Voting Theory: What We Know (and Don’t Know) About Winning and Losing Elections

October 21, 2008

Book Review
James Case

Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It). By William Poundstone, Hill & Wang, New York, 2008, 352 pages, $25.00.

Bill Poundstone writes good books. His second and fourth, The Recursive Universe and Labyrinths of Reason, were nominated for Pulitzer prizes. His classic Prisoner's Dilemma is among the best books about game theory ever written for a popular audience, and his Fortune's Formula tells the story of a little known gambling strategy that works wonders in casinos, at racetracks, and on Wall Street. His latest, on voting theory, is among his best. Moreover, it appears at a time when the political process seems receptive to reform.

Such times are few and far between. The intellectual ferment that preceded the French Revolution spawned one of them. Foreseeing that the nation's most critical decisions might soon be decided in this new and unfamiliar way, members of the French Academy of Sciences wondered what subtleties might lurk behind the bland façade of majority rule. Their conclusions were far from reassuring. Whereas ordinary (aka "plurality") voting---in which each voter has a single vote to cast, and the candidate receiving the most votes wins---seems to produce the desired result in two-candidate elections, it can easily fail when candidates are more numerous. Perhaps the most striking---and decidedly the best known---of the shortcomings of simple plurality voting was discovered by Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet, a mathematician, philosopher, politician, and savant of the revolutionary era.

Confident in the fairness of two-candidate elections, Condorcet proposed that every issue be reduced to pairwise comparisons. He did not demand a separate election between each pair of candidates, however, because the required information can be obtained from a single election in which every voter is required to rank all candidates in order of preference.

Condorcet was well aware that such elections can produce ambiguous results. It happens whenever candidate A beats candidate B, who beats someone else, who beats someone else, who . . . beats candidate A. Yet such "voting cycles" shouldn't have to occur. A single candidate who finishes first on a majority of the ballots cast seemed certain to Condorcet to represent "the people's choice." Poundstone follows Donald Saari in describing such candidates as "Condorcet winners," and candidates who finish last on a majority of ballots as "Condorcet losers." Condorcet proposed that any acceptable voting procedure should, at the very least, elect the Condorcet winner whenever there is one. Imagine his surprise upon learning that ordinary plurality voting can fail to do so!

To demonstrate this disturbing fact, Condorcet considered a three-candidate election in which:

30 voters prefer A to B to C
1 voter prefers A to C to B
10 voters prefer B to C to A
29 voters prefer B to A to C
10 voters prefer C to A to B
1 voter prefers C to B to A

It is easily verified that (i) A beats B on 41 of the 81 ballots cast, (ii) A beats C on 60 of the 81, and (iii) B wins the plurality election with 39 votes to A's 31 and C's 11, so that Condorcet winner A is not elected by plurality voting! Such is the voting paradox forever associated with Condorcet's name. Something like it can occur whenever voters are asked to rank three or more candidates in order of preference. The failing is by no means peculiar to plurality voting.

Poundstone invigorates what would otherwise be an abstract discussion of voting procedures by drawing pertinent examples from the pages of history. He points out, for instance, that apparent Condorcet loser Woodrow Wilson chanced to defeat apparent Condorcet winner Theodore Roosevelt in the presidential election of 1912, when most voters would probably have preferred the Independent Roosevelt to either the Republican incumbent William Howard Taft or the Democratic challenger Wilson. Indeed, because most would have preferred either Taft or Roosevelt to Wilson, the latter appears to have been a Condorcet loser. Yet by splitting the conservative vote, Taft and Roosevelt allowed the more liberal Wilson to steal the election.

Much the same thing seems to have happened in 1966, when rivals split the vote of (ordinarily dominant) Maryland Democrats, allowing Baltimore County Republican Spiro "Ted" Agnew to be elected governor. (Two years later, Nixon picked Agnew as his running mate in the 1968 presidential election.) Such "vote splitting" is increasingly common today because, according to Poundstone, wealthy Republican donors seem eager to finance campaigns by third-party candidates likely to draw votes from Democratic opponents.

Condorcet's investigation of voting procedures seems to have been motivated by a desire to discredit a scoring system---used for a time to elect new members to the French Academy---that had been devised by another academy member, Jean-Charles de Borda. In Borda's system for an election in which each voter is required to rank all candidates in order of preference, each candidate receives from each ballot the number of points obtained by subtracting his or her rank on that ballot from the total number of candidates. Thus, in a five-candidate election, a given ballot is worth four points to the candidate ranked highest, three points to the candidate ranked second, and so on. The candidate with the most points wins. Condorcet was delighted to discover that Borda's scoring system can also fail to elect the Condorcet winner.

Donald Saari has shown that the "Borda count" manages to avoid a truly impressive number of voting paradoxes.* Yet like every other method requiring that each voter rank all candidates in order of preference, it is not immune to the Condorcet paradox. Any such method can fail to elect a Condorcet winner, and all but one of those methods--one that no democracy could even consider employing--can produce a voting cycle in which A beats B, who beats C, who beats A. Such is the content of Kenneth Arrow's famous "impossibility theorem," proved around 1950. It has long been taken to imply that no voting method can be perfect.

Further evidence to that effect is found in the more recent Gibbard–Satterthwaite theorem, which asserts that any of the procedures considered by Arrow---and many others as well---can be derailed by voters who "game the system" by "voting strategically" for someone other than their favorite candidate. The Naderites who voted for Al Gore in 2000 in an effort to keep George W. Bush from winning the presidency, for example, were voting strategically.

Arrow's theorem does not apply to a procedure called approval voting, which does not require that each voter rank all candidates in order of preference. Apparently first proposed by Robert Weber in his 1971 Yale thesis, approval voting allows voters to check off as many names on the ballot as they wish, with a check mark indicating approval of the candidate so designated. A check mark is worth one approval point for the candidate, and the candidate with the most points wins. Like most methods, approval voting can fail to elect a Condorcet winner and is vulnerable to various forms of gaming. Poundstone devotes a good deal of space to the ongoing debate between advocates of approval voting---led by Peter Fishburne and Steven Brams, who have written an entire book† on the subject---and champions of the Borda count, led by Donald Saari, who has written several books describing the unique features of that most mathematically interesting of methods.

Instant runoff voting (IRV) is yet another voting method currently in the news. It is already in use in San Francisco and in many foreign jurisdictions. Arrow's theorem does apply to IRV, because it requires each voter to rank all candidates in order of preference. In the absence of a computer, it could work as follows: After the polls close, the ballots are physically separated into piles according to which candidate is ranked number 1. If one candidate's pile contains more than half the ballots, that candidate is declared the winner. If not, the candidates are separated into two groups, those with enough first-place votes to be included in a second round of voting and those with too few. This can be done, for example, by eliminating all who fail to win a prespecified number of first-place votes. Next, the piles corresponding to the surviving candidates are combined into one, and redistributed according to which of the candidates is ranked highest. If this produces a winner, the election is over. Otherwise, the procedure is repeated until one of the piles contains a majority of the ballots and, in so doing, designates a winner. Most voting theorists believe that IRV is better than plurality voting, but that other methods are better still.

Poundstone favors a variant of approval voting whereby voters can allot 0, 1, or any proper fraction of an approval point to each candidate, the winner being the one with the largest number of approval points. This relieves voters of the need to produce a total ordering of the candidates, while allowing them to distinguish strong from weak or non-existent preferences. Calling this the "range method," Poundstone joins mathematician Warren Smith in arguing that it avoids most of the paradoxes to which the methods studied by Arrow are vulnerable.

Although the study of range voting re-mains in its infancy, Poundstone cites a telling experiment conducted by Smith: In 144 simulated elections, range voting spawned less retrospective "Bayesian regret" than did plurality voting, instant runoff voting, Condorcet voting, the Borda count, or approval voting. An interesting aspect of Smith's experiment is that many of the simulated elections were run several times---first with scrupulously honest voters and then with varying numbers of strategic voters added to the mix. In each case, the strategic electorate experienced more combined regret than the honest one. Strategic voting seems to create a "tragedy of the commons," in which individually advantageous behavior is collectively harmful.

It is impossible to predict the outcome of future research on voting methods, but it does seem fair to say that recent developments have broken the apparent logjam created by Arrow's theorem, and could soon lead to elections that better reflect the will of the people.

*Donald G. Saari, Chaotic Elections! American Mathematical Society, Providence, Rhode Island, 2001, page 20.

†Steven Brams and Peter Fishburne, Approval Voting, Birkhäuser, Boston, 1982.

James Case writes from Baltimore, Maryland.

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