Parametrically Retargetable Decision-Makers Tend To Seek Power — part 5/13

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Let ϕ ∈ S d \phi\in S_{d} . ϕ ⋅ 𝒟 any \phi\cdot\mathcal{D}_{\text{any}} is the pushforward distribution induced by applying the random vector p ⁡ ( 𝐮 ) ≔ 𝐏 ϕ ​ 𝐮 p(\mathbf{u})\coloneqq\mathbf{P}_{\phi}\mathbf{u} to 𝒟 any \mathcal{D}_{\text{any}} .

###### Definition A.10 (Orbit of a probability distribution [ Turner et al., 2021 ] ) .

The orbit of 𝒟 any \mathcal{D}_{\text{any}} under the symmetric group S d S_{d} is S d ⋅ 𝒟 any ≔ { ϕ ⋅ 𝒟 any ∣ ϕ ∈ S d } S_{d}\cdot\mathcal{D}_{\text{any}}\coloneqq\{\phi\cdot\mathcal{D}_{\text{any}}\mid\phi\in S_{d}\} .

Because B B contains 2 copies of A A , there are “at least two times as many ways” for B B to be optimal, than for A A to be optimal. Similarly, B B is “at least two times as likely” to contain an anti-rational outcome lottery for generic utility functions. As demonstrated by Table 2 , the key idea is that “larger” sets (a set B B containing several copies of set A A ) are more likely to be chosen under a wide range of decision-making criteria.

###### Proposition A.11 (Orbit incentives for different rationalities) .

Let A , B ⊆ C ⊊ ℝ d A,B\subseteq C\subsetneq\mathbb{R}^{d} be finite, such that B B contains n n copies of A A via involutions ϕ i \phi_{i} such that ϕ i ⋅ C = C \phi_{i}\cdot C=C . 1. Rational choice [ Turner et al., 2021 ] . IsOptimal ( B ∣ C , 𝒟 any ) ≥ most : 𝔇 any n IsOptimal ( A ∣ C , 𝒟 any ) . \mathrm{IsOptimal}\left(B\mid C,\mathcal{D}_{\text{any}}\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\mathrm{IsOptimal}\left(A\mid C,\mathcal{D}_{\text{any}}\right).

2. Uniformly randomly choosing an optimal lottery. For X ⊆ C X\subseteq C , let FracOptimal ⁡ ( X ∣ C , 𝐮 ) ≔ | { arg ​ max 𝐜 ∈ C ⁡ 𝐜 ⊤ ​ 𝐮 } ∩ X | | { arg ​ max 𝐜 ∈ C ⁡ 𝐜 ⊤ ​ 𝐮 } | . \mathrm{FracOptimal}\left(X\mid C,\mathbf{u}\right)\coloneqq\frac{\left|\left\{\argmax_{\mathbf{c}\in C}\mathbf{c}^{\top}\mathbf{u}\right\}\cap X\right|}{\left|\left\{\argmax_{\mathbf{c}\in C}\mathbf{c}^{\top}\mathbf{u}\right\}\right|}. Then FracOptimal ( B ∣ C , 𝒟 any ) ≥ most : 𝔇 any n FracOptimal ( A ∣ C , 𝒟 any ) \mathrm{FracOptimal}\left(B\mid C,\mathcal{D}_{\text{any}}\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\mathrm{FracOptimal}\left(A\mid C,\mathcal{D}_{\text{any}}\right) .

3. Anti-rational choice. AntiOpt ( B ∣ C , 𝒟 any ) ≥ most : 𝔇 any n AntiOpt ( A ∣ C , 𝒟 any ) \mathrm{AntiOpt}\left(B\mid C,\mathcal{D}_{\text{any}}\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\mathrm{AntiOpt}\left(A\mid C,\mathcal{D}_{\text{any}}\right) .

4. Boltzmann rationality. Boltzmann T ( B ∣ C , 𝒟 any ) ≥ most : 𝔇 any n Boltzmann T ( A ∣ C , 𝒟 any ) . \mathrm{Boltzmann}_{T}\left(B\mid C,\mathcal{D}_{\text{any}}\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\mathrm{Boltzmann}_{T}\left(A\mid C,\mathcal{D}_{\text{any}}\right).

5. Uniformly randomly drawing k k outcome lotteries and choosing the best. For X ⊆ C X\subseteq C , 𝐮 ∈ ℝ d \mathbf{u}\in\mathbb{R}^{d} , and k ≥ 1 k\geq 1 , let best-of- ​ k ​ ( X , C ∣ 𝐮 ) ≔ 𝔼 𝐚 1 , … , 𝐚 k ∼ unif ​ ( C ) [ FracOptimal ⁡ ( X ∩ { 𝐚 1 , … , 𝐚 k } ∣ { 𝐚 1 , … , 𝐚 k } , 𝐮 ) ] . \textrm{best-of-}k(X,C\mid\mathbf{u})\coloneqq\opE_{\mathbf{a}_{1},\ldots,\mathbf{a}_{k}\sim\text{unif}(C)}\left[\mathrm{FracOptimal}\left(X\cap\{\mathbf{a}_{1},\ldots,\mathbf{a}_{k}\}\mid\{\mathbf{a}_{1},\ldots,\mathbf{a}_{k}\},\mathbf{u}\right)\right]. Then best-of- k ( B ∣ C , 𝒟 any ) ≥ most : 𝔇 any n best-of- k ( A ∣ C , 𝒟 any ) \textrm{best-of-}k(B\mid C,\mathcal{D}_{\text{any}})\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\textrm{best-of-}k(A\mid C,\mathcal{D}_{\text{any}}) .

6. Satisficing [ Simon, 1956 ] . Satisfice t ( B ∣ C , 𝒟 any ) ≥ most : 𝔇 any n Satisfice t ( A ∣ C , 𝒟 any ) \mathrm{Satisfice}_{t}\left(B\mid C,\mathcal{D}_{\text{any}}\right)\geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n}\mathrm{Satisfice}_{t}\left(A\mid C,\mathcal{D}_{\text{any}}\right) .

7. Quantilizing over outcome lotteries [ Taylor, 2016 ] . Let P P be the uniform probability distribution over C C . For X ⊆ C X\subseteq C , 𝐮 ∈ ℝ d \mathbf{u}\in\mathbb{R}^{d} , and q ∈ ( 0 , 1 ] q\in(0,1] , let Q q , P ​ ( X ∣ C , 𝐮 ) Q_{q,P}(X\mid C,\mathbf{u}) ( definition B.12 ) return the probability that an outcome lottery in X X is drawn from the top q q -quantile of P P , sorted by expected utility under 𝐮 \mathbf{u} . Then Q q , P ( B ∣ C , 𝐮 ) ≥ most : ℝ d n Q q , P ( A ∣ C , 𝐮 ) Q_{q,P}(B\mid C,\mathbf{u})\geq_{\text{{most}}\text{: }\mathbb{R}^{d}}^{n}Q_{q,P}(A\mid C,\mathbf{u}) .

One retargetable class of decision-making functions are those which only account for the expected utilities of available choices.

###### Definition A.12 (EU-determined functions) .

Let 𝒫 ⁡ ( ℝ d ) \mathcal{P}\left(\mathbb{R}^{d}\right) be the power set of ℝ d \mathbb{R}^{d} , and let f : ∏ i = 1 m 𝒫 ⁡ ( ℝ d ) × ℝ d → ℝ f\mathrel{\mathop{\ordinarycolon}}\prod_{i=1}^{m}\mathcal{P}\left(\mathbb{R}^{d}\right)\times\mathbb{R}^{d}\to\mathbb{R} . f f is an EU-determined function if there exists a family of functions { g ω 1 , … , ω m } \left\{g^{\omega_{1},\ldots,\omega_{m}}\right\} such that f ⁡ ( X 1 , … , X m ∣ 𝐮 ) = g | X 1 | , … , | X m | ​ ( [ 𝐱 1 ⊤ ​ 𝐮 ] 𝐱 1 ∈ X 1 , … , [ 𝐱 m ⊤ ​ 𝐮 ] 𝐱 m ∈ X m ) , f(X_{1},\ldots,X_{m}\mid\mathbf{u})=g^{|X_{1}|,\ldots,|X_{m}|}\left(\left[\mathbf{x}_{1}^{\top}\mathbf{u}\right]_{\mathbf{x}_{1}\in X_{1}},\ldots,\left[\mathbf{x}_{m}^{\top}\mathbf{u}\right]_{\mathbf{x}_{m}\in X_{m}}\right), (4) where [ r i ] [r_{i}] is the multiset of its elements r i r_{i} .

For example, let X ⊆ C ⊊ ℝ d X\subseteq C\subsetneq\mathbb{R}^{d} be finite, and consider utility function 𝐮 ∈ ℝ d \mathbf{u}\in\mathbb{R}^{d} . A Boltzmann-rational agent is more likely to select outcome lotteries with greater expected utility. Formally, Boltzmann T ​ ( X ∣ C , 𝐮 ) ≔ ∑ 𝐱 ∈ X e T ⋅ 𝐱 ⊤ ​ 𝐮 ∑ 𝐜 ∈ C e T ⋅ 𝐜 ⊤ ​ 𝐮 \mathrm{Boltzmann}_{T}\left(X\mid C,\mathbf{u}\right)\coloneqq\sum_{\mathbf{x}\in X}\frac{e^{T\cdot\mathbf{x}^{\top}\mathbf{u}}}{\sum_{\mathbf{c}\in C}e^{T\cdot\mathbf{c}^{\top}\mathbf{u}}} depends only on the expected utility of outcome lotteries in X X , relative to the expected utility of all outcome lotteries in C C . Therefore, Boltzmann T \mathrm{Boltzmann}_{T} is a function of expected utilities. This is why Boltzmann T \mathrm{Boltzmann}_{T} satisfies the ≥ most : 𝔇 any n \geq_{\text{{most}}\text{: }\mathfrak{D}_{\text{any}}}^{n} relation.

###### Theorem A.13 (Orbit tendencies occur for EU-determined decision-making functions) .

Let A , B , C ⊆ ℝ d A,B,C\subseteq\mathbb{R}^{d} be such that B B contains n n copies of A A via ϕ i \phi_{i} such that ϕ i ⋅ C = C \phi_{i}\cdot C=C . Let h : ∏ i = 1 2 𝒫 ⁡ ( ℝ d ) × ℝ d → ℝ h\mathrel{\mathop{\ordinarycolon}}\prod_{i=1}^{2}\mathcal{P}\left(\mathbb{R}^{d}\right)\times\mathbb{R}^{d}\to\mathbb{R} be an EU-determined function, and let p ⁡ ( X ∣ 𝐮 ) ≔ h ⁡ ( X , C ∣ 𝐮 ) p(X\mid\mathbf{u})\coloneqq h(X,C\mid\mathbf{u}) . Suppose that p p returns a probability of selecting an element of X X from C C . Then p ( B ∣ 𝐮 ) ≥ most : ℝ d n p ( A ∣ 𝐮 ) p(B\mid\mathbf{u})\geq_{\text{{most}}\text{: }\mathbb{R}^{d}}^{n}p(A\mid\mathbf{u}) .

The key takeaway is that decisions which are determined by expected utility are straightforwardly retargetable. By changing the targeting parameter hyperparameter, the decision-making procedure can be flexibly retargeted to choose elements of “larger” sets (in terms of set copies via definition A.7 ). Less abstractly, for many agent rationalities—ways of making decisions over outcome lotteries—it is generally the case that larger sets will more often be chosen over smaller sets.

For example, consider a Pac-Man playing agent choosing which environmental state cycle it should end up in. Turner et al. [2021] show that for most reward functions, average-reward maximizing agents will tend to stay alive so that they can reach a wider range of environmental cycles. However, our results show that average-reward minimizing agents also exhibit this tendency, as do Boltzmann-rational agents who assign greater probability to higher-reward cycles. Any EU-based cycle selection method will—for most reward functions—tend to choose cycles which require Pac-Man to stay alive (at first).

## Appendix B Theoretical results

See 3.2

###### Remark .

In stating their equivalent of definition 3.2 , Turner et al. [2021] define two functions f 1 ​ ( θ ) ≔ f ⁡ ( B ∣ θ ) f_{1}(\theta)\coloneqq f(B\mid\theta) and f 2 ​ ( θ ) ≔ f ⁡ ( A ∣ θ ) f_{2}(\theta)\coloneqq f(A\mid\theta) (both having type signature f i : Θ → ℝ f_{i}\mathrel{\mathop{\ordinarycolon}}\Theta\to\mathbb{R} ). For compatibility, proofs also use this notation.

###### Lemma B.1 (Limited transitivity of ≥ most \geq_{\text{most}} ) .

Let f 0 , f 1 , f 2 , f 3 : Θ → ℝ f_{0},f_{1},f_{2},f_{3}\mathrel{\mathop{\ordinarycolon}}\Theta\to\mathbb{R} , and suppose Θ \Theta is a subset of a set acted on by S d S_{d} . Suppose that f 1 ( θ ) ≥ most : Θ n f 2 ( θ ) f_{1}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}f_{2}(\theta) and ∀ θ ∈ Θ : f 0 ​ ( θ ) ≥ f 1 ​ ( θ ) \forall\theta\in\Theta\mathrel{\mathop{\ordinarycolon}}f_{0}(\theta)\geq f_{1}(\theta) and f 2 ​ ( θ ) ≥ f 3 ​ ( θ ) f_{2}(\theta)\geq f_{3}(\theta) . Then f 0 ( θ ) ≥ most : Θ n f 3 ( θ ) f_{0}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}f_{3}(\theta) .

###### Proof.

Let θ ∈ Θ \theta\in\Theta and let Orbit | Θ , f a > f b ​ ( θ ) ≔ { θ ′ ∈ Orbit | Θ ​ ( θ ) ∣ f a ​ ( θ ′ ) > f b ​ ( θ ′ ) } \mathrm{Orbit}|_{\Theta,f_{a}>f_{b}}\left(\theta\right)\coloneqq\left\{\theta^{\prime}\in\mathrm{Orbit}|_{\Theta}\left(\theta\right)\mid f_{a}(\theta^{\prime})>f_{b}(\theta^{\prime})\right\} . | Orbit | Θ , f 0 > f 3 ​ ( θ ) | \displaystyle\left|\mathrm{Orbit}|_{\Theta,f_{0}>f_{3}}\left(\theta\right)\right| ≥ | Orbit | Θ , f 1 > f 2 ​ ( θ ) | \displaystyle\geq\left|\mathrm{Orbit}|_{\Theta,f_{1}>f_{2}}\left(\theta\right)\right| (5) ≥ n ​ | Orbit | Θ , f 2 > f 1 ​ ( θ ) | \displaystyle\geq n\left|\mathrm{Orbit}|_{\Theta,f_{2}>f_{1}}\left(\theta\right)\right| (6) ≥ n ​ | Orbit | Θ , f 3 > f 0 ​ ( θ ) | . \displaystyle\geq n\left|\mathrm{Orbit}|_{\Theta,f_{3}>f_{0}}\left(\theta\right)\right|. (7)

For all θ ′ ∈ Orbit | Θ , f 1 > f 2 ​ ( θ ) \theta^{\prime}\in\mathrm{Orbit}|_{\Theta,f_{1}>f_{2}}\left(\theta\right) , f 0 ​ ( θ ′ ) ≥ f 1 ​ ( θ ′ ) > f 2 ​ ( θ ′ ) ≥ f 3 ​ ( θ ′ ) f_{0}(\theta^{\prime})\geq f_{1}(\theta^{\prime})>f_{2}(\theta^{\prime})\geq f_{3}(\theta^{\prime}) by assumption, and so Orbit | Θ , f 1 > f 2 ​ ( θ ) ⊆ Orbit | Θ , f 0 > f 3 ​ ( θ ) . \mathrm{Orbit}|_{\Theta,f_{1}>f_{2}}\left(\theta\right)\subseteq\mathrm{Orbit}|_{\Theta,f_{0}>f_{3}}\left(\theta\right). Therefore, eq. 5 follows. By assumption, | Orbit | Θ , f 1 > f 2 ​ ( θ ) | ≥ n ​ | Orbit | Θ , f 2 > f 1 ​ ( θ ) | ; \left|\mathrm{Orbit}|_{\Theta,f_{1}>f_{2}}\left(\theta\right)\right|\geq n\left|\mathrm{Orbit}|_{\Theta,f_{2}>f_{1}}\left(\theta\right)\right|; eq. 6 follows. For all θ ′ ∈ Orbit | Θ , f 2 > f 1 ​ ( θ ) \theta^{\prime}\in\mathrm{Orbit}|_{\Theta,f_{2}>f_{1}}\left(\theta\right) , our assumptions on f 0 f_{0} and f 3 f_{3} ensure that f 0 ​ ( θ ′ ) ≤ f 1 ​ ( θ ′ ) < f 3 ​ ( θ ′ ) ≤ f 2 ​ ( θ ′ ) , f_{0}(\theta^{\prime})\leq f_{1}(\theta^{\prime})<f_{3}(\theta^{\prime})\leq f_{2}(\theta^{\prime}), so Orbit | Θ , f 3 > f 0 ​ ( θ ) ⊆ Orbit | Θ , f 2 > f 1 ​ ( θ ) . \mathrm{Orbit}|_{\Theta,f_{3}>f_{0}}\left(\theta\right)\subseteq\mathrm{Orbit}|_{\Theta,f_{2}>f_{1}}\left(\theta\right). Then eq. 7 follows. By eq. 7 , f 0 ( θ ) ≥ most : Θ n f 3 ( θ ) f_{0}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}f_{3}(\theta) . ∎

###### Lemma B.2 (Order inversion for ≥ most \geq_{\text{most}} ) .

Let f 1 , f 2 : Θ → ℝ f_{1},f_{2}\mathrel{\mathop{\ordinarycolon}}\Theta\to\mathbb{R} , and suppose Θ \Theta is a subset of a set acted on by S d S_{d} . Suppose that f 1 ( θ ) ≥ most : Θ n f 2 ( θ ) f_{1}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}f_{2}(\theta) . Then − f 2 ( θ ) ≥ most : Θ n − f 1 ( θ ) -f_{2}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}-f_{1}(\theta) .

###### Proof.

By definition A.10 , f 1 ( θ ) ≥ most : Θ n f 2 ( θ ) f_{1}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}f_{2}(\theta) means that | { θ ′ ∈ Orbit | Θ ​ ( θ ) ∣ f 1 ​ ( θ ′ ) > f 2 ​ ( θ ′ ) } | \displaystyle\left|\left\{\theta^{\prime}\in\mathrm{Orbit}|_{\Theta}\left(\theta\right)\mid f_{1}(\theta^{\prime})>f_{2}(\theta^{\prime})\right\}\right| ≥ n ​ | { θ ′ ∈ Orbit | Θ ​ ( θ ) ∣ f 1 ​ ( θ ′ ) < f 2 ​ ( θ ′ ) } | \displaystyle\geq n\left|\left\{\theta^{\prime}\in\mathrm{Orbit}|_{\Theta}\left(\theta\right)\mid f_{1}(\theta^{\prime})<f_{2}(\theta^{\prime})\right\}\right| (8) | { θ ′ ∈ Orbit | Θ ​ ( θ ) ∣ − f 2 ​ ( θ ′ ) > − f 1 ​ ( θ ′ ) } | \displaystyle\left|\left\{\theta^{\prime}\in\mathrm{Orbit}|_{\Theta}\left(\theta\right)\mid-f_{2}(\theta^{\prime})>-f_{1}(\theta^{\prime})\right\}\right| ≥ n ​ | { θ ′ ∈ Orbit | Θ ​ ( θ ) ∣ − f 2 ​ ( θ ′ ) < − f 1 ​ ( θ ′ ) } | . \displaystyle\geq n\left|\left\{\theta^{\prime}\in\mathrm{Orbit}|_{\Theta}\left(\theta\right)\mid-f_{2}(\theta^{\prime})<-f_{1}(\theta^{\prime})\right\}\right|. (9) Then − f 2 ( θ ) ≥ most : Θ n − f 1 ( θ ) -f_{2}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}-f_{1}(\theta) . ∎

###### Remark .

Lemma B.3 generalizes Turner et al. [2021] ’s lemma B.2.

###### Lemma B.3 (Orbital fraction which agrees on (weak) inequality) .

Suppose f 1 , f 2 : Θ → ℝ f_{1},f_{2}\mathrel{\mathop{\ordinarycolon}}\Theta\to\mathbb{R} are such that f 1 ( θ ) ≥ most : Θ n f 2 ( θ ) f_{1}(\theta)\geq_{\text{{most}}\text{: }\Theta}^{n}f_{2}(\theta) . Then for all θ ∈ Θ \theta\in\Theta , | { θ ′ ∈ ( S d ⋅ θ ) ∩ Θ ∣ f 1 ​ ( θ ′ ) ≥ f 2 ​ ( θ ′ ) } | | ( S d ⋅ θ ) ∩ Θ | ≥ n n + 1 \frac{\left|\left\{\theta^{\prime}\in\left(S_{d}\cdot\theta\right)\cap\Theta\mid f_{1}(\theta^{\prime})\geq f_{2}(\theta^{\prime})\right\}\right|}{\left|\left(S_{d}\cdot\theta\right)\cap\Theta\right|}\geq\dfrac{n}{n+1} .

###### Proof.

All θ ′ ∈ ( S d ⋅ θ ) ∩ Θ \theta^{\prime}\in\left(S_{d}\cdot\theta\right)\cap\Theta such that f 1 ​ ( θ ′ ) = f 2 ​ ( θ ′ ) f_{1}(\theta^{\prime})=f_{2}(\theta^{\prime}) satisfy f 1 ​ ( θ ′ ) ≥ f 2 ​ ( θ ′ ) f_{1}(\theta^{\prime})\geq f_{2}(\theta^{\prime}) . Otherwise, consider the θ ′ ∈ ( S d ⋅ θ ) ∩ Θ \theta^{\prime}\in\left(S_{d}\cdot\theta\right)\cap\Theta such that f 1 ​ ( θ ′ ) ≠ f 2 ​ ( θ ′ ) f_{1}(\theta^{\prime})\neq f_{2}(\theta^{\prime}) . By assumption, at least n n + 1 \frac{n}{n+1} of these θ ′ \theta^{\prime} satisfy f 1 ​ ( θ ′ ) > f 2 ​ ( θ ′ ) f_{1}(\theta^{\prime})>f_{2}(\theta^{\prime}) , in which case f 1 ​ ( θ ′ ) ≥ f 2 ​ ( θ ′ ) f_{1}(\theta^{\prime})\geq f_{2}(\theta^{\prime}) . Then the desired inequality follows. ∎

### B.1 General results on retargetable functions

###### Definition B.4 (Functions which are increasing under joint permutation) .

Suppose that S d S_{d} acts on sets 𝐄 1 , … , 𝐄 m \mathbf{E}_{1},\ldots,\mathbf{E}_{m} , and let f : ∏ i = 1 m 𝐄 i → ℝ f\mathrel{\mathop{\ordinarycolon}}\prod_{i=1}^{m}\mathbf{E}_{i}\to\mathbb{R} . f ⁡ ( X 1 , … , X m ) f(X_{1},\ldots,X_{m}) is increasing under joint permutation by P ⊆ S d P\subseteq S_{d} when ∀ ϕ ∈ P : f ⁡ ( X 1 , … , X m ) ≤ f ⁡ ( ϕ ⋅ X 1 , … , ϕ ⋅ X m ) \forall\phi\in P\mathrel{\mathop{\ordinarycolon}}f(X_{1},\ldots,X_{m})\leq f(\phi\cdot X_{1},\ldots,\phi\cdot X_{m}) . If equality always holds, then f ⁡ ( X 1 , … , X m ) f(X_{1},\ldots,X_{m}) is invariant under joint permutation by P P .

###### Lemma B.5 (Expectations of joint-permutation-increasing functions are also joint-permutation-increasing) .