Past exam of the computer science course of the University of Oxford / 2023 / Artificial intelligence / 3 / b

Use one variable $Y_j$ per item, with constraint domain $\{1,\dots ,n\}$. Its value is the recipient agent. Every assignment automatically allocates each item exactly once, so separate coverage and disjointness constraints are unnecessary.

For every ordered pair $i\ne \ell$, imposeHere $1$ is an indicator function. The constraint involves every $Y_j$ with $v_{ij}\ne 0$; one may formally give it scope $\{Y_1,\dots ,Y_m\}$, since zero coefficients make the other variables irrelevant. These constraints state envy-free allocation using the comparing agent's additive valuation on both sides. In contrast to part (a), they generally involve more than two variables.