach, and introduce some helpful terminology, the next section re-considers the two dimensional case. Thereafer, Section 3 generalizes to n dimensions and Section 4 to n-cameralism. Section 5 offers some concluding comments.

The purpose of this paper is theoretical. We seek to identify the role of bicameralism as a distinct constitutional element rather than investigate particular constitutions that incorporate bicameralism. In order to focus on the institution of bicameralism, we abstract from other institutional arrangements. In particular, we abstract from the processes by which legislators are chosen. Our analysis focuses on a given set of legislators and views the impact of alternative decision making structures given that set of legislators. Of course, most practi- cal bicameral systems do not operate by first identifying a set of legislators and then partitioning them into two houses. Rather they link a particular method of choice of a legislator (whether that process is electoral or by appointment) with a place in a particular house. However, we believe that this abstraction is a useful first step in the analysis of bicameralism. Our analysis indicates the conditions under which the institution of bicameralism can be expected to con- tribute to the stability of the political system, clearly any particular bicameral system will realise this benefit only to the extent that the constitution as a whole (including the specification of the methods by which legislators are chosen) generates the required conditions. If the conditions are not realised, then bicameralism is of no obvious merit. We shall return to this point, briefly, in Sec- tion 5 below.

ISSUE 2 I. 2

1.1

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ISSUE 1

Figure 1. The two dimensional case.

2. Bicameralism in two dimensions

Consider a set of M legislators whose ideal points are distributed in a two dimensional policy space (as illustrated by the Issue 1/Issue 2 space of Figure 1). 1 Let the set of ideal points be S and the Pareto set relating to these points, that is, the convex hull of S, be P. We wish to construct a partition of P such that simple majority voting within each subset, together with a requirement of a majority in both subsets, supports stable equilibria in the sense of a non empty core. The trivial procedure is illustrated in Figure 1. Any straight line 1111 which partitions P into two non-empty subsets (any ideal points lying on lll l may be allocated between the subsets arbitrarily) will be a bicameral partition supporting stable equilibria. That is to say that, if the set of legislators is divid- ed into two houses by reference to the partition Ill l, then the resulting voting game in which any motion requires a simple majority in both houses will have a non empty core - so that there will be at least one policy platform that is stable in the sense that it cannot be defeated under these rules.

The proof of this proposition involves a special case of the so-called ham sandwich theorem (Cox and McKelvey, 1984). 2 For any partition 1111, there will exist a second line 1212 which contains the median element of both sub-

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sets of P. This line, which Hammond and Miller refer to as a bicameral bisec- tor, we term a mutually oriented median voter line. This terminology provides a direct link with the standard discussions of the median voter theorem.

Recall that the necessary and sufficient condition for the existence of stable equilibria in the two dimensional, uni-cameral majority voting game is the existence of a "median in all directions" (Enelow and Hinich, 1984); that is, a point that is included in all lines that partition the set of ideal points into two subsets of equal size. Many such lines exist but, in general they do not have a point in common. Each of the subsets of P created by the partition Ill I has many such median lines, but 1212 is required to be a median line for both subsets. Thus, from the perspective of point a, point b has the property of being the median voter in the relevant subset; and from the perspective of b, point a has the same property relative to the other subset. ~ This mutual orientation of the median voters provides the intuition behind the stability result. Bicamer- alism acts to fix a particular direction as being of special significance, and in this setting we no longer require the existence of a median in all directions.

Given this interpretation of 1212 it is clear that point E, the intersection of Ill 1 and 1212, is a stable equilibrium (element of the core) under bicameral majority voting. The argument is essentially identical to that employed by Hammond and Miller (1987:1157). No departure from E will be approved by a majority in a subset unless it lies on its side of the line IiI p and no departure from E can satisfy this requirement for both subsets simultaneously. It is the opposition of the mutually oriented median voters that provides the basis for the result, although the existence of a mutually oriented median voter line is not, of course, sufficient for the existence of stable equilibria. Furthermore, E is not generally unique. Depending on the precise distribution of preferences within the subsets of P, points "close" to E may also be in the core.

This argument simply reiterates the result that a bicameral structure in which the Pareto sets of the two chambers are non-overlapping will support a core of stable equilibria in two dimensional policy space, although we believe that the present formulation is both simpler and more general than that found in Hammond and Miller (1987). However, the major purpose of introducing this argument is to apply it to the n dimensional case, and it is to this that we now turn.

3. Bicameralism in n dimensions

The extension to many dimensions is most easily viewed by first considering a three dimensional case - again we shall denote the set of ideal points by S, and the convex hull of S by P. We shall argue that any plane that partitions P into two non-empty subsets (with any ideal points that lie on the plane allo-

ISSUE 2

ISSUE 3

)k 2

$,i:. ..... :: )~1

P >

ISSUE 1

Figure 2. The three dimensional case.

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cated to either subset arbitrarily) will be a bicameral partition supporting a core of stable equilibria.

This case is illustrated in Figure 2, where the partitioning plane is labelled X I. The argument then parallels the two dimensional case. The first step is to find a second (not necessarily unique) plane such that it includes the median element of both subsets of P created by the X1 partition. The existence of such a plane is again ensured by the ham sandwich theorem. A plane with this property is shown in Figure 2 and labelled X2, and the line of intersection of the two constructed planes is labelled AB.

The second stage of the argument is to project all ideal points in P onto the plane X2, with the relevant projection being parallel to Xl" The result of this projection is to reduce the problem to a two dimensional problem. By construction, A B identifies the bicameral partition, and it is now possible, by a further application of the ham sandwich theorem, to construct a (not necessarily unique) line, labelled CD in Figure 2, that contains the medians of both subsets of projected ideal points. We argue that the intersection of A B and CD, point E, is a stable equilibrium (element of the core) under bicameral majority voting. As before, E is not necessarily unique, even given the particular plane X2 and the particular line CD.

Again, the intuition relates to the opposition between two mutually oriented median voters. The line CD identifies, via the two stage process described

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above, the mutually oriented median voters in the two subsets of P, and, as in the two dimensional case, the point of intersection of this line and the partition forming the boundary between the two subsets is a stable equilibrium (element of the core) since any departure from this point will be approved by a majority in a subset only if it involves a movement to its side of the partitioning plane, and this condition can not be met for both subsets simultaneously.

A further point to note in the context of this three dimensional case is that the core of stable equilibria that is guaranteed to exist is not increased in size or dimensionality by the increase in the dimensionality of the problem. For example, in the two dimensional case, as the Pareto sets of the two chambers move apart from each other, so the core identified by the above argument expands from a point to a line segment - essentially the segment of the contract curve between the two mutually oriented median voters that is not interior to either of the Pareto sets (see Hammond and Miller, 1987, Figure 1). And the same is true in the three dimensional case. If, in Figure 2, the two subsets of P are pulled apart from each other by movements orthogonal to )~l, the core identified expands from the point E to a line segment - that segment of CD that is not interior to either Pareto set.

The move to n-dimensions should now be fairly clear. Successive dimensional reduction via repeated application of the ham sandwich theorem will, eventually, give rise to the standard two dimensional problem and, therefore, the above result. For any n-dimensionai Pareto set P, an n - 1 dimensional hyperplane A l that partitions P into two non-empty subsets (with elements of P lying on the hyperplane allocated between subsets arbitrarily) will allow the construction of a second n-1 dimensional hyperplane A 2 that contains the medians of both subsets of P created by A 1. Projecting all elements of P onto A 2 (with the relevant projection being parallel to A1) then produces an n-1 dimensional problem. Repeating this process successively will eventually construct a stable equilibrium point (element of the core) that lies at the intersection of A l and the last constructed median-including line. Thus, the original partition A 1 is revealed to be a bicameral partition that supports a core of stable equilibria.

As in the two and three dimensional cases, the core is identified as a segment of the contract line between to mutually oriented median voters, regardless of the dimensionality of the policy space.

4. N-cameralism

So far we have been concerned with one aspect of the original conjecture that increasing the dimensionality of the policy space would undermine the stabilizing influence of bicameralism, and that n-cameralism is required to generate a non-empty core in n-dimensions. We have argued that the stabilizing in-

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ISSl

(a)

IE 2

e~

92

ISSUE1

(b)

ISSUE2

C

(

S D

92

ISSUE1

Figure 3. Bicameralism and tricameralism.

fluence of bicameralism is robust to increases in the dimensionality of the policy space.

We now turn to the second aspect of the conjecture. What are the effects of increasing the number of legislative chambers? This question is of some practi- cal as well as theoretical interest, since multi-cameral systems are not uncom- mon. Any decision making system in which each of three or more groups (each of which operates under simple majority voting internally) must be satisfied simultaneously may be formally described as multi-cameral. It is therefore clear that many multi-national agreements, which must be ratified by the relevant (majority voting) body in each of the countries concerned, can be seen as the outcomes of multi-cameral processes. Examples might include GATT, and many aspects of the operation of the European Community. We shall also sug- gest below that the veto power of the US President transforms an apparently bicameral system into a form of tri-cameralism; and that any system of committees such that a policy has to gain support in both committees and formal

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legislative houses places those committees in a role very similar to the houses and so instantiates a multi-cameral system.

We may think of all multi-cameral systems as combining elements of majoritarianism (within chambers) with elements of unanimity (across chambers). The possibly surprising aspect of our earlier result is that the apparently small element of unanimity involved in bicameralism can be sufficient to generate a core of stable equilibria in n-dimensional policy space. But it should be clear that as the number of chambers rises (for a given set of legislators), the balance shifts from majoritarianism towards unanimity, so that the prospects for stability improve. The precise nature of this improvement is suggested by consideration of the extreme case - where the numbers of chambers is equal to the number of individual legislators, so that each individual has an effective veto and all trace of majoritarianism is banished. Clearly, in this case, the core is the whole of the Pareto set, so that the core is expanded in both size and dimensionality relative to the bicameral case.

This is confirmed in intermediate cases. Figure 3 illustrates for the compari- son of bicameralism and tri-cameralism in a two dimensional policy space. The bicameral situation is shown in Figure 3a where the Pareto sets of the two chambers are shown as P1 and P2, the mutually oriented median voter line is AB, and the identified core is the line segment ab. Figure 3b then shows a tri- cameral situation created by a further arbitrary partitioning of P1 into P3 and P4 (shown as the dotted line). Taking each pair of chambers in turn allows the construction of the pairwise mutually oriented median voter lines shown as AB, CD and EF. It is now straightforward to check that the identified core under tri-cameralism is abcd, which necessarily contains the line ab from Figure 3a.

It is clear that the discussion of the executive veto provided by Miller and Hammond (1990) conforms to this analysis of tri-cameralism, with the restric- tion that the third chamber includes just one individual. The expansion of the core noted by Miller and Hammond is then an example of the general impact of the increase in the number of effective chambers. The same point may be made in respect of the analysis of committees within the bicameral structure. If each committee as well as each chamber has an effective veto, we have an essentially multi-cameral system with the additional proviso that some legislators are included in more than one house (this of course limits the extent to which the Pareto sets of the various houses can be non-overlapping, however what is important is that the Pareto sets of at least some of the houses are sufficiently distinct, and it is easy to see how this possibility is enhanced in a committee system). Again, the increase in the size of the core is an example of the general impact of multi-cameralism.

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5. Concluding comments

The stabilizing potential of bicameralism extends to n-dimensional policy space. Providing that the Pareto sets of the two chambers are sufficiently dissimilar, bicameralism will ensure the existence of a non-empty core. Put in the more constructive language of this paper; for any set of legislators, there exist bicameral partitions supporting stable equilibria.

Extending the number of chambers beyond two carries the potential for ex- panding the size and dimensionality of the set of stable equilibria. But this potential is again conditional upon the chambers being sufficiently different from each other.

Both of these results indicate that the basic force underlying the stabilizing influence of multi-cameralism is the institutionalization of effective opposition between mutually oriented median voters. This view of the working of bicameralism fits well with Montesquieu's treatment:

In every state there are always some people distinguished from the rest by their birth, wealth or honours. If combined indiscriminately with the people, so that everyone counted equally, such common liberty would constitute slavery [for the distinguished]... Thus their share in legislation ought to be in proportion to the other advantages they enjoy in the state. And this can be assured only when they constitute on the one side a body that has the right to check the people's actions, and, on the other, when the people have the right to check their actions (Montesquieu, quoted from Richter, 1977: 248-249).

Here bicameralism is defended as a means of protecting the interests of particular minorities, but the mechanism of protection is essentially identical to the mechanism of stability.

This analysis of bicameralism as institutionalized opposition also suggests why Condorcet was opposed to bicameralism, arguing that it added nothing to a properly conceived constitution (Condorcet, 1787, translated in Sommer- lad and McLean, 1990). Condorcet approached politics as an exercise in the revelation of truth by sampling from individuals' beliefs that were more or less enlightened. Interests played no part in the decision making process. It is clear that when political decision making is viewed in Condorcet's terms, bicameralism amounts to splitting the sample information and results in a reduction in the effective sample size, rather than any improvement in the process.

Although multi-cameralism (including bicameralism) will interact with other aspects of any particular constitution, it is clear that a considerable degree of opposition between chambers will be necessary if multi-cameralism is to have a substantive role in generating stable equilibria. There are many forces which

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tend to reduce this degree of oppos i t ion in poli t ical systems that are formal ly

bicameral . Pa r ty discipline is perhaps the mos t obvious of these forces - if the

same poli t ical par ty controls bo th houses, there will be no effective oppos i t ion

between houses and so no stabil izing inf luence of b icameral ism.

I f the stabil izing inf luence of b icamera l i sm is to be effective, the cons t i tu t ion

as a whole mus t work to generate and m a i n t a i n the required s t ructure of oppo-

si t ion between chambers . We no ted in the i n t roduc t ion tha t our analysis ab-

stracts f rom the processes by which legislators are chosen, bu t it is clear that

these processes will be vital in de te rmin ing whether any par t icular b icamera l

system achieves the potent ia l stabil i ty benefi ts that we have analysed. For ex-

ample , if the two houses are elected by similar procedures , to represent similar

const i tuencies , then we might expect the houses to be ra ther similar in composi-

t ion so that the Pare to sets of the two houses would be very similar. In these

c i rcumstances the stabil izing value of b icamera l i sm is lost, and it is by no me-

ans clear what purpose is served. Stabil i ty is served only to the extent tha t the

processes used to al locate legislators to houses generates suff icient oppos i t ion

between those houses.

Notes

1. We adopt the standard assumption that preference is a simple function of Euclidean distance from these ideal point, so that preferences are represented by concentric, circular indifference curves centred at the ideal point. These preferences are taken to represent the legislators voting intentions and may therefore depend upon both the legislator's personal preferences and any institutional incentives arising, for example, from the process of selecting legislators.

2. The ham sandwich theorem may be stated formally as follows (Cox and McKelvey, 1984: The- orem 2):

Given any n finite measures I~l" • I~n defined on the Borel sets of R n, there exists a hyperplane H = {x • Rn[ x.v = c} with v e S n-I, c • R, where S n-1 is the n-1 sphere of unit length vectors in R n, such that for all 1 < i <_ n,

p.i(R n) I~i(R n) ~i (H +) ~ ~ and I~i (H -) -< 2

The theorem shows that for any n dimensional set, a (non unique) n - I dimensional hyperplane exists such that this hyperplane is a median hyperplane relative to all measures of the original set. The theorem derives its name from a simple illustration of the problem in which a (three dimensional) sandwich made of bread, butter and ham must be cut (by a two dimensional plane) in such a way that the two portions contain equal amounts of each of the three ingredients.

3. Of course, there may be many ideal points lying on 1212 - we do not mean to imply that a and b are unique. Equally, if the number of members of a house is even, there may be no ideal points on 1212, so that the "median voter" is hypothetical.

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References

Brennan, G. and Hamlin, A. (1990). Bicameralism and stability. University of Southampton Eco- nomics Discussion Paper No. 9002.

Condorcet, Marquis de. (1787). Letters from a Freeman of New Haven to a Citizen of Virginia. In F. Sommerlad and I. McLean (Eds. and Trans.) (1990), The political theory of Condorcet H. Oxford University Faculty of Social Studies Working Paper 2/90.

Cox, G.W. and McKelvey, R.D. (1984). A ham sandwich theorem for general measures. Social Choice and Welfare 1: 75-83.

Enelow, J. and Hinich, M. (1983). Thespatial theory of voting. Cambridge: Cambridge University Press.

Hammond, T.H. and Miller, G.J. (1987). The core of the constitution. American Political Science Review 81: 1155-1174.

Miller, G.J. and Hammond, T.H. (1990). Committees and the core of the constitution. Public Choice 66: 202-227.

Richter, M. (1977). Thepoliticaltheory of Montesquieu. Cambridge: Cambridge University Press. Sommerlad, F. and McLean, I. (Eds. and Trans.) (1990). The political theory of Condorcet H.

Oxford University Faculty of Social Studies Working Paper 2/90.

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Bicameralism and majoritarian equilibrium