State-population monotonicity: Difference between revisions
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#REDIRECT [[Vote-ratio monotonicity]] |
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{{Short description|Criterion for apportionment methods}} |
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'''State-population monotonicity''' is a property of [[Mathematics of apportionment|apportionment methods]], which are methods of allocating seats in a [[parliament]] among [[federal states]] or [[Political party|political parties]]. The property says that, if the population of a state increases faster than that of other states, then it should not lose a seat. Apportionments violating this rule are called '''population paradoxes'''. |
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In the apportionment literature, this property can sometimes simply be called '''population monotonicity'''.<ref name=":022">{{cite book |last1=Balinski |first1=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}</ref>{{Rp|Sec.4}} However, the term "population monotonicity" is more commonly however used to denote a very different property of resource-allocation rules within that realm. Specifically, as it relates to the concept of [[population monotonicity]], the term "population" refers to the set of agents participating in the division process. A population increase means that the previously-present agents are entitled to fewer items, as there are more mouths to feed. |
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Conversely, when it comes to the domain of legislative seat apportionment, the term "population" in the name relates to the population of an individual state, which determines the state's ''entitlement.'' A population-increase means that a state is entitled to more seats. The parallel property in fair division is called '''weight monotonicity''':<ref>{{cite journal |last1=Chakraborty |first1=Mithun |last2=Schmidt-Kraepelin |first2=Ulrike |last3=Suksompong |first3=Warut |date=2021-04-29 |title=Picking sequences and monotonicity in weighted fair division |journal=Artificial Intelligence |volume=301 |page=103578 |arxiv=2104.14347 |doi=10.1016/j.artint.2021.103578 |s2cid=233443832}}</ref> when an agent's entitlement increases, their utility should not decrease. |
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== Population-pair monotonicity == |
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Pairwise monotonicity says that if the ''ratio'' between the entitlements of two states <math>i, j</math> increases, then state <math>j</math> should not gain seats at the expense of state <math>i</math>. In other words, a shrinking state should not "steal" a seat from a growing state. This property is also called '''[[vote-ratio monotonicity]]'''. |
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== {{Anchor|weak}}Weak monotonicity == |
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'''Weak monotonicity''', also called '''voter monotonicity''', is a property weaker than pairwise-PM. It says that, if party ''i'' attracts more voters, while all other parties keep the same number of voters, then party ''i'' must not lose a seat. Failure of voter monotonicity is called the [[No show paradox|no-show paradox]], since a voter can help their party by not voting. The largest-remainder method with the Droop quota fails voter monotonicity.<ref name=":12">{{Citation |last=Pukelsheim |first=Friedrich |title=Securing System Consistency: Coherence and Paradoxes |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=159–183 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_9 |access-date=2021-09-02 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_9 |isbn=978-3-319-64707-4}}</ref>{{Rp|Sub.9.14}} |
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== Strong Monotonicity == |
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A stronger variant of population monotonicity requires that, if a state's ''entitlement'' (share of the population) increases, then its apportionment should not decrease, regardless of what happens to any other state's entitlement. This variant is too 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.<ref name=":022" />{{Rp|Thm.4.1}} 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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== Static population-monotonicity ("concordance") == |
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'''Static population-monotonicity<ref name=":03">{{cite book |last=Balinski |first=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}</ref>{{Rp|147}}''', also called '''concordance<ref name=":42">{{Citation |last=Pukelsheim |first=Friedrich |title=Divisor Methods of Apportionment: Divide and Round |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=71–93 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_4 |access-date=2021-09-01 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_4 |isbn=978-3-319-64707-4}}</ref>{{Rp|75}}''', says that a state with a larger population should not receive a smaller allocation. Formally, if <math>t_i > t_j</math> then <math>a_i \geq a_j</math>. |
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All apportionment methods must be concordant (by definition, to be considered an apportionment method); occassionally this requires using a "tiebreaking" rule, such as assigning ties to the largest state. |
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== References == |
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<references /> |
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[[Category:Apportionment method criteria]] |
[[Category:Apportionment method criteria]] |
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