This is the twelfth and final post in the Cartesian Frames sequence. Read the first post here.

Up until now, we have (in the examples) mostly considered agents making a single choice, rather than acting repeatedly over time.

The actions, environments, and worlds we've considered might be extended over time. For example, imagine a prisoner's dilemma where "cooperating" requires pushing a button every day for five years.

However, our way of discussing Cartesian frames so far would treat "push the button every day for five years" as an atomic action, a single element $a \in A$ .

Now, will begin discussing how to use Cartesian frames to explicitly represent agents passing through time. Let us start with a basic example.

1. Partial Observability

Consider a process where two players, Yosef and Zoe, collaboratively choose a three-digit binary number. Yosef first chooses the first digit, then Zoe chooses the second digit, then Yosef chooses the third digit. The world will be represented by the three-digit number. The Cartesian frame from the perspective of Yosef looks like this:

$C_{0} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} 000 & 010 & 000 & 010 001 & 011 & 001 & 011 000 & 011 & 000 & 011 001 & 010 & 001 & 010 100 & 110 & 110 & 100 101 & 111 & 111 & 101 100 & 111 & 111 & 100 101 & 110 & 110 & 101 \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$ .

Here, $C_{0} = (A_{0}, E_{0}, \cdot_{0})$ is a Cartesian frame over $W_{0} = {000, 001, 010, 011, 100, 101, 110, 111}$ .

The four possible environments from left to right represent Zoe choosing 0, Zoe choosing 1, Zoe copying the first digit, and Zoe negating the first digit.

The eight possible agents can be broken up into two groups of four. In the top four possible agents, Yosef chooses 0 for the first digit, while in the bottom four, he chooses 1. Within each group, the four possible agents represent Yosef choosing 0 for the third digit, choosing 1 for the third digit, copying the second digit, and negating the second digit.

Consider the three partitions $W_{1}$ , $W_{2}$ , and $W_{3}$ of $W_{0}$ representing the first, second and third digits respectively. $W_{i} = {w_{i}^{0}, w_{i}^{1}}$ , where $w_{1}^{0} = {000, 001, 010, 011}$ , $w_{1}^{1} = {100, 101, 110, 111}$ , $w_{2}^{0} = {000, 001, 100, 101}$ , $w_{2}^{1} = {010, 011, 110, 111}$ , $w_{3}^{0} = {000, 010, 100, 110}$ , and $w_{3}^{1} = {001, 011, 101, 111}$ .

Clearly, by the definition of observables, $W_{2}$ &...

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Cartesian Frames and Factored Sets on ArXiv

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