Walter, Karl, and Ferdinand are on the road to Dallas. They arrive at a fork in the road; should they turn right or left?

Type t1 believes that "we should turn right, everyone here believes that we should turn right, everyone here believes that everyone here believes that we should turn right, etc.": in other words, that type believes that turning right is called for, and believes that this is common belief among the three.

Type t₂ believes that "we should turn right, the two others believe that we should turn left, the two others believe that everyone here believes that we should turn left, the two others believe that everyone here believes that everyone here believes that we should turn left, etc.": in other words, that type believes that turning right is called for, but believes that the other two believe that they should turn left and that this fact is common belief among the three.

Type t3 does not know which way to turn, but believes that the two others know the right way to turn, and believes that the others believe that everyone knows the right way to turn: he believes "the probability that we should turn right is 1/2, and the probability that we should turn left is 1/2; if we should turn right, then the two others believe that we should turn right, and that this is common belief among everyone here, and if we should turn left, then the two others believe that we should turn left, and that this is common belief among everyone here." Walter's type is t₁, Karl's type is t₂, and Ferdinand's type is t₃.

(a) Construct a belief space in which the described situation is represented by a state of the world and indicate that state.

(b) What is the minimal belief subspace of each of the three players (at the state of the world in which Walter's type is t₁, Karl's type is t₂, and Ferdinand's type is t₃)?