 A more challenging general question is how to relate classic probabilistic approaches with their fixed-point results and ensuing equilibrium existence theorems, with the logical fixed-point approaches mentioned at several places in this entry. The latter operate with step-by-step ordinal iterations, as opposed to the gradual, approximative procedures that underlie the Brouwer or Kakutani fixed-point theorems relevant for classical game theory. A related question is just how much logic is needed to reproduce probabilistic existence theorems within a qualitative framework.