Abstract
This text reviews a recent approach to modeling “radically uncertain” behavior in strategic interactions. By rigorously rooting the approach in decision theory, we provide a foundation for applications of Knightian uncertainty in mechanism design, principal agent and moral hazard models. We discuss critical assessments and provide alternative interpretations of the new equilibria in terms of equilibrium in beliefs, and as a boundedly rational equilibrium in the sense of a population equilibrium. We also discuss the purification of equilibria in the spirit of Harsanyi.
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Notes
The literature on Knightian uncertainty in games has grown at a rapid speed in recent years. We mention here the approaches that use uncertain actions or strategies. Bade (2011) studies two player games using uncertain strategies in an Anscombe–Aumann setting where players have subjective uncertainty aversion and do not use the imprecise probabilistic information contained in strategies. She focuses on the fact that in two player games, the support of equilibrium actions is the same as in Nash equilibrium. Grant et al. (2016) introduce uncertain actions without allowing randomization in the classic sense. Then the issue of existence of an equilibrium arises as strategy sets are no longer convex. In a rich Savage-like setting, they are able to prove existence of an equilibrium. Stauber (2017) develops an interesting different interpretation of ambiguity about actions as coming from irrational behavior of players.
Lo (1996), Marinacci (2000), and Eichberger and Kelsey (2011) are examples of approaches to games in which players use pure strategies, but players hold ambiguous beliefs about the other players’ actions. We would like to refer to Riedel and Sass (2014) for a review on belief–based approaches to ambiguity.
Without being too radical, though, as you are going to see.
To the modern game theorist, it might be valuable to emphasize that we follow here the idea of von Neumann and Morgenstern that players are able to commit to certain unpredictable actions. We think that it makes sense to take such uncertainty of acts into account for modeling human beings since the complexity of their reflections combined with a certain unpredictability of their behavior might be viewed as the outcome of an Ellsberg experiment, compare also Güth and Kliemt (2007) on the role of commitment in game theory.
With the exception of cases where the players are just indifferent about any action they use.
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Center for Mathematical Economics, Bielefeld University and Department of Economic and Financial Sciences, University of Johannesburg. The author thanks Quentin Couanau and Jean–Marc Tallon for helpful comments and particularly Hartmut Kliemt for excellent refereeing. Some of his remarks and ideas have found their way into this text. Financial support through the German Research Foundation Grant Ri-1128-6-1 is gratefully acknowledged.
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Riedel, F. Uncertain Acts in Games. Homo Oecon 34, 275–292 (2017). https://doi.org/10.1007/s41412-017-0061-4
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DOI: https://doi.org/10.1007/s41412-017-0061-4