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2 2 population paradox 2 3 new states paradox 3 balinski young theorem 4 references 5 external links toggle the table of contents apportionment paradox 2 languages deutsch 한국어 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia pathological behavior by an apportionment rule a joint politics and economics series social choice and electoral systems social choice mechanism design comparative politics comparison list by country single winner methods single vote plurality methods approval voting first preference plurality fpp two round us jungle primary partisan primary instant runoff uk nz alternative vote av au preferential voting us ca ranked choice rcv party block voting plurality block voting condorcet methods condorcet irv round robin voting minimax kemeny schulze ranked pairs maximal lottery positional voting plurality el irv borda count el baldwin mdn bucklin antiplurality el coombs rated cardinal voting score voting approval voting combined approval voting majority judgment star voting proportional representation party list apportionment highest averages d hondt sainte laguë huntington hill largest remainders hare niemeyer hagenbach bischoff imperiali national remnant biproportional list type closed list localized list open list free list no list quota remainder methods single transferable vote hare droop schulze stv cpo stv quota borda approval based committees thiele s method phragmén s method expanding approvals rule method of equal shares fractional social choice direct representation interactive representation liquid democracy fractional approval voting maximal lottery random ballot semi proportional representation cumulative sntv limited voting mixed systems by results of combination mixed member majoritarian mixed member proportional by mechanism of combination non compensatory parallel superposition coexistence conditional fusion majority bonus compensatory seat linkage system uk ams nz mmp vote linkage system negative vote transfer mixed ballot supermixed systems dual member mixed proportional rural urban proportional majority jackpot by ballot type single vote double simultaneous vote dual vote paradoxes and pathologies spoiler effects spoiler effect cloning paradox frustrated majorities paradox center squeeze pathological response perverse response apportionment paradox best is worst paradox no show paradox multiple districts paradox strategic voting lesser evil voting exaggeration truncation turkey raising wasted vote paradoxes of majority rule tyranny of the majority discursive dilemma conflicting majorities paradox social and collective choice impossibility theorems arrow s theorem majority impossibility moulin s impossibility theorem mckelvey schofield chaos theorem gibbard s theorem positive results median voter theorem condorcet s jury theorem may s theorem condorcet dominance theorems harsanyi s utilitarian theorem non classical mechanisms vcg mechanism quadratic voting supermajority unanimity politics portal economics portal mathematics portal v t e an apportionment paradox is a situation where an apportionment a rule for dividing discrete objects according to some proportional relationship produces results that violate notions of common sense or fairness certain quantities like milk can be divided in any proportion whatsoever others such as horses cannot only whole numbers will do in the latter case there is an inherent tension between the desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values several paradoxes related to apportionment and fair division have been identified in some cases simple adjustments to an apportionment methodology can resolve observed paradoxes however as shown by the balinski young theorem it is not always possible to provide a perfectly fair resolution that satisfies all competing fairness criteria 1 227 235 history edit an example of the apportionment paradox known as the alabama paradox was discovered in the context of united states congressional apportionment in 1880 1 228 231 when census calculations found that if the total number of seats in the house of representatives were hypothetically increased this would decrease alabama s seats from 8 to 7 an actual impact was observed in 1900 when virginia lost a seat to maine even though virginia s population was growing more rapidly this is an example of the population paradox 1 231 232 in 1907 when oklahoma became a state new york lost a seat to maine thus the name the new state paradox 1 232 233 2 the method for apportionment used during this period originally put forth by alexander hamilton but vetoed by george washington and not adopted until 1852 1 228 was as follows first the fair share of each state is computed i e the proportional share of seats that each state would get if fractional values were allowed second each state receives as many seats as the whole number portion of its fair share third any state whose fair share is less than one receives one seat regardless of population as required by the united states constitution fourth any remaining seats are distributed one each to the states whose fair shares have the highest fractional parts the hamilton method replaced a rounding method proposed by thomas jefferson 1 228 and was itself replaced by the huntington hill method in 1941 1 examples of paradoxes edit this section needs expansion with lexmann paradox you can help by adding missing information september 2023 alabama paradox edit the alabama paradox was the first of the apportionment paradoxes to be discovered the us house of representatives is constitutionally required to allocate seats based on population counts which are required every 10 years the size of the house is set by statute after the 1880 census c w seaton chief clerk of the united states census bureau computed apportionments for all house sizes between 275 and 350 and discovered that alabama would get eight seats with a house size of 299 but only seven with a house size of 300 1 228 231 in general the term alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares a similar exercise by the census bureau after the 1900 census computed apportionments for all house sizes between 350 and 400 colorado would have received three seats in all cases except with a house size of 357 in which case it would have received two 3 the following is a simplified example following the largest remainder method with three states and 10 seats and 11 seats with 10 seats with 11 seats state population fair share seats fair share seats a 6 4 286 4 4 714 5 b 6 4 286 4 4 714 5 c 2 1 429 2 1 571 1 observe that state c s share decreases from 2 to 1 with the added seat in this example of a 10 increase in the number of seats each state s share increases by 10 however increasing the number of seats by a fixed increases the fair share more for larger numbers i e large states more than small states in particular large a and b had their fair share increase faster than small c therefore the fractional parts for a and b increased faster than those for c in fact they overtook c s fraction causing c to lose its seat since the hamilton method allocates according to which states have the largest fractional remainder the alabama paradox gave rise to the axiom known as house monotonicity which says that when the house size increases the allocations of all states should weakly increase population paradox edit see also state population monotonicity the population paradox is a counterintuitive result of some procedures for apportionment when two states have populations increasing at different rates a small state with rapid growth can lose a legislative seat to a big state with slower growth some of the earlier congressional apportionment methods such as hamilton could exhibit the population paradox in 1900 virginia lost a seat to maine even though virginia s population was growing more rapidly 1 231 232 however divisor methods such as the current method do not citation needed new states paradox edit see also coherence fairness independence of irrelevant alternatives and spoiler effect the new state paradox is a property of all largest remainder methods in which the entry of a new state or party causes a change in the apportionment between two completely unrelated parties despite not changing their combined apportionment the paradox was first noticed in 1907 when oklahoma became a state it was decided it should be given its entitled fair share of 5 seats and the house expanded by 5 seats as well with the intent of maintaining all other state delegations fixed however when the apportionment was recalculated it was discovered these were not fixed new york had lost a seat while maine had gained one this resulted in a crisis of confidence in the apportionment rules procedure as well as a bitter debate within the house as it was unclear which of these apportionments the one calculated from scratch or the one calculated after the previous census should be considered the correct one 1 232 233 2 the new state paradox gave rise to the axiom known as coherence which says that whenever an apportionment rule is applied to a subset of the states with the subset of seats allocated to them the outcome should be the same as in the overall solution for all states this can be considered the analog of independence of spoilers seen in other areas of social choice that one part of the apportionment the assignment of seats between a and b should not depend on another unrelated part of the apportionment the assignment of seats between c and d balinski young theorem edit in 1982 two mathematicians michel balinski and peyton young proved that any method of apportionment that does not violate the quota rule will result in paradoxes whenever there are four or more parties or states regions etc 4 5 more precisely their theorem states that there is no apportionment system that has the following properties for more than three parties 1 233 234 as the example we take the division of seats between parties in a system of proportional representation it avoids violations of the quota rule each of the parties gets one of the two numbers closest to its fair share of seats for example if a party s fair share is 7 34 seats it must get either 7 or 8 seats to avoid a violation any other number will violate the rule it does not have the population paradox if party a gets more votes and party b gets fewer votes no seat will be transferred from a to b it is of note that any method of apportionment free of the population paradox will always be free of alabama paradox the converse is not true however webster s method can be free of incoherence and maintain quota when there are three parties all sensible methods satisfy both criteria in the trivial two party case 4 5 they show a proof of impossibility apportionment methods may have a subset of these properties but cannot have all of them a method may be free of both the alabama paradox and the population paradox these methods are divisor methods and huntington hill the method currently used to apportion house of representatives seats is one of them however these methods will necessarily fail to always follow quota in other circumstances no method may always follow quota and be free of the population paradox 6 the division of seats in an election is a prominent cultural concern in 1876 the united states presidential election turned on the method by which the remaining fraction was calculated rutherford hayes received 185 electoral college votes and samuel tilden received 184 tilden won the popular vote with a different rounding method the final electoral college tally would have reversed 1 228 however many mathematically analogous situations arise in which quantities are to be divided into discrete equal chunks 1 233 the balinski young theorem applies in these situations it indicates that although very reasonable approximations can be made there is no mathematically rigorous way to reconcile the small remaining fraction while complying with all the competing fairness elements 1 233 in general the response from mathematicians has been to abandon the quota rule as the less important property accepting that apportionment errors may sometimes slightly exceed one seat a method may follow quota and be free of the alabama paradox balinski and young constructed a method that does so although it is not in common political use 7 references edit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 stein james d 2008 how math explains the world a guide to the power of numbers from car repair to modern physics new york smithsonian books isbn 9780061241765 1 2 caulfield michael j november 2010 apportioning representatives in the united states congress paradoxes of apportionment convergence mathematical association of america doi 10 4169 loci003163 inactive 11 july 2025 cite journal cs1 maint doi inactive as of july 2025 link bogomolny alex january 2002 the constitution and paradoxes cut the knot 1 2 balinski michel l young h peyton 1982 fair representation meeting the ideal of one man one vote new haven yale university press isbn 0 300 02724 9 1 2 balinski michel l young h peyton 2001 fair representation meeting the ideal of one man one vote 2nd ed washington dc brookings institution press isbn 0 8157 0111 x balinski michel l young h peyton september 1980 the theory of apportionment pdf working papers international institute for applied systems analysis wp 80 131 balinski michel l young h peyton november 1974 a new method for congressional apportionment proceedings of the national academy of sciences 71 11 4602 4606 bibcode 1974pnas 71 4602b doi 10 1073 pnas 71 11 4602 pmc 433936 pmid 16592200 external links edit the constitution and paradoxes apportionment balinski and young s contribution v t e paradoxes philosophical analysis buridan s bridge dream argument epicurean fiction fitch s knowability free will goodman s hedonism liberal meno s mere addition moore s nihilism omnipotence preface rule following sorites theseus ship white horse zeno s logical barber berry bhartrhari s burali forti court crocodile curry s epimenides free choice paradox grelling nelson kleene rosser liar card no no pinocchio quine s yablo s opposite day paradoxes of set theory richard s russell s socratic temperature paradox barbershop catch 22 chicken or the egg drinker entailment lottery plato s beard raven ross s unexpected hanging what the tortoise said to achilles heat death paradox olbers s paradox economic allais antitrust arrow info...