Asymptotic Max-Min Fair Allocation with Random Utilities

We investigate the asymptotic behavior of max-min fair allocations for indivisible goods under i.i.d. random utilities. For $N$ agents and $K$ goods with utilities ${\text{mU}}_{i,j}$ drawn independently from a common distribution $F$, we derive asymptotic characterizations of the max-min value in the balanced case $K=N$ (and in $K=LN$ extensions) via distributional quantiles. We then study the efficiency impact of max-min fairness by comparing the resulting total welfare with the optimal sum welfare. For distributions with sufficiently light tails, we prove that the relative efficiency loss converges to zero as the market grows, implying that max-min fairness incurs negligible welfare loss in large random instances for a broad class of distributions.