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2020 | OriginalPaper | Buchkapitel

10. Mixed Strategies and Expected Utility

verfasst von : Manfred J. Holler, Barbara Klose-Ullmann

Erschienen in: Scissors and Rock

Verlag: Springer International Publishing

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Abstract

The analysis ofmixed-strategyNash equilibrium, offered in this chapter, builds on the concept of expected utility—which has been challenged by Allais, Kahneman-Tversky, and numerous experimental results. Here, it is applied as a theoretical tool to discuss the peculiarities of the Nash equilibrium when it is mixed as, e.g., in the Inspection Game. It is demonstrated that in the Nash equilibrium the strategy of player 1 is exclusively determined by the payoffs of player 2, and vice versa. Moreover, if the Maximin Solution of this game is in mixed strategies as well, then its payoffs are identical to the payoffs of the Nash equilibrium. In general, however, the prescribed strategies are different because in the Maximin Solution the strategy of player i depends exclusively on the payoffs of i. Arthur Miller’s “The Crucible,” the Stag Hunt Game, and Peter Handke’s “The Goalie’s Anxiety at the Penalty Kick” illustrate the results of this chapter.

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In his “Primitive Games,” Schellenberg (1990) banishes the use of probabilities in his theory and therefore does not considermixed strategies.

A game is finite if each player has only a finite number of pure strategies (see Chap. 4).

In 1994, he received the Nobel Prize for economics together with John Nash and Reinhard Selten.

It seems that the play is still popular. We attended two performances, both in the German language: in March 2016 at Munich Residenztheater, directed by Tina Lanik, and later on May 7, 2017, at the Vienna Burgtheater, brought to the stage by Martin Kusej. The German title of the play is “Hexenjagd,” i.e., “witch-hunt.”

Some authors use the name saddle point not only with regard to the Maximin Solution in pure strategies but apply it to games with Maximin Solutionsinmixed strategies as well.

Peter Handke (Nobel Laureate of 2019) published his novel in 1970 as “Die Angst des Tormanns beim Elfmeter,” at Suhrkamp Verlag. A translation by Michael Roloff, i.e., “The Goalie’s Anxiety at the Penalty Kick,” followed in 1972, published by Farrar, Straus, and Giroux. Already in 1972, Handke’s friend Wim Wenders, today an acclaimed movie director, delivered a film with the same title as the book. In the film, we see Josef Bloch performing as the goalie, attacking the referee. He has to leave the field. This causes him to completely lose his bearings. In the evening, roaming the city, he makes the acquaintance of a cashier and murders her the next morning. Other details in the film are different from the book as well, but they were accepted by Handke who functioned as co-scriptwriter.

That is, Manfred J. Holler.

Another name of this game is roshambo. Of course, there are also Paper-Scissors-Rock and other orderings of the three alternatives giving a name to this game. For a number of game-theoretic applications and interpretations, see Fisher (2008).

Holler (2018) contains a substantial analysis of second-mover advantages. A common observation is: If there is a second-mover advantage, then often the “game” will not be played. This generalizes to many political, economic, and social decision situations.

Allais, M. (1953). Le comportement de l’homme rational devant le risque: Critique des postulats et axiomes de l’école américaine. Econometrica,21, 503–546.CrossRef

Andreozzi, L. (2002). Oscillations in the enforcement of law: An evolutionary analysis. Homo Oeconomicus,18, 403–428.

Andreozzi, L. (2004). Rewarding policeman increases crime: Another surprising result from the inspection game. Public Choice,121, 69–82.CrossRef

Aumann, R. J., & Maschler, M. (1972). Some thoughts on the minimax principle. Management Science,18, 54–63.CrossRef

Avenhaus, R., Canty, M., Kilgour, M., von Stengel, B., & Zamir, S. (1996). Inspection games in arms control and disarmament, invited review. European Journal of Operational Research,90, 383–394.CrossRef

Cheng, L. K., & Zhu, M. (1995). Mixed strategy Nash equilibrium based upon expected utility and quadratic utility. Games and Economic Behavior,9, 139–150.CrossRef

Fisher, L. (2008). Rock, paper, scissors: Game theory for everyday life. New York: Basic Books.

Frey, B. S., & Holler, M. J. (1997). Moral and immoral views on paying taxes. In S. Hellsten, M. Kopperi, & O. Loukola (Eds.), Taking the liberal change seriously. Aldershot: Ashgate.

Frey, B. S., & Holler, M. J. (1998). Tax compliance policy reconsidered. Homo Oeconomicus,15, 27–44.

Handke, P. (2007 [1972]). The Goalie’s anxiety at the penalty kick. New York: Farrar, Straus and Giroux.

Harsanyi, J.C. (1977). Rational behavior and bargaining in games and social situations. Cambridge: Cambridge University Press.

Holler, M. J. (1990). The unprofitability of mixed-strategy equilibria in two-person games: A second Folk-theorem. Economics Letters,32, 319–323.CrossRef

Holler, M. J. (1993). Fighting pollution when decision are strategic. Public Choice,76, 347–356.CrossRef

Holler, M. J. (2018). The economics of the good, the bad, and the ugly: Secrets, desires, and second-mover advantages. London, New York: Routledge.CrossRef

Holler, M. J., & Klose-Ullmann, B. (2008). Wallenstein’s power problem and its consequences. AUCO Czech Economic Review,2, 197–218.

Holler, M. J., & Klose-Ullmann, B. (2019). Machiavelli’s conspiracy games. Munich Social Science Review, New Series,2, 33–58.

Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica,47, 263–291.CrossRef

Kahneman, D., & Tversky, A. (1982). The psychology of preferences. Scientific American,246, 160–173.CrossRef

Nash, J. F. (1950). Equilibrium points in N-person games. Proceedings of the National Academy of Sciences,36, 48–49.CrossRef

Nash, J. F. (1951). Non-cooperative games. Annals of Mathematics,54, 286–295.CrossRef

Rubinstein, A. (1998). Similarity and decision-making under risk (is there a utility theory resolution to the Allais paradox?). Journal of Economic Theory,46, 145–153.CrossRef

Schellenberg, J. A. (1990). Primitive games. Boulder, San Francisco, Oxford: Westview Press.

Tsebelis, G. (1990). Penalty has no impact on crime: A game theoretic analysis. Rationality and Society,2, 255–286.CrossRef

Von Neumann, J. (1959 [1928]). On the theory of games of strategy. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games (Vol. 4). Translation of Zur Theorie der Gesellschaftsspiele. Mathematische Annalen, 100, 295–320.

Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press.

Walker, M., & Wooders, J. (2001). Minimax play at Wimbledon. American Economic Review, 9, 1521–1538.

Wiles, J. (2006). Mixed strategy equilibrium in tennis service (Thesis). Trinity College, Duke University, Durham, NC.

Wittman, D. (1985). Counter-intuitive results in game theory. European Journal of Political Economy,1, 77–89.CrossRef

Titel: Mixed Strategies and Expected Utility
verfasst von: Manfred J. Holler
Barbara Klose-Ullmann
Verlag: Springer International Publishing
Buch: Scissors and Rock
Print ISBN: 978-3-030-44822-6

Electronic ISBN: 978-3-030-44823-3

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-44823-3_10

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