2074:
favors state 1 vs. the probability that it favors state 2. For example, the probability that a state receiving 2 seats is favored over a state receiving 4 seats is 75% for Adams, 63.5% for Dean, 57% for Hill, 50% for
Webster, and 25% for Jefferson. The unique proportional divisor method for which
2771:
2349:
2498:
38:
if it systematically favors small parties over large parties, or vice versa. There are several mathematical measures of bias, which can disagree slightly, but all measures broadly agree that rules based on
595:
1368:
951:
640:
1886:
1816:
1746:
1676:
1185:
2635:
1506:
360:
211:
766:
723:
157:
2642:
306:
1141:
456:
680:
2068:
2207:
2022:
1978:
1938:
1602:
1562:
2561:
2380:
2177:
2524:
1419:
1280:
1231:
1103:
1002:
863:
814:
510:
1394:
977:
411:
2353:
In particular, Webster's method is the only unbiased one in this family. The formula is applicable when the house size is sufficiently large, particularly, when
2200:
2136:
1440:
1255:
1206:
1077:
1022:
838:
789:
532:
477:
384:
257:
237:
105:
85:
1564:, the set of all possible apportionments yielded by M, for all possible house sizes. Theoretically, the number of possible house sizes is infinite, but since
2387:
2079:
as defined in the "Basic requirements" section above. The same result holds if, instead of checking pairs of agents, we check pairs of groups of agents.
2799:. However, researchers have found that under other definitions or metrics for bias, the Huntington-Hill method can also be described as least biased.
1443:. Therefore, Adams' method is majorized by Dean's, which is majorized by Hill's, which is majorized by Webster's, which is majorized by Jefferson's.
2075:
this probability is always 50% is
Webster. There are other divisor methods yielding a probability of 50%, but they do not satisfy the criterion of
2567:(=coalitions). Such alliances can tip the bias in their favor. The seat-bias formula can be extended to settings with such alliances.
59:
have low levels of bias, with the differences being sufficiently small that different definitions of bias produce different results.
2990:
2963:
2916:
545:
600:
2995:
2842:
1821:
1751:
1681:
1611:
3000:
1604:
are usually rational numbers, it is sufficient to check the house sizes up to the product of their denominators
1889:
1285:
868:
2766:{\displaystyle {\text{MeanBias}}(s,k,t)={\frac {s}{n}}\cdot \left(\sum _{i=k}^{n}(1/i)-1\right)\cdot (1-nt)}
2582:
1453:
459:, which takes as input a vector of entitlements and a house-size, and returns as output an apportionment of
311:
162:
110:
23:
265:
214:
1148:
2792:
2070:). Assuming the entitlements are distributed uniformly at random, one can compute the probability that
1109:
424:
1524:
To measure the bias of a certain apportionment method M, one can check, for each pair of entitlements
728:
685:
107:
representing the number of parties to which seats should be allocated. There is a vector of fractions
2344:{\displaystyle {\text{MeanBias}}(r,k,t)=(r-1/2)\cdot \left(\sum _{i=k}^{n}(1/i)-1\right)\cdot (1-nt)}
2382:. When the threshold is negligible, the third term can be ignored. Then, the sum of mean biases is:
645:
2576:
2027:
1447:
1987:
1943:
1903:
1567:
1527:
2796:
87:(=house size), representing the total number of seats to allocate. There is a positive integer
52:
2532:
2356:
2141:
2503:
44:
1373:
956:
389:
8:
2788:
2180:
48:
2099:-th highest entitlement. Averaging this number over the entire standard simplex gives a
1399:
1260:
1211:
1083:
982:
843:
794:
490:
2564:
2185:
2121:
1425:
1240:
1191:
1062:
1007:
823:
774:
517:
462:
369:
242:
222:
90:
70:
2959:
2912:
2874:
2838:
2831:
31:
2637:, when entitlement vectors are drawn uniformly at random from the standard simplex,
2951:
2904:
2493:{\displaystyle \sum _{k=1}^{n}{\text{MeanBias}}(r,k,0)\approx (r-1/2)\cdot (n/e-1)}
2088:
2863:"Apportionment Methods for the House of Representatives and the Court Challenges"
2955:
2908:
1888:, then the method is unbiased. The only unbiased method, by this definition, is
2115:
1234:
817:
27:
2943:
2896:
2984:
2878:
2784:
259:). This is usually the fraction of votes the party has won in the elections.
1028:
2775:
In particular, Hamilton's method is the only unbiased one in this family.
2948:
Proportional
Representation: Apportionment Methods and Their Applications
2901:
Proportional
Representation: Apportionment Methods and Their Applications
40:
2862:
2087:
One can also check, for each vector of entitlements (each point in the
56:
262:
The goal is to find an apportionment method is a vector of integers
47:
are strongly biased in favor of large parties, while rules based on
26:. These are methods used to allocate seats in a parliament among
2950:, Cham: Springer International Publishing, pp. 127–147,
2903:, Cham: Springer International Publishing, pp. 149–157,
2897:"Preferring Stronger Parties to Weaker Parties: Majorization"
1508:
are also ordered by majorization, where methods with smaller
2942:
Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.),
2895:
Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.),
2833:
1900:
One can also check, for each pair of possible allocations
2082:
2795:
for comparing pairs of states, followed closely by the
1895:
2944:"Favoring Some at the Expense of Others: Seat Biases"
2645:
2585:
2535:
2506:
2390:
2359:
2210:
2188:
2144:
2124:
2030:
1990:
1946:
1906:
1824:
1754:
1684:
1614:
1570:
1530:
1456:
1428:
1402:
1376:
1288:
1263:
1243:
1214:
1194:
1151:
1112:
1086:
1065:
1010:
985:
959:
871:
846:
826:
797:
777:
731:
688:
648:
603:
548:
520:
493:
465:
427:
392:
372:
314:
268:
245:
225:
219:, that is, the fraction of seats to which some party
165:
113:
93:
73:
590:{\displaystyle \mathbf {a'} \in M'(\mathbf {t} ,h)}
420:An apportionment method is a multi-valued function
2830:
2765:
2629:
2563:, there is an incentive for small parties to form
2555:
2518:
2492:
2374:
2343:
2194:
2171:
2130:
2062:
2016:
1972:
1932:
1880:
1810:
1740:
1670:
1596:
1556:
1500:
1434:
1413:
1388:
1362:
1274:
1249:
1225:
1200:
1179:
1135:
1097:
1071:
1016:
996:
971:
945:
857:
832:
808:
783:
760:
717:
674:
634:
589:
526:
504:
471:
450:
405:
378:
354:
300:
251:
231:
205:
151:
99:
79:
1519:
635:{\displaystyle \mathbf {a} \in M(\mathbf {t} ,h)}
2982:
1105:, if for any house-size and entitlement-vector,
2829:Balinski, Michel L.; Young, H. Peyton (1982).
2529:Since the mean bias favors large parties when
2828:
2106:
2570:
16:Metric for fairness of apportionment methods
1818:equals the number of house-sizes for which
1608:For each house size, one can check whether
2941:
2894:
1881:{\displaystyle a_{1}/t_{1}<a_{2}/t_{2}}
1811:{\displaystyle a_{1}/t_{1}>a_{2}/t_{2}}
1741:{\displaystyle a_{1}/t_{1}<a_{2}/t_{2}}
1671:{\displaystyle a_{1}/t_{1}>a_{2}/t_{2}}
1054:at least as many seats as they receive in
413:is the number of seats allocated to party
1748:. If the number of house-sizes for which
1363:{\displaystyle d'(a)/d'(b)>d(a)/d(b)}
946:{\displaystyle d'(a)/d'(b)>d(a)/d(b)}
2787:census data, Balinski and Young argued
2983:
2630:{\displaystyle q_{i}=t_{i}\cdot (h+s)}
2500:, when the approximation is valid for
2083:Averaging over all entitlement-vectors
1501:{\displaystyle q_{i}=t_{i}\cdot (h+s)}
355:{\displaystyle \sum _{i=1}^{n}a_{i}=h}
206:{\displaystyle \sum _{i=1}^{n}t_{i}=1}
2860:
2856:
2854:
1512:are majorized by methods with larger
1027:This fact can be expressed using the
482:
152:{\displaystyle (t_{1},\ldots ,t_{n})}
2937:
2935:
2933:
2890:
2888:
2837:. New Haven: Yale University Press.
2824:
2822:
2820:
2818:
2816:
2814:
2812:
1896:Averaging over all entitlement-pairs
487:We say that an apportionment method
1940:, the set of all entitlement-pairs
301:{\displaystyle a_{1},\ldots ,a_{n}}
13:
2851:
1180:{\displaystyle M'(\mathbf {t} ,h)}
14:
3012:
2930:
2885:
2809:
2778:
1136:{\displaystyle M(\mathbf {t} ,h)}
451:{\displaystyle M(\mathbf {t} ,h)}
1164:
1120:
761:{\displaystyle a_{j}'\leq a_{j}}
718:{\displaystyle a_{i}'\geq a_{i}}
619:
605:
574:
551:
435:
2575:For each shifted-quota method (
239:is entitled (out of a total of
2760:
2745:
2728:
2714:
2669:
2651:
2624:
2612:
2487:
2467:
2461:
2441:
2435:
2417:
2338:
2323:
2306:
2292:
2260:
2240:
2234:
2216:
2154:
2148:
2138:seats correspond to a divisor
1520:Averaging over all house sizes
1495:
1483:
1357:
1351:
1340:
1334:
1325:
1319:
1303:
1297:
1174:
1160:
1130:
1116:
1004:favors small agents more than
940:
934:
923:
917:
908:
902:
886:
880:
675:{\displaystyle t_{i}<t_{j}}
629:
615:
584:
570:
513:favors small parties more than
445:
431:
146:
114:
1:
2991:Apportionment method criteria
2802:
2063:{\displaystyle h=a_{1}+a_{2}}
67:There is a positive integer
7:
2956:10.1007/978-3-319-64707-4_7
2909:10.1007/978-3-319-64707-4_8
2861:Ernst, Lawrence R. (1994).
2017:{\displaystyle a_{1},a_{2}}
1973:{\displaystyle t_{1},t_{2}}
1933:{\displaystyle a_{1},a_{2}}
1597:{\displaystyle t_{1},t_{2}}
1557:{\displaystyle t_{1},t_{2}}
1446:The shifted-quota methods (
1050:largest parties receive in
62:
10:
3017:
2107:Stationary divisor methods
1058:. An apportionment method
2577:largest-remainders method
2571:For shifted-quota methods
1080:majorizes another method
22:is a property describing
2996:Apportionment (politics)
2556:{\displaystyle r>1/2}
2375:{\displaystyle h\geq 2n}
2172:{\displaystyle d(a)=a+r}
24:methods of apportionment
2793:median-biased estimator
2519:{\displaystyle n\geq 5}
1984:yields the allocations
1237:with divisor functions
820:with divisor functions
2797:Huntington-Hill method
2767:
2713:
2631:
2557:
2520:
2494:
2411:
2376:
2345:
2291:
2196:
2173:
2132:
2095:of the agent with the
2064:
2018:
1974:
1934:
1882:
1812:
1742:
1672:
1598:
1558:
1502:
1436:
1415:
1390:
1389:{\displaystyle a>b}
1364:
1276:
1251:
1227:
1202:
1181:
1137:
1099:
1073:
1018:
998:
973:
972:{\displaystyle a>b}
947:
859:
834:
810:
785:
762:
719:
676:
636:
591:
528:
506:
473:
452:
407:
380:
356:
335:
302:
253:
233:
207:
186:
153:
101:
81:
3001:Apportionment methods
2768:
2693:
2632:
2558:
2521:
2495:
2391:
2377:
2346:
2271:
2197:
2174:
2133:
2065:
2019:
1980:for which the method
1975:
1935:
1883:
1813:
1743:
1673:
1599:
1559:
1503:
1437:
1416:
1391:
1365:
1277:
1252:
1228:
1203:
1182:
1138:
1100:
1074:
1031:on vectors. A vector
1029:majorization ordering
1019:
999:
974:
948:
860:
835:
811:
786:
763:
720:
677:
637:
592:
529:
507:
474:
453:
408:
406:{\displaystyle a_{i}}
381:
357:
315:
303:
254:
234:
208:
166:
154:
102:
82:
2643:
2583:
2533:
2504:
2388:
2357:
2208:
2186:
2142:
2122:
2028:
1988:
1944:
1904:
1822:
1752:
1682:
1612:
1568:
1528:
1454:
1426:
1400:
1374:
1286:
1261:
1241:
1212:
1192:
1149:
1110:
1084:
1063:
1008:
983:
957:
869:
844:
824:
795:
775:
729:
686:
646:
601:
546:
518:
491:
463:
425:
390:
370:
312:
266:
243:
223:
163:
111:
91:
71:
2181:electoral threshold
744:
701:
2867:Management Science
2763:
2627:
2553:
2516:
2490:
2372:
2341:
2192:
2169:
2128:
2060:
2014:
1970:
1930:
1878:
1808:
1738:
1668:
1594:
1554:
1498:
1448:largest-remainders
1432:
1414:{\displaystyle M'}
1411:
1386:
1360:
1275:{\displaystyle d'}
1272:
1247:
1226:{\displaystyle M'}
1223:
1198:
1177:
1133:
1098:{\displaystyle M'}
1095:
1069:
1014:
997:{\displaystyle M'}
994:
969:
943:
858:{\displaystyle d'}
855:
830:
809:{\displaystyle M'}
806:
781:
758:
732:
715:
689:
672:
632:
587:
524:
505:{\displaystyle M'}
502:
483:Majorization order
469:
448:
403:
376:
352:
298:
249:
229:
203:
149:
97:
77:
45:Jefferson's method
2965:978-3-319-64707-4
2918:978-3-319-64707-4
2873:(10): 1207–1227.
2683:
2649:
2415:
2214:
2195:{\displaystyle t}
2131:{\displaystyle a}
2118:, i.e. one where
2101:seat bias formula
1435:{\displaystyle M}
1250:{\displaystyle d}
1201:{\displaystyle M}
1072:{\displaystyle M}
1017:{\displaystyle M}
833:{\displaystyle d}
784:{\displaystyle M}
527:{\displaystyle M}
472:{\displaystyle h}
379:{\displaystyle h}
252:{\displaystyle h}
232:{\displaystyle i}
100:{\displaystyle n}
80:{\displaystyle h}
32:political parties
3008:
2975:
2974:
2973:
2972:
2939:
2928:
2927:
2926:
2925:
2892:
2883:
2882:
2858:
2849:
2848:
2836:
2826:
2789:Webster's method
2772:
2770:
2769:
2764:
2741:
2737:
2724:
2712:
2707:
2684:
2676:
2650:
2647:
2636:
2634:
2633:
2628:
2608:
2607:
2595:
2594:
2562:
2560:
2559:
2554:
2549:
2525:
2523:
2522:
2517:
2499:
2497:
2496:
2491:
2477:
2457:
2416:
2413:
2410:
2405:
2381:
2379:
2378:
2373:
2350:
2348:
2347:
2342:
2319:
2315:
2302:
2290:
2285:
2256:
2215:
2212:
2201:
2199:
2198:
2193:
2178:
2176:
2175:
2170:
2137:
2135:
2134:
2129:
2089:standard simplex
2069:
2067:
2066:
2061:
2059:
2058:
2046:
2045:
2023:
2021:
2020:
2015:
2013:
2012:
2000:
1999:
1979:
1977:
1976:
1971:
1969:
1968:
1956:
1955:
1939:
1937:
1936:
1931:
1929:
1928:
1916:
1915:
1890:Webster's method
1887:
1885:
1884:
1879:
1877:
1876:
1867:
1862:
1861:
1849:
1848:
1839:
1834:
1833:
1817:
1815:
1814:
1809:
1807:
1806:
1797:
1792:
1791:
1779:
1778:
1769:
1764:
1763:
1747:
1745:
1744:
1739:
1737:
1736:
1727:
1722:
1721:
1709:
1708:
1699:
1694:
1693:
1677:
1675:
1674:
1669:
1667:
1666:
1657:
1652:
1651:
1639:
1638:
1629:
1624:
1623:
1603:
1601:
1600:
1595:
1593:
1592:
1580:
1579:
1563:
1561:
1560:
1555:
1553:
1552:
1540:
1539:
1507:
1505:
1504:
1499:
1479:
1478:
1466:
1465:
1441:
1439:
1438:
1433:
1420:
1418:
1417:
1412:
1410:
1395:
1393:
1392:
1387:
1369:
1367:
1366:
1361:
1347:
1318:
1310:
1296:
1281:
1279:
1278:
1273:
1271:
1256:
1254:
1253:
1248:
1232:
1230:
1229:
1224:
1222:
1207:
1205:
1204:
1199:
1186:
1184:
1183:
1178:
1167:
1159:
1142:
1140:
1139:
1134:
1123:
1104:
1102:
1101:
1096:
1094:
1078:
1076:
1075:
1070:
1023:
1021:
1020:
1015:
1003:
1001:
1000:
995:
993:
978:
976:
975:
970:
952:
950:
949:
944:
930:
901:
893:
879:
864:
862:
861:
856:
854:
839:
837:
836:
831:
815:
813:
812:
807:
805:
790:
788:
787:
782:
767:
765:
764:
759:
757:
756:
740:
724:
722:
721:
716:
714:
713:
697:
681:
679:
678:
673:
671:
670:
658:
657:
641:
639:
638:
633:
622:
608:
596:
594:
593:
588:
577:
569:
558:
557:
542:, and for every
533:
531:
530:
525:
511:
509:
508:
503:
501:
478:
476:
475:
470:
457:
455:
454:
449:
438:
412:
410:
409:
404:
402:
401:
385:
383:
382:
377:
361:
359:
358:
353:
345:
344:
334:
329:
307:
305:
304:
299:
297:
296:
278:
277:
258:
256:
255:
250:
238:
236:
235:
230:
212:
210:
209:
204:
196:
195:
185:
180:
158:
156:
155:
150:
145:
144:
126:
125:
106:
104:
103:
98:
86:
84:
83:
78:
49:Webster's method
3016:
3015:
3011:
3010:
3009:
3007:
3006:
3005:
2981:
2980:
2979:
2978:
2970:
2968:
2966:
2940:
2931:
2923:
2921:
2919:
2893:
2886:
2859:
2852:
2845:
2827:
2810:
2805:
2781:
2720:
2708:
2697:
2692:
2688:
2675:
2646:
2644:
2641:
2640:
2603:
2599:
2590:
2586:
2584:
2581:
2580:
2573:
2565:party alliances
2545:
2534:
2531:
2530:
2505:
2502:
2501:
2473:
2453:
2412:
2406:
2395:
2389:
2386:
2385:
2358:
2355:
2354:
2298:
2286:
2275:
2270:
2266:
2252:
2211:
2209:
2206:
2205:
2187:
2184:
2183:
2143:
2140:
2139:
2123:
2120:
2119:
2109:
2091:), what is the
2085:
2077:proportionality
2054:
2050:
2041:
2037:
2029:
2026:
2025:
2008:
2004:
1995:
1991:
1989:
1986:
1985:
1964:
1960:
1951:
1947:
1945:
1942:
1941:
1924:
1920:
1911:
1907:
1905:
1902:
1901:
1898:
1872:
1868:
1863:
1857:
1853:
1844:
1840:
1835:
1829:
1825:
1823:
1820:
1819:
1802:
1798:
1793:
1787:
1783:
1774:
1770:
1765:
1759:
1755:
1753:
1750:
1749:
1732:
1728:
1723:
1717:
1713:
1704:
1700:
1695:
1689:
1685:
1683:
1680:
1679:
1662:
1658:
1653:
1647:
1643:
1634:
1630:
1625:
1619:
1615:
1613:
1610:
1609:
1588:
1584:
1575:
1571:
1569:
1566:
1565:
1548:
1544:
1535:
1531:
1529:
1526:
1525:
1522:
1474:
1470:
1461:
1457:
1455:
1452:
1451:
1427:
1424:
1423:
1403:
1401:
1398:
1397:
1375:
1372:
1371:
1343:
1311:
1306:
1289:
1287:
1284:
1283:
1264:
1262:
1259:
1258:
1242:
1239:
1238:
1235:divisor methods
1215:
1213:
1210:
1209:
1193:
1190:
1189:
1163:
1152:
1150:
1147:
1146:
1119:
1111:
1108:
1107:
1087:
1085:
1082:
1081:
1064:
1061:
1060:
1038:another vector
1009:
1006:
1005:
986:
984:
981:
980:
958:
955:
954:
926:
894:
889:
872:
870:
867:
866:
847:
845:
842:
841:
825:
822:
821:
818:divisor methods
798:
796:
793:
792:
776:
773:
772:
752:
748:
736:
730:
727:
726:
709:
705:
693:
687:
684:
683:
682:implies either
666:
662:
653:
649:
647:
644:
643:
618:
604:
602:
599:
598:
573:
562:
550:
549:
547:
544:
543:
519:
516:
515:
494:
492:
489:
488:
485:
464:
461:
460:
434:
426:
423:
422:
397:
393:
391:
388:
387:
371:
368:
367:
340:
336:
330:
319:
313:
310:
309:
292:
288:
273:
269:
267:
264:
263:
244:
241:
240:
224:
221:
220:
213:, representing
191:
187:
181:
170:
164:
161:
160:
140:
136:
121:
117:
112:
109:
108:
92:
89:
88:
72:
69:
68:
65:
17:
12:
11:
5:
3014:
3004:
3003:
2998:
2993:
2977:
2976:
2964:
2929:
2917:
2884:
2850:
2843:
2807:
2806:
2804:
2801:
2780:
2779:Empirical data
2777:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2740:
2736:
2733:
2730:
2727:
2723:
2719:
2716:
2711:
2706:
2703:
2700:
2696:
2691:
2687:
2682:
2679:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2626:
2623:
2620:
2617:
2614:
2611:
2606:
2602:
2598:
2593:
2589:
2572:
2569:
2552:
2548:
2544:
2541:
2538:
2515:
2512:
2509:
2489:
2486:
2483:
2480:
2476:
2472:
2469:
2466:
2463:
2460:
2456:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2409:
2404:
2401:
2398:
2394:
2371:
2368:
2365:
2362:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2318:
2314:
2311:
2308:
2305:
2301:
2297:
2294:
2289:
2284:
2281:
2278:
2274:
2269:
2265:
2262:
2259:
2255:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2191:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2127:
2116:divisor method
2108:
2105:
2084:
2081:
2057:
2053:
2049:
2044:
2040:
2036:
2033:
2011:
2007:
2003:
1998:
1994:
1967:
1963:
1959:
1954:
1950:
1927:
1923:
1919:
1914:
1910:
1897:
1894:
1875:
1871:
1866:
1860:
1856:
1852:
1847:
1843:
1838:
1832:
1828:
1805:
1801:
1796:
1790:
1786:
1782:
1777:
1773:
1768:
1762:
1758:
1735:
1731:
1726:
1720:
1716:
1712:
1707:
1703:
1698:
1692:
1688:
1665:
1661:
1656:
1650:
1646:
1642:
1637:
1633:
1628:
1622:
1618:
1591:
1587:
1583:
1578:
1574:
1551:
1547:
1543:
1538:
1534:
1521:
1518:
1497:
1494:
1491:
1488:
1485:
1482:
1477:
1473:
1469:
1464:
1460:
1431:
1409:
1406:
1385:
1382:
1379:
1359:
1356:
1353:
1350:
1346:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1317:
1314:
1309:
1305:
1302:
1299:
1295:
1292:
1270:
1267:
1246:
1221:
1218:
1197:
1176:
1173:
1170:
1166:
1162:
1158:
1155:
1132:
1129:
1126:
1122:
1118:
1115:
1093:
1090:
1068:
1013:
992:
989:
968:
965:
962:
942:
939:
936:
933:
929:
925:
922:
919:
916:
913:
910:
907:
904:
900:
897:
892:
888:
885:
882:
878:
875:
853:
850:
829:
804:
801:
780:
755:
751:
747:
743:
739:
735:
712:
708:
704:
700:
696:
692:
669:
665:
661:
656:
652:
631:
628:
625:
621:
617:
614:
611:
607:
586:
583:
580:
576:
572:
568:
565:
561:
556:
553:
534:if, for every
523:
500:
497:
484:
481:
468:
447:
444:
441:
437:
433:
430:
400:
396:
375:
351:
348:
343:
339:
333:
328:
325:
322:
318:
295:
291:
287:
284:
281:
276:
272:
248:
228:
202:
199:
194:
190:
184:
179:
176:
173:
169:
148:
143:
139:
135:
132:
129:
124:
120:
116:
96:
76:
64:
61:
34:. A method is
28:federal states
15:
9:
6:
4:
3:
2:
3013:
3002:
2999:
2997:
2994:
2992:
2989:
2988:
2986:
2967:
2961:
2957:
2953:
2949:
2945:
2938:
2936:
2934:
2920:
2914:
2910:
2906:
2902:
2898:
2891:
2889:
2880:
2876:
2872:
2868:
2864:
2857:
2855:
2846:
2844:0-300-02724-9
2840:
2835:
2834:
2825:
2823:
2821:
2819:
2817:
2815:
2813:
2808:
2800:
2798:
2794:
2791:is the least
2790:
2786:
2785:United States
2776:
2773:
2757:
2754:
2751:
2748:
2742:
2738:
2734:
2731:
2725:
2721:
2717:
2709:
2704:
2701:
2698:
2694:
2689:
2685:
2680:
2677:
2672:
2666:
2663:
2660:
2657:
2654:
2638:
2621:
2618:
2615:
2609:
2604:
2600:
2596:
2591:
2587:
2579:) with quota
2578:
2568:
2566:
2550:
2546:
2542:
2539:
2536:
2527:
2513:
2510:
2507:
2484:
2481:
2478:
2474:
2470:
2464:
2458:
2454:
2450:
2447:
2444:
2438:
2432:
2429:
2426:
2423:
2420:
2407:
2402:
2399:
2396:
2392:
2383:
2369:
2366:
2363:
2360:
2351:
2335:
2332:
2329:
2326:
2320:
2316:
2312:
2309:
2303:
2299:
2295:
2287:
2282:
2279:
2276:
2272:
2267:
2263:
2257:
2253:
2249:
2246:
2243:
2237:
2231:
2228:
2225:
2222:
2219:
2203:
2189:
2182:
2166:
2163:
2160:
2157:
2151:
2145:
2125:
2117:
2114:
2104:
2102:
2098:
2094:
2090:
2080:
2078:
2073:
2055:
2051:
2047:
2042:
2038:
2034:
2031:
2009:
2005:
2001:
1996:
1992:
1983:
1965:
1961:
1957:
1952:
1948:
1925:
1921:
1917:
1912:
1908:
1893:
1891:
1873:
1869:
1864:
1858:
1854:
1850:
1845:
1841:
1836:
1830:
1826:
1803:
1799:
1794:
1788:
1784:
1780:
1775:
1771:
1766:
1760:
1756:
1733:
1729:
1724:
1718:
1714:
1710:
1705:
1701:
1696:
1690:
1686:
1663:
1659:
1654:
1648:
1644:
1640:
1635:
1631:
1626:
1620:
1616:
1607:
1589:
1585:
1581:
1576:
1572:
1549:
1545:
1541:
1536:
1532:
1517:
1515:
1511:
1492:
1489:
1486:
1480:
1475:
1471:
1467:
1462:
1458:
1450:) with quota
1449:
1444:
1442:
1429:
1407:
1404:
1383:
1380:
1377:
1354:
1348:
1344:
1337:
1331:
1328:
1322:
1315:
1312:
1307:
1300:
1293:
1290:
1268:
1265:
1244:
1236:
1219:
1216:
1195:
1187:
1171:
1168:
1156:
1153:
1143:
1127:
1124:
1113:
1091:
1088:
1079:
1066:
1057:
1053:
1049:
1045:
1041:
1037:
1034:
1030:
1025:
1011:
990:
987:
966:
963:
960:
937:
931:
927:
920:
914:
911:
905:
898:
895:
890:
883:
876:
873:
851:
848:
827:
819:
802:
799:
778:
769:
753:
749:
745:
741:
737:
733:
710:
706:
702:
698:
694:
690:
667:
663:
659:
654:
650:
626:
623:
612:
609:
581:
578:
566:
563:
559:
554:
541:
537:
521:
514:
498:
495:
480:
466:
458:
442:
439:
428:
418:
416:
398:
394:
373:
365:
364:apportionment
349:
346:
341:
337:
331:
326:
323:
320:
316:
293:
289:
285:
282:
279:
274:
270:
260:
246:
226:
218:
217:
200:
197:
192:
188:
182:
177:
174:
171:
167:
141:
137:
133:
130:
127:
122:
118:
94:
74:
60:
58:
54:
53:Hill's method
50:
46:
42:
41:Droop's quota
37:
33:
29:
25:
21:
2969:, retrieved
2947:
2922:, retrieved
2900:
2870:
2866:
2832:
2782:
2774:
2639:
2574:
2528:
2384:
2352:
2204:
2112:
2110:
2100:
2096:
2092:
2086:
2076:
2071:
1981:
1899:
1605:
1523:
1513:
1509:
1445:
1422:
1145:
1106:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1032:
1026:
770:
539:
535:
512:
486:
421:
419:
414:
363:
362:, called an
261:
216:entitlements
215:
66:
57:Hare's quota
35:
19:
18:
1042:if for all
2985:Categories
2971:2021-09-01
2924:2021-09-01
2803:References
2113:stationary
1421:majorizes
1144:majorizes
2879:0025-1909
2752:−
2743:⋅
2732:−
2695:∑
2686:⋅
2610:⋅
2511:≥
2482:−
2465:⋅
2448:−
2439:≈
2393:∑
2364:≥
2330:−
2321:⋅
2310:−
2273:∑
2264:⋅
2247:−
2111:For each
2093:seat bias
1481:⋅
1370:whenever
1036:majorizes
953:whenever
746:≤
703:≥
610:∈
560:∈
317:∑
283:…
168:∑
131:…
30:or among
20:Seat bias
2648:MeanBias
2414:MeanBias
2213:MeanBias
1408:′
1316:′
1294:′
1269:′
1233:are two
1220:′
1157:′
1092:′
991:′
899:′
877:′
852:′
816:are two
803:′
742:′
699:′
567:′
555:′
499:′
386:, where
63:Notation
1396:, then
979:, then
2962:
2915:
2877:
2841:
2783:Using
2179:, and
1282:, and
1046:, the
865:, and
36:biased
2024:(for
1188:. If
308:with
159:with
55:, or
2960:ISBN
2913:ISBN
2875:ISSN
2839:ISBN
2540:>
1851:<
1781:>
1711:<
1641:>
1381:>
1329:>
1257:and
1208:and
964:>
912:>
840:and
791:and
660:<
597:and
538:and
2952:doi
2905:doi
1678:or
1024:.
771:If
725:or
366:of
43:or
2987::
2958:,
2946:,
2932:^
2911:,
2899:,
2887:^
2871:40
2869:.
2865:.
2853:^
2811:^
2526:.
2202::
2103:.
1892:.
1516:.
768:.
642:,
479:.
417:.
51:,
2954::
2907::
2881:.
2847:.
2761:)
2758:t
2755:n
2749:1
2746:(
2739:)
2735:1
2729:)
2726:i
2722:/
2718:1
2715:(
2710:n
2705:k
2702:=
2699:i
2690:(
2681:n
2678:s
2673:=
2670:)
2667:t
2664:,
2661:k
2658:,
2655:s
2652:(
2625:)
2622:s
2619:+
2616:h
2613:(
2605:i
2601:t
2597:=
2592:i
2588:q
2551:2
2547:/
2543:1
2537:r
2514:5
2508:n
2488:)
2485:1
2479:e
2475:/
2471:n
2468:(
2462:)
2459:2
2455:/
2451:1
2445:r
2442:(
2436:)
2433:0
2430:,
2427:k
2424:,
2421:r
2418:(
2408:n
2403:1
2400:=
2397:k
2370:n
2367:2
2361:h
2339:)
2336:t
2333:n
2327:1
2324:(
2317:)
2313:1
2307:)
2304:i
2300:/
2296:1
2293:(
2288:n
2283:k
2280:=
2277:i
2268:(
2261:)
2258:2
2254:/
2250:1
2244:r
2241:(
2238:=
2235:)
2232:t
2229:,
2226:k
2223:,
2220:r
2217:(
2190:t
2167:r
2164:+
2161:a
2158:=
2155:)
2152:a
2149:(
2146:d
2126:a
2097:k
2072:M
2056:2
2052:a
2048:+
2043:1
2039:a
2035:=
2032:h
2010:2
2006:a
2002:,
1997:1
1993:a
1982:M
1966:2
1962:t
1958:,
1953:1
1949:t
1926:2
1922:a
1918:,
1913:1
1909:a
1874:2
1870:t
1865:/
1859:2
1855:a
1846:1
1842:t
1837:/
1831:1
1827:a
1804:2
1800:t
1795:/
1789:2
1785:a
1776:1
1772:t
1767:/
1761:1
1757:a
1734:2
1730:t
1725:/
1719:2
1715:a
1706:1
1702:t
1697:/
1691:1
1687:a
1664:2
1660:t
1655:/
1649:2
1645:a
1636:1
1632:t
1627:/
1621:1
1617:a
1606:.
1590:2
1586:t
1582:,
1577:1
1573:t
1550:2
1546:t
1542:,
1537:1
1533:t
1514:s
1510:s
1496:)
1493:s
1490:+
1487:h
1484:(
1476:i
1472:t
1468:=
1463:i
1459:q
1430:M
1405:M
1384:b
1378:a
1358:)
1355:b
1352:(
1349:d
1345:/
1341:)
1338:a
1335:(
1332:d
1326:)
1323:b
1320:(
1313:d
1308:/
1304:)
1301:a
1298:(
1291:d
1266:d
1245:d
1217:M
1196:M
1175:)
1172:h
1169:,
1165:t
1161:(
1154:M
1131:)
1128:h
1125:,
1121:t
1117:(
1114:M
1089:M
1067:M
1056:b
1052:a
1048:k
1044:k
1040:b
1033:a
1012:M
988:M
967:b
961:a
941:)
938:b
935:(
932:d
928:/
924:)
921:a
918:(
915:d
909:)
906:b
903:(
896:d
891:/
887:)
884:a
881:(
874:d
849:d
828:d
800:M
779:M
754:j
750:a
738:j
734:a
711:i
707:a
695:i
691:a
668:j
664:t
655:i
651:t
630:)
627:h
624:,
620:t
616:(
613:M
606:a
585:)
582:h
579:,
575:t
571:(
564:M
552:a
540:h
536:t
522:M
496:M
467:h
446:)
443:h
440:,
436:t
432:(
429:M
415:i
399:i
395:a
374:h
350:h
347:=
342:i
338:a
332:n
327:1
324:=
321:i
294:n
290:a
286:,
280:,
275:1
271:a
247:h
227:i
201:1
198:=
193:i
189:t
183:n
178:1
175:=
172:i
147:)
142:n
138:t
134:,
128:,
123:1
119:t
115:(
95:n
75:h
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