Steven J. Brams, D. Marc Kilgour, and William S. Zwicker

1. Introduction

Aggregation paradoxes abound in the literature of statistics and social choice. Roughly speaking, they describe situations in which the sum of the parts is, in some sense, not equal to the whole, such as an election outcome that fails to mirror the `will' of the electorate. They include the paradox of voting and Arrow's impossibility theorem (Arrow, 1963), Anscombe's paradox (Anscombe, 1976, Wagner, 1983, 1984), Ostrogorski's paradox (Daudt and Rae, 1976, Deb and Kelsey, 1987), and Simpson's paradox (Simpson, 1951, Gardner, 1976, Wagner, 1982).

The new paradox we analyze here is illustrated in presidential elections, when about two-thirds of voters (those in states with a senatorial contest) choose candidates for president, senator, and representative. In 1988, for example, voters elected a Republican (R) president, but Democrats (D) prevailed in Congress, winning majorities in both the Senate and House. Thus, the outcome was divided government, whereby the president is of one party and one or both houses of Congress is controlled by the other party.[2]

We show that it is theoretically possible that not a single voter voted for the combination RDD for president, senator, and representative, respectively, in 1988. We call such an outcome a `paradox of divided government'--or, more generally, a `paradox of vote aggregation'--and give several examples of it and variants in section 2.[3] In section 3 we offer a theoretical analysis of the paradox, giving necessary and sufficient conditions for a combination to win and for the existence of the paradox; we also show how the occurrence of the paradox depends on the `incoherence' of support for the winning combination.

In section 4, we give a real-life example of the paradox, based on the choices made by 1.8 million Los Angeles county voters choosing among the 28 propositions on the 1990 California ballot. In addition, we discuss other empirical examples that are not full-blown paradoxes but, nonetheless, indicate a discrepancy between the most popular parts of a combination and the less popular whole. Thus, the 1976 presidential election was nonparadoxical (the winning combination, DDD, was the most popular one), but the 1980 election was decidedly more paradoxical (the winning combination, RRD, came in only fourth out of eight)--as was voting by members of the House of Representatives on the budget and NAFTA in 1993 (the winning combination, YY, was supported by only 18%).

In section 5 we consider two legislative examples from the House of Representatives in which there was an apparent paradox of voting (the Wilmot Proviso of 1846 and the Revenue Act of 1932). We explicate the sense in which this paradox is a special case of the more general vote-aggregation paradox. In section 6 we consider certain normative and social-choice consequences of the paradox. For example, should voters be presented with the opportunity to choose combinations on ballots? If so, should they be allowed to vote for more than one combination under approval voting, or to rank the combinations under the Borda count? Answers to these questions, and their implications for making coherent social choices, are explored in the context of democratic political theory.

2. Divided Government: An Illustration of the Paradox

In the early 1990s, the literature on divided government burgeoned. It seems to have been stimulated in part by the fact there was divided government in 20 of the 24 years between 1968 and 1992, and again in 1994, and in part by controversy about the performance of federal and state governments over this period.

We shall touch on different views of the efficacy, or lack thereof, of divided government. Then we will show that, in the extreme case, a government, whether divided or not, may be the choice of absolutely no voters.

How can this happen? In virtually all elections in the United States, votes are aggregated by office.[4] At the federal level, for example, votes for president, senator, and representative are separately tallied. Thus, a voter might vote for the Republican candidate for president, the Democratic candidate for Senate, and the Republican candidate for House--that is, the combination RDR. While the election returns tell us who won the race for each office, they do not tell us how many citizens voted for RDR.

An alternative to aggregation by office is to keep separate tallies for each of the eight possible combinations (if voters can vote D or R for each of the three offices, they have 23 = 8 possible choices, as will be illustrated later). These tallies would also seem a reasonable basis for determining the election outcome. Indeed, insofar as the federal government is conceived of as a single entity, arguments can be made that the most popular combination should win.

But most voters, it seems, do not think in terms of electing a combination, at least not at a conscious level (Fiorina, 1992, pp. 65). Yet in voting for their favorite candidates for each office, they may worry about the possible consequences of divided (or unified) government and possibly act on this concern in choosing a favorite.

In so doing, voters seem to apply different criteria in selecting presidents and legislators. For example, voters until 1994 tended to favor Democrats for the benefits, protections, and services they provided at the district and state level, but Republicans for the discipline and responsibility, especially on economic matters, that they exercised at the national level (Zuppan, 1991; Jacobson, 1992, pp. 71-75). Thereby they hedged their bets, especially if they were `sophisticated,' and opted for `balance' in the government (Alesina and Rosenthal, 1995), which, in the view of some (e.g., Conlan, 1991; Mayhew, 1991; Fiorina, 1992), seems not to have suffered unduly because it has been divided.

To most voters, there is nothing incoherent in voting for a combination like RDR. Moreover, if this combination receives more votes than any of the other combinations when votes are aggregated by office (as they are today), there is nothing paradoxical about the fact that the voters, collectively, chose divided government.

The paradox of divided government, or the vote-aggregation paradox, is said to occur when this combination, as a combination, receives the fewest votes, or is tied for the fewest. Define a combination to be an ordered triple (like RDR) in which the first element is the party of the president, the second element is the party of the senator, and the third element is the party of the representative.

In a presidential election year in which there is also a senatorial election, a voter effectively chooses one of the eight possible combinations. Fiorina (1992, p. 120) argues that these choices are more numerous than voters have in many multiparty systems; furthermore, unlike in multiparty systems, voters can `vote directly for the coalition they most prefer.' But, as the vote-aggregation paradox vividly demonstrates, this is not at all the case: a combination like RDR can win under the extant system without any voters voting directly for it, casting in doubt the two-party, separation-of-powers system in the United States as an expression of popular will.

Voting directly for coalitions would be facilitated by being able to vote for the combinations themselves, not just their components, which is a possibility we consider later. But next we illustrate the basic paradox of vote aggregation without ties for lowest, after which we will illustrate both stronger and weaker versions of this paradox:

Example 1 (basic paradox without ties for lowest: 3 offices). Suppose there are 13 voters who cast the following numbers of votes for the eight different combinations:

DDD: 3 DDR: 1 DRD: 1 RDD: 0 DRR: 1 RDR: 3 RRD: 3 RRR: 1.

We indicate the total numbers of voters voting for a Democrat or Republican for office J by D(J) and R(J), where J is one of the following: P (president), S (senator), or H (representative, or house member). The election results for each office are

R(P) > D(P), D(S) > R(S), D(H) > R (H),

each by 7 votes to 6.

We have deliberately constructed this example so that the winning combination when votes are aggregated by office, RDD, is the divided-government situation that occurred throughout most of the 1968-92 period. Observe, however, that no voters support this combination, which is the only one that receives 0 votes.[5]

This example is minimal, like those that follow, in the sense that no example with fewer voters can meet the stated conditions of the paradox. The construction depends on assigning the fewest votes (i.e., 0) to the paradoxical winner RDD, the next-fewest (i.e., 1) to combinations that agree in one office, and then finding the smallest number for the combinations that agree in two offices (i.e., 3) so as to create the paradox--that is, so that RDD (barely) wins when votes are aggregated for each office.

The paradox vividly illustrates the difference that may arise from aggregating votes by individual office, which we call office aggregation, and aggregating votes by combination, which we call combination aggregation. It also illustrates how office aggregation may leave no voter completely satisfied with the outcome: RDD is not supported by any of the 13 voters.[6]

By comparison, DDD, RDR, and RRD are tied for first place under combination aggregation. We stress, however, that this is not to say that most voters would prefer one of these combinations to the office-aggregation winner, RDD--only that RDD is not the first choice of any voters if they are sincere in their voting.

The paradox, which describes a conflict between two different aggregation procedures, does not depend on either sincere or strategic voting. Thus, voters may be perfectly sincere in voting for preferred candidates for every office, or they may be strategic (in some sense); the paradox says only that their majority choices by office aggregation receive the fewest votes when votes are aggregated by combination.

Example 1 is not the minimal example of the basic paradox if we allow ties for fewest votes. Example 2 shows that the paradox can occur with only 3 voters:

Example 2 (basic paradox with ties for lowest: 3 offices). Suppose there are 3 voters who cast the following numbers of votes for the three combinations

DDR: 1 DRD: 1 RDD: 1.

DDD is the winning combination, according to office aggregation, by 2 votes to 1 for each office; yet it receives 0 votes as a combination. But so do the other four combinations (DRR, RDR, RRD, and RRR), so in this case we can say only that the winning combination ties for lowest, not--as in Example 1--that it is the only combination with the fewest (i.e., 0) votes.

A more pathological form of the paradox can occur if there are four offices. For example, assume that a governor is being chosen in the same election in which the three federal offices are being filled, so voters must chooose one of sixteen (24) voting combinations. Then we may get what we call a complete-reversal paradox:

Example 3 (complete-reversal paradox without ties for lowest: 4 offices). Suppose there are 31 voters who cast the following numbers of votes for the sixteen different combinations:



DRRR: 1 RDRR: 1 RRDR: 1 RRRD: 1 RRRR: 5.

Then it is easy to show that D beats R for each of the four offices by 16 votes to 15, but DDDD is the only combination to receive 0 votes. The new wrinkle here is that RRRR, the `opposite' of the office-aggregation winner, DDDD, has the most votes (i.e., 5) and therefore wins under combination aggregation.

The complete-reversal paradox also occurs in the following simpler example, wherein several combinations tie for lowest:

Example 4 (complete-reversal paradox with ties for lowest: 4 offices). Suppose there are 11 voters who cast the following numbers of votes for five combinations:

DDDR: 2 DDRD: 2 DRDD: 2 RDDD: 2 RRRR: 3.

There is a complete-reversal paradox, because DDDD ties eleven other combinations for the fewest votes (0) but, nevertheless, wins according to office aggregation by 6 votes to 5 against its opposite, RRRR, the combination winner with 3 votes.

It is not difficult to show that the basic paradox cannot occur if there are only two offices, but a milder version of this paradox can arise--namely, that the winner by office aggregation may come in as low as third (out of four) by combination aggregation. We call this the two-office paradox:

Example 5 (two-office paradox without ties for lowest). Suppose there are 5 voters who cast the following numbers of votes for the four combinations

DD: 1 DR: 2 RD: 2 RR: 0.

While DD (1 vote) wins under office voting by 3 votes to 2 for each of the two offices, it is behind both DR and RD (2 votes each) under combination voting; it is ahead only of RR (0 votes), putting it in next-to-last place.

This relatively low rank for the office-aggregation winner in the two-office case may still describe a possibly serious discrepancy between the two aggregation procedures. Indeed, there is no theoretical limit on how far behind the office-aggregation winner can be from the first-place and second-place combination winners (though it is easy to demonstrate that it will always place above the fourth-place combination). The fact that the two-office paradox can arise in far more elections (at the federal level, for example, in two-thirds of nonpresidential elections) makes its possible occurrence worth investigating empirically, as we shall do later.

If there are only two offices, but we permit voters a third option of abstaining (A), then the basic paradox can occur with only two offices:

Example 6 (basic paradox with abstention and with ties for lowest: 2 offices). Suppose there are 15 voters who cast the following numbers of votes for the nine (32) different combinations:

DD: 0 DR: 3 RD: 3 DA: 3 AD: 3 RA: 1 AR: 1 RR: 1 AA: 0.

There is a paradox, because DD wins under office voting by 6 votes to 5 but, as a combination, it ties for the fewest votes (i.e., 0) with AA. The notion that AA receives 0 votes--or any other number--is somewhat misleading, however, because it is often impossible to ascertain the numbers who `chose' AA if these voters did not go to the polls (later we shall count as abstainers voters who cast ballots but abstain on the offices being considered).

It is worth noting in this example that DD wins in large part because 6 voters (the DA and AD voters) abstained on one office and supported D for the other. An A, however, is not necessarily to be interpreted as a vote against DD. A's are qualitatively different from other votes, in part because they are never a component of a winning combination.

Example 7 (complete-reversal paradox with abstention and without ties for lowest: 3 offices). Suppose there are 52 voters who cast the following numbers of votes for the twenty-seven (33) combinations:

DDD: 0 DDR: 4 DRD: 4 RDD: 4 DDA: 4 DAD: 4 ADD: 4

DRR: 1 RDR: 1 RRD: 1 DAA: 1 ADA: 1 AAD: 1

RAA: 1 RAR: 1 RRA: 1 RDA: 1 ARD: 1 DAR: 1 AAA: 5

ARR: 1 ARA: 1 AAR: 1 ADR: 1 RAD: 1 DRA: 1 RRR: 5.

There is a complete-reversal paradox, because DDD wins under office voting by 20 votes to 16 for each office but, as a combination, it has the fewest (i.e., 0) votes. On the other hand, the other two identical combinations, AAA and RRR (the latter might be considered the opposite of DDD[7]), have the most votes (i.e., 5).

The foregoing examples illustrate a range of seeming discrepancies between aggregating votes by office and aggregating them by combination. We next analyze the general conditions that give rise to these discrepancies, focusing on the basic paradox in the two-option, three-office case and the `coherence' of voter support.

3. The Coherence of Support for Winning Combinations

Having demonstrated the existence of a vote-aggregation paradox and some variants of it, we turn in this section to the analysis of conditions that give rise to it. In particular, we distinguish between voting directly for a combination and voting indirectly for it by supporting some of its parts.

This distinction is illustrated by Example 2. The three voters who vote for DDR, DRD, and RDD give D a 2-to-1 margin of victory for each office, resulting in the choice of DDD by office aggregation. But this indirect support by the three voters for DDD is indistinguishable under office aggregation from the direct support that one hypothetical voter, voting for DDD, would give to this combination.

In effect, this one voter would contribute three times as much support to DDD as does any of the three voters who `tilts' toward DDD by agreeing with it on two of the three offices. Not only is the support of this one voter more potent, but we also consider it more `coherent' because there is no question that if DDD prevails, the DDD voter supported it.

To make these ideas more precise, we define a quantitative measure Q of the support for combination XYZ. It possesses two properties:

1. It is the sum of the coherent (C) and incoherent (I) contributions of voters:

Q(XYZ) = C(XYZ) + I(XYZ). (1)

where the C and I components will be defined shortly.

2. The winning combination according to office aggregation is that which

maximizes Q (to be proved in Theorem 1).

To construct the C and I terms, let n(XYZ) denote the number of votes cast for combination XYZ. We define four differences between `opposites':

n0 = n(DDD) - n(RRR)

n1 = n(DDR) - n(RRD)

n2 = n(DRD) - n(RDR)

n3 = n(RDD) - n(DRR).

These differences are set up to favor DDD, with the positive term in each difference agreeing with DDD in more than half the offices and the negative term agreeing with DDD in fewer than half the offices.

Given these differences, we define

C(DDD) = 3n0 and I(DDD) = n1 + n2 + n3,

based on the intuition in the example just discussed that a direct vote has three times the effect of indirect votes that tilt in favor of DDD.[8] Substituting into (1),

Q(DDD) = 3n0 + n1 + n2 + n3.

Q values for combinations other than DDD are similarly defined, but they require the insertion of some minus signs to compensate for the arbitrary choices of signs in the definitions of n0, . . ., n3. For example,

Q(RRD) = -3n1 - n0 + n2 + n3,

because it is negative values of n1 and n0 that agree with RRD in more than half the offices.

Theorem 1. The winning combination according to office aggregation maximizes Q.

Proof. Assume A and B are the two candidates for the first office (e.g., president). Define the following difference (d) for this office:

d1(A > B) = no. of voters voting for A - no. of voters voting for B.

Note that d1(A > B) = - d1(B > A). Similarly, define d2 and d3 to be the differences for the second and third offices.

Given any combination, such as RRD, define the following sum (S):

S(RRD) = d1(R > D) + d2(R > D) + d3(D > R).

Note that the R or D for each office in RRD matches the R or D that is assumed greater in each di term on the right-hand side of the equation. It is apparent that RRD will win the election if and only if each of the di's is positive (we ignore here the possibility of ties and how they might be broken to determine a winner).

Assume RRD is the winning combination according to office aggregation. The S for any nonwinning combination sums the same three numbers as for RDD, but with one or more sign changes. Necessarily, at least one of the numbers for the nonwinning combinations is negative. Not only is the winning combination the only one for which each of the three di's is positive, but this combination is also the one that maximizes S.

To complete the proof, it remains only to show that S(XYZ) = Q(XYZ) for any combination XYZ. We do this for DDD and leave the other combinations for the reader to check:

d1(D > R) = n0 + n1 + n2 - n3;

d2(D > R) = n0 + n1 - n2 + n3;

d3(D > R) = n0 - n1 + n2 + n3.

When we sum the three di's given above, we get

S(DDD) = 3n0 + n1 + n2 + n3 = Q(DDD),

as desired. Q.E.D.

Theorem 1 shows that the winning combination according to office aggregation is the one with the largest Q value. The fact that this value has both a C and an I component enables us to judge the extent to which a victorious combination owes its triumph to coherent, or direct, support rather than to incoherent, or indirect (`tilt'), support.

The paradox of divided government, as given in all the examples in section 3 except Example 5 (the two-office paradox), describes the extreme case wherein all the support for each winning combination is incoherent--no voter votes for this combination. To be sure, the paradox may occur when the winning combination receives some, but fewer, votes than any other combination.

It is worth noting that the Q value bears some resemblance to the Borda count. Imagine that a person voting for the combination XYZ actually awards some points to each of the eight combinations, with the rule being that +1 point is awarded for each office on which XYZ agrees with the combination in question, and -1 point for each office on which XYZ disagrees. For example, a vote for DDD awards +1+1+1 = 3 points to DDD itself, whereas it awards -1+1+1 = 1 to RDD.

There are four possible levels of agreement and disagreement. Specifically, a vote for DDD awards

+3 points to DDD;

+1 point to DDR, DRD, RDD;

-1 point to DRR, RDR, RRD;

-3 points to RRR.

It is easy to show that if we add the total number of points awarded by all voters to a particular combination (e.g., DDD), the resulting sum equals Q(DDD).[9] Hence, the winning combination according to office aggregation--that is, the combination that maximizes Q--is the one with the highest Borda score (as we have interpreted it here).

Thus, the voting system defined by the preceding system of awarding points is fully equivalent to the system of office aggregation currently in use. This correspondence shows how our present system of electing candidates office by office presumes an underlying cardinal evaluation of combinations of candidates. Thus, a ballot cast for DDD gives, effectively, a ranking of the eight combinations--but only an incomplete ranking, truncated into four levels of agreement and disagreement separated by equal intervals of 2 points, as we just showed. Not only does a DDD ballot contribute more to, say, DDR than DRR, but it does so by the same amount that other ballots that agree in two versus one office do to other combinations.

Of course, if the standard version of the Borda count were applied directly to the combinations, a voter could give a complete ranking of all eight combinations. We consider this possibility later.

Next we consider under what conditions Q(DDD) is maximal and therefore the combination DDD is winning. Then we discuss the more stringent conditions that render this winning combination paradoxical.

Theorem 2. A necessary and sufficient condition for DDD to be winning is that

Q(DDD) > 2n0 + 2[max{n1, n2, n3}]. (Other combinations are governed by similar inequalities.)

Proof. What we need to show is that DDD maximizes the Q value, and is therefore the winning combination, if and only if the above inequality is satisfied. To do this, note that DDD is winning precisely when each term of

d1(D > R) + d2(D > R) + d3(D > R)

is positive, or

n0 + n1 + n2 > n3;

n0 + n1 + n3 > n2; and (2)

n0 + n2 + n3 > n1.

We begin by showing that the three inequalities given by (2) are all necessary. Assume that n3 [[threesuperior]] max{n1, n2 }, which we shall refer to as case 1. Then the second and third inequalities are true if the first is true: interchanging n3 and n2, thus obtaining the second inequality from the first, preserves the truth of the inequality, as does interchanging n3 and n1. But the first inequality must be true for all three to be satisfied, so all three inequalities are true if and only if the first is true.

Now the first inequality is equivalent to

n0 + n1 + n2 + n3 > 2n3,

which is equivalent to

3n0 + n1 + n2 + n3 > 2n0 + 2n3.

This is the same as

Q(DDD) > 2n0 + 2n3,

which implies

Q(DDD) > 2n0 + 2[max{n1, n2, n3}]. (3)

Inequality (3) is the condition of Theorem 2. Because it is symmetric in n1, n2, and n3, it is similarly implied by either of the other two cases, corresponding to the possibility that n2 or n1 is largest, or tied for largest, among n1, n2, and n3.

To show sufficiency, note that if inequality (3) holds, regardless of which case prevails, the steps of the earlier proof are reversible, as they are with the interchange of n2 and n3, or n1 and n3. Thereby we can establish that the three inequalities of (2) all hold, ensuring that DDD is the winning combination. Q.E.D.

To obtain further insight into the conditions of the paradox, observe that the condition of Theorem 2 is equivalent to

n0 + n1 + n2 + n3 > 2[max{n1, n2, n3}]. (4)

Inequality (4) says that the total margin by which voters favor combinations with more Ds than Rs over their opposites must be greater than twice the maximum of the margins corresponding to indirect support.

If DDD receives the fewest votes--or ties for the fewest--then n0 [[twosuperior]] 0 in inequality (4), which makes this inequality more difficult to satisfy than were n0 > 0. Let us momentarily drop the n0 term from (4) (e.g., assume n0 = 0 because DDD and RRR receive equal numbers of votes). Then inequality (4) becomes

n1 + n2 + n3 > 2[max{n1, n2, n3}]. (5)

Roughly speaking, inequality (5) is satisfied when n1, n2, and n3 are all positive (i.e., all the tilt terms favor DDD over RRR) and, in addition, n1, n2, and n3 are close to each other in size (i.e., the tilt is evenly spread).

When we go back to inequality (4), how does the presence of n0 affect this observation? The more direct support that DDD receives over its opposite, RRR (i.e., the larger n0 is), then the less even the tilt must be in order for DDD to prevail.

On the other hand, if DDD receives no votes--making a win paradoxical--the tilt must be spread fairly evenly for DDD to win. At the same time, the larger the vote for RRR (i.e., the more negative n0 is), the more evenly spread as well as larger the tilt must be to produce the paradox.

These considerations suggest two conditions sufficient to guarantee that the paradox not occur for DDD. First, if n0 is negative with absolute value either equal to or greater than the largest of n1, n2, or n3, then inequality (4) cannot be satisfied and, hence, there can be no paradox. Second, the paradox is also precluded if even one of the tilt terms, n1, n2, or n3, is less than or equal to 0, and this happens in particular if even one of the combinations--DDR, DRD, or RDD (i.e., those that tilt towards DDD)--receives no (net) votes.

To sum up, there must be more-or-less-equal positive differences between the mixed combinations that favor D (DDR, DRD, and RDD) and their opposites (RRD, RDR, and DRR) for the paradox to occur. These positive differences overwhelm the greater direct support that RRR enjoys over DDD, enabling DDD to win even though it receives fewer votes than any other combination.

We turn next to three empirical cases. The first involves voting on multiple propositions, the second has a divided-government interpretation, and the third raises questions about the coherence of legislative choices. A genuine paradox of vote aggregation occurred in the first case, which was generated by voting for different propositions on a referendum rather than for different offices in an election. Although there was no full-fledged paradox in either of the latter two cases, they illustrate situations in which there was a discrepancy between office and combination aggregation.

4. Empirical Cases

Case 1: Voting on Propositions

On November 7, 1990, California voters were confronted with a dizzying array of choices on the general election ballot: 21 state, county, and municipal races, several local initiatives and referendums, and 28 statewide propositions. We analyze here only voting results on the 28 propositions, which concerned such issues as alcohol and drugs, child care, education, the environment, health care, law enforcement, transportation, and limitations on terms of office.

The data are images from actual ballots cast by approximately 1.8 million voters in Los Angeles county (Dubin and Gerber, 1992). Voters could vote yes (Y), no (N), or abstain (A)--abstention being the residual category of voting neither Y nor N--with a proposition passing if the number of its Ys exceeded the numbers of its Ns, and failing otherwise. In Los Angeles county, 11 of the 28 propositions passed, but several of these were defeated statewide, and some of the defeated propositions in Los Angeles county passed statewide.

For the purposes of this analysis, we consider only the results for Los Angeles county and ask how many voters voted for the winning combination, NNNYNNYNYNNNNNNYNYYYNYYNNYNY, on propositions 124 - 151. The answer is that nobody did, so there was a vote- aggregation paradox.

Because there are 228 268.4 million possible Y-N combinations, however, this is no great surprise. With fewer than 2 million voters, more than 99% of the combinations must have received 0 votes, even if each of the voters voted for a different combination.

In fact, however, this was not the case. `All abstain' received the most votes (1.75%), and `all no' was a close second (1.72%). Ranking fifth (0.29%) among the combinations were the recommended votes of the Los Angeles Times, and ranking ninth was `all yes' (0.20%).[10] Thus, the effect of the Times recommendations, at least for the complete list of propositions, was marginal. Nonetheless, it was greater than what Mueller (1969) found in the 1964 California general election, in which absolutely nobody in his sample of approximately 1,300 voters backed either all, or all except one, of the Times recommendations on 19 propositions.

Although a paradox occurred in voting on all 28 propositions, it was not a complete-reversal paradox, because the opposite of the winning combination did not garner the most votes (it, too, received zero votes). Because voters really had abstain (A)--in addition to Y and N--as voting options, it seems proper to use the 328 22.9 trillion combinations, which include A as well as Y and N as choices, in asking whether anybody voted for the winning combination. While A was `selected' by between 7.1% and 16.3% of voters on each of the propositions, its choice over Y or N could never elect A but could influence whether Y or N won.

What a voter's choice of A on any proposition did preclude was that voter's voting for the winning combination, thereby decreasing the already small likelihood that the winning combination received any votes. One could, of course, count A as a vote for both Y and N, thereby increasing the number of combinations that a voter supports; alternatively, one could give each voter one vote, splitting it among all combinations that he or she supports with either a Y, an N, or an A. In fact, we investigated these two different ways of aggregating votes to determine a winner and found that they would have given different results for three related propositions--all environmental bond issues--that were on the 1990 ballot (Authors, uncited).

In the case of these three propositions, we also checked for a possible vote-aggregation paradox. The winning combination according to office (i.e., proposition) aggregation was YNY, but it was supported by fewer than 6% of the voters, placing it fifth out of the eight possible combinations. While not a full-fledged paradox, the poor showing of YNY illustrates how an unpopular compromise may defeat more popular `pure' combinations (YYY was supported by 26% of the voters, NNN by 25%). The winning combination in this case was not only incoherent in the mathematical sense used earlier but also in a more substantive sense: it was pro-environment on two bond issues, anti-environment on the third, rendering policy choices by the voters somewhat of a hodgepodge that, as we pointed out, had little direct support.

Case 2: Voting in Federal Elections

We suggested earlier that the aggregation paradox, while not specifically tied to the election of divided governments, may more often manifest itself in the form of divided government than unified government. This is because divided governments are probably less likely to benefit from the direct support of many voters than are unified governments, which often attract many straight-ticket voters.

As a case in point, a Democratic president, Senate, and House were elected in 1976, giving DDD. In the absence of reliable combination voting data for the three offices (either from actual ballots cast by individual voters for the three offices or from voter surveys), we treated the 435 congressional districts as if they were voters, which raises difficulties we shall assess shortly. We then classified the subset of districts with senatorial races according to the eight combinations, depending on which party (D or R) won each of the three offices in a district.

We caution that, unlike the hypothetical examples given in section 2, we consider the office-aggregation winner in the Senate and House to be the party that wins a majority of seats in each house, not the party with the greatest number of Senate or House votes nationwide. In the case of the House in 1980, the Democrats also won a majority of votes nationwide, but the relatively close division of the Senate in 1980 (53 Rs to 47 Ds) makes the nationwide vote winner not certain. (We have not checked which party actually won the nationwide vote in the Senate, because it is the majority-seat winner that counts politically.) Another complication is that two-thirds of senators are not up for election in any election year. Finally, as we noted in section 2, many voters base their choices less on party than on the individual candidates running, which further vitiates the interpretation of their choices as signaling a preference for either divided or unified government.

Bearing these caveats in mind in interpreting whether or not citizens voted for divided government, the results for 1976 are that the DDD combination was the most popular, being the choice of 40.8% of the 316 districts with senatorial races, whereas the next-most-popular combination, RRR, won in only 20.6% of the districts (see Table 1). ________________________________________________________________________

Table 1 about here


In short, unified government garnered more than three-fifths of the vote that year, at least as indicated by the congressional district results with senatorial races; and the most popular of these combinations, DDD, concurred with the office-aggregation winner.

By comparison, RRD was the winning combination in 1980, but it was only the fourth-most-popular voting combination (again, by congressional districts with senatorial races, of which there were 315), as shown in Table 1. As in 1976, the straight-ticket voting combinations won in the most districts (28.6% for RRR and 22.2% for DDD). Although RRD was not the least popular combination, its fourth-place finish with 14.3% seems at least semi-paradoxical. And, of course, there was divided government in 1980.

To be sure, the contrast between 1976 and 1980, with unified government coinciding with the winning combination in 1976 and divided government coinciding with the fourth-place combination in 1980, could be happenstance. Unfortunately, combination-voting data for the three federal offices seem to have been collected only for 1976 and 1980 (Gottron, 1983), so we cannot test for the paradox in other presidential election years.[11]

We have, however, analyzed combination-voting data for the two-office elections of president/senator and president/representative for the five presidential elections between 1976 and 1992. In such elections, it will be recalled from Example 5 in section 2, the winning combination by office aggregation can rank as low as third out of four. Treating the 50 states (actually, only the 33 or 34 states that had senatorial contests in each year) as if they were voters in the president/senate comparisons, and the 435 House districts as if they were voters in the president/representative comparisons, we found that in only two of the ten comparisons--the president/ representative comparisons in 1980 and 1988--did the winning combination by office aggregation (i.e., RD) come in even as low as second (RR in each of these years won according to combination aggregation).[12]

The absence of even a mildly paradoxical third-place finish of the combination winner (the two-office paradox) may well be attributable to aggregating voters by district and state and treating these large units as if they were individual voters. It seems likely that this aggregation wipes out numerous mixed combinations--one of which might win according to office aggregation with relatively few votes--that one would pick up from individual ballots.[13] The fact that the second-place finishes occur only in the president/representative comparisons and not in the president/senate comparisons is prima facie evidence that more aggregative units (i.e., states rather than congressional districts) have this wipe-out effect, decreasing the probability of observing a paradox.

Case 3: Voting on Bills in Congress

There was a two-office paradox in the case of what are generally acknowledged to be the two most important votes to come before the House of Representatives during the first two years of Bill Clinton's administration--that on the budget on August 5, 1993, and that on NAFTA on November 17, 1993. On these two bills, NY got 36%, YN got 32%, YY got 18%, and NN got 14%; the winning combination was YY. The explanation for why only 18% of the House--all Democrats--supported President Clinton on both bills lies in the fact that 84% of Democrats and no Republicans voted Y on the budget bill, whereas 40% of Democrats and 75% of Republicans voted Y on NAFTA bill.

If there is a third bill to be voted on, a paradox of voting can occur. In section 5 we show that the paradox of voting implies the vote-aggregation paradox, but not vice-versa. We illustrate this tie-in with some empirical examples of voting in Congress.

5. The Paradox of Voting

We present two examples in this section to illustrate the linkage of the vote-aggregation paradox to the paradox of voting. We do so by introducing preferences into the analysis, which up until now has been based only on numerical comparisons. The preferences we assume over alternatives enable us to create an isomorphism that renders the vote-aggregation paradox a natural generalization of the paradox of voting.

Our first example concerns the Wilmot Proviso, which prohibited slavery in land acquired from Mexico in the Mexican war. On August 8, 1846, there were several votes in the House of Representatives for attaching this proviso to a $2 million appropriation to facilitate President James K. Polk's negotiation of a territorial settlement with Mexico. The three possible outcomes were:

a. Appropriation without the proviso;

b. Appropriation with the proviso;

c. No action.

Riker (1982, pp. 223-227) reconstructs the preferences of eight different groups of House members for these outcomes, where xyz indicates a group prefers x to y, y to z, and x to z (the groups are assumed to have transitive preferences). The preferences of these groups, which comprise a total of 172 House members, are shown in Table 2.


Table 2 about here


To simplify the subsequent analysis, assume that the 8 border Democrats split 4 - 4 for each of their two possible preference scales shown in Table 2, and the 3 border Whigs split 1 1/2 - 1 1/2 for their two possible preference scales (not actually possible, of course, but the subsequent results do not depend on how we split the votes of either group). Then it is easy to show that majorities are cyclical: b beats a (as happened) by 93 to 79 votes, a beats c by 129 1/2 to 42 1/2 votes, and c beats b by 107 to 65 votes. Consequently, there is no Condorcet outcome that defeats each of the other outcomes in pairwise contests, which makes the social choice an artifact of the order of voting.

To establish an isomorphism between the paradox of voting and the vote-aggregation paradox, assume that the eight votes actually taken on the proviso in the House on August 8, 1846, can be reduced to three hypothetical pairwise contests between (1) a and b, (2) b and c, and (3) c and a. Assume further that, given its preferences, each group can answer yes (Y) or no (N) about whether it prefers the first member of each pair to the second. (We assume, as before, that the border Democrats and border Whigs split 50-50 on their preferences for second and third choices.)

Answers to these three questions give what we call an answer sequence. For example, an answer sequence of YYN indicates that the group prefers a to b, b to c, but not c to a, so its preference scale is abc. (In the remainder of this section, we assume for simplicity that preferences are strict.) Likewise, we can associate five other mixed answer sequences of Ys and Ns with the preference scales shown below each:

Preference: abc cab bca acb bac cba ? ?


The question marks indicate intransitive preferences. Thus, for a group to answer Y to all three questions indicates a preference cycle abca; to answer N to these questions reverses the direction of the cycle, giving cbac. Although we assume that groups of like-minded House members have, like individuals, transitive preferences, we shall return to this matter later.

In voting on the Wilmot Proviso, observe that the winner by combination aggregation is acb (YNN) with 66 votes, comprising 4 border Democrats, 46 Southern Democrats, and 16 Southern and border Whigs:


Votes: 11 2 1 1/2 66 52 1/2 39 0 0

By contrast, the winner by office aggregation in pairwise contests (1), (2), and (3) is NNN. Specifically, N beats Y in contest (1) by 93 to 79 votes, in contest (2) by 107 to 65 votes, and in contest (3) by 129 1/2 to 42 1/2 votes. Thus, we have a basic vote-aggregation paradox: the winner by office aggregation (NNN) ties for the fewest votes (with the other intransitive sequence, YYY).

This coincidence of the paradox of voting and the vote-aggregation paradox is no accident. If there is a paradox of voting, the outcome is cyclical majorities, which in our isomorphism translates into either YYY or NNN. But since no group with transitive preferences has these sequences, they must, according to combination aggregation, receive 0 votes. Consequently, the winning combination according to office aggregation (either YYY or NNN) when there is a paradox of voting must tie for the fewest votes (with the other intransitive sequence). Thus we have

Theorem 3. If the preferences of individual voters (or like-minded groups) are transitive with respect to pairwise contests among three or more alternatives, then a paradox of voting, based on the pairwise contests, implies a vote-aggregation paradox.

Whether the reverse implication holds turns on the number of alternatives being ranked. For three alternatives, it turns out that none of the six mixed combinations can win, according to office aggregation, and also receive 0 votes when YYY and NNN do, too. To show that there is no such example for three pairwise contests (offices), associate the following numbers of voters with the six mixed-answer sequences:


Number: 0 v w x y z

Without loss of generality, we have assumed that YYN receives the fewest votes, and that this number of votes is 0. The other numbers are all nonnegative. Now in order for Y to win the first and second offices by office aggregation, we need

v + x > w + y + z

w + y > v + x + z.

Adding these inequalities gives 0 > 2z, which is impossible since z [[threesuperior]] 0. This contradiction shows that YYN cannot win by office aggregation and receive the fewest votes.

On the other hand, if there are four alternatives we have

Example 8 (basic paradox with ties for lowest: 4 outcomes and 6 pairwise contests). Suppose there are 3 voters, whose preferences among the alternatives a, b, c, and d are as follows:

bacd: 1 cabd: 1 dabc: 1.

Then their votes on the six questions of whether their first alternative is preferred to their second for the six possible pairwise contests--a and b, b and c, c and d, a and c, a and d, and b and d--will be


Now YYYYYY is the winning combination according to office aggregation, corresponding to the transitive ordering abcd for which none of the voters voted. Thus, we have an example of a vote-aggregation paradox that does not arise, given our particular enumeration of pairwise contests, from a paradox of voting. Note that the Condorcet alternative, a, is not ranked first by any of the voters.

More generally, this example, together with our earlier argument that a transitive combination cannot win according to office aggregation when there are only three pairwise contests, yield the following strengthening of Theorem 3:

Theorem 4. Assume there are three or more alternatives over which voters have transitive preferences. Then every paradox of voting corresponds to a vote-aggregation paradox. The reverse correspondence holds for three alternatives but fails for more than three.

Theorem 4 shows that, given our isomorphism, the vote-aggregation paradox is a generalization of the paradox of voting, because whenever the latter occurs so does the former, but not vice versa if there are more than three alternatives.

We caution that Theorem 4 should not be construed as an empirical law in situations in which voters may, for a variety of reasons, not express transitive preferences and therefore not meet the condition of Theorem 4. For example, it may not be clear at the outset that they will vote in a particular sequence in three pairwise contests, so the question of being consistent is not a primary consideration.

Even if it is, voters may decide to vote YYY or NNN if such ostensibly inconsistent behavior on the part of enough voters leads to a contradiction, which in turn triggers a default option that these voters prefer. For example, assume that an NNN sequence indicates that a voter votes `no' on three pairwise contests between three levels or types of regulation; if none wins, the status quo of no regulation prevails, which the voter prefers. Then the apparent contradiction of preferring none of the three levels--when matched in pairs against each other--is really no contradiction, given a preference for the default option.

That some voters are, at least on the surface, inconsistent in this sense is observable in actual legislative contests. Blydenburgh (1971) studied the voting behavior of members of the House on Representatives in voting on three provisions of the Revenue Act of 1932: the first to delete a sales tax, the second to add an income tax, and the third to add an excise tax.

Let a be the status quo (SQ) without a sales tax, b be the SQ with an income tax, and c be the SQ with an excise tax. Based on his reconstruction of voter preferences, Blydenburgh (1971) argues that there was a paradox of voting abca, so majorities would answer YYY in each of the three pairwise contests.

In fact, however, there were no such contests, because the voting was sequential under the amendment procedure. The first contest was a versus SQ; when a passed, the second contest was a versus a plus b (i.e., SQ with both a sales and income tax); when the latter failed, the third contest was a versus a plus c (i.e., SQ and both a sales and excise tax), which passed. Thus, the winner by office aggregation in these three pairwise contests was YNY. This combination was chosen by 38 voters, ranking fifth of the eight combinations according to combination aggregation.[14]

The fact that there was no vote-aggregation paradox shows there is obviously some slippage between our theoretical results and their empirical reality. We take the fifth-place finish of the winning combination, nonetheless, as partial confirmation of a discrepancy between--if not a paradoxical aspect of--the two different ways of aggregating votes.

The significance of this discrepancy is underscored by the linkage of the vote-aggregation paradox to the paradox of voting. The paradox of voting has produced an enormous literature since the pioneering work of Black (1958) and Arrow (1963), which first appeared in the late 1940s and early 1950s, that extended and generalized the original paradox discovered by Condorcet in the late 18th century (see Black, 1958). The vote-aggregation paradox, we believe, casts the paradox of voting in a new light that illuminates, especially, its implications for making coherent social choices using different aggregation procedures.

6. Normative Questions and Democratic Political Theory

Given that the winner under office aggregation can receive the fewest votes under combination aggregation--and even that the two methods of aggregation can produce diametrically opposed social choices (when there is a complete-reversal paradox)--it is legitimate to ask which choice, if either, is the proper one. In defining `proper,' one might apply such social-choice criteria as the election of Condorcet outcomes (if they exist), the selection of Pareto-efficient outcomes, the existence of incentives to vote sincerely, and so on.[15] We shall not pursue this line of inquiry here, however, but instead ask an explicitly normative question: Is a conflict between the office and combination winners necessarily bad?

In addressing this question, we first consider whether this conflict comes as any great surprise. If there is one thing that social choice theory has taught us over the last several decades, it is that strange things may happen when we try to aggregate individual choices into some meaningful whole. Thus, the whole may lose important properties that the parts had, such as transitivity of preferences when there is a paradox of voting.

Whether the intransitivity of social preferences caused by the paradox of voting is a serious social problem has been much debated in the literature (Riker, 1982, and Miller, 1983, give representative views). The paradox of vote aggregation shows up a different aspect of this problem by drawing our attention to the discrepancy between aggregating votes by office and by combination. From a theoretical viewpoint, what is interesting about the vote-aggregation paradox is that it is a more general phenomenon than the paradox of voting--at least under our isomorphism--but we have not analyzed in detail those situations that give the vote-aggregation paradox, and not the paradox of voting, to see precisely where the differences between the two paradoxes lie.

From a practical viewpoint, we are led to ask whether, given the vote-aggregation paradox, it would be advisable for voters to vote directly for combinations rather than for offices. We have our doubts in the case of federal elections, in part because it is not clear how combination voting would work in the election of bodies like the Senate or House. In the case of the president, one could prescribe that if the winning combination includes, say, D for president, the Democrat would be elected. But if the winning combination turns out to be DDD, as occurred in 1992, what does it mean to elect a Democratic Senate and a Democratic House, and in what proportions in what states?

In voting in other arenas, such as a legislature or a referendum, we believe the choices that legislators and voters can now make substantially restrict their ability to express their preferences. Thus, legislators cannot express support for different packages of amendments, such as the amendments sequentially voted on in the 1932 Revenue Act (section 5). If they vote YYY, for example, this contributes nothing to NNN, even though this package might be their second choice. Likewise, there is no way under the present system that legislators can support exactly the six mixed combinations.

A possible solution to this problem is to use approval voting (Brams and Fishburn, 1983), whereby voters could, in the present instance, vote for as many combinations as they wish. Thus, a proponent of all the amendments or of none--assuming he or she regards these as the only acceptable packages--could indeed vote for both YYY and NNN, just as a proponent of some but not all of the amendments could vote for from one to six of the mixed combinations.

But approval voting for combinations is not the only way of expanding voter choices. Other means for producing more coherent social choices, in light of the paradox, include allowing voters to rank the combinations under a system like the Borda count. This would enable voters to make more fine-grained choices than does the variation of the Borda count, discussed in section 3, that corresponds to the present system.

To be sure, if there are more than eight or so combinations to rank, the voter's task could become burdensome. How to package combinations (e.g., of different amendments to a bill, different propositions on a referendum) so as not to swamp the voter with inordinately many choices--some perhaps inconsistent--will not be an easy practical problem to solve.

We raise these questions about packaging and voting systems not so much to provide answers but rather to show how the vote-aggregation paradox invests them with important social-choice consequences. Their ramifications, especially for reform, need to be analyzed in concrete empirical settings. As we indicated in the case of the 1990 California referendum (section 4), different ways of casting votes and counting abstentions would almost surely have led to different outcomes on a set of propositions related to the environment (Authors, 1995)

At a minimum, a heightened awareness of the vote-aggregation paradox alerts us to unintended consequences that may attend the tallying of votes. The paradox does not just highlight problems of aggregation and packaging, however, but strikes at the core of social choice--both what it means and how to achieve it--which in our view still remain unsettled. References

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Table 1

Combination Returns for 1976 and 1980 Presidential Elections, by Congressional Districts, in States with Senatorial Contests

Combination 1976 1980

1. DDD 40.8%* 22.2%

2. DDR 5.7 2.5

3. DRD 2.6 2.2

4. DRR 1.9 0.6

5. RDD 11.7 16.8

6. RDR 8.2 12.7

7. RRD 8.5 14.3*

8. RRR 20.6 28.6

Total 100.0 (n = 316) 100.0 (n = 315)

*Winner by office aggregation Table 2

Preferences of Groups of Members of the House of Representatives in Voting on the Wilmot Proviso (1846)

Group No. of Members Preferences

Northern Administration Democrats 7 abc

Northern Free Soil Democrats 51 bac

Border Democrats 8 abc or acb

Southern Democrats 46 acb

Northern prowar Whigs 2 cab

Northern antiwar Whigs 39 cba

Border Whigs 3 bac or bca

Southern and border Whigs 16 acb

Total 172

Source: Riker (1982, p. 227)



Assume that voters must choose between Democratic (D) and Republican (R) candidates for president, senator, and representative. If the winning candidates for each office are, say, R for president, D for senator, and D for representative--or RDD--the paradox of vote aggregation is that the combination RDD may receive fewer votes than any of the other eight combinations (if there are three offices, there are 23 possible combinations). Several examples of this paradox, which is interpreted as a paradox of divided government, and its variants are illustrated. Necessary and sufficient conditions for its occurrence, which depends on the `incoherence' of support for the winning combination, are given.

The paradox is shown, via an isomorphism, to be a generalization of the well-known paradox of voting. One real-life example of the vote-aggregation paradox--in voting on propositions in California, in which not a single voter voted on the winning side of all the propositions--is given. Several empirical examples of variants of the paradox that manifested themselves in federal elections, and legislative votes in the House of Representatives, are analyzed. Possible normative implications of the paradox, such as allowing voters to vote directly for combinations using approval voting or the Borda count, are discussed.


Steven J. Brams

Department of Politics

New York University

New York, NY 10003

D. Marc Kilgour

Department of Mathematics

Wilfrid Laurier University

Waterloo, Ontario N2L 3C5


William S. Zwicker

Department of Mathematics

Union College

Schenectady, NY 12308