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. The term "population monotonicity" is used in an unrelated meaning in the context of

apportionment 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 ...

of seats in the congress among states. There, the property 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. This different property is described in the page ''

state-population monotonicity State-population monotonicity is a property of apportionment methods, which are methods of allocating seats in a parliament among federal states. The property says that, if the population of a state increases faster than that of other states, the ...

''.

In fair cake cutting

In the

fair cake-cutting Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without ...

problem, classic allocation rules such as

divide and choose Divide and choose (also Cut and choose or I cut, you choose) is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource ("the cake") and two partners who have diffe ...

are not PM. Several rules are known to be PM: * When the pieces may be ''disconnected'', any function that maximizes a concave welfare function (a monotonically-increasing function of the utilities) is PM. This holds whether the welfare function operates on the absolute utilities or on the relative utilities. In particular, the Nash-optimal rule, absolute-leximin and relative-leximin rules, absolute-utilitarian and relative utilitarian rules are all PM. It is an open question whether concavity of the welfare function is necessary for PM. * When the pieces must be ''connected'', no Pareto-optimal proportional division rule is PM. The absolute- equitable rule and relative-equitable rules are weakly Pareto-optimal and PM.

In fair house allocation

In the

house allocation problem In economics and computer science, the house allocation problem is the problem of assigning objects to people with different preferences, such that each person receives exactly one object. The name "house allocation" comes from the main motivating a ...

, a rule is PM and

strategyproof In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about ...

and

Pareto-efficient Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...

, if-and-only-if it assigns the houses iteratively, where at each iteration, at most two agents trade houses from their initial endowments.

In fair item allocation

In the

fair item allocation Fair item allocation is a kind of a fair division problem in which the items to divide are ''discrete'' rather than continuous. The items have to be divided among several partners who value them differently, and each item has to be given as a whol ...

problem, the Nash-optimal rule is no longer PM. In contrast,

round-robin item allocation Round robin is a procedure for fair item allocation. It can be used to allocate several indivisible items among several people, such that the allocation is "almost" envy-free: each agent believes that the bundle he received is at least as good as ...

is PM. Moreover, round-robin can be adapted to yield

picking sequence A picking sequence is a protocol for fair item assignment. Suppose ''m'' items have to be divided among ''n'' agents. One way to allocate the items is to let one agent select a single item, then let another agent select a single item, and so on. A ...

s appropriate for agents with different entitlements. Picking-sequences based on divisor methods are PM too. However, a picking-sequence based on the quota method is not PM.

References

Fairness criteria

In fair cake cutting

In fair house allocation

In fair item allocation

See also

References