Definition. Reward-induced observational equivalence [ftip-00AF]
Definition. Reward-induced observational equivalence [ftip-00AF]
Let \(\mathcal Y\) be a finite response set and \(r:\mathcal Y\to \mathcal R\) any reward map. Two responses are observationally equivalent under \(r\), written \(y\sim _r y'\), when
\[ y\sim _r y'\quad \Longleftrightarrow \quad r(y)=r(y'). \]Equality makes \(\sim _r\) an equivalence relation. Its quotient \(\mathcal Y/{\sim _r}\) is the finite set of response classes distinguished by the reward alone. This proposed equivalence relation ignores any information in the response that is not returned by \(r\); equal rewards need not imply equal latent utility.