Public Choice 54:283-288 (1987) © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands

In search of optimum 'relative unanimity': A comment*

MICHAEL BROOKS Department o f Economics, Tasmania 7001, Australia

University o f Tasmania, GPO Box 252C, Hobart

In a recent issue of this journal Richard Cebula and Milton Kafoglis (1983) (hereafter called C&K), present a model which is designed to determine the optimal voting rule. Their analysis builds on the work of Fishkin (1979) and Rae (1975) who had offered criticisms of Buchanan and Tullock's (B&T) determination of the optimal voting rule. The purpose of this note is to indicate that C&K have left some misleading trails in their search and to rectify these false leads.

The central concern of C&K's paper is to analyse the distinct possibility that some collective outcomes may be sacrificed ' . . . because "negative minorities" may be able to block efficient decisions' (C&K, 1983: 196). To this end they ' . . . fo rmula te a simple framework for the selection of a decision rule which recognizes these costs of surrendered outcomes' (C&K, 1983: 196). Figure 1, which is reproduced here from C&K's discussion, represents in part the most reasonable depiction 1 of the costs and benefits to the collectivity of all the voting rules from simple majority to unanimity.

The benefit curve is assumed to rise as the voting rule becomes more inclusive. It is alleged that the 'free rider' problem and the related ineffi- ciency diminishes as the voting rule approaches unanimity. 2 In substance this curve does not differ markedly from that found in B&T's model. 3 The discussion by C&K of this concept is, however, confusing. Cebula and Kafoglis consistently refer to the GB curve as representing the group's gross

benefits but fail to clearly indicate what gross refers to. They note, for example, in their final paragraph that B&T's framework uses net benefits whereas their analysis refers to gross benefits. And to be sure, B&T did at one stage 4 propose to use net benefits, measured as the difference between the benefits and costs of provision (B&T, 1962: 44). Presumably C&K therefore take gross benefits to reflect the value of collective provision without any regard whatsoever to what has to be given up. But if this is so, then it is not at all clear why the benefit curve is of much analytical interest. Ninety years ago Wicksell admonished economists for their failure to take

*The helpful comments of a referee are acknowledged.

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$ GB, GC

~ GC

50 R* i00

Figure 1. Percentage of votes required for a decision

take account of the proposit ion that collective decisions about public output

ought to also address the issue of how it will be funded. C&K's procedure is a step back to the economics which Wicksell and other public choice economists have been at pains to overturn. It is therefore better to interpret

the benefit curve as net of the costs of provision. Interest centers on the cost curve, GC, because it is this concept, as C&K

are keen to note, 5 which differs f rom that found in B&T's analysis. In

B&T's model the cost curve reflects the decision making costs which will be incurred by each individual as they at tempt to reach agreement under the various rules. In Cebula and Kafoglis 's f ramework (C&K, 1983:197 and

200) the cost curve represents the value of the proposals to the group which are sacrificed as a result of a failure to come to a decision at a given voting rule. Armed with these definitions it appears that R* represents the optimal voting rule in Figure 1. It is the rule which maximizes the difference between the benefits and costs to the group of using various decision rules. The problem with this figure and its discussion is that it is misleading. The GC curve, at least as apparently defined by C&K, cannot lie above the GB curve, and the GB curve does not provide a reasonable representation of the benefits of collective action.

As a precursor to a discussion of these matters it is important to analyse C&K's treatment of the GC curve. To the extent that it represents the

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expected group benefits forgone as a result of inaction, then the GC curve

is apparently the GB curve adjusted by the probability that the collectivity will fail to come to some agreement on the collective project(s ) . That is, the total GC(R) curve can be expressed as GC(R) = Pi.GB(R) where Pi [0 < Pi -< 1] is the probability of group inaction over project i. The word apparently is used advisedly because I have not been able to discern from C&K's paper what they feel the GC curve exactly measures; they do not at any stage give a precise definition.

It might be tempting to argue that the GB curve refers to the set of projects which receive agreement and the GC curve reflects the set which are sacrificed as a result of the negative minorities. Since the two curves would refer to two different sets of projects the GC curve could not be defined, as is suggested above as an adjusted GB curve. It is worth stressing that apart from a passing reference 6 to such a position there is no further hint in C&K's argument that their cost curve refers to some subset of the potential projects. Indeed, it is instructive to note that C&K argue that the costs refer to a ' . . . diminished probability of any decisions being made' (1983: 200, their emphasis). Evidently, the costs as envisaged by C&K refer to the potential result of inaction for all the projects not just some subset. Indeed, the definition of the GC curve given here seems to be consistent with the entire tone of their argument. Their reference to the possibility that collective projects may be 'sacrificed' implies that the projects must be admitted also as potential benefits. With this in mind, I return to the central issue at hand.

Given the apparent definition of the GC curve it follows that the GC curve cannot be drawn above the GB curve. To do so would imply that the probability that the group will fail to implement the project(s) can exceed unity. How does this affect C&K's discussion? It implies that the case (C&K, 1983: 199) they make in favor of simple majority rule can not be sustained. The GC curve will not exceed GB for decision rules requiring more than a simple majority as in their Figure 2. Accordingly net benefits are not maximized with a simple majority rule. Nor can C&K appeal to the argument, which they do at one stage, that the GC lies above the GB because it alone includes the ' . . . cost of the foregone redistributions envisioned by Fishkin . . . and Rae . . . as basic to the political process ' (1983: 201) 7. If the redistribution they have in mind represents a cost when the decision is not to act, then it must also reflect a benefit if there is a possibility that the collectivity will go ahead with the collective action. The appropriate way of deriving a case for simple majority rule is discussed below. Before this is done consider again the relationship between the GB and GC curves as portrayed in Figure 1.

In the figure they draw the GB curve through the origin. There is nothing incorrect about this procedure in itself. The origin can indicate that the

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expected benefit from a simple majority rule amounts to zero, or it can be

calibrated to measure some other amount. C&K do not indicate which pro- cedure they adopted. The problem of how the origin should be interpreted

occurs when the GC curve is drawn through the origin also, as is C&K's practice. If the origin indicates a positive value, then the probability of inac- tion must be in general unity, otherwise the GC curve would lie beneath the

GB curve and cut the vertical axis below the origin, i.e., GC(R) = GB(R),

when Pi = 1. But if this is so, then there is an inconsistency in their ensuing discussion. Cebula and I(afoglis 8 are adamant that the probability of inac-

tion will increase as the voting becomes more inclusive! An alternative pro- cedure is to assume that the group benefit from a simple majority rule is

zero. The GC curve would then pass through the origin irrespective of the

probability of inaction, i.e., the origin represents zero dollars. But this defence is not totally satisfactory either. It implies, contrary to their claims, (C&K, 1983: 199) that they have not provided a case for simple majority rule, even if we momentarily allow the GC curve to exceed the GB curve as

in their Figure 2. The institutional framework C&K have in mind is one of a ' . . . vo luntary association of persons brought together for the purpose of

satisfying their common needs' (1983: 196, my emphasis). If each and every individual expected the benefit from simple majority rule to be zero, then

there is nothing which would compel the individuals to act as a collectivity in the first place. The group would be indifferent between conducting its in- period affairs under majority rule and disbanding itself at the constitutional

stage. Hardly a compelling case in favor of simple majority rule. In fact the group would disband itself if some units of a pure public good had been pro- vided prior to discussing the issue at a constitutional stage and collective

provision would completely crowd-out this market outcome. There is a way though around these problems which preserves the spirit of their argument.

In order to demonstrate that a consistent case can be made for simple

majority rule in terms of their framework, consider the situation in which the individuals at the constitutional stage expect the level of collective provision to be invariant 9 over all voting rules. This implies that the total

benefit curve is parallel to the horizontal axis. Consider Figure 2 drawn here. If the origin depicts a zero amount and there are positive benefits from simple majority rule, 10 then the GB curve will cut the vertical axis above the origin. In keeping with C&K's argument, let the probability of group inaction be relatively low at less inclusive voting rules. The GC curve will cut the vertical axis below the GB curve and will rise as the voting rule becomes more inclusive. In this particular setting the group would maximize 'net ' benefits, i.e., G B - G C , by selecting a simple majority rule.

It is relatively easy to accomodate C&K's other propositions in terms of the modified geometrical structure. The distribution of voters and cost-

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$

GB, GC

I

I t

,GB [

I ! i I

t !

!

I I I I i [ f t I

[ i ! J i J I [

- ! [

50 R* 100

Figure 2. Percen tage o f votes required for a decis ion

shares could be such that the level of the publicly provided good rises as the voting rule becomes more inclusive. This is drawn 11 as GB' in Figure 2. The optimal voting rule occurs at the relative unanimity level R*. Unlike C&K's representation there are no misleading equalities here between group benefits and costs, nor is the cost curve inappropriately depicted above the benefit curve. Similarly, the case in favor of full unanimity is also relatively easy to make in the modified framework. Say the GC curve cuts the vertical axis above the origin but below the GB curve. Unanimity will be the preferred decision rule as long as the slope of the GC curve is everywhere

less than the slope of GB. Cebula and Kafoglis's paper is a useful addition to the on-going search

for the rules of a fiscal constitution. Search is easier to carry out however when the guidelines have been appropriately set out. To this end the note here has attempted to rectify some of their false leads.

N O T E S

1. To be fair it shou ld be noted tha t Cebu la and Kafogl is (1983: 200) only go as far as s ta t ing

tha t ' . . . the GC curve . . . is the mos t r easonab le ' representa t ion .

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2. It should be noted that this result is open to question. See, for example, Brennan and Lomasky (1984) who criticize the notion that unanimous decisions are necessarily efficient ones.

3. In B&T's construction unanimity minimizes the degree of political externalities, which is just another way of saying, as C&K do, that it also results in the highest level of benefits.

4. The bulk of B&T's analysis is, of course couched in terms of decision-making costs and external costs.

5. See their discussion on page 200. 6. See their discussion in footnote 2: p.201. 7. The sort of procedure proposed here would not rescue the particulars of C&K's analysis.

Projects which were included in the benefit set at some less inclusive rule might have to be placed in the inactive set as the voting rule is made more inclusive. The two curves would not therefore refer to two mutually exclusive sets.

8. See their discussion in Section III:200. 9. See Chapter 11 of Buchanan (1967) for a discussion of the conditions under which this will

occur. This does not imply that the probability of inaction will not increase as the voting rule becomes more inclusive. As the decision rule approaches unanimity the probability increases that some individuals will hold out for more favorable 'tax' prices.

10. The benefits could be measured net of the benefits which occur under the pre-constitutional market outcome.

11. In order to simplify the geometry the GC curve has not been constructed in line with the modified GB' curve.

REFERENCES

Brennan, G., and Lomasky, L. (1984). Inefficient unanimity. Journal of Applied Philosophy 1 (1):151-163.

Buchanan, J. M. (1967). Public finance in democratic process: Fiscal institutions and individual choice. Chapel Hill: University of North Carolina Press.

Buchanan, J. M., and Tullock, G. (1962). The calculus of consent: Logical foundations of constitutional democracy. Ann Arbor: The University of Michigan Press.

Cebula, R. J., and Kafoglis, M. Z. (1983). In search of optimum 'relative unanimity'. Public Choice 40 (2): 195-201.

Fishkin, J. S. (1979). Tyranny and legitimacy: A critique of political theories. Baltimore: The Johns Hopkins Press.

Rae, D. W. (1975). The limits of consensual decision. American Political Science Review 69 (December): 1270-1294.