State-population monotonicity

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State-population monotonicity is a property of apportionment methods, which are methods of allocating seats in a parliament among federal states or political parties. The property says that if the population of State A increases faster than that of State B, then State A should not lose any seats to State B. Apportionment methods violating this rule are called population paradoxes.

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In the apportionment literature, this property can sometimes simply be called population monotonicity. [1] :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. Conversely, in the domain of legislative seat apportionment, the term "population" refers 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 [2] : when an agent's entitlement increases, their utility should not decrease.

Population-pair monotonicity

Pairwise monotonicity says that if the ratio between the entitlements of two states increases, then state should not gain seats at the expense of state . In other words, a shrinking state should not "steal" a seat from a growing state. This property is also called vote-ratio monotonicity .

Weak monotonicity

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, since a voter can help their party by not voting. The largest-remainder method with the Droop quota fails voter monotonicity. [3] :Sub.9.14

Strong Monotonicity

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. [1] :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.

Static population-monotonicity ("concordance")

Static population-monotonicity [4] :147, also called concordance [5] :75, says that a state with a larger population should not receive a smaller allocation. Formally, if then .

All apportionment methods must be concordant (by definition, to be considered an apportionment method); occasionally this requires using a "tiebreaking" rule, such as assigning ties to the largest state.

Related Research Articles

The D'Hondt method, also called the Jefferson method or the greatest divisors method, is an apportionment method for allocating seats in parliaments among federal states, or in proportional representation among political parties. It belongs to the class of highest-averages methods.

The Webster method, also called the Sainte-Laguë method, is a highest averages apportionment method for allocating seats in a parliament among federal states, or among parties in a party-list proportional representation system.

In mathematics, economics, and political science, the highest averages methods, also called divisor methods, are a class of apportionment algorithms for proportional representation. Divisor algorithms seek to fairly divide a legislature between agents. More generally, divisor methods are used to divide or round a whole number of objects being used to represent (non-whole) shares of a total.

The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with various highest averages methods.

An apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense.

Congressional districts, also known as electoral districts in other nations, are divisions of a larger administrative region that represent the population of a region in the larger congressional body. Countries with congressional districts include the United States, the Philippines, and Japan.

Apportionment is the process by which seats in a legislative body are distributed among administrative divisions, such as states or parties, entitled to representation. This page presents the general principles and issues related to apportionment. The page Apportionment by country describes specific practices used around the world. The page Mathematics of apportionment describes mathematical formulations and properties of apportionment rules.

Majority judgment (MJ) is a single-winner voting system proposed in 2010 by Michel Balinski and Rida Laraki. It is a kind of highest median rule, a cardinal voting system that elects the candidate with the highest median rating.

Biproportional apportionment is a proportional representation method to allocate seats in proportion to two separate characteristics. That is, for two different partitions each part receives the proportional number of seats within the total number of seats. For instance, this method could give proportional results by party and by region, or by party and by gender/ethnicity, or by any other pair of characteristics.

  1. Example: proportional by party and by region
  2. Then, as nearly as possible given the totals for each region and each party:

In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings of its fractional proportional share. As an example, if a party deserves 10.56 seats out of 15, the quota rule states that when the seats are allotted, the party may get 10 or 11 seats, but not lower or higher. Many common election methods, such as all highest averages methods, violate the quota rule.

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Population monotonicity (PM) is a principle of consistency in allocation problems. It says that, when the set of agents participating in the allocation changes, the utility of all agents should change in the same direction. For example, if the resource is good, and an agent leaves, then all remaining agents should receive at least as much utility as in the original allocation.

House monotonicity is a property of apportionment methods. These are methods for allocating seats in a parliament among federal states. The property says that, if the number of seats in the "house" increases, and the method is re-activated, then no state should have fewer seats than it previously had. A method that fails to satisfy house-monotonicity is said to have the Alabama paradox.

Seat bias is a property describing methods of apportionment. These are methods used to allocate seats in a parliament among federal states or among political parties. A method is biased if it systematically favors small parties over large parties, or vice versa. There are various ways to compute the bias of apportionment methods.

Coherence, also called uniformity or consistency, is a criterion for evaluating rules for fair division. Coherence requires that the outcome of a fairness rule is fair not only for the overall problem, but also for each sub-problem. Every part of a fair division should be fair.

Optimal apportionment is an approach to apportionment that is based on mathematical optimization.

Vote-ratio monotonicity (VRM) is a property of apportionment methods, which are methods of allocating seats in a parliament among political parties. The property says that, if the ratio between the number of votes won by party A to the number of votes won by party B increases, then it should NOT happen that party A loses a seat while party B gains a seat.

Balance or balancedness is a property of apportionment methods, which are methods of allocating identical items between among agens, such as dividing seats in a parliament among political parties or federal states. The property says that, if two agents have exactly the same entitlements, then the number of items they receive should differ by at most one. So if two parties win the same number of votes, or two states have the same populations, then the number of seats they receive should differ by at most one.

In mathematics and apportionment theory, a signpost sequence is a sequence of real numbers, called signposts, used in defining generalized rounding rules. A signpost sequence defines a set of signposts that mark the boundaries between neighboring whole numbers: a real number less than the signpost is rounded down, while numbers greater than the signpost are rounded up.

In apportionment theory, rank-index methods are a set of apportionment methods that generalize the divisor method. These have also been called Huntington methods, since they generalize an idea by Edward Vermilye Huntington.

References

  1. 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.
  2. Chakraborty, Mithun; Schmidt-Kraepelin, Ulrike; Suksompong, Warut (2021-04-29). "Picking sequences and monotonicity in weighted fair division". Artificial Intelligence. 301: 103578. arXiv: 2104.14347 . doi:10.1016/j.artint.2021.103578. S2CID   233443832.
  3. Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Securing System Consistency: Coherence and Paradoxes", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 159–183, doi:10.1007/978-3-319-64707-4_9, ISBN   978-3-319-64707-4 , retrieved 2021-09-02
  4. 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.
  5. Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN   978-3-319-64707-4 , retrieved 2021-09-01