Before revision: A basic test of fairness when we divide a manna  of private items  between n agents is the lowest welfare the rule guarantees to each agent, irrespective of others preferences. Two familiar examples are: the Equal Split Guarantee when the manna is divisible and preferences are convex; and 1/n of the utility of a heterogenous non atomic cake, if utilities are additive. The minMax utilityof an agent is that of her best share in the worst possible n-partition of .
After revision: A basic test of fairness when we divide a manna of private items between n agents is the lowest welfare the rule guarantees to each agent, irrespective of others ' preferences. If the manna is non atomic and utilities are continuous (not necessarily monotone or convex), we can guarantee the minMax utility: that of our agent's best share in the worst possible n-partition of .