  The Santa Claus problem is a fundamental problem in fair division: the goal
is to partition a set of heterogeneous items among heterogeneous agents so as
to maximize the minimum value of items received by any agent. In this paper, we
study the online version of this problem where the items are not known in
advance and have to be assigned to agents as they arrive over time. If the
arrival order of items is arbitrary, then no good assignment rule exists in the
worst case. However, we show that, if the arrival order is random, then for $n$
agents and any $\varepsilon > 0$, we can obtain a competitive ratio of
$1-\varepsilon$ when the optimal assignment gives value at least $\Omega(\log n
/ \varepsilon^2)$ to every agent (assuming each item has at most unit value).
We also show that this result is almost tight: namely, if the optimal solution
has value at most $C \ln n / \varepsilon$ for some constant $C$, then there is
no $(1-\varepsilon)$-competitive algorithm even for random arrival order.

Computer Science and Game Theory;Data Structures and Algorithms