Stability and competitive equilibria in multi-unit trading networks with discrete concave utility functions

  • Yoshiko T. Ikebe
  • Yosuke Sekiguchi
  • Akiyoshi Shioura
  • Akihisa Tamura
Original Paper Area 1

Abstract

Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp showed the existence of stable outcomes and competitive equilibria in a model of trading networks under the assumption that all agents’ preferences satisfy a condition called the full substitutes condition. In this paper, we extend their model by using discrete concave utility functions called twisted \(\hbox {M}^{\natural }\)-concave functions. We show that a valuation function of an agent is twisted \(\hbox {M}^{\natural }\)-concave if and only if the agent’s preference satisfies the generalized variant of the full substitutes condition. We also show that under the generalized full substitutes condition, there exist stable outcomes and competitive equilibria in the extended model and the set of competitive equilibrium price vectors forms a lattice. In addition, we discuss the connection among competitive equilibria, stability, and efficiency. Finally, we investigate the relationship among stability, strong group stability, and chain stability and verify these three stability concepts are equivalent as long as valuation functions of all agents are twisted \(\hbox {M}^{\natural }\)-concave.

Keywords

Stability Competitive equilibria Efficiency Lattice Twisted \(\hbox {M}^{\natural }\)-concave functions Generalized full substitutes condition 

Mathematics Subject Classification

91B50 91B68 

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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of Management ScienceTokyo University of ScienceTokyoJapan
  2. 2.Department of MathematicsKeio UniversityYokohamaJapan
  3. 3.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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