Welfare bounds in the fair division problem.

*(English)*Zbl 0743.90007Take 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.

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.

Reviewer: S.Dasgupta (Halifax)

##### MSC:

91B14 | Social choice |

Full Text:
DOI

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