Characterizing monotone games
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Characterizing monotone games Anne-Christine Barthel1 · Eric Hoffmann1 Received: 29 October 2018 / Accepted: 6 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Solution concepts in games of strategic heterogeneity (GSH), which include games of strategic complements as a special case, have been shown to possess very useful properties, such as the existence of highest and lowest serially undominated strategies, and the equivalence of the stability of equilibria and dominance solvability. The main result of this paper gives necessary and sufficient conditions for when a very general class of games, referred to as games of mixed heterogeneity, can be transformed into GSH in such a way so that these properties are preserved, allowing us to draw the same strong conclusions about solution sets in games that are not originally GSH. This is achieved by reversing the orders on the actions spaces of a given subset of players. Our second main result shows, rather surprisingly, that under mild conditions on the underlying ordering of action spaces, the reversal of orders is the only way in which such a transformation can be achieved. Applications to aggregate games, market games, and crime networks are given. Keywords Strategic complements · Strategic substitutes · Monotone games · Heterogeneity JEL Classification C60 · C70 · C72
1 Introduction Solution sets in games in which players’ payoffs satisfy certain monotonicity conditions have been shown to possess very useful properties. Milgrom and Shannon (1994) show that in games of strategic complements (GSC), where each agent best responds in a monotone increasing way to an increase in opponents’ strategies, there
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Eric Hoffmann [email protected] Anne-Christine Barthel [email protected]
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West Texas A&M University, 2501 4th Ave, Canyon, TX 79016, USA
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A.-C. Barthel, E. Hoffmann
exist largest and smallest serially undominated strategies a ∗ and a∗ which are also Nash equilibria, and that the limits of all adaptive learning processes eventually fall within the interval defined by [a∗ , a ∗ ]. Thus, if a unique serially undominated strategy aˆ exists, it can be guaranteed to be a globally stable Nash equilibrium in the sense that any adaptive learning processes starting from any initial strategy will converge to it. In fact, as Milgrom and Roberts (1990) show, dominance solvability is equivalent to global stability. Roy and Sabarwal (2012) show that the same can be said for games of strategic substitutes (GSS), where players best respond in a monotone decreasing manner, with the exception that a ∗ and a∗ need not be Nash equilibria. Barthel and Hoffmann (2019) generalize both GSC and GSS by defining games of strategic heterogeneity (GSH), where players may best respond either increasingly or decreasingly to an increase in the joint action choice of opponents, and show that these same aforementioned properties of solution sets hold in general GSH as well. One sensible and alternative way to generalize the notions of GSC and GSS woul
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