Histories
In a spacetime game, a history is a map from information sets to actions that is consistent with the game structure. With "consistent", we mean that exactly those information sets that are active, given the actions at other information sets, are assigned an action. No more, no less. Thus, the map may leave some information sets unassigned: those that are not activated.
Since the information sets are the same in the spacetime game and in the converted game in extensive form, the histories can be visualized on any of these two forms. This is equivalent.
Histories correspond exactly to the game outcomes: there are as many histories as leaves in the converted extensive form. Every history leads to a specific leaf by following the associated path, and every leaf can be mapped back to a history by following its path in reverse back to the root.
Let us take a former example, where we gave names to the information sets: A (yellow), X (red), Y (green), and Z (blue):
Then, the outcomes correspond to the following histories (four columns on the right). Note how for every outcome, not all information sets are assigned an action: A=a activates X and Y, A=b activates X and Z, and A=c activates Y and Z. This is known as a 3-cycle in quantum contextuality (2 measurements out of 3).
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A history can be visualized directly on the DAG by emphasizing the active information sets and the edges corresponding to the action assignments. In the spacetime form, it forms multiple paths leading to multiple leaf nodes. For example, this is history 9 (A=c, Y=0, Z=0):
And this is the correspondence of the histories with the leaves of an equivalent game in extensive form that we have obtained in the previous section:
A history can be visualized in the extensive form as a path from the root to the corresponding leaf. This is, again, history 9:
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