Computing welfare and fairness in allocating identical goods with entitlements and general utility functions

A number of goods are called identical if they provide the same level of utility to each agent. In various real-world instances of fair division scenarios, identical indivisible items are allocated to consumers and demandants with different entitlements. We assume that the utility of $t$ identical items to each agent $A$ equals $f_A(t)$, where $f_A$ is an arbitrary increasing function corresponding to $A$. We present a polynomial time algorithm that determines the maximum weighted Rawlsian and Leximin welfare for scenarios with identical goods and show that the allocation obtained by the algorithm is equitable up to any item (WEQX). Some results concerning restricted utilitarian welfare and existence of WMMS and WEFX allocations are also presented. We introduce a new quantity ``total weighted deficit," for allocations, and by which we obtain a tractable algorithm to achieve equitable allocations for scenarios with identical goods and different weights via compensation by using a minimum number of identical coins. Some result are for scenarios with $k$ goods of different types.