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uiai [2026/03/17 01:40] – [Definition: Counterfactual action] pedroortegauiai [2026/03/17 10:30] (current) – [Definition: Third-party action] pedroortega
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 Note that $k'$ is determined inside the branch and therefore the length of $\dot{a}_{t+1}$ need not match the length of the on-path $\mathcal{A}$-token written by the agent starting at $k$. Note that $k'$ is determined inside the branch and therefore the length of $\dot{a}_{t+1}$ need not match the length of the on-path $\mathcal{A}$-token written by the agent starting at $k$.
 +
 +{{ ::uiai-cf-action.png?600 |Counterfactual Action}}
  
 **Diagram note.**   **Diagram note.**  
-The intended picture is that after the shared on-path prefix ending at $a_3,o_3$, the on-path transcript has factual action $a_4$, while the counterfactual branch replaces that with the world-generated block $\dot{a}_4$ occupying the same would-be action slot.+The diagram shows that after the shared on-path prefix ending at $a_3,o_3$, the on-path transcript has factual action $a_4$, while the counterfactual action $\dot{a}_4$ is spawn from the same prefix but generated by the world.
  
 ==== Definition: Third-party action ==== ==== Definition: Third-party action ====
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 o_t = w\,\dot{a}_{t+1}\,v. o_t = w\,\dot{a}_{t+1}\,v.
 $$ $$
 +
 +{{ :uiai-third-party.png?600 |}}
  
 **Diagram note.**   **Diagram note.**  
-The intended picture is that inside a long world-written observation token, there is an embedded block $\dot{a}_4$ that occupies an $\mathcal{A}$-position under the tokenization convention, even though on-path it is still written by the world and therefore appears as evidence.+The diagram illustrates a third party action. Inside a long observation token $o_3$, one can identify an embedded block $\dot{a}_4$ that is interpreted as a third-party action. Because it is written by the world, it counts as evidence.
  
 It is not hard to see that, for a given potential index $k$, the counterfactual action $\dot{a}_{t+1}$ and the third-party action $\dot{a}_{t+1}$ are the same random block: the difference is only whether the gate sampled $\gamma_k = 1$ (counterfactual, not observed) or $\gamma_k = 0$ (third-party, observed) at position $k$. The precise distinction between the different types of $\mathcal{A}$-tokens is important; we will also refer to them as //factual// (first-person, agent-generated), //counterfactual//, and //third-party// $\mathcal{A}$-tokens. It is not hard to see that, for a given potential index $k$, the counterfactual action $\dot{a}_{t+1}$ and the third-party action $\dot{a}_{t+1}$ are the same random block: the difference is only whether the gate sampled $\gamma_k = 1$ (counterfactual, not observed) or $\gamma_k = 0$ (third-party, observed) at position $k$. The precise distinction between the different types of $\mathcal{A}$-tokens is important; we will also refer to them as //factual// (first-person, agent-generated), //counterfactual//, and //third-party// $\mathcal{A}$-tokens.
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