1311:(share of the population) increases, then its apportionment should not decrease, regardless of what happens to any other state's entitlement. This variant is extremely strong, however: whenever there are at least 3 states, and the house size is not exactly equal to the number of states, no apportionment method is strongly monotone for a fixed house size. Strong monotonicity failures in divisor methods happen when one state's entitlement increases, causing it to "steal" a seat from another state whose entitlement is unchanged.
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1567:
Chakraborty, Mithun; Schmidt-Kraepelin, Ulrike; Suksompong, Warut (2021-04-29). "Picking sequences and monotonicity in weighted fair division".
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However, it is worth noting that the traditional form of the divisor method, which involves using a
1285:, do not satisfy VRM, and thus exhibit the population paradox. For example, after the 1900 census,
757:
591:
244:
1594:
1576:
917:
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399:
183:
866:
1685:
How Math
Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics
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27:
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divisor and allowing the house size to vary, satisfies strong monotonicity in this sense.
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47:
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193:
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1739:
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771:
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259:
97:
60:
35:
1278:. In other words, a shrinking state should not "steal" a seat from a growing state.
1330:
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275:
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1538:
389:
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Palomares, Pukelsheim and
Ramirez proved that very apportionment rule that is
357:
352:
899:
1772:
Proportional
Representation: Apportionment Methods and Their Applications
1647:
Proportional
Representation: Apportionment Methods and Their Applications
1531:
Proportional
Representation: Apportionment Methods and Their Applications
1286:
608:
954:"Population-ratio monotonicity" redirects here. Not to be confused with
394:
1363:
Vote-ratio monotonicity implies that, if population moves from state
1581:
1333:
proved that an apportionment method is VRM if-and-only-if it is a
1290:
1403:
while the populations of other states do not change, then both
1293:, even though Virginia's population was growing more rapidly.
1774:, Cham: Springer International Publishing, pp. 159–183,
1649:, Cham: Springer International Publishing, pp. 159–183,
1533:, Cham: Springer International Publishing, pp. 159–183,
1177:
gains a seat. Apportionments violating this rule are called
46:
1766:
Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.),
1741:
1712:
1641:
Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.),
1616:
1525:
Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.),
1768:"Securing System Consistency: Coherence and Paradoxes"
1643:"Securing System Consistency: Coherence and Paradoxes"
1527:"Securing System Consistency: Coherence and Paradoxes"
1301:
A stronger variant of population monotonicity, called
1452:
1409:
1389:
1369:
1264:
1244:
1218:
1163:
1143:
1115:
1092:
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1109:
More formally, if the ratio of votes or populations
1738:
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1169:
1149:
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994:
1200:exhibits both population and no-show paradoxes.
1737:Balinski, Michel L.; Young, H. Peyton (1982).
1708:Balinski, Michel L.; Young, H. Peyton (1982).
1612:Balinski, Michel L.; Young, H. Peyton (1982).
1258:should not gain seats at the expense of state
935:
8:
1184:A particularly severe variant, where voting
1281:Some earlier apportionment rules, such as
942:
928:
18:
1580:
1473:
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1414:
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1305:monotonicity requires that, if a state's
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1114:
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1071:
1051:
1027:
1007:
987:
1514:
1212:between the entitlements of two states
1208:Pairwise monotonicity says that if the
34:
1520:
1518:
7:
1745:. New Haven: Yale University Press.
1716:. New Haven: Yale University Press.
1620:. New Haven: Yale University Press.
14:
16:Property of apportionment methods
1482:{\displaystyle a_{j}'\leq a_{j}}
1439:{\displaystyle a_{i}'\geq a_{i}}
911:
898:
886:
834:McKelvey–Schofield chaos theorem
480:Semi-proportional representation
112:First preference plurality (FPP)
1687:. New York: Smithsonian Books.
872:Harsanyi's utilitarian theorem
829:Moulin's impossibility theorem
794:Conflicting majorities paradox
1:
1814:Apportionment method criteria
1157:should not lose a seat while
972:population-ratio monotonicity
698:Frustrated majorities paradox
1591:10.1016/j.artint.2021.103578
1322:Relation to other properties
1204:Population-pair monotonicity
1002:grows at a faster rate than
867:Condorcet dominance theorems
807:Social and collective choice
1780:10.1007/978-3-319-64707-4_9
1655:10.1007/978-3-319-64707-4_9
1539:10.1007/978-3-319-64707-4_9
533:By mechanism of combination
304:Proportional representation
1830:
1086:should not lose a seat to
953:
731:Multiple districts paradox
462:Fractional approval voting
450:Interactive representation
1198:largest remainders method
678:Paradoxes and pathologies
527:Mixed-member proportional
522:Mixed-member majoritarian
517:By results of combination
408:Approval-based committees
1683:Stein, James D. (2008).
1360:is vote-ratio monotone.
857:Condorcet's jury theorem
658:Double simultaneous vote
633:Rural–urban proportional
628:Dual-member proportional
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546:Parallel (superposition)
438:Fractional social choice
425:Expanding approvals rule
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956:Population monotonicity
784:Tyranny of the majority
561:Fusion (majority bonus)
378:Quota-remainder methods
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1238:increases, then state
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1016:
996:
978:. It says that if the
974:is a property of some
918:Mathematics portal
824:Majority impossibility
813:Impossibility theorems
609:Negative vote transfer
430:Method of equal shares
51:
1499:Apportionment paradox
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98:Single-winner methods
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71:Comparative politics
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1297:Strong monotonicity
1231:{\displaystyle i,j}
1192:seats, is called a
1130:{\displaystyle A/B}
893:Politics portal
604:Vote linkage system
575:Seat linkage system
162:Ranked-choice (RCV)
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623:Supermixed systems
326:Largest remainders
184:Round-robin voting
52:
1789:978-3-319-64707-4
1664:978-3-319-64707-4
1548:978-3-319-64707-4
1396:{\displaystyle j}
1376:{\displaystyle i}
1283:Hamilton's method
1271:{\displaystyle i}
1251:{\displaystyle j}
1170:{\displaystyle B}
1150:{\displaystyle A}
1099:{\displaystyle B}
1079:{\displaystyle A}
1059:{\displaystyle B}
1035:{\displaystyle A}
1015:{\displaystyle B}
995:{\displaystyle A}
952:
951:
839:Gibbard's theorem
779:Dominance paradox
716:Perverse response
420:Phragmen's method
286:Majority judgment
214:Positional voting
172:Condorcet methods
40:electoral systems
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846:Positive results
741:Strategic voting
638:Majority jackpot
595:
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455:Liquid democracy
331:National remnant
321:Highest averages
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151:Alternative vote
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133:Partisan primary
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66:Mechanism design
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693:Cloning paradox
683:Spoiler effects
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467:Maximal lottery
434:
415:Thiele's method
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281:Approval voting
269:Cardinal voting
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204:Maximal lottery
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772:majority rule
770:Paradoxes of
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222:
219:
218:
217:
216:
215:
205:
202:
200:
197:
195:
192:
190:
187:
186:
185:
182:
180:
179:Condorcet-IRV
177:
176:
175:
174:
173:
163:
158:
154:
152:
147:
143:
142:
141:
138:
134:
131:
130:
128:
123:
118:
115:
113:
110:
109:
108:
106:
99:
94:
93:
86:
82:
79:
77:
74:
72:
69:
67:
64:
62:
61:Social choice
59:
58:
56:
55:
49:
45:
44:
41:
37:
36:Social choice
33:
29:
25:
21:
20:
1793:, retrieved
1771:
1761:
1740:
1732:
1711:
1703:
1684:
1678:
1668:, retrieved
1646:
1636:
1615:
1607:
1572:
1568:
1562:
1552:, retrieved
1530:
1362:
1339:
1325:
1315:
1313:
1307:
1302:
1300:
1280:
1209:
1207:
1189:
1185:
1183:
1178:
1107:
971:
968:weight-ratio
967:
965:
961:
960:
845:
844:
811:
769:
768:
753:Exaggeration
739:
738:
709:
708:
682:
646:
645:
614:Mixed ballot
569:Compensatory
567:
540:compensatory
537:
532:
516:
478:
477:
436:
435:
406:
405:
376:
375:
363:List-free PR
308:
276:Score voting
267:
266:
212:
211:
199:Ranked pairs
170:
169:
102:
1489:must hold.
1308:entitlement
980:entitlement
653:Single vote
556:Conditional
551:Coexistence
400:Quota Borda
390:Schulze STV
348:Closed list
291:STAR voting
236:Borda count
1795:2021-09-02
1670:2021-09-02
1582:2104.14347
1575:: 103578.
1554:2021-09-02
1510:References
1354:homogenous
1350:concordant
1046:more than
962:Vote-ratio
758:Truncation
487:Cumulative
310:Party-list
85:By country
76:Comparison
1599:233443832
1467:≤
1424:≥
1383:to state
1342:anonymous
665:Dual-vote
358:Panachage
353:Open list
343:List type
221:Plurality
117:Two-round
105:plurality
28:Economics
1808:Category
1493:See also
1463:′
1420:′
1358:coherent
1346:balanced
1327:Balinski
1287:Virginia
385:Hare STV
24:Politics
22:A joint
395:CPO-STV
245:Baldwin
194:Schulze
189:Minimax
107:methods
1786:
1749:
1720:
1691:
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1597:
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1356:, and
1303:strong
1196:. The
1042:grows
1022:(i.e.
260:Coombs
30:series
1595:S2CID
1577:arXiv
1331:Young
1316:fixed
1291:Maine
1210:ratio
970:, or
597:'MMP'
586:'AMS'
1784:ISBN
1747:ISBN
1718:ISBN
1689:ISBN
1659:ISBN
1622:ISBN
1543:ISBN
1446:and
1329:and
1190:lose
982:for
538:Non-
492:SNTV
81:List
38:and
26:and
1776:doi
1651:doi
1587:doi
1573:301
1535:doi
1186:for
1066:),
256:el.
241:el.
230:IRV
226:el.
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1770:,
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1593:.
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1517:^
1352:,
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592:NZ
581:UK
157:US
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122:US
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1653::
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1471:a
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1371:i
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1220:i
1165:B
1145:A
1125:B
1121:/
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1106:.
1094:B
1074:A
1054:B
1030:A
1010:B
990:A
964:,
958:.
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594::
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253:(
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238:(
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119:(
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