Multiple Exchange Property for M$^\natural$-concave Functions and Valuated Matroids
Kazuo Murota
TL;DR
This paper extends the multiple exchange property from matroid bases to valuated matroids and M$^ atural$-concave functions, using discrete convex analysis, with implications for economic conditions like SNC and GS.
Contribution
It generalizes the multiple exchange property to broader classes of functions and establishes an equivalence between economic conditions using discrete convex analysis.
Findings
Generalization of exchange property to valuated matroids and M$^ atural$-concave functions
Proof based on Fenchel-type duality in discrete convex analysis
Equivalence of SNC and GS conditions in economics
Abstract
The multiple exchange property for matroid bases is generalized for valuated matroids and M-concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities (SNC) condition of Gul and Stacchetti is in fact equivalent to the gross substitutes (GS) condition of Kelso and Crawford.
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Taxonomy
TopicsGame Theory and Voting Systems · Complexity and Algorithms in Graphs · Markov Chains and Monte Carlo Methods