Table of contents
This post treats reward functions as “specifying goals”, in some sense. As I explained in Reward Is Not The Optimization Target, this is a misconception that can seriously damage your ability to understand how AI works. Rather than “incentivizing” behavior, reward signals are (in many cases) akin to a perdatapoint learning rate. Reward chisels circuits into the AI. That’s it!
A year ago, I thought it would be really hard to generalize the powerseeking theorems from Markov decision processes (mdps); the mdp case seemed hard enough. Without assuming the agent can see the full state, while letting utility functions do as they please—this seemed like asking for trouble.
Once I knew what to look for, it turned out to be easy—I hashed out the basics during half an hour of conversation with John Wentworth. The theorems were never about mdps anyways; the theorems apply whenever the agent considers finite sets of lotteries over outcomes, assigns each outcome realvalued utility, and maximizes expected utility.
ThanksThanks to Rohin Shah, Adam Shimi, and John Wentworth for feedback on drafts of this post.
At each time step $t$, the agent takes one of finitely many actions $a_{t}∈A$, and receives one of finitely many observations $o_{t}∈O$ drawn from the conditional probability distribution $E(o_{t}∣a_{1}o_{1}…a_{t})$, where $E$ is the environment.^{1} There is a finite time horizon $T$. Each utility function $u:O_{T}→R$ maps each complete observation history to a real number (note that $u$ can be represented as a vector in the finitedimensional vector space $R_{∣O∣_{T}}$). From now on, u_{OH} stands for “utility function(s) over observation histories.”
First, let’s just consider a deterministic environment. Each time step, the agent observes a blackandwhite image ($n×n$) through a webcam, and it plans over a 50step episode ($T=50$). Each time step, the agent acts by choosing a pixel to bitflip for the next time step.
And let’s say that if the agent flips the first pixel for its first action, it “dies”: its actions no longer affect any of its future observations past time step $t=2$. If the agent doesn’t flip the first pixel at $t=1$, it’s able to flip bits normally for all $T=50$ steps.
Do u_{OH} tend to incentivize flipping the first pixel over flipping the second pixel, vice versa, or neither?
If the agent flips the first bit, it’s locked into a single trajectory. None of its actions matter anymore.
But if the agent flips the second bit—this may be suboptimal for a utility function, but the agent still has lots of choices remaining. In fact, it still can induce $(n×n)_{T−1}$ observation histories. If $n=100$ and $T=50$, then that’s $(100×100)_{49}=10_{196}$ observation histories. Probably at least one of these yields greater utility than the shutdownhistory utility.
And indeed, we can apply the scaling law for instrumental convergence to conclude that for every u_{OH}, at least $10_{196}+110_{196} $ of its permuted variants (weakly) prefer flipping the second pixel at $t=1$, over flipping the first pixel at $t=1$.
$10_{196}+110_{196} .$Choose any atom in the universe. Uniformly randomly select another atom in the universe. It’s about $10_{117}$ times more likely that these atoms are the same, than that a utility function incentivizes “dying” instead of flipping pixel 2 at $t=1$.
The general rule will be: for every u_{OH}, at least $(n×n)_{T−1}+1(n×n)_{T−1} $ of its permuted variants weakly prefer flipping the second pixel at $t=1$, over flipping the first pixel at $t=1$. And for almost all u_{OH}, you can replace “weakly” with “strictly.”
The powerseeking results hinge on the probability of certain linear functionals being “optimal.” For example, let $A,B,C⊊R_{n}$ be finite sets of vectors,^{2} and let $D_{any}$ be any probability distribution over $R_{n}$.
Definition: Optimality probability of a linear functional setThe optimality probability of $A$ relative to $C$ under distribution $D_{any}$ is
$p_{D_{any}}(A≥C)=defP_{r∼D_{any}}(a∈Amax a_{⊤}r≥c∈Cmax c_{⊤}r).$If vectors represent lotteries over outcomes (where each outcome has its own entry), then we can say that:

$A$ and $B$ each contain some of the things the agent could make happen

$C$ contains all of the things the agent could make happen ($A,B⊆C$)

Each $r∼D_{any}$ is a utility function over outcomes, with one value for each entry.
 If $x∈R_{n}$ is an outcome lottery, then $x_{⊤}r$ is its $r$expected value.

Things in $A$ are more likely$_{D_{any}}$ to be optimal than things in $B$ when $p_{D_{any}}(A≥C)≥p_{D_{any}}(B≥C)$.
 This isn’t the notion of “tends to be optimal” we’re using in this post; instead, we’re using a stronger line of reasoning that says: for most variants of every utility function, suchandsuch is true.
Nothing here has anything to do with a Markov decision process, or the world being finite, or fully observable, or whatever. Fundamentally, the powerseeking theorems were never about mdps—they were secretly about the probability that a set $A$ of linear functionals is optimal, with respect to another set $C$. mdps were just a way to relax the problem.
In terms of the pixelflipping environment:

When followed from a time step, each (deterministic) policy $π$ induces a distribution over observation histories
 These are represented as unit vectors, with each entry marking the probability that an observation history is realized
 If the environment is deterministic, all deterministic policies induce standard basis vectors (probability 1 on their induced observation history, 0 elsewhere)

Let $B$ be the set of histories available given that $π$ selects $a_{1}$ ('death’) at the first time step.
 As argued above, $∣B∣=1$—the agent loses all control over future observations. Its element is a standard basis vector.

Define $A$ similarly for $a_{2}$ (flipping pixel 2) at the first time step.
 As argued above, $∣A∣=10_{196}$; all elements are standard basis vectors by determinism.

Let $C$ be the set of all available observation histories, starting from the first time step.

There exist $10_{196}$ different involutions $ϕ$ over observation histories such that $ϕ(B)=A_{′}⊆A$ (each $ϕ$ transposing $B$’s element with a different element of $A$). Each one just swaps the deathhistory with an $a_{2}$history.
 By the scaling law of instrumental convergence, we conclude that For every u_{OH}, at least$10_{196}+110_{196} $ of its permuted variants (weakly) prefer flipping the second pixel at$t=1$, over flipping the first pixel at$t=1$.
I often give lifevsdeath examples because they’re particularly easy to reason about. But the theorems apply to more general cases of morevsless control.
For example, if $a_{1}$ restricts the agent to two effective actions at each time step (it can only flip one of the first two pixels)—instead of “killing” the agent, then $a_{2}$ is still convergently instrumental over $a_{1}$. There are $2_{49}≈5.6×10_{14}≥10_{14}$ observation histories available after taking action $a_{1}$, and so these can be embedded at least $10_{14}10_{196} =10_{182}$ times into the observation histories available after $a_{2}$. Then for every u_{OH}, at least $10_{182}+110_{182} $ of its permuted variants (weakly) prefer flipping the second pixel at $t=1$, over flipping the first pixel at $t=1$.
Let’s consider utility functions over actionobservation histories (u_{aoh}).
Since each utility function is over an aoh, each path through the tree is assigned a certain amount of utility. But when the environment is deterministic, it doesn’t matter what the agent observes at any point in time—all that matters is which path is taken through the tree. Without further assumptions, u_{aoh} won’t tend to assign higher utility to one subtree than to another.
More formally, for any two actions $a_{1}$ and $a_{2}$, let $ϕ$ be a permutation over aoh which transposes the histories available after $a_{1}$ with the histories available after $a_{2}$ (there’s an equal number of histories for each action, due to the regularity of the tree—you can verify this by inspection).
For every u_{aoh} $u$, suppose $a_{1}$ is strictly $u$optimal over $a_{2}$. The permuted utility function $ϕ⋅u$ makes $a_{2}$ be strictly $u$optimal over $a_{1}$, since $ϕ$ swaps $a_{1}$’s strictly $u$optimal history with $a_{2}$’s strictly $u$suboptimal histories.
Symmetrically, $ϕ$ works the other way around ({$a_{2}$ strictly optimal} $→$ {$a_{1}$ strictly optimal}). Therefore, for every utility function $u$, the # of variants which strictly prefer $a_{1}$ over $a_{2}$, is equal to the # of variants strictly preferring $a_{2}$ over $a_{1}$.
While I haven’t been writing in the “definitiontheoremcorollary” style, the key claims are just corollaries of the scaling law of instrumental convergence. They’re provably true. (I’m just not writing up the math here because it’s annoying to define all the relevant quantities in a nice way that respects existing formalisms.)
And even if the environment is stochastic, I think that there won’t be any kind of interesting instrumental convergence. The theorems let us reason about that case, but their applicability depends on the details of the stochasticity, and so I won’t talk about that more here.
Conclusion: Optimal policies for u_{aoh} will tend to look like random twitching. For example, if you generate a u_{aoh} by uniformly randomly assigning each aoh utility from the unit interval $[0,1]$, there’s no predictable regularity to the optimal actions for this utility function. In this setting and under our assumptions, there is no instrumental convergence without further structural assumptions.
Consider the $n=2$ pixelflipping case (with $T=50$ still). Action $a_{1}$ still leads to a single OH, while $a_{2}$ leads to $(2×2)_{49}=4_{49}≈10_{29}$ OHs. So we have instrumental convergence for $10_{29}+110_{29} $ of all u_{OH} variants.
Let’s model the pixelflipping environment as a Markov decision process (mdp), with both the timestep and alive/dead status observed at each time step in order to ensure full observability, and the final timestep observations being terminal states where the agent stays forever. Dying allows the agent access to 1 terminal state: the observation 1/0/0/0 (dead)
. But surviving via $a_{2}$ lets the agent access $2_{4}=16$ terminal states (all 16 binary strings of length 4, with ‘alive’ appended to the end).
For each reward function over states, only $16+116 =1716 $ of its permuted variants will incentivize not dying at $t=1$ (considering policies which maximize average pertimestep reward). This is a lot looser than the bound for u_{OH}. What gives?
mdps assume that utility functions have a lot of structure: the utility of a history is timediscounted additive over observations. Basically, $u(a_{1}o_{1}a_{2}o_{2}…)=def∑_{t=1}γ_{t−1}R(o_{t})$, for some $γ∈[0,1)$ and reward function $R:O→R$ over observations. And because of this structure, the agent’s average pertimestep reward is controlled by the last observation it sees. There are exponentially fewer last observations than there are observation histories. Therefore, in this situation, instrumental convergence is exponentially weaker for reward functions than for arbitrary u_{OH}.
This suggests that rolling a random u_{OH} for aixi might be far more dangerous than rolling a random reward function for an optimal reinforcement learner.
Structural assumptions on utility really do matter when it comes to instrumental convergence:
Setting  Strength of instrumental convergence 

u_{aoh}  Nonexistent 
u_{OH}  Strong 
Statebased objectives (e.g. statebased reward in mdps)  Moderate 
Environmental structure can cause instrumental convergence, but (the absence of) structural assumptions on utility can make instrumental convergence go away (for optimal agents).
Notes

Of course, you can represent u_{aoh} as u_{OH} by including the agent’s previous action in the next observation.
 But this is a different environment; whether or not this is in fact a good model depends on the agent’s action and observation encodings.

Timereversible dynamics & full observability is basically the u_{aoh} situation, since each action history leads to a unique world state at every time step.
 But if you take away full observability, timereversibility is insufficient to make instrumental convergence disappear.

For optimal agents, instrumental convergence can be extremely strong for utility functions over observation histories.

Instrumental convergence doesn’t exist for utility functions over actionobservation histories.
 i.e. optimal action will tend to look like random twitching.
 This echoes previous discussion of the triviality of coherence over actionobservation histories, when it comes to determining goaldirectedness.
 This suggests that consequentialism over observations/world states is responsible for convergent instrumental incentives.
 Approaches like approvaldirected agency focus on action selection instead of optimization over future observations.

Environmental structure can cause instrumental convergence, but (lack of) structural assumptions on utility can make instrumental convergence go away.
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alex@turntrout.com
Time to cross another item off of the list from last time; the theorems:
Quote
 assume the agent is following an optimal policy for a reward function
 I can relax this to $ϵ$optimality, but $ϵ>0$ may be extremely small
assume the environment is finite and fully observable Not all environments have the right symmetries
 But most ones we think about seem to
 don’t account for the ways in which we might practically express reward functions
 For example, stateaction versus statebased reward functions (this particular case doesn’t seem too bad, I was able to sketch out some nice results rather quickly, since you can convert stateaction mdps into statebased reward mdps and then apply my results).
Re 3), in the setting of this post, when the observations are deterministic, the theorems will always apply. (You can always involute one set of unit vectors into another set of unit vectors in the observationhistory vector space.)
Another consideration is that when I talk about “powerseeking in the situations covered by my theorems”, the theorems don’t necessarily show that gaining social influence or money is convergently instrumental. I think that these “resources” are downstream of formalpower, and will eventually end up being understood in terms of formalpower—but the current results don’t directly prove that such highlevel subgoals are convergently instrumental.

For simplicity, I just consider environments which are joint probability distributions over actions and observation. This is much simpler than the lower semicomputable chronological conditional semimeasures used in the aixi literature, but it suffices for our purposes, and the theory could be extended to lscccss if someone wanted to. ⤴

I don’t think we need to assume finite sets of vectors, but things get a lot harder and messier when you’re dealing with $sup$ instead of $max$. It’s not clear how to define the nondominated elements of an infinite set, for example, and so a few key results break. One motivation for finite being enough is: in real life, a finite mind can only consider finitely many outcomes anyways, and can only plan over a finite horizon using finitely many actions. This is just one consideration, though. ⤴