The RM principle has been studied in various division problems.
units of some homogeneous divisible resource, such as water or flour.
Society has to decide how to divide the resource among the agents, i.e, to find a vector
Two classic allocation rules are the egalitarian rule - aiming to equalize the utilities of all agents (equivalently: maximize the minimum utility), and the utilitarian rule - aiming to maximize the sum of utilities.
The egalitarian rule is always RM:[1]: 47 when there is more resource to share, the minimum utility that can be guaranteed to all agents increases, and all agents equally share the increase.
For example, suppose there are two agents, Alice and Bob, with the following utilities: The egalitarian allocation is found by solving the equation:
This is because Alice has increasing returns: her marginal utility is small when she has few resources, but it increases fast when she has many resources.
Hence, when the total amount of resource is small (specifically,
In general, the utilitarian allocation rule is RM when all agents have diminishing returns, but it may be not RM when some agents have increasing returns (as in the example).
[1]: 46–47 Thus, if society uses the utilitarian rule to allocate resources, then Bob loses value when the amount of resources increases.
This is bad because it gives Bob an incentive against economic growth: Bob will try to keep the total amount small in order to keep his own share large.
Consider a cloud server with some units of RAM and CPU.
There are two users with different types of tasks: Thus, the utility functions (=number of tasks), denoting RAM by r and CPU by c, are Leontief utilities: If the server has 12 RAM and 12 CPU, then both the utilitarian and the egalitarian allocations (and also the Nash-optimal, max-product allocation) are: Now, suppose 12 more units of CPU become available.
The egalitarian allocation does not change, but the utilitarian allocation now gives all resources to Alice: so Bob loses value from the increase in resources.
The Nash-optimal (max-product) allocation becomes: so Bob loses value here too, but the loss is less severe.
[1]: 83–84 In the fair cake-cutting problem, classic allocation rules such as divide and choose are not RM.
Several rules are known to be RM: Resource-monotonicity was studied in problems of fair division with single-peaked preferences.
[5][6] The egalitarian rule (maximizing the leximin vector of utilities) might be not RM when the resource to divide consists of several indivisible (discrete) units.
Alice gets a utility of 1 whenever she has a racket, since she enjoys playing against the wall.
But Bob and Carl get a utility of 1 only when they have two rackets, since they only enjoy playing against each other or against Alice.
Hence, Alice loses utility when the total amount of rackets increases.
In the fair item allocation problem, classic allocation procedures such as adjusted winner and envy-graph are not RM.
In contrast, round-robin item allocation is RM.
Moreover, round-robin can be adapted to yield picking sequences appropriate for agents with different entitlements; all these picking sequences are RM too.
[7] The special case in which all items are identical and each agent's utility is simply the number of items he receives is known as apportionment.
It originated from the task of allocating seats in a parliament among states or among parties.
Consider the following network of roads, where the letters denote junctions and the numbers denote distances:A---6---B--5--C--5--D---6---EThe population is distributed uniformly along the roads.
Now, there is a new junction X and some new roads (the previous roads do not change): The egalitarian rule now locates the facility at X, since it allows to decrease the maximum distance from 11 to 9 (the utilitarian and Nash rules also locate the facility at X).
The increase in resources helped most people, but decreased the utility of those living in or near C.[1]: 84–85 A monotonicity axiom closely related to resource-monotonicity appeared first in the context of the bargaining problem.
A bargaining problem is defined by a set of alternatives; a bargaining solution should select a single alternative from the set, subject to some axioms.