Iterated dominance revisited
- PDF / 870,293 Bytes
- 45 Pages / 439.37 x 666.142 pts Page_size
- 102 Downloads / 198 Views
Iterated dominance revisited Amanda Friedenberg1 · H. Jerome Keisler2 Received: 27 July 2018 / Accepted: 23 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Epistemic justifications of solution concepts often refer to type structures that are sufficiently rich. One important notion of richness is that of a complete type structure, i.e., a type structure that induces all possible beliefs about types. For instance, it is often said that, in a complete type structure, the set of strategies consistent with rationality and common belief of rationality are the set of strategies that survive iterated dominance. This paper shows that this classic result is false, absent certain topological conditions on the type structure. In particular, it provides an example of a finite game and a complete type structure in which there is no state consistent with rationality and common belief of rationality. This arises because the complete type structure does not induce all hierarchies of beliefs—despite inducing all beliefs about types. This raises the question: Which beliefs does a complete type structure induce? We provide several positive results that speak to that question. However, we also show that, within ZFC, one cannot show that a complete structure induces all second-order beliefs. Keywords Iterated dominance · Rationalizability · Rationality and common belief of rationality · Type structures · Epistemic game theory JEL Classification C70 · C72 · C79 · D81 · D89
We thank Adam Brandenburger, Pierpaolo Battigalli, Byung Soo Lee, Marciano Siniscalchi, and participants of numerous seminars. Friedenberg thanks the UCL Economics Department for its hospitality.
B
Amanda Friedenberg [email protected] H. Jerome Keisler [email protected]
1
Department of Economics, University of Arizona, Tucson, AZ, USA
2
Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA
123
A. Friedenberg, H. J. Keisler
1 Introduction Iterated deletion of strongly dominated strategies has a long tradition in game theory. Bernheim (1984) and Pearce (1984) asserted that—up to issues of correlation—the iteratively undominated (IU) strategies are the strategies consistent with rationality and common belief of rationality. In many ways, this step seems intuitive and obvious. Brandenburger and Dekel (1987) and Tan and Werlang (1988) are early treatments that provide a formal statement of this claim. (See, also, Battigalli and Siniscalchi (2002) and Arieli (2010), among many others.) Each of these papers provide epistemic conditions for IU. This paper argues that, in rather subtle ways, we have an incomplete understanding of the epistemic conditions for IU. The gap in our understanding arises because we have failed to appreciate subtleties of an epistemic framework that has become standard in the literature. We give a series of results aimed at improving our understanding of that framework and, in turn, IU. To provide foundations for IU, we must first specify a framework in which players rea
Data Loading...
