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A Necessary and Sufficient Condition for a Unique Maximum with an Application to Potential Games

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  • Finn Christensen

    (Department of Economics, Towson University)

Abstract

Under regularity and boundary conditions which ensure an interior maximum, I show that there is a unique critical point which is a global maximum if and only if the Hessian determinant of the negated objective function is strictly positive at any critical point. Within the large class of Morse functions, and subject to boundary conditions, this local and ordinal condition generalizes strict concavity, and is satisfied by nearly all strictly quasiconcave functions. The result also provides a new uniqueness theorem for potential games.

Suggested Citation

  • Finn Christensen, 2017. "A Necessary and Sufficient Condition for a Unique Maximum with an Application to Potential Games," Working Papers 2017-04, Towson University, Department of Economics, revised Oct 2017.
    • Handle: RePEc:tow:wpaper:2017-04
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    File URL: http://webapps.towson.edu/cbe/economics/workingpapers/2017-04.pdf
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    References listed on IDEAS

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    1. Varian, Hal R, 1975. "A Third Remark on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 43(5-6), pages 985-986, Sept.-Nov.
    2. Christensen, Finn & Cornwell, Christopher R., 2018. "A strong correspondence principle for smooth, monotone environments," Journal of Mathematical Economics, Elsevier, vol. 77(C), pages 15-24.
    3. Dohtani, Akitaka, 1998. "The system stability of dynamic processes," Journal of Mathematical Economics, Elsevier, vol. 29(2), pages 161-182, March.
    4. Finn Christensen, 2019. "Comparative statics and heterogeneity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(3), pages 665-702, April.
    5. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
    6. Dierker, Egbert, 1972. "Two Remarks on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 40(5), pages 951-953, September.
    7. Charles D. Kolstad & Lars Mathiesen, 1987. "Necessary and Sufficient Conditions for Uniqueness of a Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(4), pages 681-690.
    8. Timothy J. Kehoe, 1985. "Multiplicity of Equilibria and Comparative Statics," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 100(1), pages 119-147.
    9. Hefti, Andreas, 2016. "On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 83-96.
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    Cited by:

    1. Guanghui Yang & Chanchan Li & Jinxiu Pi & Chun Wang & Wenjun Wu & Hui Yang, 2021. "Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games," Mathematics, MDPI, vol. 9(1), pages 1-13, January.
    2. Barthel, Anne-Christine & Hoffmann, Eric & Sabarwal, Tarun, 2022. "Characterizing robust solutions in monotone games," Games and Economic Behavior, Elsevier, vol. 135(C), pages 201-219.
    3. Rubio-Herrero, Javier & Baykal-Gürsoy, Melike, 2020. "Mean-variance analysis of the newsvendor problem with price-dependent, isoelastic demand," European Journal of Operational Research, Elsevier, vol. 283(3), pages 942-953.
    4. Christensen, Finn & Cornwell, Christopher R., 2018. "A strong correspondence principle for smooth, monotone environments," Journal of Mathematical Economics, Elsevier, vol. 77(C), pages 15-24.
    5. Finn Christensen, 2019. "Comparative statics and heterogeneity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(3), pages 665-702, April.
    6. Christensen, Finn, 2022. "Streaming Stimulates the Live Concert Industry: Evidence from YouTube," International Journal of Industrial Organization, Elsevier, vol. 85(C).
    7. Anne-Christine Barthel & Eric Hoffmann & Tarun Sabarwal, 2021. "A Unified Approach to p-Dominance and its Generalizations in Games with Strategic Complements and Substitutes," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 202109, University of Kansas, Department of Economics.
    8. Vincenzo Scalzo, 2020. "On the uniqueness of Nash equilibrium in discontinuous ordinal and normal form games," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 163-168, April.

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    More about this item

    Keywords

    optimization; index theory; potential games.;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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