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 formula_1 units of some homogeneous divisible resource, such as water or flour. The resource should be divided among formula_2 agents with different utilities. The utility of agent formula_3 is represented by a function formula_4; when agent formula_3 receives  formula_6 units of resource, he derives from it a utility of formula_7. Society has to decide how to divide the resource among the agents, i.e, to find a vector formula_8 such that: formula_9. 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 there is more resource to share, the minimum utility that can be guaranteed to all agents increases, and all agents equally share the increase. In contrast, the utilitarian  rule might be not RM. For example, suppose there are two agents, Alice and Bob, with the following utilities: The egalitarian allocation is found by solving the equation: formula_12, which is equivalent to formula_13, so formula_14 is monotonically increasing with formula_1. An equivalent equation is: formula_16, which is equivalent to formula_17, so formula_18 too is monotonically increasing with formula_1. So in this example (as always) the egalitarian rule is RM. In contrast, the utilitarian rule is not RM. 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, formula_20), the utilitarian sum is maximized when all resources are given to Bob; but when there are many resources (formula_21), the maximum is attained when all resources are given to Alice. 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). 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. Two complementary resources, Leontief utilities. 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. Cake cutting, additive utilities. In the fair cake-cutting problem, classic allocation rules such as divide and choose are not RM. Several rules are known to be RM: 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, suppose there are formula_1 tennis rackets. 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, if there is only one racket, the egalitarian rule gives it entirely to Alice. But if there are two rackets, they are divided equally between  the agents (each agent gets a racket for 2/3 of the time). Hence, Alice loses utility when the total amount of rackets increases. Alice has an incentive to oppose growth. Different items, additive utilities. In the fair item allocation problem, classic allocation procedures such as adjusted winner and envy-graph are not RM. Moreover, even the Nash-optimal rule, which is RM in cake-cutting, is not RM in item allocation. 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. 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:A---6---B--5--C--5--D---6---EThe 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): 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. Bargaining. 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. The resource-monotonicity axiom was presented in two variants: