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

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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.

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Notes

  1. The gross substitutes condition also appears in auction theory. Gul and Stacchetti [7] proposed an ascending auction, which converges to the smallest Walrasian equilibrium prices under the gross substitutes condition. Sun and Yang [24, 25] introduced the gross substitutability and complementarity (GSC) condition, which is a generalization of the gross substitutes condition. They showed that there exists a Walrasian equilibrium in the auction model under the GSC condition.

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Authors and Affiliations

  1. Department of Management Science, Tokyo University of Science, Tokyo, 162-8601, Japan

    Yoshiko T. Ikebe

  2. Department of Mathematics, Keio University, Yokohama, 223-8522, Japan

    Yosuke Sekiguchi & Akihisa Tamura

  3. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Akiyoshi Shioura

Authors
  1. Yoshiko T. Ikebe
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  2. Yosuke Sekiguchi
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  3. Akiyoshi Shioura
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  4. Akihisa Tamura
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Corresponding author

Correspondence to Yosuke Sekiguchi.

Additional information

This research is supported by KAKENHI (24220003, 24300003, 24500002, 25106503).

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Ikebe, Y.T., Sekiguchi, Y., Shioura, A. et al. Stability and competitive equilibria in multi-unit trading networks with discrete concave utility functions. Japan J. Indust. Appl. Math. 32, 373–410 (2015). https://doi.org/10.1007/s13160-015-0175-7

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  • DOI: https://doi.org/10.1007/s13160-015-0175-7

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