State-population monotonicity or vote 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. Apportionments violating this rule are called population paradoxes.
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.
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, 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
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.
In the study of electoral systems, the Droop quota is the minimum number of votes needed for a party or candidate to guarantee they will win at least one seat in a legislature.
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. The Sainte-Laguë method shows a more equal seats-to-votes ratio for different sized parties among apportionment methods.
In mathematics, economics, and social choice theory, the highest averages or divisor methods, sometimes called divide-and-round, are a family of apportionment algorithms that aim to fairly divide a legislature between several groups, such as political parties or states. More generally, divisor methods can be used to round shares of a total, e.g. percentage points. The two names for these methods—highest averages and divisors—reflect two different ways of thinking about them, and their two independent inventions. However, both procedures are equivalent and give the same answer.
The largest remainder methods or quota methods are methods of allocating seats proportionally that are based on calculating a quota, i.e. a certain number of votes needed to be guaranteed a seat in parliament. Then, any leftover seats are handed over to "plurality" winners. They are typically contrasted with the more popular highest averages methods.
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.
In the study of apportionment, the Harequota is the number of voters represented by each legislator under an idealized system of proportional representation, where every legislator represents an equal number of voters. The Hare quota is the total number of votes divided by the number of seats to be filled. The Hare quota was used in the original proposal for a single transferable vote system, and is still occasionally used, although it has since been largely supplanted by the Droop quota.
The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.
The Huntington–Hill method is a highest averages method for assigning seats in a legislature to political parties or states. Since 1941, this method has been used to apportion the 435 seats in the United States House of Representatives following the completion of each decennial census.
In proportional representation systems, an electoral quota is the number of votes a candidate needs to be guaranteed election.
Mathematics of apportionment describes mathematical principles and algorithms for fair allocation of identical items among parties with different entitlements. Such principles are used to apportion seats in parliaments among federal states or political parties. See apportionment (politics) for the more concrete principles and issues related to apportionment, and apportionment by country for practical methods used around the world.
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.
In social choice theory, a function satisfies voter anonymity, neutrality, or symmetry if the rule does not discriminate between different voters ahead of time; in other words, it does not matter who casts which vote. Formally, this is defined as saying the rule returns the same outcome if the vector of votes is permuted arbitrarily.
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 several mathematical measures of bias, which can disagree slightly.
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.
Static population-monotonicity, also called concordance, says that a party with more votes should not receive a smaller apportionment of seats. Failures of concordance are often called electoral inversions.
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.
