A necessary and sufficient condition for a unique maximum with an application to potential games
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 funct...
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| Published in | Economics letters Vol. 161; pp. 120 - 123 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
01.12.2017
Elsevier Science Ltd |
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| 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.
•In some contexts, “∇f(x∗)=0⇒det(−D2f(x∗))>0” iff there is a unique critical point that is a global maximum.•This is an alternative to strict quasiconcavity which is only a sufficient condition.•The result is applied to potential games and yields a new uniqueness theorem.•The proof is an application of the Poincaré–Hopf Theorem from differential topology |
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| AbstractList | 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. 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. •In some contexts, “∇f(x∗)=0⇒det(−D2f(x∗))>0” iff there is a unique critical point that is a global maximum.•This is an alternative to strict quasiconcavity which is only a sufficient condition.•The result is applied to potential games and yields a new uniqueness theorem.•The proof is an application of the Poincaré–Hopf Theorem from differential topology |
| Author | Christensen, Finn |
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| Cites_doi | 10.1016/j.mathsocsci.2016.02.008 10.2307/2297489 10.2307/1911341 10.1016/S0304-4068(97)00008-6 10.1007/BF01295851 10.2307/1912091 10.2307/1885738 10.1006/game.1996.0044 |
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| References_xml | – volume: 26 start-page: 223 year: 1997 end-page: 227 ident: b11 article-title: Correlated equilibrium and potential games publication-title: Int. J. Game Theor. – volume: 54 start-page: 681 year: 1987 end-page: 690 ident: b7 article-title: Necessary and sufficient conditions for uniqueness of a cournot equilibrium publication-title: Rev. Econ. Stud. – volume: 18 start-page: 119 year: 1985 end-page: 147 ident: b6 article-title: Multiplicity of equilibria and comparative statics publication-title: Q. J. Econ. – volume: 14 start-page: 124 year: 1996 end-page: 143 ident: b10 article-title: Potential games publication-title: Games and Economic Behavior – reference: Christensen, Finn, (2017) Comparative statics and heterogeneity. Towson University Working Paper No. 2016–01. – year: 1965 ident: b9 publication-title: Topology from the Differentiable Viewpoint – volume: 43 start-page: 985 year: 1975 end-page: 986 ident: b13 article-title: A third remark on the number of equilibria of an economy publication-title: Econometrica – volume: 40 start-page: 951 year: 1972 end-page: 953 ident: b3 article-title: Two remarks on the number of equilibria of an economy publication-title: Econometrica – reference: Christensen, Finn, Cornwell, Christopher R., (2017) A strong correspondence principle for smooth, Monotone functions. Towson University Working Paper No. 2016–05. – year: 1990 ident: b8 publication-title: Introduction To Topology – volume: 29 start-page: 161 year: 1998 end-page: 182 ident: b4 article-title: The system stability of dynamic processes publication-title: J. Math. Econ. – year: 1994 ident: b12 publication-title: Mathematics for Economists – volume: 80 start-page: 83 year: 2016 end-page: 96 ident: b5 article-title: On the relationship between uniqueness and stability in sum-aggregative, symmetric, and general differentiable games publication-title: Math. Soc. Sci. – year: 1965 ident: 10.1016/j.econlet.2017.10.008_b9 – volume: 80 start-page: 83 year: 2016 ident: 10.1016/j.econlet.2017.10.008_b5 article-title: On the relationship between uniqueness and stability in sum-aggregative, symmetric, and general differentiable games publication-title: Math. Soc. Sci. doi: 10.1016/j.mathsocsci.2016.02.008 – volume: 54 start-page: 681 issue: 4 year: 1987 ident: 10.1016/j.econlet.2017.10.008_b7 article-title: Necessary and sufficient conditions for uniqueness of a cournot equilibrium publication-title: Rev. Econ. Stud. doi: 10.2307/2297489 – year: 1994 ident: 10.1016/j.econlet.2017.10.008_b12 – volume: 43 start-page: 985 issue: 5/6 year: 1975 ident: 10.1016/j.econlet.2017.10.008_b13 article-title: A third remark on the number of equilibria of an economy publication-title: Econometrica doi: 10.2307/1911341 – year: 1990 ident: 10.1016/j.econlet.2017.10.008_b8 – volume: 29 start-page: 161 year: 1998 ident: 10.1016/j.econlet.2017.10.008_b4 article-title: The system stability of dynamic processes publication-title: J. Math. Econ. doi: 10.1016/S0304-4068(97)00008-6 – volume: 26 start-page: 223 year: 1997 ident: 10.1016/j.econlet.2017.10.008_b11 article-title: Correlated equilibrium and potential games publication-title: Int. J. Game Theor. doi: 10.1007/BF01295851 – volume: 40 start-page: 951 issue: 5 year: 1972 ident: 10.1016/j.econlet.2017.10.008_b3 article-title: Two remarks on the number of equilibria of an economy publication-title: Econometrica doi: 10.2307/1912091 – ident: 10.1016/j.econlet.2017.10.008_b2 – volume: 18 start-page: 119 year: 1985 ident: 10.1016/j.econlet.2017.10.008_b6 article-title: Multiplicity of equilibria and comparative statics publication-title: Q. J. Econ. doi: 10.2307/1885738 – ident: 10.1016/j.econlet.2017.10.008_b1 – volume: 14 start-page: 124 year: 1996 ident: 10.1016/j.econlet.2017.10.008_b10 article-title: Potential games publication-title: Games and Economic Behavior doi: 10.1006/game.1996.0044 |
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| Title | A necessary and sufficient condition for a unique maximum with an application to potential games |
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