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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 trans…nite elimination, we show a general existence of the iterated elimination procedure. Inspired by the seminal work of Gilboa, Kalai and Zemel (1990), we identify a fairly weak su¢ cient 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 di¤erent forms of iterated elimination procedures used in (in)…nite 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 (1992) idea of "boundedness," and show the iterated elimination procedure is order independent in the class of CD* games. In …nite games, we also formulate and show an "outcome"order-independence result suitable for Marx and Swinkels's (1997) notion of nice weak dominance. JEL Classi…cation: C70, D70.

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