Exterior Penalty Policy Optimization with Penalty Metric Network Under Constraints

Abstract

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(sqrt{r}), where r denotes the ratio between the two weights.