At any time t, full rationality requires that individual i is aware of the
history up to that time, insofar as it determines the truth value of proposi-
tions in Ni. That is, full rationality in a bounded domain precludes imperfect
recall.
Thus, the event observed by individual i at time t is summed up by the
truth value of elementary propositions pn where n E N = Ni ∩ Nt . We have
a corresponding representation of: the history
hi (t) = ∑ 2~(n+1) (un + 1) ;
n∈Ntl
the associated compound proposition pt, consisting of the conjunction
pit = Λ pn Λ -pn ;
{n∈Np.vn=1} {n∈Nt⅛"=-1}
and the corresponding event E^.
Given full rationality in a bounded domain, any proposition p may be
written from the external viewpoint as pi Λ p i where pi E Pi ,pt E P i.
In particular, the proposition pt characterizing the time t event Et may be
written as
pt = pit λ ptl.
5 Probabilities
Suppose that we are given a measure μ on Ω, which may be taken to represent
the prior beliefs that would be held by the decisionmaker in the absence
of bounds on rationality, including bounds on the set of propositions under
consideration. Given such a measure, the structure of the state space derived
above is sufficient to give a complete characterization of Bayesian updating.
For any event E, we have, at time t, a derived measure
μ (E ∩ Et)
μ (Et)
Note that we can now derive μt+1 (E) in two ways. First, we can repeat the
definition above, yielding
μt+ι (E) =
μ (E ∩ Et+i)
μ (Et+i)