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.

Suppose society has ${\displaystyle t}$ units of some divisible resource (e.g. wood, medication, etc.). The resource should be divided among ${\displaystyle n}$ agents with different utilities. The utility of agent ${\displaystyle i}$ is represented by a function ${\displaystyle u_{i}}$; when agent ${\displaystyle i}$ receives ${\displaystyle y_{i}}$ units of resource, he derives from it a utility of ${\displaystyle u_{i}(y_{i})}$. Society has to decide how to divide the resource among the agents, i.e, to find a vector ${\displaystyle y_{1},\dots ,y_{n}}$ such that: ${\displaystyle y_{1}+\cdots +y_{n}=t}$.

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