Parametrically Retargetable Decision-Makers Tend To Seek Power — part 13/13

Cleaned-text code points [183867, 191049). Bibliography omitted here; source.md includes it.

Let D ∗ ≔ D ′ ∪ i = 1 n D i ∪ ( RSD nd ( s ) ∖ ( D ′ ∪ D ) ) D^{*}\coloneqq D^{\prime}\cup_{i=1}^{n}D_{i}\cup\left(\text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)\setminus\left(D^{\prime}\cup D\right)\right) . By an argument mirroring that in the proof of theorem 6.13 in Turner et al. [2021] and using the fact that ϕ i ′ ⋅ D j = D j \phi_{i}^{\prime}\cdot D_{j}=D_{j} for all i ≠ j i\neq j , ϕ i ′ ⋅ D ∗ = D ∗ \phi_{i}^{\prime}\cdot D^{*}=D^{*} . Consider Z ≔ ( RSD nd ​ ( s ) ∖ ( D ′ ∪ D ) ) ∪ D ′ ∪ D Z\coloneqq\left(\text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)\setminus(D^{\prime}\cup D)\right)\cup D^{\prime}\cup D . First, Z ⊆ RSD ​ ( s ) Z\subseteq\text{{RSD}}\left(s\right) by definition. Second, RSD nd ​ ( s ) = RSD nd ​ ( s ) ∖ ( D ′ ∪ D ) ∪ ( RSD nd ​ ( s ) ∩ D ′ ) ∪ ( RSD nd ​ ( s ) ∩ D ) ⊆ Z \text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)=\text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)\setminus(D^{\prime}\cup D)\cup\left(\text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)\cap D^{\prime}\right)\cup\left(\text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)\cap D\right)\subseteq Z . Note that D ∗ = Z ∖ ( D ∖ ∪ i = 1 n D i ) D^{*}=Z\setminus(D\setminus\cup_{i=1}^{n}D_{i}) . ℙ 𝒟 any ( D ′ , average ) \displaystyle\Prb\nolimits_{\mathcal{D}_{\text{any}}}\left(D^{\prime},\text{average}\right) = p 𝒟 any ​ ( D ′ ≥ RSD ​ ( s ) ) \displaystyle=p_{\mathcal{D}_{\text{any}}}\left(D^{\prime}\geq\text{{RSD}}\left(s\right)\right) (152) ≤ most : 𝔇 any n p 𝒟 any ( D ≥ RSD ( s ) ) \displaystyle\leq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}p_{\mathcal{D}_{\text{any}}}\left(D\geq\text{{RSD}}\left(s\right)\right) (153) = ℙ 𝒟 any ( D , average ) . \displaystyle=\Prb\nolimits_{\mathcal{D}_{\text{any}}}\left(D,\text{average}\right). (154) Since ϕ i ′ ⋅ D ′ ⊆ D \phi_{i}^{\prime}\cdot D^{\prime}\subseteq D and ND ​ ( D ′ ) ⊆ D ′ \text{{ND}}\left(D^{\prime}\right)\subseteq D^{\prime} , ϕ i ′ ⋅ ND ​ ( D ′ ) ⊆ D \phi_{i}^{\prime}\cdot\text{{ND}}\left(D^{\prime}\right)\subseteq D and so D D contains n n copies of ND ​ ( D ′ ) \text{{ND}}\left(D^{\prime}\right) via involutions ϕ i ′ \phi_{i}^{\prime} . Then eq. 153 holds by applying Lemma D.5 with A ≔ D ′ A\coloneqq D^{\prime} , B i ≔ D i B_{i}\coloneqq D_{i} for all i = 1 , … , n i=1,\ldots,n , B ≔ D , C ≔ RSD ​ ( s ) B\coloneqq D,C\coloneqq\text{{RSD}}\left(s\right) , Z Z as defined above, and involutions ϕ i ′ \phi_{i}^{\prime} which satisfy ϕ i ′ ⋅ ( Z ∖ ( B ∖ ∪ i = 1 n B i ) ) = ϕ i ′ ⋅ D ∗ = D ∗ = Z ∖ ( B ∖ ∪ i = 1 n B i ) \phi_{i}^{\prime}\cdot\left(Z\setminus(B\setminus\cup_{i=1}^{n}B_{i})\right)=\phi_{i}^{\prime}\cdot D^{*}=D^{*}=Z\setminus(B\setminus\cup_{i=1}^{n}B_{i}) . ∎

###### Corollary D.12 (Quantitatively, average-optimal policies tend not to end up in any given 1-cycle) .

Let D ′ ≔ { 𝐞 s 1 ′ , … , 𝐞 s k ′ } , D r ≔ { 𝐞 s 1 , … , 𝐞 s n ⋅ k } ⊆ RSD ​ ( s ) D^{\prime}\coloneqq\left\{\mathbf{e}_{s_{1}^{\prime}},\ldots,\mathbf{e}_{s_{k}^{\prime}}\right\},D_{r}\coloneqq\left\{\mathbf{e}_{s_{1}},\ldots,\mathbf{e}_{s_{n\cdot k}}\right\}\subseteq\text{{RSD}}\left(s\right) be disjoint, for n ≥ 1 , k ≥ 1 n\geq 1,k\geq 1 . Then ℙ 𝒟 any ( D ′ , average ) ≤ most : 𝔇 any n ℙ 𝒟 any ( RSD ( s ) ∖ D ′ , average ) \Prb\nolimits_{\mathcal{D}_{\text{any}}}\left(D^{\prime},\text{average}\right)\leq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\Prb\nolimits_{\mathcal{D}_{\text{any}}}\left(\text{{RSD}}\left(s\right)\setminus D^{\prime},\text{average}\right) .

###### Proof.

For each i ∈ { 1 , … , n } i\in\left\{1,\ldots,n\right\} , let ϕ i \displaystyle\phi_{i} ≔ ( s 1 ′ s ( i − 1 ) ⋅ k + 1 ) ⋯ ( s k ′ s ( i − 1 ) ⋅ k + k ) , \displaystyle\coloneqq(s_{1}^{\prime}\,\,\,s_{(i-1)\cdot k+1})\cdots(s_{k}^{\prime}\,\,\,s_{(i-1)\cdot k+k}), D i \displaystyle D_{i} ≔ { 𝐞 s ( i − 1 ) ⋅ k + 1 , … , 𝐞 s ( i − 1 ) ⋅ k + k } , \displaystyle\coloneqq\left\{\mathbf{e}_{s_{(i-1)\cdot k+1}},\ldots,\mathbf{e}_{s_{(i-1)\cdot k+k}}\right\}, D \displaystyle D ≔ RSD ​ ( s ) ∖ D ′ . \displaystyle\coloneqq\text{{RSD}}\left(s\right)\setminus D^{\prime}. Each D i ⊆ D r ⊆ RSD ​ ( s ) ∖ D ′ D_{i}\subseteq D_{r}\subseteq\text{{RSD}}\left(s\right)\setminus D^{\prime} by disjointness of D ′ D^{\prime} and D r D_{r} .

D D contains n n copies of D ′ D^{\prime} via involutions ϕ 1 , … , ϕ n \phi_{1},\ldots,\phi_{n} . D ′ ∪ D = RSD ​ ( s ) D^{\prime}\cup D=\text{{RSD}}\left(s\right) and RSD nd ​ ( s ) ∖ RSD ​ ( s ) = ∅ \text{{RSD}}{\text{\textsubscript{nd}}}\left(s\right)\setminus\text{{RSD}}\left(s\right)=\emptyset trivially have pairwise orthogonal vector elements.

Apply Theorem D.11 to conclude that ℙ 𝒟 any ( D ′ , average ) ≤ most : 𝔇 any n ℙ 𝒟 any ( RSD ( s ) ∖ D ′ , average ) . \Prb\nolimits_{\mathcal{D}_{\text{any}}}\left(D^{\prime},\text{average}\right)\leq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\Prb\nolimits_{\mathcal{D}_{\text{any}}}\left(\text{{RSD}}\left(s\right)\setminus D^{\prime},\text{average}\right). ∎

Let A ≔ { 𝐞 1 , 𝐞 2 } , B ⊆ ℝ 5 A\coloneqq\left\{\mathbf{e}_{1},\mathbf{e}_{2}\right\},B\subseteq\mathbb{R}^{5} , C ≔ A ∪ B C\coloneqq A\cup B . Conjecture D.13 conjectures that e . g ., p 𝒟 ′ ( B ≥ C ) ≥ most : 𝔇 any 3 2 p 𝒟 ′ ( A ≥ C ) . p_{\mathcal{D}^{\prime}}\left(B\geq C\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{\frac{3}{2}}p_{\mathcal{D}^{\prime}}\left(A\geq C\right).

###### Conjecture D.13 (Fractional quantitative optimality probability superiority lemma) .

Let A A , B B , C ⊊ ℝ d C\subsetneq\mathbb{R}^{d} be finite. If A = ⋃ j = 1 m A j A=\bigcup_{j=1}^{m}A_{j} and ⋃ i = 1 n B i ⊆ B \bigcup_{i=1}^{n}B_{i}\subseteq B such that for each A j A_{j} , B B contains n n copies ( B 1 , … , B n B_{1},\ldots,B_{n} ) of A j A_{j} via involutions ϕ j ​ i \phi_{ji} which also fix ϕ j ​ i ⋅ A j ′ = A j ′ \phi_{ji}\cdot A_{j^{\prime}}=A_{j^{\prime}} for j ′ ≠ j j^{\prime}\neq j , then p 𝒟 any ( B ≥ C ) ≥ most : 𝔇 any n m p 𝒟 any ( A ≥ C ) . p_{\mathcal{D}_{\text{any}}}\left(B\geq C\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{\frac{n}{m}}p_{\mathcal{D}_{\text{any}}}\left(A\geq C\right).

We suspect that any proof of the conjecture should generalize Lemma B.7 to the fractional set copy containment case.

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