In addition, some works consider a setting in which a benevolent third-party is willing to subsidize the allocation, but wants to minimize the amount of subsidy subject to envy-freeness.
Svensson[1] first proved that, when all agents are Archimedean, an envy-free allocation exists and is Pareto-optimal.
Demange, Gale and Sotomayor[2] showed a natural ascending auction that achieves an envy-free allocation using monetary payments for unit demand agents.
Maskin[3] proved the existence of a Pareto-optimal envy-free allocation when the total money endowment is more than (n-1)V. The proofs use competitive equilibrium.
Tadenuma and Thomson[4] study several consistency properties of envy-free allocation rules.
Connectivity of the undirected envy graph characterizes the extreme points of this polytope.
The first procedure for fair allocation of items and money was invented by Bronislaw Knaster and published by Hugo Steinhaus.
[8][9] This auction works as follows, for each item separately: The utility of every agent is at least 1/n of the value he attributes to the entire set of objects, so the allocation is proportional.
Some researchers analysed its performance when agents play strategically: Knaster's auction has been adapted to fair allocation of wireless channels.
Their procedure allows arbitrary constraints on bundles of items, as long as they are anonymous (do not differentiate between partners based on their identity).
When there are many item and complex constraints, the initial step - finding a maxsum allocation - may be difficult to calculate without a computer.
The authors say: Some works assume that a benevolent third-party is willing to subsidize the allocation, but wants to minimize the amount of subsidy subject to envy-freeness.
Meertens, Potters and Reijnierse[20] prove the existence of envy-free and Pareto-optimal allocations under very mild assumptions on the valuations (not necessarily quasilinear).
Cavallo[21] generalizes the traditional binary criteria of envy-freeness, proportionality, and efficiency (welfare) to measures of degree that range between 0 and 1.
When selling objects to buyers, the sum of payments is not fixed in advance, and the goal is to maximize either the seller's revenue, or the social welfare, subject to envy-freeness.
Mu'alem presents a general framework for optimization problems with envy-freeness guarantee that naturally extends fair item allocations using monetary payments.