Epistemic conditions for iterated admissibility

Iterated admissibility (weak dominance) is a long-standing and attractive solution concept, making strong predictions in many games, e.g. the forward induction path in signalling games and the backward-induction path in perfect-information trees. Yet its logical basis has remained unclear. The difficulty appears to be this. The 'philosophy' behind iterated admissibility is that a player, Ann say, should consider everything possible; in particular, she should assign strictly positive probability to each of Bob's strategies. Now turn to Bob. If he assumes that Ann adheres to the foregoing 'philosophy,' then he should rule out the possibility that Ann will choose an inadmissible strategy, i.e., he should assign zero probability to these of Ann's strategies. But if Bob, like Ann, adheres to the everything-is-possible philosophy, he should give all of Ann's strategies positive probability. We seem to have reached some kind of contradiction.

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