Variation of the electorate: Veto and purge

ROY GARDNER*

Abstract

In the paradigm social choice problem, there is a fixed set of alternatives and a fixed set of voters. This essay considers variations of the electorate when some subset of voters has a special voice on some subset of alternatives. We formalized such a situation by means of the veto function. We focus on stable veto functions, exhibit a stable liberal social choice function and promotion mechanism, and investigate a notion of stability for groups whose membership itself is the social state.

1. Introduction

In the p a r a d i g m social choice p rob lem, there is a fixed set of al ternat ives, A,

and a fixed set of voters, N. W e add s t ructure to the p rob lem by impos ing

res t r ic t ions on admiss ib le profiles of voters ' preferences or on the charac ter -

istics of the social choice function. If one al lows for unrest r ic ted d o m a i n of

preference profiles and requires only tha t social choice be non-empty , then

essential ly no s t ructure is added.

This essay considers social choice functions, all of which have a var iable

e lec tora te structure: different groups of voters have a special voice on different

subsets of al ternat ives. Examples of such s i tuat ions a b o u n d in practice. When

the member s of a society de l ibera te on an extension of the franchise, those

present ly enfranchised have a special voice on the enfranchisement of others.

P r o p e r t y r ights give an agent a special voice a b o u t those social states tha t

concern his proper ty . Rank often gives an agent a special voice on quest ions of

p r o m o t i o n in a hierarchy.

N o t w i t h s t a n d i n g tha t such social choice functions are impor tan t , they have

not been much s tudied in the formal l i terature, a l though two i m p o r t a n t

* The author wishes to thank P. Aranson, F. Breyer, A. Denzau, S. Matthews, R. Rosenthal, and two anonymous referees for their helpful comments. The errors which remain are the author's own. A preliminary version was presented at the 1981 Public Choice Meetings.

Department of Economics, Iowa State University, Ames, IA 50011.

Public Choice 40:237-247 (1983). © 1983 Martinus NijhoffPublishers, The Hague. Printed in the Netherlands.

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exceptions to this generalization should be cited. First, the literature on liberal social choice functions (Sen, 1976) studies a special kind of electorate variat- ion, akin to property rights. Second, Klevorick and Kramer (1973) and Gardner (1982) give examples of convergent sequences of social choice, in which, at each step of the sequence, an agent's voting strength is social-state dependent. This essay proposes a general formalism for analyzing var iab le electorates, the veto function. Interest focuses in the next section on stable veto functions, those that lead to non-empty social choice and have non-empty cores. Section 3 applies stable veto functions to the question of liberal social choice. Here, we prove the existence of a strategically consistent liberal social function, essentially Gibbard's (1974) system of first-order rights. We also exhibit a stable veto function within a rank hierarchy. Section 4 examines electorate expansion and contraction. A stable veto function is upheld by a stable group, that is, one whose members wish neither to purge existing members nor to recruit new ones. We show the existence of stable groups and introduce a dynamic adjustment process for the variations in the membership of unstable groups. The concluding section places these results in perspective, both with respect to known results and with respect to promising avenues for further work.

2. The veto function

Let A be a fixed set ofm social alternatives, A = {xl,x2 . . . . , Xm}, with m > 3. Let N be a fixed set ofn votes, N = {1, 2 . . . . , n}, with n > 2. Each person i e N has an irreflexive, transitive, and complete ordering, R~, of the alternatives in A. R = (R1, R2 . . . . . Rn) denotes a profile of preferences.

Denote by • the empty set and by 2 A the power set of A. Thus, 24 = { U: U c A}. In particular, 4~ e 2 A, and A e 2 A. A social choice function, F, is a map

F: F I R, - , 2 4 -

To formalize the notion that a certain subset of voters, S c N, has a special voice on a certain subset of alternatives, B c A, one introduces the veto function, V. B ~ V(S) means that the members of S have the power to veto the set of alternatives, B, from the social choice set. The full veto power ofcoalition S, V(S), is the collection of all subsets of alternatives over which S has veto power. The veto function can be considered the dual of the effectivity function of Moulin and Peleg (1982). We place four conditions on the veto function.

Standardness. For all S e 2 N, • s V(S) A ¢ V (S). A coalition may be powerless (~ = V(S)), but no coalition may stymie the social choice (A ~ V(S)).

Pareto. V(N) --- 2 A - {A}. The grand coalition has veto power consistent with standardness.

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Monotonicity. If B ~ V(S) and B = B', then B' ~ V(S). In a coalition can veto ,B, then it can veto any subset of B.

Superadditivity. 1 If $1 and $2 are disjoint coalitions, BI ~ V(SO and B 2 e V(S2) ,

then B1 w B E ~ V(S 1 ~ S2). Disjoint coalitions do not lose veto power when they join forces.

Given these four conditions, one can readily deduce another monotonicity condition: if B ~ V(S) and S c T, then B ~ V(T), the veto power of an enlarged coalition does not shrink. 2

A simple example of a veto function is the representation of the Pareto- extension rule. V(S)is empty unless S = N, in which case V(S)is given by the Pareto condition. More generally, for any social choice mechanism in which a coalition is either winning (has the veto power of the grand coalition) or losing

(has no veto power), one has

~ V(N) if S is winning V(S) = ~ {~} otherwise.

For example, such structures would apply to majority rule and oligarchical rule. However, such simple examples do not involve any variation of the electorate.

A more complicated example, which does involve variation of the elector- ate, is based on the notion of an ideal over A a, IA, which is a collection of subsets of A satisfying the conditions:

~ I a , A~Ia; if B ~ Ia, and B' c B, then B' ~ IA; if B ~ I ~, and B' ~I~, then B ~ B' ~ I ~.

For each coalition S ¢ N, let V(S) be an ideal over A, denoted Ia(S). For disjoint S, S' for which S ~ S' ~ N, let V(S ~ S') be the minimal ideal contain- ing IA(S) and Ia(S'). Finally, V(N) is given by the Pareto condition. However, note that 2 A - {A} is not an ideal over A. Such a construction satisfies the conditions for a veto function and will be used repeatedly in the sequel.

A coalition must have both the will and the power to exercise its veto. In this regard, a rather conservative criterion governs the will to veto: no member of a coalition ever regrets the exercise of that coalition's veto. Formally, given a fixed profile of preferences, R, let x > (S) denote the set {y e A: yPix for all i ~ S}. Then S exercises its veto over x if and only if A -- x>(S)~ V(S).

An alternative is stable relative to R if no coalition exercises its veto against it. A veto function, V, is stable if the set of stable alternatives is non-empty for every R. Stable veto functions have two interpretations, first as social choice

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functions, and second as cooperative games in normal form. In the game interpretation, the strategy set of each coalition consists of its veto options. In particular, the core of a stable veto function is non-empty. A stable veto function is strategically consistent social choice, since outcomes sincerely revealed by preferences are in the core of the corresponding cooperative game. Stable veto functions thus form the focus of the sections that follow.

3. Liberal veto functions

This section considers the veto function representation of social choice satisfy- ing a liberalism condition. A minimal liberalism condition is introduced, which is common to both Sen's (1970) condition L and Gibbard's (1974) condition L2. The main result is that the veto function satisfying minimal liberalism and the ideal-based construction is stable. However, stronger liberalism conditions jeopardize stability.

The minimal condition of liberalism is that for each person, i, there is a social state, x~, over which i has veto power. Suppose that no two persons have

veto power over the same social state; then we can think of each person's veto state, x~, as his protected sphere. In terms of the veto function, one has:

Minimal liberalism. For all i N, V({i}) = {4~, {x,}}.

Under minimal liberalism, each person's veto function is an ideal. The veto function for groups is based on the ideal construction of the last section. Thus, the veto function of {i,j} is the ideal

V({i,j}) = {~, {xl}, {xj}, {xi, x2}}.

To satisfy the standardness condition, alternatives must outnumber voters; otherwise, V(S) contains A for some S ¢ N.

Under the no-regret hypothesis, we can now show:

Proposition 1. Let the veto function, V, satisfy minimal liberalism and the ideal-based construction. If alternatives outnumber voters, then V is stable.

Proof. Since alternatives outnumber voters, there is at least one alternative against which no person has a veto; call this alternative w. If w is Pareto optimal, then N does not exercise a veto against it. I fN does not exercise a veto against w, then neither does any subset, S, of N; for if S does exercise its veto against w, then this implies that some member of S has a veto against w, which is a contradiction. Thus, if w is Pareto optimal, w is stable. Suppose that w is not Pareto optimal. Then there exists w', such that w'Piw for all i. At most, one person, i, has a veto against w'. However, i cannot exercise his veto against w',

Table 1. Stable outcomes, m in ima l l ibera l i sm veto function, m --- 3, n = 2

Preferences, Preferences, person 2 person 1

$ x z y x y z y x z y z x z y x z x y

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x y z x x x , y x , y x , y , z x , z

x z y x x x , y x , y , z x , z x , z

z x y x , z x , z x , y , z y, z z z

z y x z y , z y , z y , z z z

y z x z y y y y, z z

y x z x x , y y y y, z x , z

nor can any coalition containing i do so, because the set of alternatives A - w'>({i}) contains w, against which no person has a veto. Therefore, if w' is Pareto optimal, w' is stable. If not, then there exists w", which is Pareto superior to w'. By finiteness of A, repeating the argument must lead to a stable alternative. Hence, V is stable.

To illustrate, suppose m = 3, n = 2, x 1 = x, and x2 = y. The stable out- comes of the veto function, V, are found in Table 1, where x l x z x 3 means x ~ t P x z P x 3. Note in the table that since no person has a veto against z, it appears as a stable outcome, unless it is Pareto inferior to some other alternative.

The veto function satisfying minimal liberalism is mainly of theoretical interest, because the liberalism conditions proposed in the literature are substantially stronger. However, it is not hard to extend minimal liberalism along such lines. Thus, Sen's (1970) condition L requires that for each person, i, there exist a pair of alternatives (xil, xn) such that i is decisive over the pair in either direction of preferences. The veto function representation of this con- dition is

V({i} = {~, {xi~}, {xn} }.

Condition L allows i to veto either member of the pair but not both simultaneously. However, the veto function for a person is no longer an ideal. There is a variety of ways to extend the veto function so defined to groups, the most obvious being

y ( s ) = y({i}),

when S ¢ N. Proposition 1 continues to hold for the liberalism function so extended. 4

To see the effect of this extension on the example just considered, let voter 1 have the pair (x, z), while voter 2 has the pair (x, y). Table 2 shows the difference

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Table 2. Stable outcomes, Sen liberalism veto function, m = 3, n = 2

Preferences, Preferences, person 2 person 1

$ x z y x y z y x z y x z z y x z x y

x y z x x x , y y y, z x , z

x z y X X X Z Z X , Z

z x y X , Z X , Z X , Z Z Z Z

z y x z y, z y , z y, z z z

y z x z y y y y , z z

y x z x x , y y y y, z x , z

that condition L makes to the outcomes. The social choice function in Table 2 has already been mentioned in the literature: it accords what Gibbard (1974: 403) calls a system of first-order rights. Proposition 1 implies that for such a system of rights, the sincere outcome is also in the core of the associated cooperative game. For systems according higher-order rights, in particular for systems satisfying Gibbard's condition L2, this strategic consistency no longer obtains. A sincere outcome need not be in the core (Gardner, 1980). Thus, there is an inevitable conflict between the strength of a libertarian claim and the demand for strategic consistency.

To see how such a veto function might arise in practice, suppose that 1 and 2 belong to a hierarchy in which 1 outranks 2. In particular, 1 can veto 2's promotion. However, either 1 or 2 may veto the recruitment of an outsider (person 3) to the hierarchy. Thus, the social states are:

x = promote 2, recruit 3 y = not promote 2, recruit 3 z = promote 2, not recruit 3 w= not promote 2, not recruit 3,

w being the status quo. Person l's veto function consists of {x, y, z} and all of its subsets, by virtue of his rank. Person 2's veto function consists of {x, y} and its subsets. In this case, the status quo is the alternative against which no person has a veto. Again applying the argument of Proposition 1, this veto function, which is a mixture of minimal and Sen liberalism, is stable. A person is promoted or recruited only if no higher ranking person vetoes his pro- motion or recruitment. A person who succeeds in getting promoted or re- cruited is thus blackballproof.

4. Veto functions for recruitment and purge

We now take up the question of variation in the group of voters itself. We

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consider two kinds of variations: recruitments (expansions) and purges (con- tractions). These variations require an amendment of the conditions for the ve- to function. To see why, let N O be the current set of voters and N the set of all logically possible voters. The group membership agenda becomes 2 ~ = A, and the empty set sannot be excluded (the group may want to dissolve). Thus, the standardness and Pareto conditions now read:

Standardness'. For all S = No, • e V(S).

Pareto'. V(No) = 2 ~.

The concept of individual preferences also poses some problems, since the alternatives being ordered are themselves sets of persons. Here, although it is not necessary for the results, the presentation is greatly simplified by assum- ing that preferences are measurable. Let R~ be an ordering of 2 N and u~ its numerical representation, u~ is measurable when

i(s) = E u,(j). .i~S

Agent i thinks that j is good if ui(j) > 0 and bad, if ui(j) < 0. Agent i thinks that the current group is bad if ui(No) < 0. The empty set itself has no value: u,(~) = 0. 5 Such preferences are measurable in the sense that u~(j) measures the utility difference between a group containingj or not.

Preferences are selfish if each person thinks himself good. Denote by G(/) the set of members j that i thinks are good:

G(O = (j ~ N: u~(j) > 0}.

For any group S, G(S) denotes the set of persons unanimously considered good by S:

= G S

One can also define the analogous bad sets, B(i) and B(S). With selfish preferences, G(i) is never empty.

In view of the complications that the preference side of the model raised, we restrict attention to veto functions of the form

~2 N if S is winning V(S) = ( ~ otherwise.

Here, the notion of winning is relative to N o. Let W be the set of winning

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coalitions relative to No.

W = {S c No: S is winning}.

Finally, let C(No) denote the group that is part of every winning coalition:

C(No) = A S. S ~ W

C(No) is non-empty if W represents an oligarchy or collegial polity (Brown, 1975).

Let the group membership question, both purge and recruitment, be con- sidered simultaneously. Proposition 2 gives a simple condition under which the group does not dissolve.

Proposition 2. Suppose preferences are selfish. Then G(No) ~ C(No) is con- tained in the outcome of the group membership question.

Proof. Restrict attention to a winning coalition, S. G(S)~ G(No). By the hypothesis of measurability of utility, for any group S', the members of S unanimously prefer G(No) u S' to $'. Thus, if G(No) c No, they are not purged, and if they are not members, they are recruited.

Again, if preferences are selfish, then every member i of C(No) prefers No to N O - {i}. Since i is a member of every winning coalition, no purge of /can take

place. Under the Pareto extension rule, No = C(No). If preferences are selfish, no

current member is purged. We can compare this result to those of the last section, since a natural liberal condition in this case would be that no agent be purged unless he judges himself bad (ui(i) < 0). Such a condition, although it would guarantee that the group not dissolve, would nevertheless conflict with the principle of majority rule, however qualified.

If the outcome of the group membership question is the group itself, then the group is stable. Its members wish to carry out no purges, nor do they wish to recruit further members. Even if N O is not stable, one would expect there to be some other set that is stable. Proposition 3 shows the existence of a stable group under majority rule.

Proposition 3. If at least one agent's preferences are selfish, then there exists a stable group relative to majority rule.

Proof. We shall prove a slightly stronger result, that there exists a maximal stable group.

Let V be an n-dimensional square matrix of l's and - l's:

v u = 1 if ui(j) > 0 and - 1 if u~(j) < O.

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Let x be an n-dimensional row vector representing group membership:

x i = 1 if i ~ S and 0 otherwise.

The maximal stable group is defined by the linear programming problem (LP). Maximize ~ xi subject to (1) xs v >_ 0; (2) - X~_s -< 0; and (3) 0 < xl < 1, such that Vs = {columns V~ in V: j ~S}.

The ,constraint (1) corresponds to the property that in a stable group no member is purged (has a majority opposed); constraint (2), to the property that nu further member is recruited (has a majority in favor). The problem is feasible, since x = 0 is a feasible solution. The problem is bounded, since ~ x~ < n. Hence, there exists an optimal solution. As long as V has one positive element, the 0-vector is not optimal, and the maximal group is not empty.

We required only a very weak form of measurability to establish this result. Proposition 3 can also be extended to the cases in which C(No) is not empty, simply by adding constraints to the problem (LP). The proposition gives a simple test for discerning whether the largest possible group, N, is stable, namely that the column sums of V all be non-negative.

Given the existence of stable groups, one is led to define the group member- ship dynamics Nt+ 1 = f(Nt), such that N~ is the group at decision stage t and f(Nt) is the outcome of the group membership decision at t. Indeed, ifS = N~ is stable, then it is a fixed point of the dynamics, S = f(S). One might suppose that these dynamics are stable, in the sense that they converge to a stable group. However, this need not be the case, as a simple example shows.

Let N = { 1, 2, 3, 4, 5}, and suppose that the V matrix satisfies

¸V=

1 - 1 - 1 --1 1 ) - 1 1 - - 1 - -1 1

- -1 - 1 1 - -1 1 .

- -1 - 1 - -1 1 1

1 1 1 1 1

Suppose No = {1,2,3,4}. A strict majority wins. Then {1,2,3} purges 4, {2, 3, 4} purges 1, { 1, 3, 4} purges 2, { 1, 2, 4} purges 3, and all recruit 5. So {5} = f(No). 5 in turn recruits everyone, so N = f(N1). At this point, the purges recommence. Thus, the membership oscillates between N and {5}, neither of which is stable.

We conclude the discussion with an example which is stable. N is as before, but now player 1 has a veto. Player 1 and any two other players win. N o = {1, 2, 3}, and V satisfies

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V =

1 - 1 - 1 1 1 ) - 1 1 1 - 1 - 1

1 - 1 1 1 1 .

1 - 1 - 1 1 1

1 - 1 - 1 1 1

f ( N o ) = {1, 3,4}, since {1, 3} purges 2 and recruit 4. f ( N 1 ) = {1,4, 5}, since { 1, 4} purges 3 and recruit 5. The group is now stable, and the dynamics have converged.

5. Conc lus ion

This essay explains how social choice theory can handle some questions of variation in the electorate. The basic formalism for analyzing the extra structure found in electorate variation is the veto function. Veto functions for liberal social choice problems and for group membership questions have been developed. A useful corollary to the liberal veto function construction is the discovery of a strategically consistent liberal social choice function. This construction aids in modeling questions of promotion in a hierarchy. The veto function for group membership explains how to define stability for groups and dynamics for group membership.

A variety of practical applications of the veto function suggest themselves. Certain historical episodes, such as the fifteenth and nineteenth amendments to the U.S. Constitution, 6 or the great purges in the Soviet Union, might prove interesting for applying veto functions. The results of Aranson (1982) on the reapportionment question might be formalized by a veto function. We leave these questions for further research.

The results presented reveal another side of the conflict between liberal and majoritarian values. As far as membership in a group is concerned, liberal principles would seem to imply that only the person has the right to resign or to retain membership. Just the opposite conclusion follows from majoritarian principles. What is clear is that unbridled majoritarianism can lead the group to dissolve. The veto function approach brings this conflict to the level of logical irreconcilability.

NOTES

1. I am grateful to A. Denzau and R. Rosenthal for correcting an earlier misstatement of this condition.

2. To show this condition, consider the coalitions S and T - S. B ~ V(S) by hypothesis, while ~ ~ V ( T - S) by standardness. By superadditivity, ~ u B = B ~ V(S w T - S) = V(T).

3. Brown (1975) first introduced this notion into social choice theory.

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4. A somewhat stronger extension is the following. For each voter i, denote by n(/) has assigned

pair of alternatives. Denote by H(S) = I~ n(i), the Cartesian product of the n(i). Define the veto i~S

function for S as

V(S) = {B: B c II(S') for some S' c S}.

5. According to von Wright (1972), this proposal was first suggested by G. E. Moore. 6. A. Denzau pointed out this possibility. For further discussion, see Aranson (1981).

REFERENCES

Aranson, P. H. (1981). American government: Strategy and choice. Aranson, P. H. (1982). Political inequality: An economic approach. In P. Ordeshook and K.

Shepsle (Eds.), Political equilibrium. Brown, D. (1975). Aggregation of preferences. Quarterly Journal of Economics 89: 456-469. Gardner, R. (1980). The strategic inconsistency of Paretian liberalism. Public Choice 35(2): 241-

252. Gardner, R. (1983). 2-transfer value and fixed price equilibrium in 2-sided markets. In P. K.

Pattanaik and M. Salles (Eds.), Social choice and welfare. London: North-Holland. Gibbard, A. (1974). A Pareto-consistent libertarian claim. Journal of Economic Theory 7: 388-410. Klevorick, A. K., and Kramer, G. H. (1973). Social choice on pollution management: The

Genossenschaften. Journal of Public Economics 2: 101-146. Moulin, H., and Peleg, B. (1982). Cores of effectivity functions and implementation theory.

Journal of Mathematical Economics 10:115-145. Sen, A. K. (1970). Collective choice and social welfare. San Francisco: Holden-Day. Sen, A. K. (1976). Liberty, unanimity, and rights. Economica 43: 217-245. von Wright, G. H. (1972). The logic of preference reconsidered. Theory and Decision 3: 140-169.