Existence and uniqueness of maximal reductions under iterated strict dominance

Iterated elimination of strictly dominated strategies is an order dependent procedure. It can also generate spurious Nash equilibria, fail to converge in countable steps, or converge to empty strategy sets. If best replies are well-defined, then spurious Nash equilibria cannot appear; if strategy spaces are compact and payoff functions are uppersemicontinuous in own strategies, then order does not matter; if strategy sets are compact and payoff functions are continuous in all strategies, then a unique and nonempty maximal reduction exists. These positive results extend neither to the better-reply secure games for which Reny has established the existence of a Nash equilibrium, nor to games in which (under iterated eliminations) any dominated strategy has an undominated dominator.