嚴格決定博弈

在博弈論中，一個有兩方參與的零和博弈被稱為嚴格決定博弈，當在雙方使用純策略的情況下有納許均衡。嚴格決定博弈的值（博弈的結果）等同於該均衡給出的值。^[1]^[2]^[3]^[4]^[5]

嚴格決定博弈的一個例子是國際象棋。

定義

記博弈的支付矩陣為 $A=(a_{i,j})$ 。博弈被稱為嚴格決定的，當矩陣中的一個值 $a_{h,k}$ 同時是其所在行的最小值、所在列的最大值。在這種情況下， $a_{h,k}$ 被稱為博弈的值，該行（記作 $R_{h}$ ）被稱為行玩家（row player）的最優行為或最優選擇，該列（記作 $C_{k}$ ）被稱為列玩家（column player）的最優行為或最優選擇。^[6]

在這裏的支付矩陣代表的是不同情況下付給行玩家的收益，通常寫在矩陣左側的行玩家有 $R_{1},R_{2},R_{3},...R_{m}$ 行行為以供選擇，而寫在矩陣上方的列玩家有 $C_{1},C_{2},C_{3},...C_{n}$ 列行為以供選擇。^[6]

另見

已解遊戲

參考文獻

^ Waner, Stefan (1995–1996). "Chapter G Summary Finite". Retrieved 24 April 2009.
^ Steven J. Brams (2004). "Two person zero-sum games with saddlepoints". Game Theory and Politics. Courier Dover Publications. pp. 5&ndash, 6. ISBN 9780486434971.
^ Saul Stahl (1999). "Solutions of zero-sum games". A gentle introduction to game theory. AMS Bookstore. p. 54. ISBN 9780821813393.
^ Abraham M. Glicksman (2001). "Elementary aspects of the theory of games". An Introduction to Linear Programming and the Theory of Games. Courier Dover Publications. p. 94. ISBN 9780486417103.
^ Czes Kośniowski (1983). "Playing the Game". Fun mathematics on your microcomputer. Cambridge University Press. p. 68. ISBN 9780521274517.
^ ^6.0 ^6.1 Susanna S. Epp. Discrete Mathematics with Applications. Cengage Learning. ISBN 978-0495391326.

[1] Waner, Stefan (1995–1996). "Chapter G Summary Finite". Retrieved 24 April 2009.

[2] Steven J. Brams (2004). "Two person zero-sum games with saddlepoints". Game Theory and Politics. Courier Dover Publications. pp. 5&ndash, 6. ISBN 9780486434971.

[3] Saul Stahl (1999). "Solutions of zero-sum games". A gentle introduction to game theory. AMS Bookstore. p. 54. ISBN 9780821813393.

[4] Abraham M. Glicksman (2001). "Elementary aspects of the theory of games". An Introduction to Linear Programming and the Theory of Games. Courier Dover Publications. p. 94. ISBN 9780486417103.

[5] Czes Kośniowski (1983). "Playing the Game". Fun mathematics on your microcomputer. Cambridge University Press. p. 68. ISBN 9780521274517.

[:0-6] 6.0 ^6.1 Susanna S. Epp. Discrete Mathematics with Applications. Cengage Learning. ISBN 978-0495391326.

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