where N = {1, 2, ....n,....} is supposed to be a finite or countably infinite set,
indexing a family of nodes. Each node represents either an act of nature or
a decision by individual i, and is associated with a specific time t (n) , and
an elementary proposition pn.
Each elementary proposition pn is a statement such as ‘The winner of the
2008 US Presidential election is Hillary Clinton’ or ‘Individual i votes for the
Republican candidate in 2008’. The negation of pn is denoted by -ιpn.
At time t (n) , the proposition takes the truth value υn, which will be
denoted 1 (True) or —1 (False). The set of time periods is a finite or countably
infinite set of the form T = 0,1,.....1Without loss of generality, we will
assume that the elements of N are ordered so that n > n' ^ t (n) ≥ t (n') .
Conversely, we may denote by N (t) the subset N Ç N = {n : t (n) = t} .
An exhaustive description of the state of the world, including the decisions
made by individual i, consists of an evaluation of each of the elementary
propositions pn, n ∈ N. From the viewpoint of a fully informed observer, any
state of the world can therefore be described by a real number ω ∈ Ω Ç
[0,1] 2, given by
ω = ∑ 2~(n+1) (vn + 1) .
n∈N
An elementary proposition pn is true in state ω if and only if ωn = 1,
where ωn ∈ {0,1} is the nth element in the binary expansion of ω. Hence,
corresponding to any elementary proposition pn is an event
Epn = {ω : ωn = 1}
3 Propositions, histories and events
Now consider the perspective of an external observer at time t, with full
knowledge of the state space Ω and of the history up to time t, given by the
values υn : for Nt = {n : t (n) ≤ t} . The history at time t may be numerically
represented by
h (t) = ∑ 2~(n+1) (vn + 1) .
n∈Nt
The history h (t) may be viewed in three distinct, but equivalent ways.
First, as the name implies, it is an element of a sequence h (1) , h (2) ...h (t) ,
1For simplicity, we will focus on the case when both N and T are finite, but we will
not rely on this assumption in any essential fashion.
2If some propositions may be true in all states of the world, Ω may be a proper subset of
[0,1] . Alternatively, Ω may be set equal to [0,1] with some states having zero probability
in all evaluations.