Voting Power in US Presidential Elections under a Modified Popular Vote Plan

Voting Power in US Presidential Elections under a Modified Popular Vote Plan Steven J. Brams Department of Politics New York University New York, NY 10012 USA steven.brams@nyu.edu D. Marc Kilgour Department of Mathematics Wilfrid Laurier University Waterloo, Ontario N2L 3C5 CANADA mkilgour@wlu.ca March 2013

2 Abstract Under the National Popular Vote (NPV) plan for US presidential elections, states that join an interstate compact pledge to cast all their electoral votes for the national popular-vote winner rather than for the state winner. The compact becomes effective only after states with a majority of electoral votes enact it, which would then ensure the election of the national popular-vote winner. So far, eight states and the District of Columbia, controlling 49 percent of the 270 electoral votes needed for NPV to become law, have passed NPV. We propose a Modified Popular Vote (MPV) plan whereby states, independent of what other states do, pledge to cast all their electoral votes for the national popular vote winner. We argue that this measure would benefit the 41 states in the 2012 presidential election including the three largest, CA, TX, and NY that received virtually no attention from the candidates because they were not considered battleground, or competitive, states. Voters in these states would increase their Banzhaf voting power by up to half what they would have if they were battleground states. Of course, once states with a majority of electoral votes adopt MPV, it becomes NPV, giving all voters nationwide the same voting power. Prospects for the adoption of MPV are briefly discussed.

3 Voting Power in US Presidential Elections under a Modified Popular Vote Plan 1 1. Introduction Under the National Popular Vote (NPV) plan for US Presidential elections, states that join an interstate compact pledge to cast all their electoral votes for the national popular-vote winner, rather than for the state winner. The compact becomes effective only after states with a majority of electoral votes enact it, which would then ensure the election of the national popular-vote winner. So far, eight states and the District of Columbia, controlling 49 percent of the 270 electoral votes needed for NPV to become law, have passed NPV (NPV websites, 2013). We propose a modified version of NPV (MPV), which drops the stipulation that the plan becomes effective only after states with a majority of electoral votes pass it. Unlike NPV, MPV does not require an interstate compact saying when it becomes law it becomes effective immediately for any state that enacts it. We argue that this reform would have benefited the 41 states in the 2012 presidential election including the three largest states, CA, TX, and NY that received virtually no attention from the candidates because they were not considered battleground states, in which either candidate might win. Under MPV, states would become appealing for their popular votes, because a vote gained in an MPV state would have just as much effect on the national popular vote as a vote gained in a battleground state. Because it would be the national popular vote that would determine how the MPV states cast their electoral votes, they would become 1 We gratefully acknowledge the computer-programming assistance of Thai Pham and Eli Ross and the comments of Paul H. Edelman.

4 significant players even though they are not battleground states. NPV has been adopted by eight states and the District of Columbia, which have the following numbers of electoral votes: CA 55; IL 20; NJ 14; WA 12; MA 11; MD 10; HI 4; VT 33; DC 3. Unsurprisingly, none of them are battleground states. As well, they all voted Democratic in 2012. The primary source on NPV, besides the aforementioned websites, is Koza et al. (2011); Bennett (2001) and Amar and Amar (2001) were the first to propose some form of NPV. Recent critiques of NPV include Miller (2012) and Williams (2011, 2012), who questions, among other things, the constitutionality of interstate compacts (for recent exchanges on NPV, see Brams et al., 2012, and Second MIT Presidential Election Conference, 2012). Because MPV does not require an interstate compact, we present it as a procedure that any state could adopt to determine how to cast its electoral votes in a presidential election. Today, all but two states (ME and NE) cast all their electoral votes for the popular-vote winner in their states. In this paper, we analyze the voting power of individual voters in states, not the voting power of states in two-candidate presidential elections. True, it is the members of the Electoral College, whose behavior is governed by state law, who determine the winner. But it is the voters in each state who determine who the winner is in their states and in congressional districts, as well, in ME and NE and who, ultimately, wins in the Electoral College. We measure a voter s voting power using the well-known relative, or normalized, Banzhaf index of voting power, which measures the ability of a voter, by changing his or her vote, to change the outcome in his or her state, and for that state, in turn, to change

5 the outcome in the Electoral College. We define this index in section 2, where we describe what it measures. In section 3 we apply this index to two simple 3-state examples first assuming winner-take-all, and then MPV for one or more of the states asking who benefits, and by how much. As we show, the size of a state matters, but its voters are invariably hurt if the state adopts MPV. In section 4 we calculate the voting power of voters in the nine battleground states in 2012, assuming they possess all the voting power and the 41 nonbattleground states possess none. Generally speaking, we find that the voting power of an individual voter increases with the size of a state, echoing earlier studies. If one of the battleground states switches to MPV, the voting power of its voters would decrease, giving no battleground state reason to switch to MPV. The story is very different for nonbattleground states, because their voters begin with no voting power, so they can only go up in voting power if their state adopts MPV. But by how much? If the election is close, voters in the smallest nonbattleground states (3 electoral votes) would have less than 1/5 of the voting power they would have if their state were a battleground state, but voters in the largest nonbattleground state (CA, with 55 electoral votes), would have almost 1/2 as much. In section 5, we draw two main conclusions: 1. Nonbattleground states have good reason to switch to MPV to avoid being taken for granted by the candidates and, after the election, by the winner but in doing so they risk supporting a candidate who was rejected by a majority of their own voters.

6 2. Once states that switch to MPV have a majority of electoral votes, they implement direct popular vote without an interstate compact or a constitutional amendment to abolish the Electoral College. Should MPV render the Electoral College nugatory in this way, its constitutionality will doubtless be decided by the Supreme Court. 2. The Banzhaf Index of Voting Power John Banzhaf proposed an index of voting power in Banzhaf (1965), which he later applied to the Electoral College (Banzhaf, 1968). This index measures the ability of a voter, by changing his or her vote, to change the outcome in his or her state and, in turn, for that state to change the outcome in the Electoral College. 2 To describe the Banzhaf index, define a coalition to be a subset of voters. A coalition is winning if it provides a candidate with enough votes to win, and losing otherwise (for now, we ignore the possibility of ties). Of course, whether a coalition is winning or losing in a presidential election depends on the exact rules under which the election is conducted. Define a winning coalition to be vulnerable if, among its members, there is at least one member whose defection (i.e., exit from the coalition) would cause the coalition to be losing. Call such a member critical. If there is only one critical voter, that voter is uniquely powerful in the winning coalition. 2 The Banzhaf index, and a related index due to Johnston (1978), have been applied to members of other federal institutions, including the President, Representatives, and Senators (Brams, Affuso, and Kilgour, 1989). Different voting-power indices, such as the Shapley-Shubik (1954) index, are analyzed and compared in Felsenthal and Machover (1998). These indices have been applied to many voting bodies, as well as other institutions connected by a set of rules, including, for example, the European Union Council of Ministers and the European Parliament (Cichocki and Zyczkwoski, 2010).

7 Denote the set of all vulnerable coalitions by V. For each coalition c V and each voter i, let v i (c) = 1 if i is critical in c; v i (c) = 0 if i is not critical in c. (Note that voter i is not critical in c if either i does not belong to c, or if i does belong to c but c remains winning even after i leaves c.) The Banzhaf power of voter i is, where n is the total number of voters. Thus, B(i) is the number of critical defections of player i divided by the total number critical defections of all voters, or voter i s proportion of critical defections 3 Equally, B(i) can be interpreted as the probability that a randomly chosen critical defection is cast by voter i when all critical defections are assumed to be equiprobable. The Banzhaf power of voter i measures the relative degree to which he or she can influence the outcome in a presidential election via the outcomes in i s state and, ultimately, in the Electoral College. Battleground states are states that are considered competitive, whereas nonbattleground states are certain to support a one candidate or the other. Thus, voters in 3 The absolute Banzhaf index of voter i is the total number of its critical defections the numerator of the right side of B(i) divided by the total number of winning coalitions, or 2 n 1 (exactly half of the 2 n coalitions are winning if there are no ties). It gives the probability that voter i will be critical, assuming that all winning coalitions are equally likely. The relative Banzhaf index normalizes the absolute Banzhaf index, so the voting power of all voters sums to 1. We use the relative index here, because it better highlights how voting power depends on the size of a state and whether or not it is MPV.

8 nonbattleground states can never be critical and, hence, have no power. Consequently, the Banzhaf index is applicable only to voters in battleground states unless nonbattleground states adopt MPV, as we will next show. 3. Two Examples To illustrate the calculation of the Banzhaf index, assume there are three states, (A, B, C), with (3, 2, 2) electoral votes. The set of voters in state A are {a 1, a 2, a 3 }, the set in state B are {b 1, b 2 }, and the set in state C are {c 1, c 2 }. We assume that (i) each of the 7 voters votes, (ii) the electoral votes of each state equal its number of voters, and (iii) a simple majority of at least 4 out of 7 electoral votes is sufficient to win. All States Winner-Take-All We begin by calculating the voting power of a voter in each state when all states are winner-take-all (WTA), which we call all-wta. Under all-wta, a candidate receives all the electoral votes of a state if and only if a majority of voters in that state support that candidate, which in this example means at least 2 voters in each of states A, B, and C. A winning coalition of at least 4 voters comprises (i) either 2 or 3 a s and 2 b s, (ii) either 2 or 3 a s and 2 c s, or (iii) 2 b s and 2 c s. Recall that an a, b, or c voter is critical in a winning coalition if, by defecting from that coalition, he or she causes it to be losing (i.e., to receive fewer than 4 electoral votes). Under all-wta, there are 10 distinct forms of vulnerable coalition, which are shown in the 1 st column of Table 1. Note that the forms differ according to the numbers of a, b, and c voters. There may be more than one vulnerable coalition for each form. For example, there are two coalitions of form 2, aaabbc, namely {a 1, a 2, a 3, b 1, b 2, c 1 } and

9 {a 1, a 2, a 3, b 1, b 2, c 2 }. Any coalition of form 2 is vulnerable because if one of the b s defects, the coalition will produce only 3 electoral votes from the 3 a s (actually, 2 a s would suffice). But if one of the 3 a s defects, or if the one c defects, the coalition will remain winning, because it would still have 5 electoral votes. By contrast, the form aabbcc does not appear in Table 1, because any coalition of this form is winning but is not vulnerable: If any one of its members defects, it would still have at least 4 electoral votes from the five nondefectors. Table 1 about here In Table 1, we indicate, for each of the 10 forms of vulnerable coalition, the number of distinct coalitions of that form (2 nd column), and who the critical members are (3 rd column). In the next three columns, we indicate the sum, over all instances of each given form, of the number of critical defections (CDs) of each of the a s, each of the b s, and each of the c s. To illustrate the calculation of CDs, consider form 6, aabbc. There are 3 ways of choosing two a s from the three available, and 2 ways of choosing one c from the two available (and there is only 1 way of choosing two b s), which makes for a total of 3 2 = 6 different instances of the form aabbc. Each b will belong to all 6, because no b is ever excluded, whereas each a will belong to 4 of the 6, because each way of choosing two of the three a s excludes one of them. For the 6 distinct instances of a form 6 vulnerable coalition, each a has a total 4 CDs, and each b has a total of 6 CDs (c is never critical). Thus, in each instance of a form 6 vulnerable coalition, 4 members (2 a s and 2 b s) are critical.

10 In the last row of Table 1, we give the total numbers of vulnerable coalitions ( = 28) and CDs of each a (12), each b (16), and each c (also 16). To find the Banzhaf power of each player, we calculate the total number of critical defections by summing the CDs of all 7 players, which yields = 3(12) + 2(16) + 2(16) = 100. Therefore, the Banzhaf power values are B(a) = 12/100 = 3/25 = 0.12; B(b) = B(c) = 16/100 = 4/25 = 0.16. Thus, each of the a s has less power than each of the b s and c s, indicating that a voter in the large state (A) is disadvantaged under WTA. This will not always be the case, as we will see later in the Electoral College. Large State (A) MPV; Two Small States (B and C) WTA We next look at the case in which the large state, A, is MPV, whereas states B and C remain WTA. As Table 2 shows, there are now 11 forms of vulnerable coalition, with most duplicating those under all-wta. But forms 9 and 10 are new, and form 9 in Table 1, abbcc, is absent from Table 2. Table 2 about here Why is a coalition of form 10, abcc, included in Table 2, but not in Table 1? Its 2 c voters (from WTA state C) contribute 2 votes, but the 1 b voter from WTA state B is not sufficient to deliver any votes from state B. However, because A is an MPV state, and abcc has a total of 4 votes (i.e., a majority), A will deliver 3 votes to abcc, producing a total of 3 + 2 = 5 electoral votes, all from states A and C.

11 The defection of any of the 4 members of abcc will render it losing, so all its members are critical. On the other hand, if A, along with B and C, were WTA states (Table 1), abcc would not be a winning coalition, because it would command only 2 electoral votes (from state C). Observe that we do not include in Table 2 abbcc (form 9 in Table 1), because the defection of any of its members still leaves the coalition winning when A is MPV and B and C are WTA. Summing the CDs of the 7 voters yields = 3(8) + 2(22) + 2(22) = 112, giving Banzhaf powers of B(a) = 8/112 = 1/14 = 0.071; B(b) = B(c) = 22/112 = 11/56 = 0.196. Again, each of the a s has less power than each of the b s and c s, so voters in the large state are again disadvantaged. But this time state A hurts its voters even more by becoming MVP (0.083 vs. 0.12 in Table 1). One Small State (C) MPV; Two Other States (A and B) WTA Finally, we look at the case in which one of the two small states, C (it could as well be B), is MPV, whereas states A and B are WTA. As Table 3 shows, there are now 9 forms of vulnerable coalitions, and = 38 vulnerable coalitions altogether. Even though there are fewer forms than in Table 1 (10 forms) and Table 11 (11 forms), not all forms in Table 3 duplicate those in Tables 1 and 2. For example, form 1 in Table 3, aaab, is not a vulnerable coalition in Tables 1 and 2, but it is vulnerable in Table 3, because if any of its members defected, it would go from 5 electoral votes (3 from A and 2 from C) to 3. Table 3 about here

12 There are a total of = 3(22) + 2(20) + 2(14) = 134 critical defections, giving Banzhaf powers of B(a) = 22/134 = 11/67 = 0.164; B(b) = 20/134 = 10/67 = 0.149; B(c) = 14/134 = 7/ 67 = 0.104. This time the Banzhaf index gives more power to each of the a s than to each of the b s and c s, so now voters in the large state (A) are advantaged. But, as earlier, voters in the MPV state (C) do the worst: Compared with voters in the WTA state of the same size (B), they have Banzhaf power of 0.104 in C (vs. 0.149 in B). We need not examine individually the cases in which two of the three states, or all three, adopt MPV, because these states would then have a majority of electoral votes. For reasons given earlier, this situation is tantamount to direct popular vote, so voters in all three states would have equal Banzhaf voting power of 1/7 = 0.143. Summary for (3, 2, 2) Example In Table 4 we give the Banzhaf values for the three scenarios we have discussed. Note that voters in a state never gain when the state switches from WTA to MPV: A voters would drop by 69 percent (from 0.12 to 0.071), and B and C voters by 54 percent (from 0.16 to 0.104), so the bigger the state, the bigger the drop. Furthermore, if one state switches to MPV in hopes of making it attractive for a second state also to switch, it will not succeed, because voters in the two remaining WTA states do better sticking with WTA than switching to MPV and giving all voters the same voting power of 0.143. Table 4 about here

13 (7, 5, 3) Example Consider the case in which states (A, B, C) have (7, 5, 3) voters and electoral votes, respectively. A simple majority of 8 out of 15 electoral votes is sufficient to win and, if a state is WTA, a majority within that state is decisive. In Table 5 we give the Banzhaf powers of voters under all-wta and when each of the three states now all different in size adopts MPV, comparing it with the situation in which at least two states adopt MPV and thereby implement direct popular vote. Table 5 about here As in the (3, 2, 2) example, a state that adopts MPV alone always hurts its voters by 22 percent in the case of A (7 votes), by 40 percent in the case of B (5 votes), and by 67 percent in the case of C (3 votes). Unlike the (3, 2, 2) example, the smaller the state, the bigger the drop. For the two remaining WTA states, it is always advantageous to be a voter in the smaller WTA state, whereas this was not always the case in the (3, 2, 2) example: Voters in the 3-vote WTA state do better (0.164) than voters in the 2-vote WTA state (0.149) when the other 2-vote state (0.104) is MPV. Thus, the size of the state has different effects in the two examples, illustrating that in small examples, large size may either help or hurt voters in both WTA and MPV states. In the case of the Electoral College, as we will show shortly, the size of a state has more consistent effects, benefiting voters who live in large states. The two small examples agree that if one state adopts MPV, it (i) hurts its voters and (ii) never attracts any other state to follow suit and give all voters equal power. In a presidential election, however, the situation is very different for the nonbattleground

14 states, whose voters always benefit but in varying degrees, depending on the size of their states if their state adopts MPV, even if no other state does so. 4. The Power of Battleground and Nonbattleground States In the 2012 presidential election, there was a consensus among campaign analysts, pollsters, and political pundits that nine states were up for grabs. These were the socalled battleground states, to which both candidates allocated almost all their campaign resources in the final weeks of the campaign. These states were all WTA, and all except one (NC) voted for Barack Obama. 4 In calculating the Banzhaf power of voters in these states, we assume that the number of their voters equals the number of their electoral votes, which is an approximation. This approximation is dictated by the fact that the number of winning coalitions increases exponentially with the number of voters. Counting the number of winning coalitions that are vulnerable, and which voters are critical in each, is computationally infeasible without drastically reducing the numbers of voters from tens of millions to much smaller numbers. As points of comparison, in the (3, 2, 2) example, there are 7 voters and 2 7 coalitions, because each of the 7 voters is either in or not in a coalition. Exactly half the coalitions, or 2 6 = 64, are winning because, with an odd number of voters, there can be no ties. Similarly, in the (3, 5, 7) example, there are 15 voters and 2 14 = 16,364 winning 4 The fact that Obama won in almost all the battleground states does not invalidate the perception before the election that the battleground states had the tightest races and could swing the election. Indeed, the perception of tightness turned out to be the reality: These states had the nine smallest margins of victory for the presidential candidates, ranging from 0.88 percent in FL to 6.94 percent in WI.

15 coalitions. But this number is dwarfed by the number of winning coalitions in our next examples, necessitating a computer-based calculation of Banzhaf voting power. 5 The Nine Battleground States The nine battleground states (see Table 6) have a total of 110 electoral votes (and, in our model, voters). Assume, initially, that a candidate needs a simple majority of at least 56 of these electoral votes to win; a 55-55 tie is neither winning nor losing. Later we modify this assumption to reflect the fact that the nonbattleground states did not split evenly in their electoral votes, substantially favoring Barack Obama over Mitt Romney. Table 6 about here The Banzhaf values of voters in the nine battleground states, given the decision rule is 56, are shown as nonitalicized figures in Table 6. In the first row, we assume that the states are all WTA; in the next nine rows we make one state starting with FL and ending with NH the sole MVP state. The column headings give the number of electoral votes of each battleground state. First consider the Banzhaf values in the all-wta case, which are shown in the first row and are shaded. 6 They go from a high of 0.01402 for FL to a low of 0.00405 for NH, indicating that a voter in FL is 3.46 times more powerful than a voter in NH, with the Banzhaf values decreasing monotonically with the size of a state. 5 To simplify the calculations, a generating function for the Banzhaf index, given in Brams and Affuso (1976) and coded in Bilbao (2000), was considered, but it is not directly applicable, because the players are the voters, rather than states, in the weighted-voting game that we analyze. 6 Ignore for now the Banzhaf values in the first row (and along the diagonal) that are in italics and also shaded. These are based on the assumption that Obama and Romney did not split evenly the electoral votes in the nonbattleground states and will be discussed later.

16 This ratio is remarkably close to the 3.31 ratio that Banzhaf (1968), making an approximate calculation, found for the power of a NY voter (43 electoral votes) to that of Washington, DC, voter (3 electoral votes), based on the 1960 census when NY was the largest state. After CA became the largest state in the 1970 census, Owen (1975), using a different approximation, found that the ratio of the Banzhaf power of a CA voter (45 electoral votes) to a Washington, DC, voter was 3.18. Clearly, our ratio of the power of a FL voter to that of a NH voter for the battleground states (3.46) closely approximates the ratios giving voters in the largest states more than a 3:1 power advantage over voters in the smallest states found by earlier researchers for all 50 states. 7 Next compare the power of a voter when a state is WTA and when it is MPV. If FL, and only FL, were to become an MPV state, the power of its voters would go from 0.01402 to 0.00708, a decrease of 50 percent of its WTA value. For NH, the decrease is even greater, going from 0.00405 to 0.00120, so its voters would retain only 30 percent of the power they had when NH was WTA. The other battleground states all drop to less than half of their WTA values, confirming that it would be foolish for any battleground state to give up its status as a WTA state if it wants to maximize the power of its voters in a presidential election. In the 2012 election, Barack Obama started with an electoral-vote advantage over Mitt Romney in the nonbattleground states of 237 to 191 electoral votes. To win with 270 electoral votes, Obama needed only 33 of the 110 electoral votes from the battleground states, whereas Romney needed 79 electoral votes. The italicized figures in 7 Unlike Banzhaf s and Owen s calculations, which are approximate, ours are exact for the nine battleground states, but our calculations are based on the assumption that the number of voters is equal to the number of electoral votes in a state, which makes our later MPV calculations feasible.

17 Table 6 give the Banzhaf power values of voters in the battleground states for this more realistic decision rule. 8 As earlier, we focus on the shaded figures for all-wta (first row) and for the single MTV state (along the diagonal) in Table 6. For all-wta, they show that the Banzhaf powers of voters generally decrease the smaller the state, but not monotonically: WI has one more electoral vote than CO (10 vs. 9), but the power of a voter in CO is slightly greater than a voter s power in WI (0.00659 vs. 0.00634). Overall, however, the Banzhaf values for the decision rule of 33 (Obama) and 79 (Romney) electoral votes are quite similar to their values when both candidates need a simple majority of 56 electoral votes to win. As shown in italics along the diagonal, when the decision rule for Obama and Romney to win is different, the power of a voter in an MPV state does decrease monotonically with the size of the state, with a voter in FL having almost eight times as much power as a voter in NH when each state is MPV (0.00442 vs. 0.00058). The MPV values, however, are considerably lower sometimes by almost 50 percent for the decision rule of 33 (Obama) and 79 (Romney) than when both candidates need 56 electoral votes to win in the battleground states. Evidently, voters in an MPV state benefit when a race is tied or very close in the nonbattleground states, because then they can more easily influence the outcome through the national popular vote. Benefit to Voters in a Nonbattleground State If It Becomes MPV 8 This decision rule gives Obama many more opportunities than Romney to win. Because the Banzhaf index counts only vulnerable winning coalitions, however, it does not take into account the fact that many more of Obama s winning coalitions, compared to Romney s, are invulnerable. Rather, the power of voters rests on their ability to render the vulnerable winning coalitions of either Obama or Romney losing.

18 So far we have shown that voters in battleground states lose power, usually by 50 percent or more, if their state switches to MPV. On the other hand, voters in nonbattleground states can only go up (from zero voting power) by being able to affect the national popular vote. But by how much? It turns out that voters in the largest battleground states, CA (55 electoral votes) for the Democrats and TX (38 electoral votes) for the Republicans, have considerably more power than voters in the smallest Democratic and Republican states, VT and WY (each with 3 electoral votes) when these states switch to MPV. In the 2012 election, we illustrate how this calculation is made for the two largest states, and then give the results for both the largest and the smallest states. If CA switched to MPV, Obama would have 237 55 = 182 electoral votes from the nonbattleground states, so he would need 270 182 = 88 of the 110 + 55 = 165 electoral votes from the nine battleground states (all WTA) and CA (now MPV) to win. If TX switched to MPV, Romney would have 191 38 = 153 electoral votes from the nonbattleground states, so he would need 270 153 = 117 of the 110 + 38 = 148 electoral votes from the nine battleground states (all WTA) and TX (now NPV) to win. An analogous calculation can be made for the two smallest states. In Table 7, we show the Banzhaf power of voters in the two largest and two smallest Democratic and Republican states under MPV, and how it would change if each were the tenth battleground state under WTA. Manifestly, voters have substantially less power if their states are MPV rather than WTA battleground states. Still, having from 15.2 percent (WY) to 48.0 percent (CA) of the power of a WTA battleground state is better than having no power at all (as nonbattleground states now do).

19 Table 7 about here If one lives in a nonbattleground state that switches to MPV, it is much better to live in a large state than a small one: A Democratic voter has 8.5 (0.0052278/0.0006130) times more voting power in CA than VT, and a Republican voter has 4.6 times (0.0017456/0.0003768) more voting power in TX than WY. Were these states battleground WTA states rather than nonbattleground MPV states, the comparable ratios are both 3.08 (0.0108857/0.0035372 and 0.0076531/0.0024831), mirroring the 3:1 bias in favor of large states that we discussed earlier. Whether a voter in a tenth nonbattleground MPV state can swing the national popular vote in favor of his or her preferred candidate will depend on (i) the split in electoral votes of the other nonbattleground states and (ii) the split of the popular votes in all states. The 2012 election was not close on either count, so even a voter in a large nonbattleground MPV state had no chance of being critical. Our analysis is applicable even when the split of the electoral votes in the nonbattleground states is not 50-50 as long as the split in the popular vote is approximately 50-50. To illustrate, if voters in the battleground states tend slightly to favor one candidate (say, the Democrat, as was true in 2012), and the nonbattleground states favor the Republican by a more-or-less-equal margin (which was not true in 2012), then a 50-50 overall split in popular and electoral votes is likely. Thereby voters in MTV nonbattleground states may be critical, even though they are not critical in their individual states. There was almost a 50-50 split in popular votes in the 1960 presidential election (John Kennedy defeated Richard Nixon by a margin of 0.17 percent). In the 2000

20 presidential election, there was almost a 50-50 split in electoral votes; in fact, if a 4- electoral vote state that voted for George Bush not just Florida, though that was the closest state had instead voted for Al Gore, Gore would have won (he did win the popular vote by 0.51 percent). While the combination of very close splits in both electoral votes and popular votes is rare, these are the conditions most likely to enhance the power of voters in nonbattleground MPV states. Of course, any model that assumes a close enough split in electoral and popular votes that a voter s vote can make a difference is an idealization, which has an infinitesimal chance of happening in practice. But insofar as the assumptions of our model approximate voting in close elections, the model gives insight into how a voter s voting power depends on the size of his or her state and whether it is WTA or MPV. Besides calculations of voting power, a resource-allocation model of presidential campaigning, developed by Brams and Davis (1973, 1974), reinforces some of our findings. This model posits a 2-person, zero-sum game between the two major-party candidates. To maximize their expected electoral vote under WTA, the candidates should expend resources in battleground states according to the 3/2 s power of the electoral votes of states, which makes voters in the largest states about three times more attractive campaign targets than voters in small states. Although this strategy is only a local equilibrium when candidates expend the same resources in each of the battleground states (without this assumption, the candidates optimal strategies are mixed, which are difficult to interpret), it is a realistic insofar as candidates share the same perception of which states are the battleground

21 states. Indeed, data on the campaign expenditures of candidates in different presidential campaigns provide empirical support for the 3/2 s rule (Brams, 1978, ch. 3). If the candidates are assumed to maximize their expected probability of winning a majority of electoral votes (instead of their expected electoral vote), Lake (1979) showed that the local equilibrium is to spend their resources in battleground states according to the Banzhaf index, which also favors voters in large states by a ratio of about 3:1. It is remarkable that the cooperative n-person game-theoretic model on which the Banzhaf index is based, and the noncooperative 2-person game-theoretic model of Brams and Davis and of Lake, give essentially the same result for the attractiveness of voters in different-sized battleground states. 5. Conclusions The eight states and the District of Columbia that have adopted NPV will not benefit until enough other states have joined the interstate compact to implement direct popular vote. By contrast, under MPV, the citizens of nonbattleground states will be empowered immediately in close elections both before an election by the attention they receive, and after an election if their votes help the elected candidate to win which should encourage these states to adopt this election reform, which requires no interstate compact. The last state to enact NPV, CA, did so on August 8, 2011, some 18 months ago. Since then, NPV seems to have lost its early momentum. We believe the idea underlying NPV can be reinvigorated by a switch to MPV, because it benefits the citizens of each

22 state that enacts it independent of what other states do and may well create a bandwagon in favor of passage that expedites direct popular-vote election of a president. 9 To be sure, the dark side of MPV is that a state may be forced to vote against the wishes of a majority of its citizens if this majority differs from the national majority. When this happens, the injunction to do the right thing vote as the nation does may ring hollow. But once states with a majority of electoral votes adopt MPV, all voters will have equal sway, which seems difficult to oppose even if not all states sign onto MPV. Moreover, those states that do not can still express their opposition by casting their electoral votes against the national popular vote winner. Until MPV states have a majority of electoral votes, voters in battleground states always do better as WTA states, and by a factor of at least two. Thus, it is likely that the battleground states will try to preserve their privileged status by resisting MPV. But today these states represent only about 20 percent of the 538 electoral votes. Should MPV be adopted by enough states to implement direct popular vote and perhaps earlier its constitutionality will undoubtedly be challenged. The Supreme Court will be the final arbiter. But should the Court rule against MPV, this may well give impetus to a constitutional amendment to abolish the Electoral College, which the Court cannot overrule. Anticipating such an amendment, the Court might think twice before ruling against MPV. 9 As more states enact MPV short of having a majority of electoral votes it becomes more likely that the national popular-vote winner will be the electoral-vote winner, as Bennett (2006, pp. 166-170) argues.

23 Table 1 All-WTA: Number of Critical Defections (CDs) of Each of 3 a s, 2b s, and 2 c s Form of Vulnerable Coalition Number Critical Members CDs of each a CDs of each b CDs of each c 1. aaabb 1 2 b s 0 1 0 2. aaabbc 2 2 b s 0 2 0 3. aaabcc 2 2 b s 0 0 2 4. aaacc 1 2 c s 0 0 1 5. aabb 3 2 a s, 2 b s 2 3 0 6. aabbc 6 2 a s, 2 b s 4 6 0 7. aabcc 6 2 a s, 2 b s 4 0 6 8. aacc 3 2 a s, 2 c s 2 0 3 9. abbcc 3 2 a s, 2 c s 0 3 3 10. bbcc 1 2 a s, 2 c s 0 1 1 Total 28 12 16 16

24 Table 2 A - MPV; B & C - WTA: Number of Critical Defections (CDs) of Each of 3 a s, 2 b s, and 2 c s Form of Vulnerable Coalition Number Critical Members CDs of each a CDs of each b CDs of each c 1. aaabb 1 2 b s 0 1 0 2. aaabbc 2 2 b s 0 2 0 3. aaabcc 2 2 c s 0 0 2 4. aaacc 1 2 c s 0 0 1 5. aabb 3 2 a s, 2 b s 2 3 0 6. aabbc 6 2 b s 0 6 0 7. aabcc 6 2 c s 0 0 6 8. aacc 3 2 a s, 2 c s 2 0 3 9. abbc 6 a, 2 b s, c 2 6 3 10. abcc 6 a, b, 2 c s 2 3 6 11. bbcc 1 2 b s, 2 c s 0 1 1 Total 37 8 22 22

25 Table 3 A & B WTA; C MPV: Number of Critical Defections (CDs) of Each of 3 a s, 2 b s, and 2 c s Form of Vulnerable Coalition Number Critical Members CDs of each a CDs of each b CDs of each c 1. aaab 2 3 a s, b 2 1 0 2. aaac 2 3 a s, c 2 0 1 3. aabb 3 2 a s, 2 b s 2 3 0 4. aabc 12 2 a s, b, c 8 6 6 5. aabcc 6 2 a s 4 0 0 6. aacc 3 2 a s, 2 c s 2 0 3 7. abbc 6 a, 2 b s, c 2 6 3 8. abbcc 3 2 b s 0 3 0 9. bbcc 1 2 b s, 2 c s 0 1 1 Total 38 22 20 14

26 Table 4 Banzhaf Powers of Voters in States (A, B, C) with (3, 2, 2) Electoral Votes for Four Scenarios under WTA and MPV Scenario for States A - B - C A B C 1. WTA WTA - WTA 0.12 0.16 0.16 2. MPV WTA - WTA 0.071 0.196 0.196 3. WTA WTA - MPV 0.164 0.149 0.104 4. At Least Two States MPV 0.143 0.143 0.143

27 Table 5 Banzhaf Powers of Voters in States (A, B, C) with (7, 5, 3) Electoral Votes for Five Scenarios under WTA and MPV Scenario for States A - B - C A B C 1. WTA WTA - WTA 0.0562 0.0674 0.0899 2. MPV WTA - WTA 0.0460 0.0824 0.0886 3. WTA MPV - WTA 0.0741 0.0483 0.0800 4. WTA WTA - MPV 0.0696 0.0703 0.0538 5. At Least Two States MPV 0.0667 0.0667 0.0667

28 Table 6 Banzhaf Powers of Voters in Nine Battleground States, Going from All-WTA ( None ) to Each State Being a Single MPV State* MPV State FL (29) OH (18) NC (15) VA (13) WI (10) CO (9) IA (6) NV (6) NH (4) None.01402.01327 FL.00708.00442 OH.01292.01496 NC.01315.01486 VA.01310.01427 WI.01323.01374 CO.01329.01378 IA.01349.01347 NV.01349.01347 NH.01346.01306.00926.00974.01053.01284.00445.00291.01018.00986.01009.01024.00976.00991.00962.00979.00953.00968.00953.00968.00949.00981.00835.00911.01026.01234.00963,00960.00377.00260.00901.00894.00894.00943.00892.00934.00861.00939.00861.00939.00875.00928.00810.00817.01001.01120,00952.00926.00894.00819.00347.00227.00879.00848.00872.00845.00857.00812.00857.00812.00832.00818.00659.00634.00954.00951.00852.00764.00817.00738.00800.00731.00272.00157.00769.00650.00711.00678.00711.00678.00698.00656.00627.00659.00934.00977.00819.00790.00797.00771.00779.00770.00754.00691.00236.00151.00710.00707.00710.00707.00677.00722.00483.00502.00888.00832.00749.00653.00689.00678.00688.00620.00622.00620.00617.00619.00168.00094.00527.00581.00534.00553.00483.00502.00888.00832.00749.00653.00689.00678.00688.00620.00622.00620.00617.00619.00527.00581.00168.00094.00534.00553.00405.00353.00849.00684.00707.00590.00661.00565.00636.00533.00568.00475.00557.00530.00498.00420.00498.00420.00120.00058 *The figures that are not italicized assume that both candidates need 56 of the 110 electoral votes to win, whereas those that are italicized assume that Obama needs 33, and Romney 79, electoral votes to win, reflecting the breakdown of the nonbattleground states that favored Obama. The shaded figures are for all-wta (first row) and when each state is the single MPV state (diagonal).

29 Table 7 Banzhaf Powers under MPV and WTA, and Their Ratios, in Largest and Smallest Democratic and Republican States in the 2012 Presidential Election State MPV WTA MPV/WTA CA (55) Dem. 0.0052278 0.0108857 0.480 TX (38) Rep. 0.0017456 0.0076531 0.228 VT (3) Dem. 0.0006130 0.0035372 0.173 WY (3) Rep. 0.0003768 0.0024831 0.152

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