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Welfare bounds in the fair division problem. (English) Zbl 0743.90007
Take a problem of division between $$n$$ agents of given amounts of $$K$$ goods, $$n\cdot w$$ where $$w$$ in $$R^ K_{++}$$ is the per capita endowment. $$D$$ is the set of all ordinal utility functions $$(R^ K\to R)$$ satisfying certain standard properties. A mechanism $$F$$ denotes a correspondence, mapping each $$(n,u,w)$$ into the corresponding set of feasible allocations $$A(n,u,w)\equiv\{z=(z_ 1,\dots,z_ n)\mid$$ $$z_ i$$ in $$R^ K_ +$$ and $$\sum_ i z_ i=n\cdot w\}$$. Here $$u\in D^ n$$. Let $$S(n,u,w)\equiv\{(u_ 1(z_ 1),\dots,u_ n(z_ n))$$ in $$R^ n\mid (z_ 1,\dots,z_ n)\in F(n,u,w)\}$$. Let $$S_ i(n,u,w)$$ be the set which is the projection of $$S$$ in the $$i$$th coordinate. Let $$F$$ be Pareto optimal, anonymous and goods neutral [Pareto optimal means: if $$z$$ is in $$F(\cdot)$$ then there is no other $$z'$$ in $$A(\cdot)$$ s.t. $$u_ i(z_ i')\geq u_ i(z_ i)$$ for all $$i$$ with $$F>$$ for some $$i$$; eschewing lengthy technicalities, anonymity means that if $$u$$, $$u'$$ in $$D^ n$$ are two utility profiles in which the only difference is that $$i$$th and $$j$$th utility functions are interchanged then their payoffs sets under $$F$$, $$S_ i$$ and $$S_ j$$ are interchanged leaving others unchanged; goods neutral means a change in the numbering of goods or in their units of measurent leaves $$F$$ essentially unchanged]. Let $$N$$ be the set of positive integers.
Define the lower and upper bound functions of $$F$$ as follows: for all $$u_ i\in D$$, all $$w\in R^ K_{++}$$, $$\overline S_ i\equiv\{x\mid$$ $$x\in S_ i(n,u,w)$$, $$n\in N$$, $$u\in D^ n$$ s.t. $$i$$th coordinate of $$u=u_ i\}$$; $$lb(u_ i,w)=\inf\overline S_ i$$; $$ub(u_ i,w)=\sup\overline S_ i$$. Given any $$w\gg 0$$ also define $$p$$ by $$p^ k=1/w^ k$$ for each $$k=1,\dots,K$$ and the simplex $$\Omega(w)\equiv\{z$$ in $$R^ K_ +\mid$$ $$p\cdot z=p\cdot w=K\}$$. The following is the principal theorem: for all $$u_ 0\in D$$, all $$w\gg 0$$; $$lb(u_ 0,w)\leq u_ 0(w)$$ and $$\hbox{Max}_{z\in\Omega(w)} u_ 0(z)\leq ub(u_ 0,w)$$. In the absence of a priori knowledge regarding the distribution of agents’ utility functions and the number of agents, intuitively a mechanism is more egalitarian the smaller is the gap between these lower and upper bounds. The theorem asserts that the smallest gap is achieved by a mechanism that (i) guarantees to everyone his equal split utility $$u_ 0(w)$$ and (ii) never gives more to anyone than provided by the opportunity to exchange $$w$$ at the canonical price $$p$$, $$\hbox{Max}_{z\in\Omega(w)} u_ 0(z)$$.
An example of a mechanism which achieves both bounds is provided. Some other mechanisms in the literature are examined from this standpoint. Some well-known normative axioms suggested in the literature such as no envy, guanranteed equal split etc. are examined for compatibility or lack thereof. This is an interesting well-written paper which is very readable.

##### MSC:
 91B14 Social choice
##### Keywords:
fair division; welfare bounds; growth paradox; no envy; equal split
Full Text:
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