[1]We thank Kenneth F. McCue for providing the Los Angeles county ballot data used in this study, and Gary King and Katri Saari for facilitating our use of these data. Hali Marko provided valuable research assistance, and Bob Ellsworth was unstinting in the computer assistance he gave us. Steven J. Brams thanks the C. V. Starr Center for Applied Economic Research for its support.

[2]Insofar as voters anticipate this problem, they may feel regret before the fact as well.

[3]Similarly, the voter's preference on the first proposition depends on the outcome of voting on the second--that is, dependence does not depend on the order in which the propositions are voted upon.

[4]A method for determining the number of independent preferences, and identifying which they are, is given in Kilgour and Brams (1995). Independent preferences are equivalent to both separable and lexicographic preferences when n = 2; when n > 2, the latter categories are more restrictive. Thus, when n = 3, 96 preference orders are separable, and 48 are lexicographic; moreover, lexicographic preferences are a proper subset of separable preferences, and separable preferences are a proper subset of independent preferences. Lacy and Niou (1994) analyze referenda in which voters have nonseparable preferences, arguing that they may produce inefficient outcomes, whereas Benoit and Kornhauser (1994) show that the election of legislatures may lead to inefficient outcomes even when voters have separable preferences.

[5]Lacy and Niou (1994) give a real-life example of such a dilemma facing North Carolina voters in 1993.

[6]The same problem afflicts a voter's choosing among politicians running for different offices (Brams, Kilgour, and Zwicker, 1994). For instance, a voter today cannot indicate that his or her highest priority is that the same party control both the Senate and the House--that is, that the outcome be either DD (both houses Democratic) or RR (both houses Republican) over the mixed outcomes, DR or RD. This is because the voter who votes, say, DD is implicitly saying that the `compromise' of DR or RD is better than RR, as we will show quantitatively through our analysis of the standard-aggregation scoring method in section 3.

[7]Note that, consistent with most election procedures, an outcome cannot be abstention, even if abstention gets the most votes.

[8]It is, of course, the ranking implicit in these ratings that may be an inaccurate reflection of a dependent voter's preferences. For example, the score of 0 that a (1,1) voter gives to outcome (1,-1) may not reflect the fact that this YY voter actually thinks that YN is the worst possible outcome rather than a middling outcome.

[9]But it is the same as `yes-no voting' (Brams and Fishburn, 1993) when interpreted in the following way: a Y vote for a proposition means that any combination approved of must include Y for that proposition; an N vote against a proposition means that any combination approved of must include N against that proposition; and an A on a proposition means that any combination approved of may include either a Y or an N for that proposition.

[10]This is similar to what Brams, Kilgour, and Zwicker (1994) called a `paradox of vote aggregation': the winner under standard aggregation is the combination that receives the least direct support. We will illustrate this paradox, and milder variations, with the California referendum data in section 5.

[11]We will say shortly which of these voters probably had the most dependent preferences.

12Because approximately 0.02% of the ballots were spoiled, the figures in Table 4a do not sum to 1,684,786, the number of nonabstaining voters, but instead to 1,672,793.

[13]A genuine form of the paradox of vote aggregation occurred in voting on all 28 propositions by the Los Angeles county voters. The winning combination was NNNYNNYNYNNNNNNYNYNYNYYNNYNY on propositions 124 - 151 but, excluding all voters who abstained on any proposition, nobody made this winning choice. Hence, this combination received the fewest votes; however, more than 99% of all 228 = 268.4 million possible Y-N combinations must have received 0 votes--even if each of the voters voted for a different combination--so there is nothing particularly surprising about this result. More surprising, and perhaps paradoxical, is that when there are as few as two propositions with abstention allowed, the winning combination can come in last of the 32 = 9 combinations when there are only 15 voters; when there are three propositions and abstention is not allowed, the winning combination can come in last of the 23 = 8 combinations when there are just 3 voters (Brams, Kilgour, and Zwicker, 1994).

[14]If Y is the outcome of the first election, this voter will unconditionally prefer that Y be the outcome of the second election; and if Y is the outcome of the second election, this voter will unconditionally prefer that Y be the outcome of the third election. In our empirical case, this says that a pro-environment voter wants more of the environmental propositions to pass, period. But insofar as this voter has intermediate preferences for which one or two propositions he or she would most prefer be passed, this voter's preferences will be dependent, as we illustrate next in the text.

[15]Thereby approval aggregation may help positions that have few or no direct supporters, leading to a combination about which nobody is enthusiastic (like the winning NNN combination in Example 1).

[16]For a discussion of this and other proposed solutions to a special case of the dependency problem (preferences are not separable), see Lacy and Niou (1994).