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2015 | OriginalPaper | Chapter

2. Strategic Dominance

Author : Takako Fujiwara-Greve

Published in: Non-Cooperative Game Theory

Publisher: Springer Japan

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Abstract

This chapter formulates normal-form (strategic-form) games, explains strategic dominance, and introduces an equilibrium concept through iterative elimination of strictly dominated strategies.

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previous chapter Games in Game Theory

next chapter Nash Equilibrium

Alternatively, it is written as “Prisoners’ Dilemma”. This book follows Luce and Raiffa [4]. Also notice that the players are not yet prisoners.

If there is a natural order among players in the description of the game, the product is usually taken that way. In this case the alphabetical order of the names suggests that A is the first coordinate. One can equivalently formulate the product and the game with player B as the first player.

For the extended definition of strict dominance under “mixed strategies”, see Sect. 3.6.

However, if there is a strategy that strictly dominates all other strategies (of the relevant player), then this strategy (called the dominant strategy) should be chosen by a rational player. In the Prisoner’s Dilemma, the strategy that is not strictly dominated by some other strategy coincides with the dominant strategy, and thus it is easy to predict an outcome by the dominant strategy. In general, there are few games with dominant strategies. Hence we do not emphasize the prediction that a player chooses the dominant strategy. Note also that, if there is a strategy combination such that all players are using a dominant strategy, the combination is called a dominant-strategy equilibrium.

Its matrix representation is symmetrical to the one in Table 2.1.

Strictly speaking, this definition is Strong Efficiency. There is also Weak Efficiency, which requires that there is no \((s'_1, s'_2, \ldots , s'_n)\in S\) such that \(u_i(s_1,s_2,\ldots , s_n) <u_i(s'_1,s'_2,\ldots , s'_n)\) for all \(i\in \{1,2,\ldots ,n\}\).

For a more detailed explanation of common knowledge, see, for example, Aumann [1], Chap. 5 of Osborne and Rubinstein [7], and Perea [8].

For a formal proof, see Ritzberger [9], Theorem 5.1.

The definitions of maxmin value and minmax value in this section are within the “pure” strategies. A general definition of the minmax value is given in Sect. 5.7.

Aumann R (1976) Agreeing to disagree. Ann Stat 4(6):1236–1239MATHMathSciNetCrossRef

Fan K (1953) Minimax theorems. Proc Nat Acad Sci 39:42–47MATHMathSciNetCrossRef

Kohlberg E, Mertens J-F (1986) On the strategic stability of equilibria. Econometrica 54(5):1003–1037MATHMathSciNetCrossRef

Luce D, Raiffa H (1957) Games and decisions. Wiley, New YorkMATH

von Neumann J (1928) Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100:295–320 (English translation (1959): On the theory of games of strategy. In: Tucker A, Luce R (eds) Contributions to the theory of games. Princeton University Press, Princeton, pp 13–42

von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton, NJ

Osborne M, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge, MAMATH

Perea A (2001) Rationality in extensive form games. Klouwer Academic Publishers, Dordrecht, the NetherlandsCrossRef

Ritzberger K (2002) Foundations of non-cooperative game theory. Oxford University Press, Oxford, UK

Title: Strategic Dominance
Author: Takako Fujiwara-Greve
Publisher: Springer Japan
Book: Non-Cooperative Game Theory
Print ISBN: 978-4-431-55644-2

Electronic ISBN: 978-4-431-55645-9

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-4-431-55645-9_2