A quick sketch. Suppose the dumbest players engage in 0 steps of thought, so they just choose strategies uniformly at random. Then one can inductively define a hierarchy of ever “smarter” players who execute increasing steps of thinking: a k-step player is unaware of the existence of players with equal or higher k (Elster’s “younger sibling syndrome”), but knows the relative proportion of players with lower k, and their moves are best responses to that perceived, truncated k distribution. All but the 0-step players are therefore rational, because their moves are optimal given what they believe, but because of incomplete information the game’s outcome can deviate from the usual equilibria when some of the players are dumb.
Camerer, Ho & Chong’s “cognitive hierarchy theory” of one-shot games might be of interest.
A quick sketch. Suppose the dumbest players engage in 0 steps of thought, so they just choose strategies uniformly at random. Then one can inductively define a hierarchy of ever “smarter” players who execute increasing steps of thinking: a k-step player is unaware of the existence of players with equal or higher k (Elster’s “younger sibling syndrome”), but knows the relative proportion of players with lower k, and their moves are best responses to that perceived, truncated k distribution. All but the 0-step players are therefore rational, because their moves are optimal given what they believe, but because of incomplete information the game’s outcome can deviate from the usual equilibria when some of the players are dumb.