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Suppose You Want to Vote Strategically

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Suppose You Want to Vote Strategically DONALD SAARI Suppose You Want to Vote Strategically e honest. There have been times when you voted strate- To check, suppose five voters prefer the candidates Anita, Bgically to try to force a personally better election result; Bonnie, and Candy in that order, denoted by ABC, six oth- I have. The role of manipulative behavior received brief ers prefer CBA, and the last four prefer ACB. While it doesnt attention during the 2000 US Presidential Primary Season seem like anything can go wrong, lets check. when the Governor of Michigan failed on his promise to • By voting for one candidate, the commonly used plural- deliver his states Republican primary vote for George Bush. ity system, Anita wins with a 60% landslide; the ACB out- His excuse was that the winner, John McCain, strategically come has the 9:6:0 tally. attracted cross-over votes of independents and Democrats. • Bonnie failed to receive a single plurality vote, yet she McCains strategy was just the accepted behavior of en- wins when each voter votes for her top two candidates couraging supporters who can vote, to vote. But lets pursue where the BCA outcome has the 11:10:9 tally. this issue further; lets question whether the power of math- • Candy? She wins with the procedure offering 5 and 4 ematics can help identify when and how you can strategically points, respectively, to a voters first and second choices; alter the election outcome of your fraternity, sorority, social the CAB outcome has a 46: 45: 44 tally. group, or department to force a personally better conclusion. This example illustrates the worrisome reality that rather First, a disclaimer. I am not interested in training a gen- than representing the views of the voters, an election outcome may eration of manipulative strategiststhe economists do a more accurately reflect the choice of the election procedure. On the much better job of this. Instead, my goal is to demonstrate other hand, this setting presents strategic opportunities to unusual applications of elementary mathematics while alert- a young aspiring Machiavelli to select a procedure which ing the reader to subtle political actions which affect all of yields his preferred winner. While it may seem difficult to us. As we will see, each component of an electionthe pro- discover which procedure Machiavelli should select, the cedure, voting, and processoffers manipulative opportu- analysis is mathematically simple when there are only three nities. So, I describe the mathematics behind the strategy or four candidates. of selecting an election procedure, casting a ballot, and To introduce terminology, a positional voting method assigns debating or introducing amendments. weights (w1, w2, 0) to candidates according to how a voter positions them on a ballot. As w1 and w2 points are assigned Choice of a procedure to a voters top- and second-ranked candidates, the weights must satisfy w1 ³ w2 ³ 0. Thus, the standard plurality vote, Remember those boring debates about the choice of a voting where each voter votes for one candidate, corresponds to procedure? Someone wants to use the standard vote for one (1, 0, 0). The Borda Count (BC) named after the eighteenth- approach; others prefer to vote for two, while still others wish century mathematician J.C. Borda, is given by (2, 1, 0). A to distinguish between a voters top- and second-ranked can- polite way to vote against one candidate is to vote for the didate by assigning, respectively, five and four points. Who other two; this antiplurality method is defined by (1, 1, 0). cares? Lets just vote. Wont the answers be essentially the same? Each w1 and w2 choice defines a procedure, so there are an uncountable number of them. But notice, when voting for one candidate, the election ranking is the same if a can- didate is assigned one point, or 500 points, for each vote. This suggests normalizing the tallying procedures so that When the article was written, DONALD SAARI was the Pancoe Pro- fessor of Mathematics as Northwestern. He now is at the Univer- w1 = 1. By doing so, the points become sity of California, Irvine. ws = (1, s, 0), 0 £ s £ 1 © Mathematical Association of America Math Horizons November 2000 5 1 18 where s = w2 /w1. So, s = 0, ¤2, 1 represent, respectively, the teams can differ drasticallyn = 20 allows up to 2 ´ 10 plurality vote, the BC, and the antiplurality method. The ear- different rankingsdepending upon how the ballots are lier system giving five and four points, respectively, to a voters tallied. Yet, not everything can happen; e.g., try as hard as 4 first- and second-ranked candidates becomes w4/5 = (1, ¤5, 0). I may, I cannot find weights which would rank the North- To find personally beneficial procedures, tally the bal- western University football team at the top. lots to learn which s values deliver an outcome to our lik- By the way, this strategic behavior is not restricted to ing. By doing so, the above example yields voting. For instance, suppose the head of a company pro- Number Preferences ABC ducing a particular product wants his product to be the best in a statistical comparison. Can an appropriate choice 6 CBA 06s 6 of a statistical method skew the answer? Maybe; Deanna 5 ABC 55s 0 Haunsperger showed how a single data set can define a 4 ACB 404s surprisingly wide array of different rankings by varying the Total 911s 6 + 4s choice of seemingly excellent statistical procedures. Which procedure, that is, which s should be used? For an election ranking to change, the tallies must pass through a tie. So, set pairs of tallies equal and solve for those s values Strategic Voting which cause {A, B}, {A, C}, {B, C} election ties. For instance, As it is difficult to continually change procedures with each an AC tie requires 9 = 6 + 4s, so it occurs with s = 3¤4. The results, in Figure 1, prove that these seemingly innocuous election, lets examine ways to be strategic in the privacy of preferences generate seven different election outcomes where the voting booth. This is commonly done. During the March four of them have no ties. Moreover, as each candidate wins 2000 primaries, for instance, a Keyes supporter from the with some procedure, this setting provides fruitful oppor- State of Michigan confided that since Keyes had no realis- tunities for our young Machiavelli! He just needs to justify tic chance of winning, he voted strategically for Bush in an using an appropriate choice of weights which ensures his unsuccessful attempt to prevent McCain from winning. This preferred outcome. Since most people dont worry about behavior, which illustrates the commonly used Dont waste which procedure is used, this offers no serious challenge. your vote! strategy, alters the legitimacy of any message The same approach identifies all outcomes for all speci- based on election outcomes. As such, it is worth wondering fied preferences for any number of candidates. As a chal- whether a procedure can be invented where it never is in a lenge, use this approach to find all (1, s, t, 0), 1 ³ s ³ t ³ 0, voters best interest to be strategic. election outcomes for the following ten-voter example. (In No such method exists. In the early 1970s, Alain Gibbard graphing the outcomes, replace the marks with lines; 18 and Mark Satterthwaite independently proved that with different rankings without ties emerge.) three or more alternatives, all reasonable election proce- dures (e.g., not a dictatorship) provide opportunities for Number Preference Number Preference someone to strategically obtain a personally more favor- 2 ABCD 2 CBDA able outcome. But while this result ensures that opportuni- 1 ACDB 3 DBCA ties exist for our Machiavelli, it does not explain how he 2 ADCB can recognize when they arise or how to take advantage of A strategist might worry whether only rare, unlikely in- them. Not much help comes from the extensive literature stances of voter preferences allow this phenomenon where on this topic as it favors existence theoremsproving that the election ranking can switch with the weights. No; as Maria strategic opportunities existover the pragmatic issue of Tataru and I showed, even with conservative assumptions describing how to find them. So, let me indicate how to about the distribution of voters preferences, about 69% of solve the more general problem for all procedures. the time a three-candidate election ranking changes with the weights. Then, this likelihood and the number of possible Notation Designate each voter type with a number as follows. election outcomes escalate with the number of candidates. Type Ranking Type Ranking For instance, examples can be constructed where, by choos- 1 ABC 4 CBA ing different weights for n alternatives, as many as n! (n 1)! 2 ACB 5 BCA different election rankings emerge without ties. This sug- 3 CAB 6 BAC gests that the ranking of the top twenty collegiate football If pj represents the number of voters of the jth type, then our introductory example has p1 = 5, p2 = 4, p4 = 6 and pj = CAB CBA 0 for all other voter types. These voter preferencesa pro- ACB BCA filecreate the six-dimensional vector p = (5, 4, 0, 6, 0, 0). 0 3 9 6 1 4 11 7 Changing the outcome When Bob, a type-four voter, votes A-C tie A-B tie B-C tie as though he is type six, there is one fewer type-four voter Figure 1 and one more type-six voter. This change is represented by 6 Math Horizons November 2000 © Mathematical Association of America Illustration by Loel Barr Loel by Illustration v = (0, 0, 0, 1, 0, 1).
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