Iterated elimination procedures

We study the existence and uniqueness (i.e., order independence) of any arbitrary form of iterated elimination procedures in an abstract environment. By allowing for a transfinite elimination, we show a general existence of the iterated elimination procedure. Inspired by the seminal work of Gilboa et al. (OR Lett 9:85–89, 1990), we identify a fairly weak sufficient condition of Monotonicity* for the order independence of iterated elimination procedure. Monotonicity* requires a Monotonicity property along any elimination path. Our approach is applicable to different forms of iterated elimination procedures used in (in)finite games, for example iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and so on. We introduce a notion of CD* games, which incorporates Jackson’s (Rev Econ Stud 59:757–775, 1992) idea of “boundedness,” and show the iterated elimination procedure is order independent in the class of CD* games. In finite games, we also formulate and show an “outcome” order-independence result suitable for Marx and Swinkles’s (Games Econ Behav 18:219–245, 1997) notion of nice weak dominance.