How Proper is Sequential Equilibrium

We show, that Strategy profile of a normal form game is proper if and only if it is quasiperfect in every extensive form (with that normal form). Thus, properness requires optimality along a sequence of suppol ting trembles, while sequentiality only requires optimality in the limit. A decision-theoretic implementation of sequential rationality, strategic independence respecting equilibrium (SIRE), is then defined and compared to proper equilibrium, using lexicographic probability systems. SIRE and proper equilibrium difler in which indifference over strategies are appealed to higher level beliefs in a player's lexicographic sequence. Finally, we give tremble based characterizations of the rankings of strategies that underlie proper equilibrium and SIRE that do not involve structural futures of the game.