This paper elucidates the logic behind recent papers which show that a
unique equilibrium is selected in the presence of higher order uncert
ainty, i.e., when players lack common knowledge. We introduce two new
concepts: belief potential of the information system and p-dominance o
f Nash-equilibria of the game, and show that a Nash-equilibrium is uni
quely selected whenever its p-dominance is below the belief potential.
This criterion applies to many-action games, not merely 2 x 2 games.
It also applies to games without dominant strategies, where the set of
equilibria is shown to be smaller and simpler than might be initially
conjectured. Finally, the new concepts help understand the circumstan
ces under which the set of equilibria varies with the amount of common
knowledge among players.