An allocation of objects is called proportional (PROP) if every agent i values his bundle at least 1/n of the total.
Nevertheless, such a division exists with high probability for indivisible items under certain assumptions on the valuations of the agents.
Conitzer, Freeman and Shah[6] proved that, in the context of fair public decision making, a PROP1 allocation that is also PE.
Barman and Krishnamurthy[7] presented a strongy-polynomial-time algorithm finding a PE+PROP1 allocation for goods (objects with positive utility).
Aziz, Caragiannis, Igarashi and Walsh[9] extended the condition of PROP1 to mixed valuations (objects can have both positive and negative utilities).
With additive valuations: The following maximin-share approximations are implied by PROP*(n-1):[12]: Lem.2.7 An allocation is called proportional up to the worst item (PROPx) if for every agent i, for any subset with one item not allocated to i, if the subset is given to i, then it brings his value to at least 1/n of the total.
Formally, for all i:[13] An equivalent definition is: the value of each agent i is at least (1/n of the total) minus (the least valuable item not assigned to i): Obviously, PROPx is stronger than PROP1.