Public Choice 42:225-234 (1984). © 1984 Martinus Ni jhof f Publishers, The Hague. Printed in the Netherlands.

Social choice and the status quo

JEFFREY T. RICHELSON*

1. Introduction

One option usually available to voters in an election is the maintenance of

the status quo. In many instances the status quo may be represented by an

incumbent office holder or set of office holders. While elections sometimes

involve candidates none of whom is the incumbent, the large majority of

elections in the United States, Great Britain and Canada involve in-

cumbents. Even in those instances where the incumbent office holder is not

running for re-election, voters may perceive the proposed successor

nominated by the incumbent's party as simply the new representative of the

' incumbent party. ' Further, in many countries (for example, Israel and the

Federal Republic of Germany), voters choose among party lists rather than

among specific candidates. Hence, there is always a status quo alternative.

The status quo option is always available in a legislature, whether the op- tion is a law or program, or the absence of a law or program. But most

social choice procedures, do not take any special account of the existence

of a status quo. Thus, the theoretical outcome of an election using the

plurality of rank-order method does not especially reflect that one of the

feasible alternatives is the status quo. In general, that it is n e u t r a l among alternatives - that no alternative is

advantaged by its label (for example, 'status quo') - is regarded as a

positive feature of a social choice rule. One possible consequence of using

a social choice procedure that does not distinguish between the status quo and its challengers is that a majority of voters may wind up worse off than

they were before the election. For example, suppose that there are nine voters and four alternatives - a, b, c and the status quo, s. If three voters

* The author would like to thank several anonymous referees and Peter Aranson for helpful comments and suggestions.

School of Government and Public Administration, The American University, Washington, DC 20016.

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prefer a to s to b to c, four prefer c to s to a to b, and two prefer b to s to c to a, then even though a majori ty prefers s to c (and to every other alternative) the plurality system would chose c.

This essay examines two procedures that insure that the alternative selected is not majori ty inferior to the status quo. The procedures achieve this result at the cost o f weakening neutrality, but they come very close to satisfying Arrow's conditions as applied to the restricted domain. Section 2 introduces some basic definitions and concepts. Section 3 considers the two procedures. Section 4 examines the dynamic aspects of the procedures. Section 5 offers some concluding remarks.

2. Definit ions and notations

I assume a set v = [ 1 . . . . . n I of voters and a set X of conceivable alter- natives with [ X [ = m. The status quo is seX. The set of potential fea- sible subsets of X,X, consists of all those subsets Y of X such that se Y. That is, X = [ Y~_X: seYI .

Each person ie V has a linear preference ordering over X. Thus, X1PiX2 means i prefers X~ to X2, for any X1, X2eX. Pi is complete (X~PiX2 or XzPiXI ), irreflexive ( - (Xt PiXy)), and transitive (Xi PiX2 . . . PiXr, im- plies X~ PiX,n). The set o f individual orderings is designated D = (P~ . . . . P~) while D is the set o f all possible D.

The social preference relation, R, such that X~RX2 means X~ is at least as good X2, summarizes the result of t ransforming individual preferences into a social preference. I f R is based on majori ty rule, then

X, RX2 ~ I l i: x~PiX~l I>-I l i : x z P i X ~ l [

Thus, XI is socially preferred to X2 (XIPX2) if and only if X1 gets more votes than X2. I f XIRX2 and X2RXI then X~IX2.

R generates a weak ordering over X if and only if: X~RX2 or X2RX1 (completeness), X~RX~ (R is reflexive), and XIRX2 . . . X,~_ ~RX,~ im- plies X~RX, n (transitivity of R). R generates a quasi-transitive ordering over X if and only if R is complete and reflexive and P is transitive so that X~PX2 . . . X,n-~PXm implies XI Pgm. Transitivity thus implies quasi- transitivity.

A social choice function F is a mapping F: X × D --, X that selects for each feasible Y and each D a non-empty subset F(Y, D) of Y as the social choice set. Thus F(Y, D) ~ ¢ and F(Y, D) _~ Y. One means of ascertaining a social choice set is to generate a social preference ordering based on R and let F(Y, D) consist of the maximal elements of that ordering. We then have F(Y, D) = {XjeY: XjRXk for all XkeY}.

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Finally, let n(X1, X2) represent the number of persons who prefer X~ to X2 so that n(X~, X2) = [ 1i: X~PiXz ] I and let X~MX2 denote that

I{i: S~PiXzl l > [ l i : S z P i S l I I •

3. The Ft~ M procedures

The argument in Section 1 suggests that we should avoid choice procedures that select an alternative that is majority inferior to the status quo. We can formalize this as a desire to satisfy the No Worse-Off Principle,

NWOP: If s M X j then Xj~F(Y, D).

We might also demand that if there are majority preferred alternatives to the status quo then the alternative chosen should be among them. Thus, we might require that the social choice procedure satisfy the Majority Im- provement Principle,

MIP: If XjeY and XjMs then F(Y, D)~_ [X~eY: XjMs}

We now examine two procedures that satisfy NWOP and MIP. Consider the social choice procedure based on the social preference rela-

tionship, ROM (where DM stands for differential maximization), defined as follows:

XjRDMXk ~ n(Xj, s)-n(s, Xj)>.n(XI¢, s)-rt(s, X~).

Using RDM, we can then define the social choice function FnM whose choice set consists of those alternatives that maximize the non-negative dif- ferential of votes that they get against the status quo versus the votes the status quo gets against each alternative. That is,

FDM(Y, D) = [XjeY: X j R D M X k , for all X , eYI.

Hence, social preference is defined according to how alternatives do against the status quo, and not necessarily (unless one is the status quo) how they do against each other. With Xk = s the social preference is based on a binary comparison and

SRDMXj ~ O>?/(Xj, s)-tt(s, Xj).

Thus, the status quo will be in the choice set if and only if none of the Xje Y defeats it. It will be the unique social choice only if it defeats all others in

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majori ty contests. I f it loses to any alternative then the social choice set will consist of those alternatives that maximize the vote against the status quo. Thus, if there is some majori ty in favor of moving away f rom the status

quo, the FoM rule will choose the alternative that commands the greatest support in terms of votes for such a move. 1 In addition to maximizing sup- port, allowing the size of an alternative's majori ty against the status quo to affect the choice also serves to narrow down the choice set. Clearly, Fn~t will satisfy both N W O P and MIP. It will also satisfy three of the four con- ditions that are used in Arrow's impossibility theorem including both tran- sitivity and independence f rom irrelevant alternatives (IIA). 2

Before proceeding further, attention should be given to the alternative domain. Since the R relation that determines the FnM choice set is based on comparisons of alternatives with the status quo, the only feasible sets (Y's) are those for which seY. Arrow assumes that all subsets of X are potential feasible sets. Hence, it is important to make sure that the condi- tions stated as Arrow's are based on the appropriate translations to a situa- tion in which se Y is required for Y to be feasible.

There is no problem with the choice set theoretic statement of IIA.

IIA: D = D ' on Y -~ F(Y, D)o= F(Y, D' ) .

Thus, IIA simply says that the choice set over the set Y depends only on individual preferences over Y and is not affected by preferences for non- candidates (the members of X - y).3 It should be clear that FnM satisfies IIA .

Likewise, the Rn~t relation by which the Fn~t choice set is determined will generate a transitive ordering. Hence, FnM will satisfy transitivity.

Transitivity: F(Y, D) is rationalizable by a weak ordering. I f Xj gets more votes against the status quo than Xk and Xk gets more

than Xt then Xi will get more than Xt .4 The Fn~t function is thus unusual - choice procedures satisfying Arrow's restrictions on m, n and X usually conform to the theorem by violating either IIA or transitivity, whether or not they violate any other conditions. Rare exceptions are the flat or in- verse dictatorial functions (Wilson, 1972), which are significantly less 'democrat ic ' than the Fo~t function.

In addition, the Fu~t function satisfies Anonymity: F(Y, (P1 . . . . . P~)) = F(Y, (P~(1) . . . . . P~(n))), which im-

plies Arrow's Non-Dictatorship condition (there is no person such that XjPiXk implies XjPXk) , and which says that changing the preference orderings among persons will not change the social choice.

The only condition that Arrow uses in his impossibility result that FOM fails to satisfy is the Binary Pareto (BP) condition. 5

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BP: S j P i S k for all i -~ X~PX~.

IT also fails with respect to Neutrality. Neutrality: XF(Y, D) = F(XF, D×),

such that X is a permutat ion of the set of alternatives and D × is a cor- responding permutat ion of preferences. To illustrate, if alternatives a, b, and c are relabeled so that ~,(a) = b, X(b) = c, X(c) = a, and if preferences are changed in a corresponding manner so that the preference ordering a b c becomes b c a, then if the social choice was a then it should become b()~F(Y, D) = { b]).

As noted earlier, neutrality, in that it requires all alternatives to be

treated equally, is generally considered desirable. The thrust of the argu- ment here is that it is not as desirable as insuring that N W O P and MIP are satisfied. And, as it happens, the FDM function satisfies the following not very weakened version of neutrality.

( m - l ) Neutrality: kin- ~F(Y, D) = F(k,n- 1 Y, DX,~- 1),

such that Xm-1 represents any permutat ion on m - 1 alternatives and the identity permutat ion X(X,,) = Xm on the m th alternative, where Xm = s. 6 Thus, ( m - 1) neutrality says that we can permute the names in any way we like among ( m - l) of the m alternatives and have the choice set change ac- cordingly. Thus, ( m - 1 ) neutrality is only one step short of full neutrality.

The most disturbing aspect of FoM is its violation of binary Pareto. However, we can correct this flaw if we are willing to settle for quasi- transitivity instead of transitivity. Consider the following modification of FOM, FROM, based on the modified binary relation R~M, such that

X~R;~X~ ~ n(Xj, s)-n(s, Xj) >_ n(X~, s)-n(s, X , )

and - (XkPiXj) for all i,

and

F~M (Y, D) = [XjeY: X~R~MXt for all X t e Y I .

The modified binary relation R~M is the same as RnM when two alter- natives get different numbers of votes against the status quo. However, if they get the same number and one Pareto-dominates the other, then the one that Pareto-dominates is socially preferred to the other. Thus, Pareto

domination is used only to break ties among alternatives that have equal numbers of votes against the status quo. However, this is sufficient to allow FOM* to satisfy the Binary Pareto Principle. It is easy to check that

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if Xj gets as many votes against the status quo as Xk and Xk Pareto- dominates Xj, then X~ gets at least as many votes against the status quo. Thus, it is impossible for Xj to get the maximum number of votes against s and be Pareto-dominated by something that gets less. Hence, the binary Pareto principle will be satisfied.

As with Fo~t, IIA, anonymity, NWOP, and MIP will also be satisfied. However, transitivity is now violated. Consider this set of orderings:

1. XI X2 X3 X4 s 2. X4 X1 X2 X3 s 3. X2 X~ X4 X~ s

Since n(Xj, s) = 3 for j = 1, 2, 3, 4 but X3 is Pareto-dominated by X2, we have * * XII~MX2PDMX3, but X~I~tX3, so R ~ t is not transitive. 7

R*, however, is quasi-transitive, - its P* components being transitive. To show this let P~' mean P* results from one alternative having more votes against the status quo than another and let P~ mean that P* results from one alternative having the same number of votes as another alternative but

P1 P 1 , 1 r E , 2 and Pareto-dominating it. The reader can check that * * P*'~* P~', P* P~P'{ all result in transitive P*.

Two final observations should be made concerning the FD~t and F~ta functions. First, the Fo~t (but not the F~a4) function has an important similarity with approval voting (Brams and Fishburn, 1978): voters are not required to order the alternatives. Using s as a benchmark, each voter need only partition the other alternatives into two sets, those preferred to s and those to which s is preferred. Hence, voters might find the FD~t voting system as simple as the approval voting system (which does not satisfy NWOP or MIP).

Second, both Fo~t and F ~ t violate strong monotoricity: if X and Y are tied as the choice set and one or more voters who preferred Y to X moves X up past Y without jumping over s, then X and Y are still tied after this change.

4. Dynamic aspects

A natural question to ask, in light of work by Simpson (1969), Kramer (1977), and McKelvey (1979) is what happens over a series of elections using the FD~t rule. Does the trajectory consisting of successive vote max- imizing (against the status quo) winners enter and remain in (or always return to, after departures from) the minmax or other set such as the M- dominant, GOCHA or uncovered sets? 9

The following example conclusively answers the question. Let Y = [ t, a, b, c , x , y , z l a n d D be

1¸0

I

. ,~ . . . ._ . . . . , . -~f

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Figure 1. Graph for preference profile.

1. t c a b z x y (3 voters)

2. t a b c z x y ( 2 ) 3. t a b c x y z ( 1 ) 4. t a c b x y z ( 3 ) 5. a c b x y z t ( l ) 6. x y z b c a t ( 1 ) 7. z y x b c a t ( l )

8. y z x b c a t ( 5 ) .

This tableau yields the graph in Figure 1, with an ar row indicating dominance and the associated number o f persons in the majori ty . Note that t is a Condorce t winner and hence the only member o f the minmax and M - d o m i n a n t sets.

Now, if s = t FD~t selects t as the initial winner the process will remain at t, since no alternative will be able to obtain a vote-maximizing major i ty

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against t. However, if the process begins at any point other than t (that is, if s # t) then not only will t not be selected in the initial selection, it will

never be selected no mat ter h o w many elections are held. Thus, the trajec- tory of vote maximizing (against the status quo) winners will never enter either the minmax or M-dominant sets.

To demonstrate why this is so, consider Figure 1. Note that t defeats each of a, b, and c by a nine-to-eight margin. However, for every member of [ a, b, c] there is another member of I a, b, c] that defeats it by a ten- to-seven margin - the cycle among [ a, b, c] consisting solely of ten-to- seven margins. Hence, no matter which member of I a, b, c} is the status quo, another member will be the alternative that maximizes votes against it. Therefore, the trajectory will only include a, b, and c regardless of the number of elections involved.

The same reasoning applies to Ix , y, z} with respect to [a , b, c, t ] . Each member of Ix , y, z] loses to t by a nine-to-eight margin. Each member of [ a, b, c] defeats each member of { x, y, z} by a ten-to-seven margin. Since the cycle among { x, y, z ] is based on eleven-to-six margins, the alternative that maximizes votes against a member of [ x, y, z ] will also be a member of [ x, y, z ] . Hence, if it beings in { x, y, z ] , the trajectory would always remain in Ix , y, z ] .

This example suggests that one could construct a profile with different cycles such that under majority rule each member of the top cycle defeats every alternative outside the top cycle, in which every member of the next- to-top cycle defeats every alternative outside the top and next-to-top cycles, and so on yet never does the trajectory associated with the FoM rule move out of the cycle in which it begins. Clearly, then, the initial designation of the status quo could heavily influence the initial and all subsequent results of using the FDM rule.

5. Concluding remarks

1 have investigated the social choice problem from the perspective of seek- ing to insure that a majority of voters are not worse off after the election, such that worse off means selecting an alternative that would lose to the status quo in a majority contest. This investigation led rather naturally to using the status quo as a benchmark against which to judge challengers.

Both the FDM and F~M procedures have attractive properties. Both satisfy the Not Worse-Off and Majority Improvement Principles. The Fo~t rule is rationalizable by a transitive R and allows for ease of voter response, while the FoM. rule satisfies binary Pareto. Both satisfy I l A , anonymity and ( m - 1 ) neutrality.

Examination of the results of repeated applications of the F~9~ pro-

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cedure reveals that these results are heavily influenced by the starting point, with there being no guarantee that the process will move into the minmax, M-dominant, or similar solution set. Whether this is considered acceptable will depend on whether one believes the social choice should be a subset of these sets regardless of the identity of the status quo.

NOTES

1. Plott (1976) investigates a method for finding alternatives to the status quo that a group

will unanimously support. But his framework uses a continuous alternative space.

2. Arrow (1963) proves that if m ~ 3 , n_>2 and both are finite and X = P(X)-~I, (all non-

empty subsets of X) and all possible D are permissible, then there is no social choice func-

tion that satisfies transitivity, 1IA, non-dictatorship, and binary pareto optimality.

3. FDM certainly violates the requirement that: if D = D ' on Y then X j R X k ~, X j R ' X k , which is simply Arrow's IIA restricted to two-element sets when all two element sets are

feasible. However, when not all two-element sets are feasible, as in this essay, IlA no

longer implies this requirement. Since our concern is with what happens on feasible sets,

IIA is the appropriate condition as well as being the one that Arrow uses.

4. Note that I am not making the common mistake (perpetrators will remain anonymous) of

determining R from the choices over the entire feasible set, which when done with respect to any procedure that is equivalent to simple majority results in an intransitive (Arrovian)

R. Rather, we are following the Arrow tradition by determining the relation between Xj and Xk on the basis of the smallest potential feasible-subjet ([ Xj, Xk, s] ) of X of which

Xj and X~ are members. See Fishburn (1974).

5. FDM certainly satisfies the requirement that XjPi Yg for all i --, F([ Xj, Xk }, D) = [ Xj }, which when all two-element sets are feasible is the same as BP. However, in our case for

{ Xj , Xk I to be feasible, one o f the two must be the status quo. Hence, this requirement

is rather trivial and BP is more appropriate. 6. The same alternative X,~ is the subject of the permutat ion ~,(Xm) = X,~ for all possible

permutat ions of X1 . . . . Xm - 1 • 7. There is no impossibility theorem that says that given n _> 2, m _> 3, and all preference order-

ings are permissible, that transitivity, IIA, binary Pareto and non-dictatorship are incom-

patible if seY is required for Y to be feasible. Grether and Plott (1982) establish a result from which such a result follows if transitivity is replaced by the weak axiom of revealed

preference (WARP). Unfortunately, WARP two and three-element subsets of X, regardless of whether they contain the status quo.

8. The following discussion also applies to the Fo~t function.

9. The minmax set is obtained by ascertaining for each alternative the max imum number of votes any other alternative can receive against it in a binary context. The alternatives with

min imum maximums constitute the minmax set. The M-dominant set is the smallest set of alternatives such that everything in the set is majority (M) preferred to everything outside the set. The GOCHA and uncovered sets are subsets of the M-dominant set.

Usually, instead of the M-dominant set discussion involves the P-dominant set such that P indicates social preference and may or may not be determined by majority rule.

However, since the FoM rule allows social preference to be relative to the status quo so does the P-dominant set - there is no single P-dominant set that exists regardless of which alter- native is designated s. Hence, it is both more appropriate and more interesting to consider

the M-dominant set.

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REFERENCES

Arrow, K.J. (1963). Social choice and individual values, second edition. New Haven: Yale University Press.

Blair, D. (1979). On variable majority rule and Kramer's dynamic competitive process. Review of Economic Studies 46: 667-673.

Brams, S.J., and Fishburn, P.C. (1978). Approval voting. American Political Science Review 72:831-847.

Fishburn, P.C. (1974). Impossibility theorems without the social completeness axion. 2Econometrica 42: 695-706.

Grether, D.M., and Plott, C.R. (1982). Nonbinary social choice: An impossibility theorem. Review of Economic Studies 49: 143-149.

Kramer, G. (1977). A dynamic model of political equilibrium. Journal o f Economic Theory 16: 310-334.

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Simpson, P. (1969). On defining areas of voter choice: Professor Tullock on stable voting. Quarterly Journal of Economics 83: 478-490.

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