極小極大定理與兩人零和之動態對局

Minimax Theorem and Two-Person Zero-Sum Dynamic Games

游詔雅(Chao-Ya Yu)

指導教授 ： 賴漢卿

中原大學/理學院/應用數學研究所/博士(2012年)

https://doi.org/10.6840/cycu201200275

摘要

本篇論文是討論有關極小極大定理與兩人零和之動態對局關係。
在此動態對局系統中，我們建構伴隨著轉移機率之損失和獲益的總期望值函數，它將表現出極小極大問題的性質。再者，我們證明對於遵循一個動態規則的二人零和策略空間，極小極大定理亦將存在，並且證明鞍值函數在合理條件下存在，使得此動態對局系統達到平衡點。除了非分數型態之對局，亦推廣至分數型態的對局在合理條件下，極小極大定理仍成立。

關鍵字

總期望值函數 ； 鞍值函數 ； 極小極大定理 ； 動態對局 ； 零和

並列摘要

The dissertation is aim to consider the minimax problem on a two-person zero-sum dynamic game. Let X and Y be the stochastic strategy spaces of players I and II, respectively, in a two-person zero-sum dynamic game. The establishment
of the total value functions of losses and gains with transition probabilities in the game system will perform the property for minimax problem.
Further the minimax theorem is proved for the strategy spaces of the two-person zero-sum game if it follows a law of motion. It is also established that the saddle value function exists under certain conditions so that the equilibrium point exists in the game system. In addition to nonfractional type game, it is
proved that the fractional function of the total conditional expectations of players I and II which satisfies Ky Fan type minimax theorem under some reasonable conditions.

並列關鍵字

minimax theorem ； dynamic game ； total expectation function ； zero-sum ； saddle value function

參考文獻

[1] J.C. Chen, H.C. Lai, and S. Schaible. Complex fractional programming and the Charnes-Cooper transformation. Journal of Optimization Theory and Applications, 126(1):203–213, 2005.

[2] Ky Fan. Minimax theorems. Proceedings of the National Academy of Sciences of the United States of America, 39:42–47, 1953.

[3] R. Isaacs. Differential Games. Google Books, Dover, 1999.

[6] H.C. Lai and T.Y. Huang. Optimality conditions for a nondifferentiable minimax programming in complex spaces. Nonlinear Analysis, 71:1205–1212,2009.

[7] H.C. Lai and T.Y. Huang. Complex analysis methods related on Optimization problem with complex variable. European Journal of Pure and Applied Mathematics, 3(6):989–1005, 2010.