This commodity reviews some of the basal institutional elements and empiric regularities of the U.S. balloter academy system. Historical acclamation after-effects affirm the accepted ascendancy of the winner-take-all allocation of electors aural anniversary state. Of greater absorption is the arrangement of accompaniment preferences for the balloter academy accustomed a civic accepted vote as the advised alternative.
It is all-important to be bright about what is meant by a state’s preference. The abstraction explored actuality is sometimes referred to as a state’s a priori preference. A accompaniment prefers a priori the arrangement that after-effects in the greater abeyant access of its voters, as represented by the state’s popular-vote majority, on the aftereffect of an election. In this ambience the appellation “a priori” agency in application of alone the arrangement itself, afterwards (a posteriori) considerations of absolute or acceptable outcomes in accurate elections (see Felsenthal and Machover 2004.) In addition, as discussed below, a accompaniment can be said to adopt an academy alone in application of and about to a specific another institution.
The assay actuality demonstrates (contrary to the absolute bookish literature) that able-bodied over bisected of the states — in accurate the abate citizenry states — adopt the balloter college.1 The accord amid accompaniment citizenry and balloter academy another derives from the axiological methods — some declared in the Constitution, others in century-old laws — for free the cardinal of aldermanic assembly (and appropriately the cardinal of balloter votes) assigned to anniversary state. In addition, the afflicted accord amid accompaniment admeasurement and balloter academy preference, as able-bodied as the majority-of-states another for the system, accept existed throughout the absolute history of U.S. presidential elections. Those characteristics are assertive to abide for the accountable future.
The acclamation of the U.S. admiral is bent by a simple majority vote of the balloter college, the associates of which are appointed by the states.2 The Constitution directs that the cardinal of electors for anniversary accompaniment shall be “equal to the able Cardinal of Senators and Assembly to which the Accompaniment may be advantaged in the Congress.” The Constitution additionally directs that the methods for appointing electors and for administering how those electors casting their votes are to be bent by accompaniment legislatures.3 At first, electors were called anon by accompaniment legislatures, but the alteration to some anatomy of accepted vote to adjudge the affair of electors proceeded steadily. By 1836 about all states had afflicted from accompaniment assembly another of electors to some anatomy of accepted vote another (see Ross 2012: chap. 2; Miller 2012).
The aboriginal balloter college, for the acclamation of 1788, consisted of 69 electors aloft the 10 states that by again had ratified the Constitution. The balloter academy added in admeasurement monotonically, in accordance with the access in the cardinal of senators and representatives, extensive its accepted admeasurement of 538 with the acclamation of 1964. The cardinal of senators has been at 100 back the acceptance of Hawaii and Alaska to the Union in 1959. The cardinal of assembly has been capped at 435 back the Agreement Act of 1911.4 The Commune of Columbia has neither senators nor representatives, but was accepted three balloter votes in accordance with Alteration XXIII to the Constitution, ratified in 1961. Hence, the accepted absolute of 538 balloter votes.
Because every accompaniment has two senators and at atomic one representative, every accompaniment has at atomic three balloter votes. As of 2016 anniversary of the seven aboriginal states, as able-bodied as the Commune of Columbia, has three balloter votes; the better state, California, has 55. The average is eight balloter votes. Unless a new accompaniment is added or Congress revises the cardinal of assembly (something it has not done in over 100 years), the accepted absolute of 538 balloter votes will prevail.
Regardless of who was accomplishing the selecting — accompaniment legislatures or some subset of the citizenry — electors accept been directed, with few exceptions, to casting their state’s balloter votes for admiral unanimously: champ booty all. From 1788 through 2016, there accept been 58 presidential elections and 2,238 accompaniment balloter vote tabulations for the appointment of admiral of the United States; 2,186 of the 2,238 tabulations, 97.7 percent, accept been unanimous. The abridgement of accordance in the 2.3 percent of vote tabulations could accept resulted from an absolute nonunanimous allocation adjustment called by a state, or from a alleged agnostic elector: one who casts a vote in abuse of a state’s voting rules.5
In any case, the ascendancy of a winner-take-all allocation should not be surprising. Once the majority’s another is determined, that majority would apparently appetite to aerate the state’s access on the final outcome, which is able with the winner-take-all allocation of balloter votes.
Nonetheless, Maine (four balloter votes) and Nebraska (five balloter votes) accept autonomous out of champ booty all, in favor of the “congressional district” method. Beneath that adjustment the accepted vote champ in anniversary aldermanic commune receives that district’s balloter vote; the accepted vote champ in the accompaniment receives the two “senatorial” balloter votes. Maine adopted the aldermanic commune adjustment in beforehand of the 1972 acclamation and Nebraska in beforehand of the 1992 election.6
When a accompaniment splits its balloter votes (as both Maine and Nebraska accept done), the access of the state’s majority on the aftereffect of the acclamation is diminished. Thus, whatever the affidavit for those two states to accept adopted the aldermanic commune method, it is not acceptable that their archetype will be followed by any cogent cardinal of added states. In sum, the attainable account of accordance explains why so few states accept anytime called the aldermanic commune or any added nonunanimous adjustment for the allocation of their balloter votes.
Let ei represent the balloter votes in accompaniment i, which are all allocated to the champ of the accepted vote in the accompaniment (with few exceptions, as noted). Thus, anniversary state’s proportional accession to the balloter vote is:
conditional alone on the champ of the accepted vote in the state. I will use Ei as the admeasurement of a state’s balloter vote influence. (Other measures of access are advised below, but as a starting point I focus on shares.)
Now accede the administration of accepted vote access aloft the states. At first, a state’s accepted vote access ability assume to be an bogus construct. Anniversary vote counts the aforementioned against the civic accepted vote behindhand of the voter’s abode of residence. Nonetheless, the accompaniment charcoal the accordant locus because whether a accompaniment prefers the balloter academy can alone be bent by the accompaniment and in application of an alternative. In particular, I assay the best frequently adapted and believable another — namely, whether a accompaniment would adopt to accept the acclamation bent by the balloter academy or by the civic accepted vote. To acknowledgment that catechism it is affected that a accompaniment will adopt the arrangement that maximizes the abeyant access of its voters, as represented by the state’s accepted vote majority, on the aftereffect of an election.
With attention to a state’s accepted vote influence, it will advice to ascertain some basal measures. Start with two political parties, appointed as the D affair and the R party.
For anniversary accompaniment the accepted vote allowance in allotment agreement is the aberration in the vote amid the two parties, about to the absolute votes cast:
it follows that the aberration in the absolute vote aural a accompaniment about to the absolute vote in the country is accustomed by
Thus, every state’s accession to the civic accepted vote allowance is the accepted vote allowance in the accompaniment assorted by the about admeasurement of the state. In agreement of the accepted vote, the accompaniment allowance and accompaniment admeasurement are multiplicative. And not surprisingly, abstracting from the allowance (an a posteriori concept), anniversary state’s allotment of the civic citizenry (βi) is a absolute admeasurement of its abeyant (a priori) access on the acclamation beneath a civic accepted vote.
The balloter vote shares and the citizenry shares for anniversary accompaniment are apparent in Table 1.7 The correlations for the assorted metrics are apparent in Table 2.8 Citizenry shares and balloter shares are about altogether correlated. The added absorbing value, however, is the arrangement of the two shares. That ratio, with the balloter vote allotment as the numerator, can be declared as the balloter academy another or artlessly the “electoral preference”:
An balloter another greater than one agency that the state’s balloter vote allotment is greater than its accepted vote share, and carnality versa. This arrangement offers a simple and absolute indicator of a state’s a priori another for the balloter college. In 2016, 32 states and the Commune of Columbia had an balloter another greater than one. The average balloter another was 1.08. The administration of balloter preferences aloft states is apparent in Table 1 and Figure 1.
The states with an balloter another greater than one are overwhelmingly the abate states. Indeed, balloter another is abundantly a action of accompaniment population. The rank alternation amid citizenry shares and balloter another is 20.988; the two measures aftermath about identical (inverse) rankings. That accord amid balloter another and citizenry share, apparent in Figure 2, after-effects in the adapted abstract form.
The afflicted accord amid citizenry and balloter another derives from the adjustment acclimated to actuate the cardinal of aldermanic assembly per state, which determines the cardinal of electors per state. As already described, anniversary accompaniment has two senators and at atomic one representative, so anniversary accompaniment has at atomic three balloter votes. The actual 385 assembly — and their balloter votes — are allocated amid the actual states based on population, as declared in the aing section.
In accordance with the Constitution, the Abode of Assembly is reapportioned every 10 years (see U.S. Constitution, Commodity I, Sections 2 and 3). The reapportionment occurs with the added abounding aldermanic appellation afterward the advertisement of the decennial census. For example, the agreement of the 112th Congress (2011-12 term) was the aftermost apportioned according to the 2000 census. The 113th Congress (2013-14 term) was the aboriginal apportioned according to the 2010 census. As noticed earlier, the absolute cardinal of assembly has been capped at 435 back 1911. The 385 added assembly (beyond the minimum of one per state) are apportioned by accompaniment population, according to the agenda aloof described.
Several altered appointment methods accept been acclimated back the aboriginal demography in 1790. The accepted adjustment has been in abode back 1941 and is accepted as the “equal accommodation method.” The adjustment appears to accept been adapted by the mathematician Edward Huntington and by Joseph Hill, Chief Statistician of the Bureau of Census. Sometimes referred to as the Huntington-Hill method, it is activated as follows.9
Each added adumbrative (house seat) is accustomed a weight:
Where n represents the cardinal of seats a accompaniment would accept if it acquired a seat. Because all states automatically accept one seat, the aboriginal bench acquired is bench two. The weight for bench two is
the weight for bench three is
and so on. The weights are apparent in Figure 3. The weighting blueprint reflects, absolutely it can be said to generate, the accord amid balloter another and citizenry apparent in Figure 2.
The weights are assorted by the accompaniment populations (Pi) to actualize a alternation of ethics (Hn,i) for bench n, for anniversary accompaniment i:
A acceptable cardinal of the Hn,i values, say 99, are affected for anniversary state: Alabama H(2) through Alabama H(100); Alaska H(2) through Alaska H(100); and so on. (The cardinal of ethics created for anniversary accompaniment needs alone to beat by one the cardinal of seats eventually accepted to the better state.) The constant ethics are again sorted from accomplished to lowest. The aboriginal 385 ethics of that sorted arrangement actuate the added seats accustomed by anniversary state.
The aloft blueprint for allocating those 385 added representatives, forth with the three-vote minimum for anniversary state, generates the aftereffect declared here: baby states will accept aerial balloter shares about to their citizenry shares, while the afflicted holds for ample states.
While the affair is not consistently acutely presented, the accepted accord in the bookish abstract appears to be that the balloter academy arrangement reflects and maintains the inherent advantage of ample states.10 That assay begins with the astute acumen amid voting allotment and voting ability in systems with diff voting shares, such as the balloter college.11
Voting ability derives from the likelihood that a aborigine will be influential, in the faculty that alteration aloof that one vote would change the aftereffect of the election. That likelihood is bent not aloof by a voter’s share, but by the administration of all voting shares. While voting ability will about be awful activated with voting share, the two concepts are not necessarily according and can, beneath assertive circumstances, be absolutely different. As a simple example, if a distinct article controls added than bisected of the voting shares in a simple majority election, again alone that entity’s vote matters. Its voting ability is abstinent as one, and any added entity’s voting ability is abstinent as zero, behindhand of the specific shares.
Two frequently acclimated measures of voting ability are the Banzhaf basis (sometimes referred to as the Penrose-Banzhaf index) and the Shapley-Shubik index. The Banzhaf basis for a aborigine is affected by creating all attainable acceptable coalitions, of any size, and again artful the allotment of those acceptable coalitions in which the aborigine at affair casts an affecting vote — a vote that if afflicted would change the aftereffect of the election. Because any accurate affiliation can accept assorted affecting voters, the Banzhaf basis charge be normalized to sum to one.
The Shapley-Shubik basis for a aborigine is affected by creating all attainable sequences of all of the voters and again artful the allotment of those sequences in which the aborigine at affair casts the chief vote, d that all above-mentioned voters in the arrangement voted in the aforementioned way (that is, as a coalition). Every arrangement will accept alone one chief voter, the one who “tips the scales” back the votes are archival sequentially.12
We now yze these two indices with the citizenry allotment calculations from Table 1. Table 3 displays the correlations amid (1) the balloter vote shares and the two balloter vote ability indices, (2) the citizenry shares and the two citizenry ability indices, and (3) the balloter another ratios acquired from the shares and the two ability indices.13
When it comes to the balloter academy and accompaniment populations, shares and ability indices are awful activated (the everyman accepted alternation is 0.977) and tend to aftermath agnate rankings (the everyman rank alternation is 0.988). Although some mid-size states about-face their balloter another from one ancillary of 1.0 to the other, based on allotment against ability index, the balloter another ethics are additionally awful correlated. The everyman accepted alternation is 0.954; the everyman rank alternation is 0.860.
In addition, the average balloter another is 1.08 based on shares, 1.17 based on the Banzhaf index, and 1.09 based on the Shapley-Shubik index. The cardinal of states (and the Commune of Columbia) with an balloter vote another greater than 1.0 is 33 based on shares, 36 based on the Banzhaf index, and 34 based on the Shapley-Shubik index. Back it comes to barometer balloter academy influence, accepted vote influence, or the balloter another of the states, it makes no absolute aberration whether we use shares or one of the ability indices.
In any case, because balloter voting shares and balloter voting ability both favor the beyond states, the absolute abstract about sees the balloter academy as absorption and advancement the inherent large-state advantage that admeasurement confers.14 That aboveboard ascertainment about the advantage of admeasurement is, however, uninformative about a state’s another for one arrangement over the other.
Recall the catechism that we are attempting to answer: Bold that a accompaniment wants to aerate the a priori access of its voters, as represented by the accepted vote majority, would that accompaniment adopt to accept the acclamation bent by the balloter academy or by the civic accepted vote? The absolute abstract gets the acknowledgment amiss because it takes the greater access of ample states — inherent in both the balloter academy and the civic accepted vote — as dispositive, afterwards authoritative the adapted allegory amid the two systems. That is, the absolute abstract fails to yze the state’s balloter academy access about to the access the accompaniment would accept in a civic accepted vote. The assay actuality measures that about access and thereby identifies which arrangement would be adopted a priori by the voters in any accustomed state.
The accepted administration of populations and balloter votes implies that the majority of states — in particular, the abate ones — prefers the balloter academy system. Has that arrangement consistently prevailed? Computing the aforementioned ethics (as apparent in Table 1 for the 2016 election) over all of the presidential elections is straightforward. To access at populations for an acclamation year, I admit amid decennial demography figures. The 1790 demography is acclimated to admeasurement citizenry shares for 1788. The 2010 demography is acclimated to admeasurement citizenry shares for 2012.15
Measuring the balloter academy shares from appear tabulations and the citizenry shares from the demography abstracts generates the after-effects apparent in Table 4. In alone two of the 58 presidential elections (1876 and 1880) has the average balloter another been lower than one. In all of the added elections the majority of the states have, by this measure, adopted the balloter academy system. In addition, the able abrogating accord amid balloter another and citizenry has consistently prevailed; both the simple and the rank correlations amid balloter another and citizenry accept consistently been decidedly negative. The empiric after-effects announce that the accepted adjustment of agreement (in aftereffect back 1941) did not fundamentally adapt the accord amid accompaniment citizenry and balloter preference; that accord has consistently existed.
It should additionally be acclaimed that there is annihilation assured about the majority of states preferring the balloter college. It is attainable to call a administration of populations such that the majority of states would adopt a accepted vote. Such an aftereffect would occur, for example, if the administration of accompaniment populations were bimodal, with a accumulation of agnate baby citizenry states and a accumulation of agnate ample citizenry states, with the cardinal of baby states beneath than the cardinal of ample states. For example, accede if there were 20 states with baby populations (20 states like, say, Wyoming) such that anniversary was accorded the minimum cardinal of electors, with the actual citizenry and electors disconnected amid the 30 added states (30 states like, say, Florida). That administration of populations and the adumbrated administration of balloter votes could accomplish a majority of states in favor of a accepted vote system.
There is a able acumen for the chain of the balloter college: a majority of states — overwhelmingly the abate ones — adopt it, which has been the case throughout U.S. history. Furthermore, abandoning the balloter academy can alone appear through built-in amendment, which requires the acceptance of not aloof a majority but three-fourths of the states (see U.S. Constitution, Commodity V). Accustomed the anatomy of the balloter academy and the administration of accompaniment populations, such an aftereffect is doubtful in the accountable future.
The administration of aldermanic assembly (and appropriately of electors) was not happenstance, but rather the aftereffect of a bent accommodation amid the states, in application of the advanced aberration in accompaniment populations. The Constitution embodies several such accommodations, so as to anticipate a baby cardinal of ample states from assertive political outcomes. As one of those accommodations, the balloter academy is acceptable to abide durable.
Abbott, D. W., and Levine, J. P. (1991) Amiss Winner: The Coming Debacle in the Balloter College. Westport, Conn.: Praeger.
Banzhaf, J. F. (1965) “Weighted Voting Doesn’t Work.” Rutgers Law Review 19: 317-43.
— — — — — (1968) “One Man, 3.312 Votes: A Mathematical Assay of the Balloter College.” Villanova Law Review 13: 304-32.
Bhattacharyya, G. K., and Johnson, R. A. (1977) Statistical Concepts and Methods. New York: Wiley.
Boylan, T. S. (2008) “A Built-in Defense of the Balloter Academy and the Acclamation of the American President.” The Open Political Science Journal 1: 50-58.
Farrand, M., ed. (1937) The Records of the Federal Convention of 1787. New Haven, Conn.: Yale University Press.
Felsenthal, D. S., and Machover, M. (2004) “A Priori Voting Power: What Is It All About?” Political Studies Review 2: 1-23.
— — — — — , eds. (2012) Balloter Systems: Paradoxes, Assumptions, and Procedures. New York: Springer.
Gelman, A.; Katz, J. N.; and Bafumi, J. (2004) “Standard Voting Ability Indices Do Not Work: An Empiric Analysis.” British Journal of Political Science 34: 657-74.
Gelman, A.; Katz, J. N.; and Tuerlinckx, F. (2002) “The Mathematics and Statistics of Voting Power.” Statistical Science 17: 420-35.
Gelman, A.; Silver, N.; and Edlin, A. (2012) “What Is the Anticipation Your Vote Will Make a Difference?” Economic Inquiry 50: 321-26.
Gibson, C., and Jung, K. (2002) “Historical Demography Statistics on Citizenry Totals by Race, 1790 to 1990, and by Hispanic Origin, 1790 to 1990, for the United States, Regions, Divisions, and States.” Washington: U.S. Demography Bureau.
Gregg, G. L., ed. (2001) Securing Democracy: Why We Accept an Balloter College. Wilmington, Del.: ISI Books.
Hamilton, A.; Madison, J.; and Jay, J. ( 1961) The Federalist Papers. New York: New American Library.
Huntington, E. V. (1921) “The Mathematical Theory of the Agreement of Representatives.” Proceedings of the Civic Academy of Sciences 7: 123-27.
— — — — — (1928) “The Agreement of Assembly in Congress.” Transactions of the American Mathematical Society 30: 85-110.
Katz, J. N.; Gelman, A.; and King, G. (2004) “Empirically Evaluating the Balloter College.” In A. N. Crigler et al. (eds.) Rethinking the Vote: The Politics and Prospects of American Acclamation Reform. Oxford: Oxford University Press.
Kimberling, W. C. (1992) “The Balloter College.” Federal Acclamation Commission.
Kuroda, T. (1994) The Origins of the Twelfth Amendment: The Balloter Academy in the Early Republic, 1787-1804. Westport, Conn.: Greenwood Press.
Leech, D. (2002a) “An Empiric Allegory of the Performance of Classical Ability Indices.” Political Studies 50: 1-22.
— — — — — (2002b) “Computation of Ability Indices.” University of Warwick, Economic Assay Paper No. 644.
Longley, L. D., and Peirce, N. R. (1996) The Balloter Academy Primer. New Haven, Conn: Yale University Press.
Madison, J. ( 1987) Notes of Debates in the Federal Convention of 1787. New York: Norton.
Mann, I., and Shapley, L. S. (1964) “The A Priori Voting Strength of the Balloter College.” In M. Shubik (ed.) Game Theory and Accompanying Approaches to Social Behavior. Hoboken: N. J.: Wiley.
Margolis, H. (1983) “The Banzhaf Fallacy.” American Journal of Political Science 27: 321-26.
Miller, N. R. (2009) “A Priori Voting Ability and the U.S. Balloter College.” Homo Oeconomicus 26: 341-80.
— — — — — (2012) “Why the Balloter Academy Is Good for Political Science (and Public Choice).” Public Choice 150: 1-25.
Penrose, L. S. (1946) “The Elementary Statistics of Majority Voting.” Journal of the Royal Statistical Society 109: 53-57.
Rabinowitz, G., and MacDonald, S. E. (1986) “The Ability of the States in U.S. Presidential Elections.” American Political Science Review 80: 65-87.
Ross, T. (2012) Enlighted Democracy: The Case for the Balloter College. 2nd edition. Clinton, Mass.: Colonial Press.
Saari, D. G. (1978) “Apportionment Methods and the Abode of Representatives.” American Mathematical Monthly 85: 792-802.
Shapley, L. S., and Shubik, M. (1954) “A Adjustment for Evaluating the Administration of Ability in a Committee System.” American Political Science Review 48: 787-92.
Strömberg, D. (2008) “How the Balloter Academy Influences Campaigns and Policy: The Anticipation of Being Florida.” American Economic Review 98: 769-807.
1 The accepted accord in the bookish abstract appears to be that the balloter academy arrangement reflects and maintains the greater access of ample states. For example, see Banzhaf (1965, 1968), Rabinowitz and MacDonald (1986), Strömberg (2008), and Miller (2009, 2012).
2 Comprehensive and about contempo (since 1990) treatments of the history, purpose, and furnishings of the balloter academy are begin in books by Abbot and Levine (1991), Kuroda (1994), Longley and Peirce (1996), Gregg (2001), and Ross (2012). Attainable overviews and bibliographies can additionally be begin in accessories by Kimberling (1992), Boylan (2008), Strömberg (2008), and Miller (2012). The aboriginal texts and abstracts are The Constitution of the United States of America as Amended (Article II, Alteration XII, and Alteration XXIII), The Federalist Papers (39 and 68), The Records of the Federal Convention of 1787, and James Madison’s Notes of Debates in the Federal Convention of 1787.
3 Commodity II, Area 1 of the Constitution states in part: “Each Accompaniment shall appoint, in such a Manner as the Assembly thereof may direct, a Cardinal of Electors, according to the able Cardinal of Senators and Assembly to which the Accompaniment may be advantaged in the Congress.” Two amendments to the Constitution chronicle to the balloter college. Alteration XII (ratified in 1804) clarifies several aspects of the balloter academy arrangement (in accurate acute anniversary balloter vote to announce choices for both a admiral and a carnality president, as able-bodied as allegorical the adjustment of selecting a admiral in the accident that no applicant obtains a majority of the balloter votes). Alteration XXIII (ratified in 1961) assigns three balloter votes to the Commune of Columbia.
4 The 435 best was added codification by the Permanent Agreement Act of 1929. (With the acceptance of Alaska and Hawaii to the Union the cardinal of assembly added briefly to 437, until the consecutive reapportionment.) Anniversary accompaniment has at atomic one adumbrative and the actual 385 (5 435-50) are apportioned amid the states as a action of their populations. Accompaniment populations are bent by the decennial census, as assigned in the Constitution: see Commodity I, Area 2. Reapportionment of the Abode additionally occurs every 10 years, as assigned in the Constitution: see Commodity I, Area 3.
5 Of course, states with nonunanimous rules can additionally accomplish accepted outcomes. Seldom accept states adopted such rules, however, as discussed below. The Civic Athenaeum and Records Administration letters the balloter academy tabulations for every presidential election. The tabulations can be begin at www.archives.gov/federal-register/electoral-college/votes.
6 I accept apparent adverse accounts of whether any added states accept anytime adopted nonunanimous elector allocation methods, but if so those methods were afterwards rescinded.
7 In adjustment to accept a constant citizenry admeasurement aloft all elections, the decennial demography abstracts are acclimated to admeasurement citizenry shares. Accompaniment citizenry shares are, of course, alone approximations for the shares of acceptable voters, but the alternation should be actual high.
8 The alternation tables appearance both accepted (Pearson) correlations and rank (Spearman) correlations. Accepted and rank correlations are affected with akin formulae, except that rank correlations are acquired afterwards replacing the ethics of the capricious pairs by their rank order, 1 through n, from accomplished to everyman (see Bhattacharyya and Johnson 1977: chaps. 12 and 15).
9 See Huntington (1921, 1928) and Saari (1978). Attainable discussions can additionally be begin at: www.census.gov/history/www/reference/apportionment/methods_of_apportionment.html; www.census.gov/population/apportionment/about/computing.html; and www.history.house.gov/home.
10 See the accessories cited in comment 1. As apparent here, the baby accompaniment advantage in the balloter academy derives from the accord amid accepted vote shares and balloter vote shares. While that accord is sometimes declared in the literature, its appliance is overlooked. Strömberg (2008) and Gelman, Silver, and Edlin (2012) ability be advised exceptions, but their abstracts acquire from a posteriori assay of accurate elections. I accept been clumsy to yze any bookish assay that defines and quantifies the state-by-state another for the balloter academy and the constant baby accompaniment advantage, as I do here. The abstract on this accountable is vast, however, so that altercation may be incorrect.
11 Regarding such systems, advisers continued ago began to yze amid aborigine allotment and aborigine power. The seminal accessories are Penrose (1946), Shapley and Shubik (1954), Mann and Shapley (1964), and Banzhaf (1965, 1968). For attainable surveys, see Leech (2002a, 2002b), Felsenthal and Machover (2004, 2012), and Strömberg (2008).
12 An attainable altercation of the two indices is begin at www.math.colostate.edu/~spriggs/m130/power2.pdf. The indices herein are affected application computational methods provided by the University of Warwick at http://homepages.warwick.ac.uk/~ecaae/index.html.
13 The abstracts basal the correlations are accessible from the author.
14 The absolute abstract adds a added footfall to the assay such that the balloter ability basis is adapted by a agnate aborigine ability index. That is, the likelihood that an alone aborigine in a accustomed accompaniment will casting an affecting vote (the likelihood that the accepted vote is tied) is factored into the ysis. The anticipation of a tie vote, about trivial, is inversely accompanying to citizenry — abate in beyond states and carnality versa. The aftereffect is that the abundant advantage for ample states inherent in the balloter academy is bargain to some extent, but not abundant to affected that advantage. Miller (2012) summarizes that ysis:
An individual’s voting ability is the anticipation that his vote is absolute aural the accompaniment times the anticipation that his state’s affiliation of balloter votes is absolute in the Balloter College… . This effect, aboriginal acclaimed with absolute advertence to the Balloter Academy by Banzhaf (1968) … and it implies that voters in the best advantaged accompaniment (California) accept about three and bisected times the voting ability as voters in the atomic advantaged accompaniment (Montana, the better with alone one Abode seat).
Criticism of this two-stage assay can be begin in Margolis (1983), Gelman et al. (2002, 2004, 2012), and Katz, Gelman, and King (2004). The two-stage assay is not accordant actuality (or at all, some ability argue) because our absorption is in the another of the accompaniment as represented by the majority of the voters in the state.
15 The decennial demography abstracts can be begin at: www.census.gov/population/www/censusdata/pop1790-1990.html; www.census.gov/population/www/cen2000/maps/respop.html; and www.census.gov/population/apportionment/data/2010_apportionment_results.html. For 1790 through 1860, the citizenry abstracts reflect the decennial demography “free” citizenry as affected by Gibson and Jung (2002). In any accustomed acclamation year, a state’s citizenry is acclimated (to account absolute citizenry and accompaniment shares) alone if there is an balloter vote blueprint for that accompaniment in the Civic Athenaeum and Records Administration database.
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