It is known that certain simultaneous games, say, matching pennies with no pure strategy equilibrium, have a mixed strategy equilibrium.
Is simulaneous move at some stage a necessary ingredient of games that don't have a pure strategy equilibirium but have a mixed strategy equilibrium?
It seems to me by induction, if a (finite or infinite)game of perfect information(with complete information) have a equilibrium, then it must have a pure strategy.
What about (finite or infinite) games of perfect information but incomplete information?