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Generalized scoring rules and the frequency of coalitional manipulability
 In Proceedings of the Ninth ACM Conference on Electronic Commerce (EC
, 2008
"... We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors—more specifically, only on the order (in terms of score) of the sum’s components. This clas ..."
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Cited by 61 (18 self)
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We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors—more specifically, only on the order (in terms of score) of the sum’s components. This class is extremely general: we do not know of any commonly studied rule that is not a generalized scoring rule. We then study the coalitional manipulation problem for generalized scoring rules. We prove that under certain natural assump), then tions, if the number of manipulators is O(n p) (for any p < 1 2 the probability that a random profile is manipulable is O(n p − 1 2), where n is the number of voters. We also prove that under another set of natural assumptions, if the number of manipulators is Ω(n p) (for any p> 1) and o(n), then the probability that a random pro2 file is manipulable (to any possible winner under the voting rule) is 1 − O(e −Ω(n2p−1)). We also show that common voting rules satisfy these conditions (for the uniform distribution). These results generalize earlier results by Procaccia and Rosenschein as well as even earlier results on the probability of an election being tied.
Estimating the probability of events that have never occurred: when is your vote decisive
 Journal of the American Statistical Association
, 1998
"... Researchers sometimes argue that statisticians have little to contribute when few realizations of the process being estimated are observed. We show that this argument is incorrect even in the extreme situation of estimating the probabilities of events so rare that they have never occurred. We show h ..."
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Cited by 29 (14 self)
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Researchers sometimes argue that statisticians have little to contribute when few realizations of the process being estimated are observed. We show that this argument is incorrect even in the extreme situation of estimating the probabilities of events so rare that they have never occurred. We show how statistical forecasting models allow us to use empirical data to improve inferences about the probabilities of these events. Our application is estimating the probability that your vote will be decisive in a U.S. presidential election, a problem that has been studied by political scientists for more than two decades. The exact value of this probability is of only minor interest, but the number has important implications for understanding the optimal allocation of campaign resources, whether states and voter groups receive their fair share of attention from prospective presidents, and how formal "rational choice" models of voter behavior might be able to explain why people vote at all. We show how the probability of a decisive vote can be estimated empirically from statelevel forecasts of the presidential election and illustrate with the example of 1992. Based on generalizations of standard political science forecasting models, we estimate the (prospective) probability of a single vote being decisive as about 1 in 10 million for close national elections such as 1992, varying by about a factor of 10 among states. Our results support the argument that subjective probabilities of many types are best obtained through empirically based statistical prediction models rather than solely through mathematical reasoning. We discuss the implications of our findings for the types of decision analyses used in public choice studies.
The empirical frequency of a pivotal vote
 Public Choice
, 2003
"... Empirical distributions of election margins are computing using data on 16,577 U.S. Congressional and 40,036 state legislator election returns. One of every 100,000 votes cast in U.S. elections, and one of every 15,000 votes cast in state elections, “mattered ” in the sense that they were cast for a ..."
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Cited by 13 (0 self)
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Empirical distributions of election margins are computing using data on 16,577 U.S. Congressional and 40,036 state legislator election returns. One of every 100,000 votes cast in U.S. elections, and one of every 15,000 votes cast in state elections, “mattered ” in the sense that they were cast for a candidate that officially tied or won by one vote. Very close elections are more rare than a binomial model predicts. The evidence also suggests that recounts, and other marginspecific election procedures, are quite relevant determinants of the frequency of a pivotal vote. Although moderately close elections (winning margin of less than ten percentage points) are more common than landslides, the distribution of moderately close U.S. election margins is approximately uniform. In contrast, the distribution of state legislature election margins is clearly monotonic, with closer margins more likely, except for very close and very lopsided elections. The frequency of one vote elections is compared with total votes, and with the frequencies suggested by some theoretical models of voting. Roughly one of every 30,000 elections with 100,000 votes are decided by one vote. For elections with 5,000 or 20,000 votes, the frequencies are 1/1500 or 1/6000, respectively. This inverse relationship between election size and the frequency of one vote margins
The Mathematics And Statistics Of Voting Power
 STATISTICAL SCIENCE
, 2002
"... In an election, voting powerthe probability that a single vote is decisive is affected by the rule for aggregating votes into a single outcome. Voting power is important for studying political representation, fairness, and strategy, and has been much discussed in political science. Although ..."
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Cited by 9 (3 self)
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In an election, voting powerthe probability that a single vote is decisive is affected by the rule for aggregating votes into a single outcome. Voting power is important for studying political representation, fairness, and strategy, and has been much discussed in political science. Although power indexes are often considered as mathematical definitions, they ultimately depend on statistical models of voting. Mathematical calculations of voting power have usually been performed under the model that votes are decided by coin flips. This simple
Empirically Evaluating the Electoral College
 IN RETHINKING THE VOTE: THE POLITICS AND PROSPECTS OF AMERICAN ELECTION REFORM
, 2002
"... The 2000 U.S. presidential election has once again rekindled interest in possible electoral reform including the possible elimination of the Electoral College. Most arguments against ..."
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Cited by 3 (3 self)
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The 2000 U.S. presidential election has once again rekindled interest in possible electoral reform including the possible elimination of the Electoral College. Most arguments against
How Much Does A Vote Count?  Voting Power, Coalitions, And The Electoral College
, 2001
"... In an election the probability that a single voter is decisive is affected by the electoral systemthat is, the rule for aggregating votes into a single outcome. Under the assumption that all votes are equally likely (i.e., random voting), we prove that the average probability of a vote being d ..."
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Cited by 2 (2 self)
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In an election the probability that a single voter is decisive is affected by the electoral systemthat is, the rule for aggregating votes into a single outcome. Under the assumption that all votes are equally likely (i.e., random voting), we prove that the average probability of a vote being decisive is maximized under a popularvote (or simple majority) rule and is lower under any coalition system, such as the U.S. Electoral College system, no matter how complicated. Forming a coalition increases the decisive vote probability for the voters within a coalition, but the aggregate effect of coalitions is to decrease the average decisiveness of the population of voters. We then review results on voting power in an electoral college system. Under the random voting assumption, it is well known that the voters with the highest probability of decisiveness are those in large states. However, we show using empirical estimates of the closeness of historical U.S. Presidential elections that voters in small states have been advantaged because the random voting model overestimates the frequencies of close elections in the larger states. Finally, we estimate the average probability of decisiveness for all U.S. Presidential elections from 1960 to 2000 under three possible electoral systems: popular vote, electoral vote, and winnertakeall within Congressional districts. We find that the average probability of decisiveness is about the same under all three systems.
Division Of The Humanities And Social Sciences
"... In an election the probability that a single voter is decisive is a#ected by the electoral systemthat is, the rule for aggregating votes into a single outcome. Under the assumption that all votes are equally likely (i.e., random voting), we prove that the average probability of a vote being de ..."
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In an election the probability that a single voter is decisive is a#ected by the electoral systemthat is, the rule for aggregating votes into a single outcome. Under the assumption that all votes are equally likely (i.e., random voting), we prove that the average probability of a vote being decisive is maximized under a popularvote (or simple majority) rule and is lower under any coalition system, such as the U.S. Electoral College system, no matter how complicated. Forming a coalition increases the decisive vote probability for the voters within a coalition, but the aggregate e#ect of coalitions is to decrease the average decisiveness of the population of voters. We then review results on voting power in an electoral college system. Under the random voting assumption, it is well known that the voters with the highest probability of decisiveness are those in large states. However, we show using empirical estimates of the closeness of historical U.S. Presidential elections that voters in small states have been advantaged because the random voting model overestimates the frequencies of close elections in the larger states. Finally, we estimate the average probability of decisiveness for all U.S. Presidential elections from 1960 to 2000 under three possible electoral systems: popular vote, electoral vote, and winnertakeall within Congressional districts. We find that the average probability of decisiveness is about the same under all three systems.
by
, 2000
"... t.borgersucl.ac.uk I would like to thank Tim Feddersen and an anonymous referee for their comments. What are good voting rules if voting is costly? We analyse this question for the case that an electorate chooses among two alternatives. In a symmetric private value model of voting we show that major ..."
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t.borgersucl.ac.uk I would like to thank Tim Feddersen and an anonymous referee for their comments. What are good voting rules if voting is costly? We analyse this question for the case that an electorate chooses among two alternatives. In a symmetric private value model of voting we show that majority voting with voluntary participation Paretodominates majority voting with compulsory participation. We also demonstrate the potential advantages of asymmetric voting rules. We consider three types of such rules: Rules which do not allow all individuals to vote, rules which rely on an arbitrary status quo which can only be overturned if a majority of individuals participates in the voting process, and sequential voting rules. 2 How should group decisions be organized when participation in the decision making process is costly? Should participation be voluntary, or should it be compulsory? Should everyone be invited to participate, or should only a small sample of those involved be invited to participate? These and related questions will be addressed in
On the Rational Choice Theory of Voter Turnout
"... Abstract. I consider a twocandidate pluralityrule election in which there is aggregate uncertainty about the popularity of each candidate, where voting is costly, and where participants are instrumentally motivated. The unique equilibrium predicts significant turnout under reasonable parameter con ..."
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Abstract. I consider a twocandidate pluralityrule election in which there is aggregate uncertainty about the popularity of each candidate, where voting is costly, and where participants are instrumentally motivated. The unique equilibrium predicts significant turnout under reasonable parameter configurations, and greater turnout for the underdog offsets the expected advantage of the perceived leader. I also present clear predictions about the response of turnout and the election outcome to various parameters, including the importance of the election; the cost of voting; the perceived popularity of each candidate; and the accuracy of preelection information sources, such as opinion polls. THE TURNOUT PARADOX Why do people vote? For supporters of different candidates, how is turnout likely to vary? Is the election result likely to reflect accurately the pattern of preferences throughout the electorate? These questions are central to the study of democratic systems; they are questions which have attracted the attention of the most insightful theorists and the most careful empirical researchers. Nevertheless, the turnout question (why do people vote?) has proved problematic for theories based on instrumental actors. In an oftquoted question based on a statement made by Fiorina (1989), Grofman (1993) provocatively asked: is turnout the paradox that ate rational choice theory? The paradox is this: people do vote and yet it is alleged that any “reasonable ” rationalchoice theory suggests that they should not. 1 Although this is a very recent paper, I have spoken with many colleagues about writing it over a long period of time. I thank of all of them for their comments, conversation, encouragement, criticism, and