A New Method for Congressional Apportionment (Congress/representation/fair division/u.s. Constitution) M. L. BALINSKI AND H. P.

Transcription

1 Proc. Nat. Acad. Sci. USA Vol. 71, No. 11, pp , November 1974 A New Method for Congressional Apportionment (Congress/representation/fair division/u.s. Constitution) M. L. BALINSKI AND H. P. YOUNG Ph.D. Program in Mathematics, Graduate School and University Center, The City University of New York, 33 West 42 Street, New York, N.Y Communicated by R. E. Gomory, August 6, 1974 ABSTRACT The problem of Congressional apportionment is explained together with a brief history of the methods used or considered for its solution. Reasons are given for rejecting the presently used method of equal proportions and for accepting a new method, the quota method, which is the unique method satisfying three essential axioms. 1. The apportionment problem The Fourteenth Amendment to the Constitution of the United States stipulates "Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed," while Article I guarantees ". each State shall have at least one Representative" and that ".. enumeration shall be made... every years." If fractional Representatives were permitted, or if the numbers fortuitously turned out to be whole, the theoretically perfect solution could readily be calculated: the number of seats accorded to a particular State would be to the total number of seats in the House as the population of that State is to the total population of the United States. But equality in these proportions has never been achieved and clearly cannot ever be expected to be satisfied. This has caused decennial controversies over the "rounding" problem, indeed it prompted the first use of a Presidential veto by George Washington in 1792, and has led to considerable discussion concerning the specific meaning that should be attached to the phrase "according to their respective numbers." This, then, is the apportionment problem, as distinct from the districting problem (often called the "reapportionment problem"), which is that of determining geographical election districts having nearly equal populations within a State after the number of Representatives apportioned to the State is determined. The districting problem has received considerable attention in recent years due to the "one-man one-vote" Supreme Court rulings. 2. Formulation and brief history Let p (p1,..., p,) be the population of s states and h 2 0 the number of seats in the house. An apportionment for h is an s-tuple of nonnegative integers (ai,..., a,) satisfying 2 a j h. An apportionment solution is a function f, which for any p and every integer h 2 0, assigns a unique apportionment for h, at fj(ph) 2 0, Zfj (ph) h. An apportionment method M is a set of apportionment solutions f E M. (A method may yield Abbreviations: MF, method of major fractions; EP, method of equal proportions several solutions if there are "ties"; for example, if two states have the same populations, it is clear that a method has more than one solution.) The exact quota of state i in a house having h seats is q1 qt(p,h) pih/(2;pj). If qf were an integer for all i and h, the desired size of the house, at qj is clearly the "perfect solution." Otherwise, each state i should certainly receive at least as many seats as its lower quota [qjj (the largest integer less than or equal to qj) and certainly no more than its upper quota [qtl (the smallest integer greater than or equal to qt), since these result from "rounding" the exact quota qi down or up. In general, an apportionment method is said to satisfy lower quota if, for each of its solutions f, ft(p,h) _ lqj(ph) J, to satisfy upper quota if fj(p,h). [qj(p,h) 1, and to satisfy quota if it satisfies both lower and upper quota. The Vinton method of 1850 (or the Hare quota, ref. 1) was used to apportion Congress* from 1850 through It assigns, first, [qj] seats to each state i; then, it assigns one additional seat to each of the first h -218 [q11 states, ranked according to the largest fractional remainders qj - [qj]. Ties in the fractional remainders lead to multiple solutions, although, of course, the probability of this event is extremely small. Therefore, any Vinton solution satisfies quota. However, in 1881 it was discovered to possess a completely unacceptable property. Namely, in apportioning a House of 299 seats after the census of 1880 Alabama received eight seats, but in a House of 300 seats Alabama received only seven seats. This "Alabama paradox" is not an anomaly of the Vinton method. It caused Representative Littlefield of Maine to declare in 1901 "God help the State of Maine when mathematics reach for her... " after seeing Maine's delegation fluctuate up and down some six times as the House size increased from 383 to 400. An apportionment solution is said to be house monotone if f(ph + 1). f (ph) for all h and p, that is, if no state delegation can decrease as the house increases with fixed populations. An apportionment method is house monotone if all its solutions are. House monotonicity is a second seemingly essential property of any acceptable apportionment method. In 1910 Willcox (3) devised the method of major fractions (MF) which was used to apportion Congress in 1911 and (For various political reasons the apportionment of 1911' remained in force for 20 years.) The idea is simply that the quota of each state should be rounded to the nearest integer. In general, such an assignment will not necessarily arrive at an apportionment of precisely h seats. However, it is possible * Before 1850 the size of the house was not fixed in advance (2).

2 Proc. Nat. Acad. Sci. USA 71 (1974) (save for "ties") to find a number h and to define "pseudoquotas" jt qi(p,h) such that if each state satisfying qt 1qd - > l/2 gets [qi I seats and each state satisfying qi - [i < 1/2 gets [qtj seats, then the sum of the seats apportioned is exactly h. The method of major fractions (MF) does not satisfy quota; but it is house monotone, as is easily verified. In 1921, E. V. Huntington (4) first proposed the method of equal proportions (EP) and devised an approach in which to embed this method. The permanent apportionment section of the Census Act of 18 June 1929 provided that apportionment be computed by (i) the method used in the preceding census, (ii) MF, and (iii) EP, and that all be presented to Congress. EP and MF happened to give identical results for the 1930 census. But in 1941, MF gave Arkansas and Michigan 6 and 18 seats, respectively, while EP gave them 7 and 17, respectively. Hotly contested bearings (5, 6) and other discussion led to the definitive acceptance of EP (Public Law 291, H.R. 2665) for the apportionment of 1941 and thence forth. 3. The method of equal proportions and Huntington methods Huntington's essential ideas were, firstt, (ref. 4, p. 123): "In a satisfactory apportionment between two states (pi greater than pj), we shall agree that pj/ai and pj/aj should be as nearly equal as possible; also at/pt and aj/pj; also pt/p1 and aj/aj; also pj/pt and aj/at." In particular, pt/at represents the average number of inhabitants per Congressional district in state i, and (at/pt) * 108 the average number of Representatives per million inhabitants in state i. In a perfect apportionment each of these pairs (and many others) would be equal. Second, (ref. 4, p. 124): "In a satisfactory apportionment, there should be no pair of states which is capable of being 'improved' by a transfer of a representative within the pair-the word 'improvement' being understood in the sense implied by the test [of difference] adopted " This asks for a solution that is stable in the sense that no inequality difference according to the measure chosen can be reduced by transferring one seat from one state delegation to another. Huntington advanced as the best test, or measure, of "nearly equal" the relative difference [the relative difference between x and y is (lx - yl/min(x,y)] between pt/at and pjlaj or between at/pt and aj/pj or between any of the pairs mentioned above, since all these relative differences are precisely equal. In trying various differences (7) it was found that either the test does not guarantee the existence of a stable solution or one of five distinct solutions result, one of which is EP and corresponds to the relative difference test and another of which is MF. Each of the five methods can be characterized by a ranking function r(a,p) as follows: if f(p,h) {ft(p,h) } is a solution for house size h, thenft(p,h + 1) f,(p,h) for i # k and fk(p,h + 1) fk(ph) + 1, where k is some one state satisfying r(akpk) 2 r(at,pi) for all i, is a solution for house size h + 1. Any apportionment method that is so specified by a ranking function is called a Huntington method. Clearly any Huntington method is house monotone. The five methods alluded to above are Huntington methods, and are commonly referred to as the "modern workable methods" because they avoid the Alabama paradox. Table 1 defines them by listing the measure t The citations are precise except for notational changes. A New Method for Congressional Apportionment 4603 TABLE 1. The five "modern workable methods" for Congressional apportionment Stable for test T (where pi/ai _ Ranking function Method pilai) r(a,p) Smallest divisors T,: a, - ai(pj/pi) p/a Harmonic mean T2: pi/ai - pjaj p/{ 2a(a + 1)/(2a + 1)} Equal proportions (EP) T3: pia,/pjai - 1 p/{a(a + 1 )1 1/2 Major fractions (MF) T4: a,/pj - ajpj p/(a + 1/2) Greatest divisors* T5: aj(pi/pj) - a, p/(a + 1) * Also known as the method of d'hondt. of difference or stability criterion and the associated ranking function for each. There appears to be one traditional two-phase argument for accepting EP. As stated in a report to the National Academy of Sciences (1), "there are five methods of apportionment now known which are unambiguous and should be considered... In the present state of knowledge... these [are] the only methods of apportionment avoiding the so-called Alabama paradox which require consideration at this time" (8). (2) The five methods are listed above in the order in which they tend to favor small states, the method of smallest devisors tending to most favor small state, that of greatest divisors to most favor large states. The report goes on to say EP "... is preferred... because it occupies a neutral position with respect to emphasis on larger and smaller states" (8). Thus, EP is supported because it is house-monotone, and of the known house-monotone methods it "occupies the central position among the five methods" (ref. 7, p. 103). A 1948 National Academy of Sciences Report (9) sustains EP as the best compromise with an additional argument. If the tests T2, T3, and T4 are accepted as the most natural ones of the five, then EP can be measured by them as against each of the other four methods. Clearly the method of harmonic mean is always best by T2; EP is always best by T3; MF is always best by T4; but, otherwise, EP is either (and usually) better than every method or not comparable in general by each test. Incredibly, all modern contributors and judges simply disregard the fact that EP does not satisfy quota. "Now it is a common misconception that in a good apportionment the actual assignment should not differ from the exact quota by more than one whole unit." [ref. 7, p. 94]. "The proper apportionment... may differ by several units from the number obtained by simple proportion" (8). Although the fact was recognized, the implicit inference of the examples illustrating it (7) is that the event is rare and at worse minor in magnitude. But, "... as a proper method of apportionment must meet every conceivable variation in population no matter how fantastic" (ref. 2, p. 73), consider the following two, which are very much in the tradition of the many lovely examples given by Huntington (7). If p (pi,...,ps), s 33, pi 60,272, pi i for 2 < i < 33, thus Zipi 100,000, and h 102; then qi , q ,..., q , are the exact quotas, but the unique EP solution is a, 68, a2. a3l 1, a32 a33 2. On the other hand, if s 21, pi 68,010, pi i, 2 < i < 21, thus Zips 100,000, and h 98; then q, , q ,..., q2l 1.577, are

3 4604 Political Science: Balinski and Young the exact quotas, but the unique EP solution is a, 60, a2 a3 1, a4... a2l 2. These are, of course, "fantastic" artificial examples. Table 2, however, contains two examples with "realistic" (possible) data for the United States "censuses of 1984", called, respectively, 1984 A and 1984 B, for which exact quotas and the EP solutions are given. In 1984 A the exact quotas and unique EP solution [written here as (qi,ai) for state i] are for California (42.960, 45), for New York (39.939, 42), and for Pennsylvania (24.974, 26); while in 1984 B they are for California (43.167, 41), for New York (39.031, 37) and for Pennsylvania (25.138, 24). This possibility surely makes EP unacceptable. As the constitutional authority Z. Chafee, Jr. pointed out, "the preservation of a respect for the law will in the long run be best obtained by the adoption of the plan which is least likely to product a sense of unfairness in those who are forced to obey legislation" (ref. 10, pp ). More generally, the following result obtains: THEOREM 1. There exists no Huntington method satisfying quota. Of the five "modern workable" methods, only one, that of smallest divisors, satisfies upper quota; and only one, that of greatest divisors, satisfies lower quota. 4. The quota method Q Given a method M, a solution restricted to h, fh, is a solution ofm for which only the values f(p,0),...,f (p,h) are specified. A completion of fh is any solution of M agreeing with fh. There may be several completions of a given f'. Given an apportionment as fj(p,h) the set of states eligible at h + 1, E(h + 1) { i; as < pi(h + 1)/2pi) } are those states that can receive an additional seat in a house h + 1 without violating upper quota. Let p* and p be the populations of some two states, and suppose that by some solution f E M, where M is house-monotone, the star-state is eligible at h for its (d* + 1)st seat and the bar-state is eligible at h for its (a + 1)st seat, but f gives the hth seat to the star-state. Then the star-state is said to have weak-priority by M over the bar-state at p*, p, a*, and a. Since both states were eligible and the star-state received the extra seat, its claim to the extra seat is certainly as good as that of the bar-state. A natural requirement for any method M is that the relative claims for an extra seat between two states should depend only upon their respective populations p* and p and current apportionments a* and a. To be precise, suppose that the star-state has weak priority by M over the bar-state, at p*, p, a*, and a. Let g E M be a solution for some population vector q, which contains a pair of states having populations p* and p, and suppose these states are, respectively, eligible for their (a* + 1)st and (a + 1)st seats at h', but that g gives the h'th seat to the bar-state rather than the star-state. Then M is said to be consistent if g"'-', the restriction of g up to h'- 1, has an extension by which the h'th seat is given to the star-state. That is, a method M is consistent if it never switches priorities at p*, p, a*, a unless the two states have equal claim to the extra seat. Clearly, any Huntington method is consistent, in fact is consistent even if the condition of eligibility is dropped. The clear and obvious question is: does there exist a consistent, house-monotone method satisfying quota? These are the three properties which, in view of "fairness" and the history of Congressional apportionment in the United States, are essential. Further, is there a unique such method, in the sense that any solution satisfying the given requirements must also be obtainable by this particular method? THEOREM 2. There exists a unique consistent, house-monotone method Q Q (p) satisfying qaota. This Q, called the quota method, is the set of all solutions so obtained recursively as follows: W(i) i(pao O. 1 _< i < s (ii) Given aj vi(ph) and E(h + 1) {i; a, < pi (h + 1)/ 2npj) },let k E E(h + 1) be some one state satisfying Pk/(ak + 1) > pj/(aj + 1) for all i E E(h + 1). Then, (Pk(p,h + 1) ak + 1, Vp(p,h + 1) a all i $ k. It is obvious that an apportionment solution satisfying quota cannot in general meet the Constitution's requirement that each state is to receive at least one seat. For example, in a house of 50 seats the 1970 exact California quota is One of the requirements habitually cited to support EP (and also applicable to the methods of smallest divisor and harmonic mean) is that it automatically provides (for h > s) at least one seat per state. This is, in fact, another weakness of EP. For, if p (108, 1, I,1) is a 50-tuple of populations, all EP solutions give f(p,.50) (1,1,..,1), in clear violation of any reasonable notion of fair division. Further, minimum requirements may be different from 1, as for example in France where the minimum number of "deputes" per "department" is 2. Therefore, minimum requirements must be explicitly recognized. To this end the various definitions previously stated need to be generalized (the same words will be used to avoid a repetitive qualifier such as "generalized.", since this can cause no confusion). Let r (r,,,r,), where ri is the minimum number of representatives permitted in state i, be the requirement vector. A house size h is said to be feasible if h _ 2 ri h. An apportionment solution is a function f(p,r) which for any p,r and every feasible h assigns an s-tuple of integers as f(p,r,h), 1 _ i. s, satisfying as _ ri and Mai h. An apportionment method M, is, for each p,r, a set of apportionment solutions f (p,r) Proc. Nat. Acad. Sci. USA 71 (1974) E M (p,r). Let I {I, Is} be the set of all states. If phi! (2;ip). ri, then state i deserves in a house of size h at most ri seats, while it is required to receive at least rj. Thus, it should receive exactly rj seats. When these states are eliminated from consideration, the number of seats each of the remaining states deserves may be determined, and so on. Define, then, for any h, Jo I, ho h, J i E Jo; piho/(yjopj) > ri}, and h, ho - Xjoj, ri. Any state i E J, deserves pjh,/(zt1p1) seats, so if ri is at least this large then i should receive exactly ri seats. Let J2 i E J,; pih,/(2 j,p1) > rj}, h2 h, - 2sl-2 ri, and continue. This produces a nested sequencei Jo D Ji D D J. and h ho > h,>... > h. such that either J,, p or qj(p,r,h) pih,2/(tjspj) > ri for all i C JU. J, 1., (p,r,h) is called the slack set at h since any other state Z J,, should receive exactly ri seats. Define, therefore,

4 Proc. Nat. A cad. Sci. USA 71 (1974) A New Method for Congressional Apportionment 4605 TABLE 2. Apportionments of 435 seats by the method of equal roportin and the quota method for 1970, 1984A and 984B. Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming Totals A 1984 B p q EP Q p q EP Q p q EP Q 3,475, ,659, ,608, , , , ,787, ,184, ,885, ,942, ,176, ,948, ,098, ,839, ,944, ,226, ,664, ,410, ,050, ,158, ,438, , , , ,855, ,081, ,671, ,627, ,112, ,053, , , , , , , ,184, ,947, ,290, ,228, ,610, ,570, ,846, ,161, ,958, ,265, ,675, , ,246, ,657, ,465, ,672, ,140, ,968, ,006, ,078, ,298, ,953, ,131, ,978, ,726, ,085, ,086, ,937, ,438, ,489, ,833, ,129, ,003, ,233, ,679, ,421, ,718, ,123, ,108, , , , ,496, ,643, ,834, , , , , , , ,208, ,573, ,676, ,026, ,182, ,313, ,338, ,303, ,842, ,125, ,614, ,552, , , , ,730, ,437, ,735, ,585, ,675, ,901, ,110, ,182, ,385, ,884, ,696, ,779, , ,131, ,316, ,617, ,674, ,905, , , , ,961, ,133, ,031, ,298, ,176, ,228, ,067, ,197, ,360, , , , ,690, ,098, ,026, ,443, ,648, ,519, ,763, ,691, ,854, ,447, ,631, ,519, , , , ,053, ,138, ,138, p populations; q exact quotas; EP equal proportions solution; Q quota solution with minimum requirements. the lower quota l(p,r,h) of state i to be rj if i Z.J,,(p,r,h) and jq(p,r,h) J otherwise; and the upper quota ui(p,r,h) of state i to be max{rj, [pjh/1pji1 } if i Z.J,,(p,r,h) and [pih/(2lpj) 1 otherwise. An apportionment method is said to satisfy quota if for every solution f(p,r), l(p,r,h) < ft(p,r,h) < ut(p,r,h). This generalized definition of a method satisfying quota results simply from the necessity of guaranteeing to each state i at least ri seats. The upper quota is not changed by imposing requirements, except for the case where a state's requirement exceeds its upper quota. An apportionment method M(p,r) is house-monotone if for every solution f(p,r) E M(p,r), f(p,r,h + 1) _ f(p,r,h). An apportionment method M(p,r) is consistent if it satisfies precisely the same definition as that given earlier. The require-

5 4606 Political Science: Balinski and Young ments in a problem p, r are said to be unbiased if pi. pj implies pj/ri _ p1/rj. That is, if state i is larger than or equal to state j, then state i's minimum allocation does not give it an advantage over state j's minimum allocation. THEOREM 3. There exists a unique consistent, house-monotone method for unbiased requirements Q Q(p,r) satisfying quota. Any solution Q of the quota method Q is obtained recursively for unbiased requirements as follows [if the requirements are biased the theorem obtains, but the method is slightly more complicated because E(h + 1) must be replaced by E(h + 1) n.j,, (p,r,h + 1) ]: (i) (pj(p,r~h) ri, 1 :! i S5 s (ii) h _ h. Given as.ri(p,r,h) and E(h + 1) {i; aj < pi(h + W)/(Zp1)}, let k E E(h + 1) be some one state satisfying pk/(ak + 1) _ pi/(aj + 1) for-all i E E(h + 1). Then, (Pk(pr,h + 1) ak+ 1 #Pi(p,r,h + 1) a for all i 5 k. The unique solutions of Q Q(p,r), wherer (1,,1) and p (the apportionment populations) are given by the censuses of 1970, 1984 A, and 1984 B, are displayed, together with the unique equal proportion solutions, in Table Conclusion The virtue of the quota method is that it unites the two basic apportionment principles-house-monotonicity and quotainto a single method. It replaces Huntington's artificial "measures of inequality" with a more fundamental measure of fairness, the exact quota, and does this without introducing the Alabama paradox. Moreover, subject to mathematical consistency, it is the only apportionment method with these two properties. Proc. Nat. Acad. Sci. USA 71 (1974) Therefore, Congress should consider replacing Public Law 291, H.R with a law declaring that, from the apportionment act of 1981 on, the quota method Q be used to deternine the distribution of seats in the U.S. House of Representatives. This work was supported by the Army Research Office under Contract no. DA ARO(D) Hoag, C. G. & Hallet, G. H. (1926) in Proportional Representation (The MacMillan Company, New York), pp Schmeckebier, L. F. (1941) Congressional Apportionment (The Brookings Institution, Washington, D.C.). 3. Willcox, W. F. (1916) Amer. Econ. Rev. VI, Supplement, Huntington, E. V. (1921) Proc. Nat. Acad. Sci. USA 7, Hearings before the Committee on the Census, (1940), House of Representatives, 76th Congress, Third Session, on the Apportionment of Representatives in Congress, 27, 28 February and 1, 5 March, Hearings before a Subcommittee of the Committee on Commerce, (1941), United States Senate, 77th Congress, First Session on H.R. 2665, 27,28 February and 1 March Huntington, E. V. (1928) Amer. Math. Soc. Trams. 30, Bliss, G. A., Brown, E. W., Eisenhart, L. P. & Pearl, R. (1929) Report to the President of the National Academy of Sciences, 9 February 1929 (in Hearings before the Committee on the Census, (1940), House of Representatives, 76th Congress, Third Session, on the Apportionment of Representatives in Congress, 27, 28 February and 1, 5 March, 1940, pp Morse, M., von Neumann, J. & Eisenhart, L. P. (1948) Report to the President of the National Academy of Sciences, 28 May 194S. 10. Chafee, Z., Jr. (1929) Harvard Law Rev. XLI,