Decomposition Theorem and its Applications

Authors

  • Victor V. Rozen Saratov State University

Abstract

In the article a complete proof of decomposition theorem is given. This theorem concerns the so called canonical extension of the order relation on the set of probabilistic measures. Here we study a structure for an extension of the order relation given on some set on the generated vector space R A . Corresponding description of the extension with help of stochastic matrices is found (Theorem 2). Decomposition theorem reveals the most significant properties of the canonical extension of orders. In particular the consequences of the theorem are two important statements: 1. The coincidence of the canonical extension of any order with its convex hull and 2. Truth of Choquet condition for the canonical extension (see the corollary 1 and the corollary 2 in the section 3.1). The complete proof of decomposition theorem is quite complicated. As the first step for the proof of this theorem we prove an assertion of existence of optimal surplus vector (Theorem 1). This theorem having game-theoretical interpretation also can be formulated in economic terms (Remark 1). A geometric interpretation of decomposition theorem is given (example 1).

Keywords:

extension of order on the set of probabilistic measures, extension of order on the generated vector space, decomposition theorem, stochastic matrix

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References

Rozen, V. (1976). Mixed extensions of games with ordered outcomes (in Russian). Journal Vych. Matem. i Matem. Phis., 6, 1436–1450.

Rozen, V. (2010). Nash Equilibrium points in Games with ordered outcomes. Contribution to game theory and management, vol. III. Collected papers presented on the Third International Conference Game Theory and management – SPb.: Graduate School of Management, SPbU, pp. 368–386.

Rozen, V. (2011). Decision Making under Many Quality Criteria. Contribution to game theory and management, vol. IV. Collected papers presented on the Third International Conference Game Theory and management – SPb.: Graduate School of Management, SPbU, pp. 407–420.

Rozen, V. (2014). Ordered vector spaces and its applications (in Russian). Saratov State University: Saratov.

Kiruta, A. Y., A. M. Rubinov and E. B. Yanovskaya (1980). The optimal choice of distributions in complex social-economic problems (in Russian). "Nauka L.O.": Leningrad.

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Published

2022-05-24

How to Cite

Rozen, V. V. (2022). Decomposition Theorem and its Applications. Contributions to Game Theory and Management, 8. Retrieved from https://gametheory.spbu.ru/article/view/13465

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Articles