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 -) for the transition from non-existence to existence, where 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 (sqrtr), where r denotes the ratio between the two weights.
Manurangsi et al. (Mon,) studied this question.