Balance or balancedness is a property of

apportionment methods The legal term apportionment (french: apportionement; Mediaeval Latin: , derived from la, portio, share), also called delimitation, is in general the distribution or allotment of proper shares, though may have different meanings in different c ...

, which are methods of allocating identical items between among agens, such as dividing seats in a

parliament In modern politics, and history, a parliament is a legislative body of government. Generally, a modern parliament has three functions: Representation (politics), representing the Election#Suffrage, electorate, making laws, and overseeing ...

among

political parties A political party is an organization that coordinates candidates to compete in a particular country's elections. It is common for the members of a party to hold similar ideas about politics, and parties may promote specific ideological or pol ...

federal states A federation (also known as a federal state) is a political entity characterized by a union of partially self-governing provinces, states, or other regions under a central federal government (federalism). In a federation, the self-governi ...

. 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. Ideally, agents with identical entitlements should receive an identical number of items, but this may be impossible due to the indivisibility of the items. Balancedness requires that the difference between identical-entitlement agents should be the smallest difference allowed by the indivisibility, which is 1. For example, if there are 2 equal-entitlement agents and 9 items, then the allocations (4,5) and (5,4) are both allowed, but the allocations (3,6) or (6,3) are not - a difference of 3 is not justified even by indivisibility.

Definitions

There is a ''resource'' to allocate, denoted by

h

. For example, it can be an integer representing the number of seats in a ''h''ouse of representatives. The resource should be allocated between some

n

''agents'', such as states or

parties A party is a gathering of people who have been invited by a host for the purposes of socializing, conversation, recreation, or as part of a festival or other commemoration or celebration of a special occasion. A party will often feature fo ...

. The agents have different ''entitlements'', denoted by a vector

t_1,\ldots,t_n

. For example, ''t_i'' can be the fraction of votes won by party ''i''. An ''allocation'' is a vector

a_1,\ldots,a_n

with

\sum_^n a_i = h

. An ''allocation rule'' is a rule that, for any

h

and entitlement vector

t_1,\ldots,t_n

, returns an allocation vector

a_1,\ldots,a_n

. An allocation rule is called balanced if

t_i = t_j

implies

, a_i-a_j, \leq 1

for all ''i,j''. Equivalently,

t_i = t_j

implies

a_i\geq a_j-1

for all ''i,j''.

Properties

All known apportionment methods are balanced. In particular, both

Highest averages method A highest-averages method, also called a divisor method, is a class of methods for allocating seats in a parliament among agents such as political parties or federal states. A divisor method is an iterative method: at each iteration, the number ...

s and

Largest remainder method The largest remainder method (also known as Hare–Niemeyer method, Hamilton method or as Vinton's method) is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with various h ...

s are balanced. Every apportionment method that is anonymous, exact and coherent, is also balanced.{{Rp, 112

References

Apportionment method criteria