Resource monotonicity
Resource monotonicity
- Resource monotonicity (RM; aka aggregate monotonicity**) is a principle of [[fair-division]]. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM principle has been studied in various division problems.
Allocating divisible resources ### Single homogeneous resource, general utilities Suppose society has <math>m</math> units of some homogeneous divisible resource, such as water or flour. The resource should be divided among <math>n</math> agents with different utilities. The utility of agent <math>i</math> is represented by a function <math>u_i</math>; when agent <math>i</math> receives <math>y_i</math> units of resource, he derives from it a utility of <math>u_i(y_i)</math>. Society has to decide how to divide the resource among the agents, i.e, to find a vector <math>y_1,\dots,y_n</math> such that: <math>y_1+\cdots+y_n = m</math>.
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: When the pieces must be connected, no Pareto-optimal [[proportional-division]] rule is RM. The absolute-[[equitable-cake-cutting|equitable]] rule is weakly Pareto-optimal and RM, but not proportional. The relative-equitable rule is weakly Pareto-optimal and proportional, but not RM. The so-called rightmost mark* rule, which is an variant of [[divide-and-choose|divide-and-choose]], is proportional, weakly Pareto-optimal and RM - but it works only for two agents. It is an open question whether there exist division procedures that are both proportional and RM for three or more agents.
Single-peaked preferences Resource-monotonicity was studied in problems of fair division with single-peaked preferences.
Allocating discrete items ### Identical items, general utilities The egalitarian-rule (maximizing the leximin vector of utilities) might be not RM when the resource to divide consists of several indivisible (discrete) units.
For example,
Identical items, additive utilities 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. Therefore, it is often called house monotonicity.
Facility location Facility location is the social choice question is where a certain facility should be located. Consider the following network of roads, where the letters denote junctions and the numbers denote distances:<blockquote>A---6---B--5--C--5--D---6---E</blockquote>The population is distributed uniformly along the roads. People want to be as close as possible to the facility, so they have "dis-utility" (negative utility) measured by their distance to the facility.
In the initial situation, the egalitarian rule locates the facility at C, since it minimizes the maximum distance to the facility, which is 11 (the utilitarian and Nash rules also locate the facility at C).
Now, there is a new junction X and some new roads (the previous roads do not change): :::B--3--X--3--D :::..........|......... :::..........4......... :::..........|......... :::..........C.........
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.