The name is absent

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 ω_n = 1,
where ω_n ∈ {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 N_t = {n : t (n) ≤ t} . The history at time t may be numerically
represented by

h (t) = ∑ 2~⁽ⁿ⁺¹⁾ (vⁿ + 1) .

n∈N_t

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.

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