Egalitarian random assignment

I argue that envy-freeness can obstruct fairness in the random assignment problem and I propose a new fairness criterion called even-handedness. Loosely speaking, a random assignment that maximises the position of the least advantaged agent is even-handed. Rules of random assignment that are stochastic-dominance efficient cannot be both even-handed and envy-free for groups of four or more. I define new rules called positive equality, prudent equality and balanced equality that are even-handed and stochastic-dominance efficient. The positive equality rule is envy-free for groups of three and average-envy-free for groups of any size. I present a general method of extending rules from the domain of strict preference to that of weak preference. That method is applicable to the equality rules, the serial rule and others. I also define a network flow algorithm for the positive equality rule.