Power Balance and Apportionment Algorithms for the United States Congress Lane A. Hemaspaandra, University of Rochester and Kulathur S. Rajasethupathy, SUNY{Brockport and Prasanna Sethupathy, Stanford University and Marius Zimand, Georgia Southwestern State University
We measure the performance, in the task of apportioning the Congress of the United States, of an algorithm combining a heuristic-driven (simulated annealing) search with an exact-computation dynamic programming evaluation of the apportionments visited in the search. We compare this with the actual algorithm currently used in the United States to apportion Congress, and with a number of other algorithms that have been proposed. We conclude that on every set of census data in this country's history, the heuristic-driven apportionment provably yields far fairer apportionments than those of any of the other algorithm considered, including the algorithm currently used by the United States for Congressional apportionment. Categories and Subject Descriptors: J.4 [Computer Applications]: Social and Behavioral Sciences; K.4 [Computing Milieux]: Computers and Society General Terms: Algorithms, Experimentation Additional Key Words and Phrases: power indices, apportionment algorithms, simulated-annealing
The rst author was supported in part by grants NSF-CCR-9322513 and NSF-INT9513368/DAAD-315-PRO-fo-ab, and the fourth author was supported in part by grants NSFCCR-8957604 and NSF-CCR-9322513. This work was done in part while the rst author was visiting Friedrich Schiller-Universitat and the Univ. of Amsterdam, and while the fourth author was at the Univ. of Rochester. Email:
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[email protected]. Addresses: Lane A. Hemaspaandra, Dept. of Computer Science, Univ. of Rochester, Rochester, NY 14627; Kulathur S. Rajasethupathy, Dept. of Computer Science, SUNY{Brockport, Brockport, NY 14420; Prasanna Sethupathy, POB 12315, Stanford Univ., Palo Alto, CA 94309; Marius Zimand, School of Computer & Applied Sciences, Georgia Southwestern State Univ., Americus, GA 31709. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or direct commercial advantage and that copies show this notice on the rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior speci c permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or
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1. MOTIVATION AND OVERVIEW How should the seats in the House of Representatives of the United States be allocated among the states? The Constitution stipulates only that \Representatives shall be apportioned among the several states according to their respective numbers, counting the whole numbers of persons in each State : : :." The obvious implementation of this requirement would almost always yield fractional numbers of seats. The issue of how to achieve fair integer seat allocations has been controversial in this country virtually since its founding. In fact, many of the apportionment algorithms we will discuss have been proposed and debated by famous historical gures, including John Quincy Adams, Alexander Hamilton, Thomas Je erson, and Daniel Webster. The debate is far from over. In fact, the relative fairness of two of the algorithms we will discuss in this paper was argued before the Supreme Court in 1992 [Supreme 1992].1 We propose an apportionment method consisting of a simulated-annealing search that is aimed at maximizing the fairness of the resultant apportionments. Even though the complexity of the algorithm is high, our implementation shows that the method is feasible for the cases of interest. Indeed, we have been able to run it on the data conducted during all the census years in US history and the results are conclusive: In all cases our method was provably superior with respect to widely agreed fairness criteria to the most prominent apportionment algorithms that have been used or proposed earlier. Balinski and Young [1982] (see also [Balinski and Young 1985]) performed a detailed comparative study, for six historical algorithms of Congressional apportionment, of the degree to which the algorithms' allocations matched states' \quotas," i.e., their portion of the population times the House size. Mann and Shapley [Mann and Shapley 1960; Mann and Shapley 1962] and others studied, for the actual usedin-Congress seat allocations, the power indices (in the Electoral College) of each state. This paper attempts to combine the strengths of these two research lines. In particular, we agree with Balinski and Young both that allocations should be \fair," and that, in light of 200 years of debate (colorfully recounted and analyzed by Balinski and Young [1982]), obtaining new insights into the merits and weaknesses of the six historical algorithms should be a priority. On the other hand, many feel that \fairness" should be de ned by a tight match between power and quotas, rather than between allocations and quotas. Our feeling is very much in harmony with modern political science theory, where it is widely recognized that allocations Brie y put, the Supreme Court ruled that Congress acted within its authority in choosing the currently used algorithm (the \Huntington-Hill Method"). However, the Supreme Court's decision left open the possibility that Congress would be acting equally within its authority if it chose to adopt some other algorithm. The court ruled that \the constitutional framework...delegate[s] to Congress a measure of discretion broader than that accorded to the States [in terms of choosing how to apportion], and Congress's apparently good-faith decision to adopt the [Huntington-]Hill Method commands far more deference, particularly as it was made after decades of experience, experimentation, and debate, was supported by independent scholars, and has been accepted for half a century [Supreme 1992]." Regarding the decision, we mention only that the power balancing issues discussed in this paper were not brought before the court, but that nonetheless the experimental results of this paper suggest that among the two algorithms being discussed by the court in the case, the Huntington-Hill algorithm in fact gives fairer apportionments in terms of power balancing.
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do not necessarily directly correspond to power, and that power indices (see the detailed de nitions later in this paper) provide a potentially more accurate gauge of power (see, e.g., the discussions in [Shapley 1981; Riker and Ordeshook 1973]).2 So, in this light, we repeat the comparative study of Balinski and Young, but we replace allocation comparisons with power index comparisons. Of course, the historical algorithms were designed (in part|the full story is more complex and political, and indeed led to the rst presidential veto (see [Balinski and Young 1982])) to achieve some degree of harmony and fairness between allocations and quotas. This is not surprising, given that power indices had not yet been invented. However, given that in this study our comparisons are based on power indices, it seems natural to add to the six historical algorithms3 an algorithm tailored to achieve harmony between power indices and quotas. We have used a heuristic based on the simulated annealing paradigm (see [Aarts and Korst 1989; Metropolis et al. 1953; Kirkpatrick et al. 1983]), which nds an apportionment by seeking to achieve a local minimum of the distance between normalized power indices and quotas (the attribute \local" is with respect to a natural neighborhood relation between apportionments). This heuristic yields results that are fairer to those obtained by all the historical algorithms. We report in Section 3 the results for the last census, 1990, but we have obtained similar results for all the census years, 1790, 1800, . . . , 1980, and 1990. The function class #P ( rst de ned by Valiant [Valiant 1979a; Valiant 1979b]) is the counting version of NP. #P is the class of all functions f such that, for some nondeterministic polynomial-time Turing machine N , for all inputs x it holds that f (x) is the number of accepting computation paths of N (x). One problem in studying power indices is that power indices are typically #P-complete [Prasad and Kelly 1990; Garey and Johnson 1979] and, consequently, we perform a combinatorial search that invokes at each iteration numerous #P-complete computations. Fortunately, a dynamic programming approach, rst proposed by Mann and Shapley [1962], yields a pseudo-polynomial algorithm for computing the power indices, i.e., an algorithm whose running time is polynomial in the size of the House and in the number of states. Since these quantities have had reasonable values throughout US history (the maximum values have been 435 and 50, which are also the current values), we have been able to exactly compute the needed power indices.
2 For those unfamiliar with why allocations may not correspond to power, consider the following typical motivating example. Suppose we have states A, B, and C with 6, 2, and 2 votes respectively. Note that though between them states B and C have 40 per cent of the seat allocation, nonetheless it is the case that in a majority-rule vote on some polarizing issue on which states have di ering interests and so their delegations vote as blocs, B and C have no power at all as A by itself is a majority. 3 By the use of \the" in the context \the six historical algorithms" we do not mean to suggest that no other algorithms have been proposed. Other algorithms have indeed been proposed (e.g., the algorithm Condorcet suggested in 1793 ([Condorcet 1847], see [Balinski and Young 1982, p. 63])). However, these six algorithms have been the key contenders in the apportionment discussion in the United States.
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2. DISCUSSION In this section, we justify a number of decisions made in designing this study, and we describe in more detail the background of the study. 2.1 Study Design Our computer program takes as its input a list of states, hS0 ; ; S ?1 i, populations, hp0 ; ; p ?1 i, a House size, h, and some other parameters regarding random number generation and the implementation of the simulated annealing algorithm. The program computes, for each state S , the appropriate quota n
n
i
q = Pp h p : 0 i
i
i
