Asymptotic analysis of weighted fair division

Several resource allocation settings involve agents with unequal entitlements represented by weights. We analyze weighted fair division from an asymptotic perspective: if $m$ items are divided among $n$ agents whose utilities are independently sampled from a probability distribution, when is it likely that a fair allocation exist? We show that if the ratio between the weights is bounded, a weighted envy-free allocation exists with high probability provided that $m=\Omega (n\log n/\log \log n)$, generalizing a prior unweighted result. For weighted proportionality, we establish a sharp threshold of $m=n/(1-\mu )$ for the transition from non-existence to existence, where $\mu \in (0,1)$ denotes the mean of the distribution. In addition, we prove that for two agents, a weighted envy-free (and weighted proportional) allocation is likely to exist if $m=\omega \left(\sqrt{r}\right)$, where $r$ denotes the ratio between the two weights.