Equitable allocation of divisible goods

Abstract The existence of equitable allocations of divisible goods is established. The methods used give divisions of a good into geometrically simple sets, such as simplexes or polyhedral convex cones. This economic problem is solved by applying David Gale's covering lemma. An extension of Gale's lemma and its dual versions are established; the proofs are based on Ky Fan's fundamental theorem on coincidence of two set-valued functions.

[1]  R. Bapat.  D1AD2 theorems for multidimensional matrices , 1982 .

[2]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[3]  Tatsuro Ichiishi,et al.  On the Knaster-Kuratowski-Mazurkiewicz-Shapley theorem , 1981 .

[4]  Ethan Akin,et al.  Vilfredo Pareto cuts the cake , 1995 .

[5]  D. Gale.  Equilibrium in a discrete exchange economy with money , 1984 .

[6]  Douglas R. Woodall,et al.  Dividing a cake fairly , 1980 .

[7]  K. Fan.  A Covering Property of Simplexes. , 1968 .

[8]  L. Shapley.  On Balanced Games Without Side Payments , 1973 .

[9]  Lars-Gunnar Svensson.  Large Indivisibles: An analysis with respect to price equilibrium and fairness , 1983 .

[10]  R. D. Anderson.  Symposium on Infinite Dimensional Topology , 1972 .

[11]  W. Thomson,et al.  On the fair division of a heterogeneous commodity , 1992 .

[12]  Tatsuro Ichiishi,et al.  Closed covers of compact convex polyhedra , 1991 .

[13]  Tatsuro Ichiishi,et al.  Theorems on closed coverings of a simplex and their applications to cooperative game theory , 1990 .

[14]  Tatsuro Ichiishi.  Alternative version of Shapley’s theorem on closed coverings of a simplex , 1988 .

[15]  N. Alon.  Algebraic, Extremal and Metric Combinatorics, 1986: Some Recent Combinatorial Applications of Borsuk-Type Theorems , 1988 .

[16]  D. Weller,et al.  Fair division of a measurable space , 1985 .

[17]  T. Raghavan,et al.  An extension of a theorem of Darroch and Ratcliff in loglinear models and its application to scaling multidimensional matrices , 1989 .