Statistical Science 2002, Vol. 17, No. 4, 420–435 © Institute of Mathematical Statistics, 2002 The Mathematics and Statistics of Power Andrew Gelman, Jonathan N. Katz and Francis Tuerlinckx

Abstract. In an , voting power—the 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 usually have been performed under the model that votes are decided by coin flips. This simple model has interesting implications for weighted , two-stage elections (such as the U.S. Electoral College) and coalition structures. We discuss empirical failings of the coin-flip model of voting and consider, first, the implications for voting power and, second, ways in which votes could be modeled more realistically. Under the random voting√ model, the standard deviation of the average of n votes is proportional to 1/ n√, but under more general models, this variance can have the form cn−α or a − b log n. Voting power calculations under more realistic models present research challenges in modeling and computation. Key words and phrases: Banzhaf index, decisive vote, elections, electoral college, Ising model, political science, random walk, trees.

1. INTRODUCTION settings, weighted voting can lead to unexpected re- sults. For example, consider an election with four vot- To decide the outcome of a U.S. Presidential elec- ers representing constituencies of unequal size, who tion, your vote must be decisive in your own state, and are given weights of 12, 9, 6 and 2, respectively, for a then your state must be decisive in the Electoral Col- weighted majority vote. The voter with 2 votes has zero lege. The different states have different populations, voting power in that these 2 votes are irrelevant to the different numbers of electoral votes and different vot- election outcome, no matter what the other three voters ing patterns, so the probability of casting a decisive do, and the other three voters have equal power in that vote will vary between states. Even in much simpler any two of them can determine the outcome. The voter with 12 votes has no more power than the voter with 6. Andrew Gelman is Professor of Statistics and Po- In fact, this system is equivalent to assigning the voters litical Science at Columbia University, New York, 1, 1, 1 and 0 votes. New York 10027 (e-mail: [email protected]; Examples such as these have motivated the mathe- http://www.stat.columbia.edu/∼gelman/). Jonathan N. matical theory of voting power, which is generally de- Katz is an Associate Professor of Political Science in fined in terms of the possibilities that a given voter or the Division of the Humanities and Social Sciences of set of voters can affect the outcome of an election. the California Institute of Technology, Pasadena, Cali- Ideas of voting power have been applied in a range fornia 91125 (e-mail: [email protected]; http://jkatz. of settings, including committee voting in legislatures, caltech.edu). Francis Tuerlinckx is in the Faculty of weighted voting (as in corporations) and hierarchical Psychology and Educational Sciences, Katholieke Uni- voting systems, such as the U.S. Electoral College and versiteit Leuven, Belgium (e-mail: francis.tuerlinckx@ the European Council of Ministers. We restrict our- psy.kuleuven.ac.be). selves in this paper to discussion of weighted majority

420 THE MATHEMATICS AND STATISTICS OF VOTING POWER 421

voting in two-party systems, with the Electoral College where the representatives casting the weighted vote are as the main running example. themselves elected by majority vote in separate dis- The major goals of voting power analysis are (1) to tricts. This is the structure of the U.S. Electoral Col- assess the relative power of individual voters or blocs lege and, implicitly, the European Council of Ministers of voters in an , (2) to evaluate the (where each minister represents an individual country system itself in terms of fairness and maximizing whose government is itself democratically elected). average voting power, (3) to assign weights so as to Voting power measures have been developed in sta- achieve a desired distribution of voting power and tistics, mathematics, political science and legal lit- (4) to understand the benefits of coalitions and bloc eratures for over 50 years. Key references are Pen- voting. rose (1946), Shapley and Shubik (1954) and Banzhaf This paper reviews some of the mathematical re- (1965). Felsenthal and Machover (1998) gave a re- search on voting power and presents some new find- cent review. This article focuses on weighted and two- ings, placing them in a political context. Many of the stage voting systems such as the Electoral College best-known results in this field are based on a conve- (which has been analyzed by Mann and Shapley, 1960; nient but unrealistic model of random voting, which Banzhaf, 1968; Brams and Davis, 1974; and Gelman, we describe in Section 3. Section 4 compares to em- King and Boscardin, 1998, among others). Power in- pirical election results and in Section 5, we move to dexes have strong and often controversial implications more complicated models of public opinion and eval- (see, e.g., Garrett and Tsebelis, 1999), and so it is im- uate their effects on voting power calculations. One of portant to understand their mathematical and statistical the challenges here is in understanding which of the re- foundations. sults on voting power are robust to reasonable choices Voting power has also been applied to more compli- of model. Section 6 concludes with an assessment of cated voting schemes, which we do not discuss here, the connections between voting power and political such as legislative elections with committees, multi- representation. ple chambers, vetoes and filibusters (Shapley and Shu- The main results presented in this article are the bik, 1954). Luce and Raiffa (1957) discussed power following: indexes as an application of game theory, following Shapley (1953). The most interesting problems of leg- 1. For the random voting model, we review well- islative voting arise when considering coalitions that known findings on voting power in weighted and can change, so that individual voters are motivated to two-stage elections. form coalition structures that increase their power. This 2. We critically evaluate the random voting model, game-theoretic structure differs from the examples in both in theory and in relation to data from elections, this article, where the system of coalitions is fixed and and discuss the general relevance of voting power votes are simply added at each level. in politics and the mistaken recommendations that have arisen from simplistic voting power calcula- 2.2 Definitions of Voting Power tions. Voting power can and has been defined in a variety 3. We review recent work and present new results on of ways (see Straffin, 1978; Saari and Sieberg, 1999; probability models for voters on trees. These results Heard and Swartz, 1999), and fairness of an electoral are mathematically interesting and point the way system can be defined even more generally (see, e.g., to further work connecting multilevel probability Beitz, 1989; Gelman, 2002). In this article we shall models of voters, empirical data on the variability use the definition based on the probability that a of elections and political science models of public vote affects the outcome of the election. Consider an opinion. electoral system with n voters. We use the notation i for an individual voter, vi =±1 for his or her vote, 2. BACKGROUND v = (v1,...,vn) for the entire vector of votes and R = R(v) =±1 for the rule that aggregates the n individual 2.1 Areas of Application of Voting Power votes into a single outcome. It is possible for R to There are a variety of voting situations in which be stochastic (because of possible ties or, even more votes are not simply tallied, and so voting power is a generally, because of possible errors in vote counting). nontrivial concept. We consider weighted majority vot- We shall assume various distributions on v and rules ing and the closely related setting of two-level voting, R(v) which then together induce distributions on R. 422 A. GELMAN, J. N. KATZ AND F. TUERLINCKX

Voting power can also be defined without any prob- is decisive in an election is also relevant in studying ability distributions, simply as the number of combina- the responsiveness of an electoral system to voter pref- tions of the other n − 1 voters for which vote i will be erences and the utility of voting (see Riker and Or- decisive, but, as we note at the beginning of Section 3, deshook, 1968; Ferejohn and Fiorina, 1974; Aldrich, this is equivalent to the probability definition under the 1993; Edlin, Gelman and Kaplan, 2002). model that all vote combinations are equally likely. The probability of a vote being decisive is important The probability that the change of an individual vote directly—it represents your influence on the electoral vi will change the outcome of the election is outcome, and this influence is crucial in a democracy— and also indirectly, because it could influence cam- = poweri power of voter i paigning. For example, one might expect campaign ef- (1) = Pr(R =+1ifvi =+1) forts to be proportional to the probability of a vote being decisive, multiplied by the expected number − Pr(R =+1ifvi =−1). of votes changed per unit of campaign expense, al- If your voting power is zero, then changing your vote though there are likely strategic complications since from −1to+1 has no effect on the probability of either both sides are making campaign decisions. Thus, cam- candidate winning. paigning strategies have been studied in the political The probabilities in (1) refer to the distribution of the science literature as evidence of voting power (Brams other n − 1 voters, unconditional on vi .Votingpower and Davis, 1974, 1975; Colantoni, Levesque and Or- represents the causal effect of changing your vote with deshook, 1975; Stromberg, 2002). all the other votes held constant—the direct effect on Perhaps the most widely publicized normative po- R of changing vi , not the information that vi might litical claim from the voting power literature is that, convey about the other n − 1votes. in two-stage voting systems with proportional weight- We shall consider the following situations: ing (that is, wj ∝ nj ), voters in larger jurisdictions have disproportionate power (Penrose, 1946; Banzhaf, 1. For simple weighted voting, we assume n voters 1965). Under a simple (and, in our judgment, inap- = with weights wi,i 1,...,n, andanaggregation n propriate) model, the voting power√ in such systems is rule R(v) = sign( = w v ); that is, the winner is i 1 i i approximately proportional to nj (see Section 3.2). determined by weighted majority. Weighted voting This has led scholars to claim that the U.S. Electoral is important for its own sake (e.g., in public corpo- College favors large states (Banzhaf, 1968), a claim rations) and also as the second level of two-stage that we and many others have disputed (see Section 4). voting. In political science these theories have been checked 2. For two-stage voting,weassumethevotersare with empirical data in various ways. Most directly, vot- divided into J jurisdictions, with the winner within ing power depends on the probability that an election each jurisdiction decided by majority vote. Each is a tie. This probability is typically so low that it is jurisdiction j has a weighted vote of wj at the difficult to estimate directly; for example, in the past second level, and the overall winner is decided 100 years, there have been about 20,000 contested elec- by the weighted majority of the winners in the tions to the U.S. House of Representatives, and none jurisdictions. of them has been tied. However, the probability of a tie can be estimated by extrapolating from the empir- We assume that ties at all levels are decided by coin ical frequency of close elections (see Mulligan and flips. Hunter, 2001); for example, about 500 of the afore- 2.3 Claims in the Literature and Connections to mentioned House of Representatives elections were de- Empirical Data cided within 1,000 votes. If we define fV to be the dis- tribution of the difference V in vote proportions be- Social scientists have studied both the theoretical and tween the two leading candidates, then the empirical implications of voting power, mostly in f (0) political applications, but also in areas such as cor- (2) Pr(tie election) ≈ V porate governance (Leech, 2002). Researchers have n looked at fairness to individuals and also at the ef- in an election with n voters. Regression-type forecast- fects of unequal voting power on campaigning and ing models for V have been used to estimate voting the allocation of resources (see Snyder, Ting and An- power for specific elections (see Section 4). In elec- solabehere, 2001). The probability that a single vote tions with disputed votes and possible recounts, so that THE MATHEMATICS AND STATISTICS OF VOTING POWER 423

no single vote can be certain to be decisive, the proba- can calculate poweri for each state (assuming random bility of affecting the outcome of the election can still voting) and√ compare to the linear approximation, ≈ be identified with the probability of a tie to a very close poweri 2/π(wi/W). The linear fit has a relative approximation (see the Appendix of Gelman, Katz and erroroflessthan10%forallstates. Bafumi, 2002). As this calculation for the Electoral College illus- trates, voting power paradoxes such as illustrated in the 3. THE RANDOM VOTING MODEL first paragraph of this article are unlikely to occur ex- cept in the context of the discreteness of very small vot- We begin with the assumption that votes are deter- ing systems. Such situations have occurred (see Felsen- mined by independent coin flips, which we call ran- thal and Machover, 2000), but in our opinion they are dom voting. As we discuss in Section 4, the random fundamentally less interesting than the results on two- voting model is empirically inappropriate for election stage voting and coalitions that we review below. In data. We devote some space to this model, however, weighted voting settings where the w ’s display central because it is standard in the voting power literature. i n limit theorem-type behavior, voting power (given ran- Under random voting, all 2 vote configurations are dom voting) is approximately proportional to weight, equally likely, and so the power of voter i is simply as one would intuitively expect. 2−(n−1) times the number of configurations of the other n − 1 voters for which voter i is decisive (and 3.2 Two-Stage Voting counting semidecisive configurations, in which votes In two-stage voting, one must first compute the are exactly tied, as 1/2). Voting power calculations can voting power of each jurisdiction j and then the power thus be seen as combinatorical. However, we see the of each of the nj voters within a jurisdiction. As probabilistic, rather than the counting, derivation as described above, if the number of jurisdictions is large fundamental. and none is dominant, it is reasonable to approximate 3.1 Weighted Voting the voting power of jurisdiction j as proportional to wj under random voting. To calculate the power of voter i with weight wi in For an individual voter i in jurisdiction j,letV−i be a simple weighted majority voting system, we define the sum of the other nj − 1 votes in the jurisdiction. = the total weighted vote of all the others as V−i Then the probability that vote i is decisive within w v and we define the sum of the squares of k =i k k jurisdiction j is 2 = n 2 all the weights as W k=1 wk . Then, for weighted = + 1 | |= Pr(V−i 0) 2 Pr( V−i 1) voting,   − = | | + 1 | |=  nj 1 (3) poweri Pr( V−i

+ 2Pr(R =−1andvi =−1) − 1 in the direction of realism. (assuming random voting) 4.1 Closeness of Elections, the Number of Voters andVotingPower = 2Pr(voter i is satisfied) − 1, The random voting model makes predictions that are and so, under random voting, maximizing average not valid for real elections. For example, in an election voting power is equivalent to maximizing the average with 1 million voters, the random voting model implies probability of satisfaction. Only voters on the winning that the proportional vote margin should have a mean side will be satisfied and so, conditional on the total of 0 and a standard deviation of 0.001. In reality, large vote, average satisfaction is maximized by assigning elections are typically decided by much more than 0.1 the winner to the side supported by more voters, which of a percentage point. is simply majority rule. This theorem can also be seen Before discussing potential model improvements, as a corollary of more general results in graph theory we consider here the voting power implications of (see Lemma 6.1 of Friedgut and Kalai, 1996). empirical problems with the random voting model. THE MATHEMATICS AND STATISTICS OF VOTING POWER 425

FIG.1. An example of four different systems of coalitions with nine voters, with the probability of decisiveness of each voter computed under the random voting model. Each is a “one person, one vote” system, but they have different implications for probabilities of casting a decisive vote. Joining a coalition is generally beneficial to those inside the coalition but hurts those outside. The average voting power is maximized under A, the popular-vote rule with no coalitions. From Katz, Gelman and King (2002). 426 A. GELMAN, J. N. KATZ AND F. TUERLINCKX

FIG.2. Voting power for individuals in the 2000 U.S. Presidential election, by state under the random voting model and under an empirical model based on randomly perturbing the actual election outcomes with variation at the national, regional, state and local levels. Voters in large states have disproportionate power under the random voting model, but not under the empirical model.

Clearly, real elections are less close, and thus individ- turnout varies slightly between states). The random ual voters have less power, than predicted under ran- voting model then implies [see (4)] that an individual’s dom voting, but how are the results of relative voting voting power should be approximately proportional to power affected? the square root of the population of his or her state; see If the number of voters nj in a district is moderate the left panel of Figure 2. or large, we can use (2) to approximate the probability Thus, the general conclusion in the voting power that a single voter is decisive within district j as literature is that the Electoral College benefits voters fj (0) in large states. For example, Banzhaf (1968) claimed (5) Pr(a vote is decisive within district j)≈ , to offer “a mathematical demonstration” that it “dis- nj criminates against voters in the small and middle-sized where fj is the distribution of the proportional vote states by giving the citizens of the large states an = nj differential V j (1/nj ) i=1 vi within district j.If excessive amount of voting power,” and Brams and fj has a mean of 0 and a fixed distributional form (e.g., Davis (1974) claimed that the voter in a large state normality), then fj (0) ∝ 1/sd(V j ) and so “has on balance greater potential voting power ...than Pr(a vote is decisive within district j) a voter in a small state.” Mann and Shapley (1960), Owen (1975) and Rabinowitz and Macdonald (1986) (6) 1 ≈ . came to similar conclusions. This impression has also nj sd(V j ) made its way into the popular press; for example, Noah Thus, when comparing the probability of decisiveness (2000) stated, “the distortions of the Electoral College ...favorbigstatesmorethantheydolittle ones.” It has for voters in districts of different sizes nj (as in similarly been claimed that if countries in the European Section 3.2), the behavior of sd(V j ) as a function√ of nj is crucial. Under random voting, sd(V j ) ∝ 1/ nj , Union were to receive votes in its council of ministers but for actual elections this is not generally true. proportional to their countries’ populations, then voters in large countries would have disproportionate power 4.2 U.S. Presidential Elections (Felsenthal and Machover, 2000). √ We focus on the most widely discussed example, 4.3 The 1/ n Rule the relative power of voters in different states in electing the President of the United States. In the The above claims all depend on the intermediate re- Electoral College, each state gets two electoral votes sult, under the random voting model, that the proba- plus a number approximately proportional to the state’s bility√ of decisiveness within a state is proportional to population. Except for the smallest states, this means 1/ nj . The extra power of voters in large states de- that wj is approximately proportional to nj (voter rives from the assumption that elections in these states THE MATHEMATICS AND STATISTICS OF VOTING POWER 427

a calculation for the 2000 Presidential election based on perturbing the empirical election results. Uncer- tainty in the election outcome is represented by adding normally distributed random errors ε at the national, regional, state and local levels: for Congressional dis- tricts i in state j within region k, the election outcomes are simulated 500 times by perturbing the observed obs outcome, V i : sim = obs + nation + region + state + district (7) V i V i ε εk εj εi . sim The V i values are then summed within each state j to get a simulation of the state-level vote differen-

FIG.3. The margin in state votes for President as a function tials, V j . The error terms in the simulation (7) rep- of the number of voters nj in the state. Each dot represents a resent variation between elections, and the hierarchi- different state and election year from 1960 to 2000. The margins cal structure of the errors represents observed corre- are proportional; for example, a state vote of 400,000 for the lations in national, regional and state election results Democratic candidate and 600,000 for the Republican would be (Gelman, King and Boscardin, 1998). For the simu- recorded as 0.2. Lines show the lowess (nonparametric regression) √ lation for Figure 2b, the error terms on the vote pro- fit, the best-fit line proportional to 1/ nj and the best-fit line of α portions have been assigned normal distributions with the form cnj . As shown by the lowess line, the proportional√ vote differentials show only very weak dependence on nj .The1/ nj standard deviations 0.06, 0.02, 0.04 and 0.09, which line, implied by standard voting power measures, does not fit the were estimated from election-to-election variation of data. vote outcomes at the national, regional, state and Con- gressional district levels. (The pattern of results of the are much more likely to be close. This assumption can voting power simulation are not substantially altered be tested with data. For example, Figure 3 shows the by moderate changes to these variance parameters.) absolute proportional vote differential |V j | as a func- Under the simulation, the probability that a voter is tion of number of voters nj for all states (excluding decisive within state j is given by (5), which we evalu- the District of Columbia) for all Presidential elections ate from the normal density function. We then compute from 1960 to 2000.√ in two steps the probability that the state’s electoral We test the 1/ nj hypothesis by fitting three differ- votes are decisive for the nation. First, we update the ent regression lines to |V j | as a function of nj .First, distributions of the national and regional error terms we use the lowess procedure (Cleveland, 1979) to fit a nation region ε and εk using the multivariate normal distrib- nonparametric regression√ line. Second, we fit a curve ution given the condition V = 0. Second, we simulate = j of the form y c/ nj , using least squares to find the the vector of election outcomes for all the other states best-fitting value of c. Third, we find the best-fitting under this condition and estimate the probability that = α − curve of the form y cnj . The best-fit α is 0.16 the wj electoral votes of state j are decisive in the na- (with a standard error of 0.03), which suggests that tional total. This last computation could be performed elections tend to get slightly closer for larger√ nj but by counting simulations, but is made more computa- with a relationship much weaker than 1/ nj .Asthe tionally efficient by approximating the proportion of electoral votes received by either candidate as a beta scale√ of the graph makes clear, it is impossible for the 1/ nj rule to hold in practice, as this would mean ex- distribution, as in Gelman, King and Boscardin (1998). treme landslides for low nj or extremely close elec- Comparing to the result under the random voting tions for high nj , neither of which, in general, will model, the empirical calculation shows much more hold. variation between states (because some states, like New Mexico, were close, and others, like Massa- 4.4 Empirical Estimates of Voting Power in chusetts, were not) and no strong dependence on state Presidential Elections √ size. In reality, but not in the random voting model, Given that the 1/ nj rule is not appropriate for real large states are not necessarily extremely close, and elections, how should voting power be calculated for thus voters in large states do not have disproportion- two-stage elections? The right panel of Figure 2 shows ate voting power. 428 A. GELMAN, J. N. KATZ AND F. TUERLINCKX

values displayed in Figure 2b), Katz, Gelman and King (2002) computed average voting power for Presidential elections under three electoral systems: popular vote, electoral vote and a hypothetical system of winner- take-all by Congressional district. They found average voting power to vary dramatically between elections (depending on the closeness of the national election); within any election, however, changing the voting rule had little effect on the average probability of decisiveness. 4.6 Empirical Evidence from Other Elections Analyses of electoral data from U.S. Congress, U.S. state legislatures and European national elections also FIG.4. The average probability of a decisive vote as a function of the number of electoral votes in the voter’s state, for each U.S. have found only very weak dependence of the close- Presidential election from 1952 to 1992 (excluding 1968, when a ness of elections on the number of voters (Mulligan and third party won in some states). The probabilities are calculated Hunter, 2001; Gelman, Katz and Bafumi, 2002). This√ based on a forecasting model that uses information available persistent empirical finding, contradicting the 1/ n two months before the election. This figure is adapted from Gelman, rule implied by random voting, casts strong doubt on King and Boscardin (1998). The probabilities vary little with state size, with the most notable pattern being that voters in the very the recommendations by mathematical analysts from smallest states are, on average, slightly more likely to be decisive. Penrose (1946) to Felsenthal and Machover (2000) to apply standard power indexes to weighted voting. The best approach for assigning weighted votes is A slightly different empirically based method of still unclear, however. We have seen that assigning computing voting power is described by Gelman, King weights to equalize voting power under random voting and Boscardin (1998). A hierarchical linear regression is inappropriate in real-world electoral systems that model, based on standard election forecasting proce- √ do not follow the 1/ n rule. However, equalizing dures used in political science, was used to obtain prob- empirical voting power is also problematic, as the abilistic forecasts for Presidential elections by state. weights would then have to depend on local political The models were then used to compute the probabil- conditions. For example, in the 2000 election, it would ity of decisive vote; Figure 4 shows the resulting av- be necessary to lower the weight of New Mexico and erage probability of decisiveness of voters in a state, raise the weight of Massachusetts (see Figure 2b), as a function of the number of electoral votes in the but then these weights might have to change in the state, for each election. The clearest pattern is that the future if New Mexico moved away or Massachusetts smallest states have slightly higher voting power, on moved toward the national average. A reasonable average; this is a result of the two “free” votes that default position is to assign weights in proportion each state receives in the Electoral College, so that the to population size or perhaps population size to the smallest states have disproportionate weights. 0.9 power (see Gelman, Katz and Bafumi, 2002), but 4.5 The Electoral College and Average these too are based on particular empirical analyses. Voting Power In general, this fits in with the literature on evaluating voting methods and power indexes based on their As discussed in Section 3.3, under random voting, performance in actual voting situations (see, e.g., average voting power is maximized under a popular Felsenthal, Maoz and Rapoport, 1993; Heard and vote system. However, this result is highly sensitive Swartz, 1999; Leech, 2002). to the assumption, under random voting, that all vote outcomes are equally likely (see Natapoff, 1996). Thus, 5. STOCHASTIC PROCESSES FOR VOTERS if the question of average voting power is of practical interest, it is important to address it using actual The simplest generalization of random voting is for electoral data. votes to be independent but with probability p,rather Using the model (7) based on perturbing district- than 1/2, of voting +1. This is not a useful model; by-district outcomes (as was used to calculate the although it corrects the mean vote, it still predicts a THE MATHEMATICS AND STATISTICS OF VOTING POWER 429

standard deviation that is extremely small for large elections (Beck, 1975). It is necessary to go further and allow each voter to have a separate pi and to model the distribution of these pi’s. If the probabilities pi are given any fixed distribution not depending on n,then the distribution of the average vote, for large n,will converge to√ the distribution of the pi’s. The empirically falsified 1/ n rule then goes away, to be replaced by the more general expression (5), because the binomial variation from which it derives is minor compared to FIG.5. Notation for the derivation of the standard deviation of any realistic variation among the probabilities pi .(This (d) d was noted by Good and Mayer, 1975; Margolis, 1977, the average vote differential, V , for the Ising models on k vot- ers in a tree structure. This tree, unlike that in Figure 1, repre- 1983; and Chamberlain and Rothschild, 1981.) sents dependence between voters, not a coalition structure. Here, The next step is give a dependence structure to the z represents the unobserved ±1 variable at the root of the tree, voters’ probabilities pi. It makes sense to build this de- x represents the unobserved ±1 variable at one of the k branches (d−1) pendence upon existing relationships among the voters. at the next level and V is the average vote differential for the The most natural starting point is a tree structure based kd−1 voters in this branch. The derivation proceeds by determin- (d) on geography; for example, the United States is divided ing the mean and variance of V , conditional on z, in terms of (d−1) into regions, each of which contains several states, the mean and variance of V , conditional on x. each of which is divided into Congressional districts, counties, cities, neighborhoods and so forth. When 5.1.1 A stochastic model on a tree of voters. To get modeling elections, it makes sense to use nested com- a sense of how these models work, we derive some munities for which electoral data are available (e.g., basic results for the Ising model on symmetric trees. states, legislative districts and precincts). It might also = d be appropriate to include structures based on nonnested Suppose we have n k voters arrayed in a tree of predictors. For example, voters have similarities based depth d with k branches at each node. At each node of ± on demographics as well as geography; nonnested the tree is a variable that equals 1(seeFigure5).For models also have been used to capture “small-world” the leaf nodes, this variable represents a vote; at the phenomena in social networks (Watts, Dodds and New- other nodes, the variable is unobserved and serves to man, 2002). induce a correlation among the leaves. The probability Section 5.1 discusses an approach based on directly model states that, conditional on a parent node, the modeling the dependence of individual votes and Sec- k children are independent, each with probability π of tion 5.2 presents a model using correlated latent vari- differing from the parent. We further assume that the ables. It is interesting to see the different implications marginal probabilities of +1and−1 are equal. for observable outcomes that are obtained by these two 5.1.2 The distribution of the average of n votes. families of probability models. One way to understand this model and to see how it 5.1 Discrete Modeling of Dependence of Votes differs from random voting is to study its implications for the probability that an individual voter is decisive One idea for modeling votes, taken from the math- in a coalition or district j of n voters. As discussed ematical literature and based on the Ising model j in Section 4, the probability of decisiveness is linked from statistical physics, is to model dependence of to the standard deviation of the proportional vote the votes v via a probability density proportional to i differential V in the district. The random voting exp(− c v v ),wherec represents the strength j ij ij i j ij ∝ −0.5 of the connection between any two voters i and j. model predicts sd(V j ) nj , whereas empirical Under this model, voters who are connected are more electoral data show much weaker declines on the order ∝ −α likely to vote similarly. If the voters form a tree struc- of sd(V j ) nj , with powers estimated at lower = ture, then the model implies correlation of the votes in values such as α 0.15 (see Figure 3). the same local community. If the correlation is higher This model can easily be simulated starting at the than a certain critical value, then the properties of av- top of the tree and working downward. However, this is erage votes can be qualitatively different than in the computationally expensive for large values of n,sowe (d) random voting model, as we prove below (see Evans, use analytic methods to evaluate sd(V ) as a function Kenyon, Peres and Schulman, 2000). of n = kd ,aswellastheparametersk and π that 430 A. GELMAN, J. N. KATZ AND F. TUERLINCKX determine the stochastic process. The results we prove with the second term being the variance of a random here also appear in Bleher, Ruiz and Zagrebnov (1995) variable that equals µd−1 with probability 1 − π and Evans et al. (2000). or −µd−1 with probability π. We can expand the The setup for our derivation is diagrammed in Fig- recursion in (10) and insert (9) to obtain ure 5. For an Ising-model tree of depth d,letz =±1 d  be the equally likely values at the root and let V d be 2 = − −d − 2 i d σd 4π(1 π)k (1 2π) k , the average value at the k leaves. The variance of V d i=0 can be decomposed, conditional on the root value z,as and combining with (9) into (8) yields (d) = (d)| + (d)| var V var E V z E var V z d  2 (d) = d + − −d i = 1 (d)| =+ − (11) var V ξ (1 ξ)k (ξk) , 2 E V z 1 0 i=0 (d) 2 + E V |z =−1 − 0 where, for convenience, we have defined (8) (d) + 1 var V |z =+1 ξ = ( − π)2. 2 1 2 (d) + var V |z =−1 We evaluate (11) separately for three cases, depend- = 2 + 2 ing on whether the factor ξk in the power series is less µd σd , than 1, equal to 1 or greater than 1. For each, we focus where µd and σd are the conditional means and on the limit of large d—that is, large n—since we are variances of V d given z =+1: interested in modeling elections of thousands or mil- = (d)| =+ 2 = (d)| =+ lions of voters. µd E V z 1 ,σd var V z 1 . • This decomposition is useful because we can evaluate If ξk <1, then we can expand (11) as 2 d µd and σd recursively. (d) 1 − (ξk) = V = ξ d + ( − ξ)k−d At d 0,therootofthetreeisthesameastheleaf, var 1 − (0) 1 ξk and V = z. Thus, (12)   1 − ξ 1 (0) ≈ for large n. µ0 = E V |z =+1 = 1, 1 − ξk n

(0) Thus, for ξ<1/k, the standard deviation of the σ 2 = var V |z =+1 = 0. √ 0 average of n votes is proportional to 1/ n,just

For d ≥ 1 we note that V d is the average of k as in the random voting model but with a different identically distributed random variables V d−1 that are proportionality constant. This will not be useful for independent conditional on z (see Figure 5). Then us in modeling data with more gradual power-law we can use the symmetry of the underlying model to behavior such as displayed in Figure 3. obtain • If ξk = 1, then (11) becomes

d (d) d −d µd = E E V |x |z =+1 var V = ξ + (1 − ξ)k d

= (1 − π)µd−1 + π(−µd)     (9) 1 1 = (1 − 2π)µ − (13) = 1 + 1 − log n d 1 n k k = ( − π)d,   1 2 1 1 ≈ 1 − log n for large n. with the last step being a recursive calculation starting k n k with µ = 1. We evaluate σ 2 using the conditional 0 d Thus, for ξ = 1/k, the standard deviation of the variance decomposition and the fact that it is the mean  average vote margin is proportional to (logk n)/n. of k independent components, • If ξk >1, then (11) becomes 1 d 2 = | | =+ d σd var E V x z 1 (d) (ξk) − 1 k var V = ξ d + (1 − ξ)k−d + d | | =+ ξk− 1 (10) E var V x z 1 (14) 1  k − 1 = 4π(1 − π)µ2 + σ 2 , ≈ ξ d for large n. k d−1 d−1 k − 1/ξ THE MATHEMATICS AND STATISTICS OF VOTING POWER 431

= d = −2α Since d logk n, we can write ξ n ,where public opinion in the short term (Gelman and King, α is a power less than 1/2; more specifically, α = 1993) and long term (Page and Shapiro, 1992). Further − 0.5logk ξ. We then can reexpress (14) as work is needed to study how the votes in these mod-   (d) k − 1 − els aggregate and their implications for voting power. (15) var V ≈ n 2α for large n. k − k2α Mathematically, this is related to models in spatial sta- Evans et al. (2000) generalized these power laws to tistics and their implications for the sampling distribu- nonregular trees. tion of spatial averages (Whittle, 1956; Ripley, 1981), with the additional analytical difficulties that arise from 5.1.3 Fitting the model to electoral data. In real the nonlinear probit transformation. elections, one can approximate the standard deviation −α of V j for districts j as proportional to nj ,whereα 5.2.1 A stochastic model on a tree of voters. Astart- is some power less than 1/2 (see Section 4). As we just ing point for theoretical exploration, by analogy to the have shown, this corresponds to the Ising model with Ising models discussed in Section 5.1, is to apply the ξ>k. √ additive model to a regular tree structure, with each Given α and k, we can solve for π = 1 (1 − ξ)= 2 node of the tree having a continuous value z. We con- 1 (1 − k−α). For example, the Presidential election 2 sider a simple but nontrivial random walk model, in data illustrated in Figure 3 can be fitted by α = ε ∼ ( ,τ2) 0.16. For k = 2, 3, 10, 100, the best-fitting π’s are which independent error terms N 0 are as- 0.05, 0.08, 0.15, 0.26. signed to each node of the tree, and then, for each node, the value z is defined as the sum of the ε’s for that node Next, one can imagine setting the parameter k and all the nodes above it in the tree. so that the coefficient (k − 1)/(k − k2α) from (15) We work with a tree of depth D with k branches at matches the coefficient c in the fitted curve, sd(V j ) ≈ D − each node, thus representing N = k voters. The value cn α. However, this second fitting step is not so j z for a node at depth d of the tree is then the sum of effective, because the coefficient in (15) has a narrow + 2 range of possibilities. For example, for α = 0.16, d 1 independent N(0,τ ) terms starting at the root and working down to the node. − − 2α (k 1)/(k k ) ranges from a maximum of 1.15 For any of the kD leaf nodes i (i.e., voters), the (at k = 2) to a minimum of 1 (as k →∞), but the probability pi = Pr(Vi =+1) is set to (zi). In our estimate of c from the data in Figure 3 is 1.74. model, the values zi at the leaves marginally have Even if the estimated c from this data set happened N(0,(D + 1)τ 2) distributions but with a correlation to be in the range (1, 1.15), we would not want to take the Ising model too seriously as a description of voters. structure induced by the tree. Such a model has Our purpose in developing such a stylized model three parameters: D, k and τ , and we can explore here is primarily to show how simple conditions of the variation of average votes at different levels of connectedness can induce power laws that go beyond aggregation, as a function of these parameters. the random voting model. In the Ising model, we did not need to consider the depth D of the larger tree in evaluating the properties 5.2 Continuous Modeling of Latent Underlying Preferences of average votes in subtrees of depth d. In contrast, the distribution of the votes in the random walk model The other natural approach to modeling variation depends on the higher branches of the tree. This in opinion, deriving from preference models in social makes sense from a political standpoint, because state- science, is to think of the votes v as independent but i level votes, for example, are affected by national and with structure on the probabilities pi . A natural starting point is an additive model on the logit or probit scale: regional as well as statewide and local swings. for example, in a hierarchical structure, 5.2.2 The distribution of the average of n votes. As = + + + +··· (d) pi (αnation βregioni γstatei γdistricti ). in Section 5.1, we shall determine the variance of V , the proportional vote differential based on averaging More generally, a nonnested model has the form pi = = d ((Xβ)i), with geographic and demographic predic- n k voters. For this model, we compute the variance tors X. This sort of model is consistent with under- by counting the number of pairs of the n voters that are standing swings in votes (Gelman and King, 1994) and a distance 0, 1, 2,...,d apart in the tree, 432 A. GELMAN, J. N. KATZ AND F. TUERLINCKX   d we can compare to the exact formula (17) at any point. (d) 1 δ−1 (16) var V = 1 + (k − 1)k ρδ , Approximating arcsin(x) by x yields kd  δ=1 d (d) 1 2 − var V ≈ 1 + (k − 1) kδ 1 where ρδ is the correlation between the votes vi at two d k π = leaves a distance δ apart in the tree. In deriving (16), δ 1   we have used the fact that, from the symmetry of the (D + 1 − δ)τ2 =± × model, each vi 1 has a marginal mean of 0 and (D + 1)τ 2 + 1 variance of 1. 1 2 τ 2 For each δ, the correlation ρδ can be determined ≈ + based on the bivariate normal distribution: if voters i kd π (D + 1)τ 2 + 1   and j are at a distance δ, then we can write − k × D(1 − k d ) + − d . k − 1 ρδ = Pr(vi =+1andvj =+1) −d = + =− =− We further simplify by ignoring terms of order k Pr(vi 1andvj 1) 1/n, to obtain   − =+ =− 2 Pr(vi 1andvj 1) (d) 2 τ k (18) var V ≈ D + − d . + 2 + − − Pr(vi =−1andvj =+1) π (D 1)τ 1 k 1 This is simply a linear function in log n (recall that = 2A − (1 − 2A) n = kd ,sologn = d log k),

= 4A − 1, (d) (19) var V ≈ a − b log n, where A is the area in the positive quadrant of the where the parameters a and b are determined by D, bivariate normal distribution with mean 0 and variance k and τ , the parameters of the underlying stochastic matrix model. For numbers of voters n in the thousands to   millions (as in our electoral data), we compared (19) (D + 1)τ 2 + 1 (D − δ + 1)τ 2   . to (17) and found the approximation to be essentially (D − δ + 1)τ 2 (D + 1)τ 2 + 1 exact. 5.2.3 Fitting the model to electoral data. The model The extra “+1” term in the variance here corresponds (19) is much different from a power law but it ac- to the latent N(0, 1) error term in the probit model, =+ = tually behaves similarly over a fairly wide dynamic Pr(vi 1) (zi). Evaluating the area of the range of n. For example, the Presidential votes by normal distribution, we obtain state displayed in Figure 3 have n ranging from 60,000   2 (D + 1 − δ)τ2 to 10 million. The best-fit line of the form (19) to ρ = arcsin . (d) = − δ + 2 + thesedataisvar(V ) 0.20 0.0113 logn. Assum- π (D 1)τ 1 (d) ing√ a normal distribution, this implies E(|V |) = Thus, 0.8 0.20 − 0.0113 logn, which we display in Figure 6  d along with the previously fitted power-law curve. The (d) 1 2 − var V = 1 + (k − 1) kδ 1 two lines look almost identical, and it would be close kd π δ=1 to hopeless to try to distinguish between them from the (17)   (D + 1 − δ)τ2 data. × arcsin . We can map the fitted values a = 0.020 and b = + 2 + (D 1)τ 1 0.0113 to D, k and τ in (18). Since we are fitting three Because of the power of k, the last terms (with parameters to two, we can set one of them arbitrarily; for simplicity, we set k = 2 for a binary tree. Then higher values of δ) will dominate in the summa- the fitted values are N = 12 million and τ = 0.175. tion in (17). For these higher terms, the expression As with the Ising model in the previous section, we + − 2 + 2 + ((D 1 δ)τ )/((D 1)τ 1) will be close to 0, do not want to take these parameters too seriously; and so the arcsine can be approximated by the iden- these estimates are merely intended to give insight into tity function. We use this approximation to gain un- the sort of model that could predict patterns of vote (d) derstanding of the behavior of var(V ), knowing that margins that occur in real electoral data. THE MATHEMATICS AND STATISTICS OF VOTING POWER 433

as, “Is it desirable for the average voting power to be increased?” After the 2000 election, some commenta- tors suggested that it would be better if close elections were less likely, even though close elections are as- sociated with decisiveness of individual votes, which seems like a good thing. The issue of the desirability of close elections raises a conflict between two political principles: on one hand, democratic process would seem to require that every person’s vote has a nonzero chance (and, ideally, an equal chance) of determining the election outcome. On the other hand, very close elections such as Florida’s damage the legitimacy of the process, and so it might seem desirable to reduce the probability of FIG.6. The proportional margin in state votes for President as ties or extremely close votes. a function of the number of voters nj in the state, repeated from α − No amount of theorizing will resolve this diffi- Figure 3. The best-fit lines of the form cnj and a b log nj are displayed. The power law is consistent with the Ising model culty, which also occurs in committees and leads to described in Section 5.1 and the logarithmic form is consistent with legitimacy-protecting moves such as voting with an in- the random walk model√ described in Section 5.2. Both fit the data formal straw poll. The official vote that follows is then much better than the 1/ nj curve predicted by the random voter often close to unanimous as the voters on the losing model (see Figure 3). side switch to mask internal dissent. This article’s the- oretical findings on the benefits of coalitions imply that 6. DISCUSSION such behavior is understandable but in a larger context can reduce the average voting power of individuals. Voting power is important for studying political rep- resentation, fairness and strategy, and has been much 6.2 Limitations of Individualistic Measures of discussed in political science. Although power indexes Group Power are often considered as mathematical definitions, they We must also realize that individual measures of po- ultimately depend on statistical models of voting. As litical choice, even if aggregated, cannot capture the we have seen in Section 3, even the simplest default structure of group power. For one thing, groups that can of random voting is full of subtleties in its implica- mobilize effectively are solving the coordination prob- tions for voting power. However, as seen in Sections 4 lem of voting and can thus express more power through and 5, more realistic data-based models lead to dras- the box (Uhlaner, 1989). For an extreme exam- tically different substantive conclusions about fairness ple, consider the case of Australia, where at one time and voting power in important electoral systems such Aboriginal citizens were allowed, but not required, to as the U.S. Electoral College. Further work is needed vote in national elections, while non-Aboriginal cit- to develop models of individual voters in a way consis- izens were required to vote. Unsurprisingly, turnout tent with available data on elections and voting, and to was lower among Aboriginals. Who was benefiting understand the implications of these models for voting here? From an individual-rights standpoint, the Abo- power. riginals had the better deal, since they had the freedom We conclude with a discussion of the fundamental to choose whether to vote, but as a group, the Abo- connections between individual voting power and po- riginals’ lower turnout would be expected to hurt their litical representation. representation in the government and thus, probably, 6.1 Fundamental Conflict between Decisiveness of hurt them individually as well. Having voting power is Votes and Legitimacy of Election Outcomes most effective when you and the people who share your opinions actually vote. Our mathematical and empirical findings do not di- rectly address normative questions such as, “Which ACKNOWLEDGMENTS electoral system should be used?” or, in a legisla- ture, “How should committees or subcommittees be as- We thank Yuval Peres for alerting us to the Ising signed?” Let alone more fundamental questions such model on trees described in Section 5.1 and suggesting 434 A. GELMAN, J. N. KATZ AND F. TUERLINCKX its application to votes. We also thank Peter Dodds, FEREJOHN,J.andFIORINA, M. (1974). The paradox of not voting: Amit Gandhi, Hal Stern, Jan Vecer, Tian Zheng and A decision theoretic analysis. American Political Science two referees for helpful discussions and comments. Review 68 525–536. This work was supported in part by NSF Grants SES- FRIEDGUT,E.andKALAI, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 9987748 and SES-0084368. 2993–3002. GARRETT,G.andTSEBELIS, G. (1999). 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