[1]We thank Aaron Epstein, Jean-Pierre Benoit, Morris P. Fiorina, Richard F. Potthoff, and Alan D. Taylor for valuable discussion and communications, and Gary King and Kenneth F. McCue for providing the Los Angeles county ballot data analyzed later as well as helpful comments. The research assistance of [unnamed] is gratefully acknowledged.

[2]Sometimes `divided government' is used to mean that only the House of Representatives is controlled by a different party from that of the president, but the exact definition is not important. For a recent discussion of the merits and demerits of divided government, see `Symposium: Divided Government and the Politics of Constitutional Reform' (1991).

[3]Some readers will not view this result as paradoxical because, once illustrated and explained, the apparent contradiction disappears. Nonetheless, the idea that a winning combination can receive zero votes seems surprising and counterintuitive to most people on first hearing, which conforms with the informal sense in which `paradoxical' is used in political science (Brams, 1976; Maoz, 1990).

[4]Divided government is not purely an American phenomenon, but it often takes different forms abroad (Laver and Shepsle, 1991; Pierce, 1991).

[5]By relabeling, the paradox could be that DDD wins and gets 0 votes, so that it is somewhat misleading to call the paradox one of divided government. Generically, the paradox is of one of vote aggregation, as we suggested earlier, but we will use `divided government' as the appellation here because later data suggest that its most likely empirical manifestation is the following: an undivided combination (e.g., DDD) is the most popular--more voters vote for this combination than any other--but a divided one (e.g., RDD) wins when votes are aggregated by office.

[6]We say that the paradox occurs if the winning combination receives the fewest votes, but not necessarily zero, as in the example. Therefore, not all paradoxes will leave no voter unsatisfied. Indeed, some people may regard any situation in which the winning combination by office aggregation does not receive the most votes by combination aggregation as paradoxical--and hence worthy of explanation--but this is not the meaning of `paradox' that we use here.

[7]`Opposite' is ambiguous, of course, when there are more than two options. In the case of the three options postulated in Example 7, for instance, the opposite of DDD might be not only RRR but also the 7 other combinations that do not include any Ds. It turns out that not even a basic paradox can be constructed when each of the 8 non-D combinations must have more votes than each of the 19 D combinations (with, say, 0 votes each)--and one of the latter combinations must also be the office-aggregation winner. Necessary and sufficient conditions for the paradox in the three-office case, but without abstention, are given in section 3.

[8]As a measure of the tilt toward DDD, I(DDD) is analogous to the `spin,' or cyclic component, of the total vote (Zwicker, 1991).

[9]The Borda socre for DDD can be seen as identical to Q(DDD) and S(DDD), with points grouped differently. Generalizations of this scoring system to elections with any number of offices and with abstention allowed are given in Authors (uncited).

[10]The total for all abstain, all no, and all yes is only 3.7%. Thus, the vast majority of voters (96.7%) were not `pure' types but discriminated among propositions by choosing mixed combinations that included both Ys and Ns.

[11]Inexplicably, voting returns reported in Congressional Quarterly's Congressional Districts in the 1990s: A Portrait of America (CQ Press, 1993) for the 1988 presidential election are for the 1992 congressional districts, based on the the 1990 census, so they do not accurately reflect the results of the 1988 election. While Congressional Quarterly's annual Politics in America and National Journal's annual Almanac of American Politics give presidential-election returns by congressional district and state, senatorial returns for each congressional district are not available (except in 1976 and 1980). Although senatorial returns are broken down by county in Congressional Quarterly's America Votes series, congressional districts often divide counties, requiring that one use precinct-level data to determine senatorial results by district. While such data for approximately 190,000 precincts have been collected for the period 1984-90 by a now-defunct group called `Fairness for the 90s,' an officially nonpartisan and nonprofit organization, the data are not presently in a form that makes them ready for computer analysis (King, 1993).

[12]In the president/senator comparison in 1988, there was a tie for first, according to combination aggregation, between RD and RR (RD was the office-aggregation winner that year). Although the Republican presidential candidate (George Bush) prevailed in both combinations, this fact does not ensure that such a candidate, who may win in a majority of states, would win a majority of either popular or electoral votes should these states be mostly small.

[13]In particular, a district that roughly reflects the nation might appear to vote for the national winning combination, even though that vote in fact represented a paradoxical combination that individual ballot data would have revealed.

[14]The winning combination was YYY with 85 votes. Because the pairwise contests were not among a, b, and c but partially overlapping sets of alternatives (see text), it is not inconsistent for individual voters to have a preference order associated with YYY in this case.

[15]Benoit and Kornhauser (1994) focus on the inefficiency of assemblies elected by office aggregation, given that voters have separable preferences over all possible combinations of candidates for the assembly. (An inefficient assembly is one in which the candidates elected by office aggregation are worse for all voters than some other assembly--possibly elected by combination aggregation--and so might receive zero votes when pitted against it.) Although the paradox of vote aggregation is based purely on numerical comparisons, it may be explicitly linked to preference-based models like that of Benoit and Kornhauser, as we illustrated in the case of the paradox of voting through the answer-sequence isomorphism. See also Lacy and Niou (1994), who analyze referenda in which voters have nonseparable preferences, and Authors (uncited), who analyze referenda in which voters have dependent preferences, a broader class than nonseparable preferences.