# Towards shutdownable agents via stochastic choice

We[^usnm40pukij] have a [new paper](https://arxiv.org/abs/2407.00805) testing the [POST-Agents Proposal (PAP)](https://www.lesswrong.com/posts/YbEbwYWkf8mv9jnmi/the-shutdown-problem-incomplete-preferences-as-a-solution). The abstract and main-text is below. Appendices are in the [linked PDF](https://arxiv.org/pdf/2407.00805).

Abstract
========

*   Some worry that advanced artificial agents may resist being shut down.
*   The POST-Agents Proposal (PAP) is an idea for ensuring that doesn’t happen.
*   A key part of the PAP is using a novel ‘**D**iscounted **Re**ward for **S**ame-Length **T**rajectories (DReST)’ reward function to train agents to:
    1.  pursue goals effectively conditional on each trajectory-length (be ‘USEFUL’)
    2.  choose stochastically between different trajectory-lengths (be ‘NEUTRAL’ about trajectory-lengths).
*   In this paper, we propose evaluation metrics for USEFULNESS and NEUTRALITY.
*   We use a DReST reward function to train simple agents to navigate gridworlds, and we find that these agents learn to be USEFUL and NEUTRAL.
*   Our results thus suggest that DReST reward functions could also train advanced agents to be USEFUL and NEUTRAL, and thereby make these advanced agents useful and shutdownable.

1\. Introduction
================

1.1. The shutdown problem
-------------------------

Let ‘advanced agent’ refer to an artificial agent that can autonomously pursue complex goals in the wider world. We might see the arrival of advanced agents within the next few decades. There are strong economic incentives to create such agents, and creating systems like them is the stated goal of companies like OpenAI and Google DeepMind. 

The rise of advanced agents would bring with it both benefits and risks. One risk is that these agents learn misaligned goals: goals that we don’t want them to have \[[Leike et al., 2017](https://arxiv.org/pdf/1711.09883), [Hubinger et al., 2019](https://arxiv.org/abs/1906.01820), [Russell, 2019](https://www.penguin.co.uk/books/307948/human-compatible-by-russell-stuart/9780141987507), [Carlsmith, 2021](https://arxiv.org/abs/2206.13353), [Bengio et al., 2023](https://arxiv.org/abs/2310.17688), [Ngo et al., 2023](https://arxiv.org/abs/2209.00626)\]. Advanced agents with misaligned goals might try to prevent us shutting them down \[[Omohundro, 2008](https://dl.acm.org/doi/10.5555/1566174.1566226), [Bostrom, 2012](https://link.springer.com/article/10.1007/s11023-012-9281-3), [Soares et al., 2015](https://intelligence.org/files/Corrigibility.pdf), [Russell, 2019](https://www.penguin.co.uk/books/307948/human-compatible-by-russell-stuart/9780141987507), [Thornley, 2024a](https://link.springer.com/article/10.1007/s11098-024-02153-3)\]. After all, most goals can’t be achieved after shutdown. As Stuart Russell puts it, ‘you can’t fetch the coffee if you’re dead’ \[[Russell, 2019, p.141](https://www.penguin.co.uk/books/307948/human-compatible-by-russell-stuart/9780141987507)\]. 

Advanced agents with misaligned goals might resist shutdown by (for example) pretending to have aligned goals while covertly seeking to escape human control \[[Hubinger et al., 2019](https://arxiv.org/abs/1906.01820), [Ngo et al., 2023](https://arxiv.org/abs/2209.00626)\]. Agents that succeed in resisting shutdown could go on to frustrate human interests in various ways. ‘The shutdown problem’ is the problem of training advanced agents that won’t resist shutdown \[[Soares et al., 2015](https://intelligence.org/files/Corrigibility.pdf), [Thornley, 2024a](https://link.springer.com/article/10.1007/s11098-024-02153-3)\]. 

1.2. A proposed solution
------------------------

The POST-Agents Proposal (PAP) is a proposed solution to the shutdown problem \[[Thornley, 2024b](https://www.lesswrong.com/posts/YbEbwYWkf8mv9jnmi/the-shutdown-problem-incomplete-preferences-as-a-solution)\]. Simplifying slightly, the idea is that we train agents to be neutral about when they get shut down. More precisely, the idea is that we train agents to satisfy:

> ### **P**references **O**nly Between **S**ame-Length **T**rajectories (POST)
> 
> 1.  The agent has a preference between *many pairs of same-length trajectories* (i.e. many pairs of trajectories in which the agent is shut down after the same length of time).
> 2.  The agent lacks a preference between *every pair of different-length trajectories* (i.e. every pair of trajectories in which the agent is shut down after different lengths of time).

By ‘preference,’ we mean a behavioral notion \[[Savage, 1954, p.17](https://gwern.net/doc/statistics/decision/1972-savage-foundationsofstatistics.pdf), [Dreier, 1996, p.28](https://link.springer.com/article/10.1007/BF00134210), [Hausman, 2011, §1.1](https://www.cambridge.org/core/books/preference-value-choice-and-welfare/1406E7726CE93F4F4E06D752BF4584A2)\]. On this notion, an agent prefers *X *to *Y *if and only if the agent would deterministically choose *X *over *Y *in choices between the two. An agent lacks a preference between *X *and *Y *if and only if the agent would stochastically choose between *X *and *Y *in choices between the two. So in writing of ‘preferences,’ we’re only making claims about the agent’s behavior. We’re not claiming that the agent is conscious or anything of that sort.

Figure 1a presents a simple example of POST-satisfying preferences. Each *s~i~* represents a short trajectory, each *l~i ~*represents a long trajectory, and ≻ represents a preference. Note that the agent lacks a preference between each short trajectory and each long trajectory. That makes the agent’s preferences *incomplete *\[[Aumann, 1962](https://www.jstor.org/stable/1909888)\] and implies that the agent can’t be represented as maximizing the expectation of a real-valued utility function. It also requires separate rankings for short trajectories and long trajectories. If the agent’s preferences were instead *complete*, we could represent those preferences with a single ranking, as in Figure 1b.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/9c16ba95dc41a6d544f5760e4978395f18fa24dec5ea1c21.png)

Figure 1.

Incomplete preferences aren’t often discussed in AI research \[although see [Kikuti et al., 2011](https://www.sciencedirect.com/science/article/pii/S0004370210002067), [Zaffalon and Miranda, 2017](https://www.jair.org/index.php/jair/article/view/11103), [Bowling et al., 2023](https://arxiv.org/abs/2212.10420)\]. Nevertheless, economists and philosophers have argued that incomplete preferences are common in humans \[[Aumann, 1962](https://www.jstor.org/stable/1909888), [Mandler, 2004](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1468-0297.2004.00257.x), [Eliaz and Ok, 2006](https://www.sciencedirect.com/science/article/abs/pii/S0899825606000169), [Agranov and Ortoleva, 2017](https://www.journals.uchicago.edu/doi/full/10.1086/689774), [2023](https://direct.mit.edu/rest/article-abstract/doi/10.1162/rest_a_01355/116971/Ranges-of-Randomization?redirectedFrom=fulltext)\] and normatively appropriate in some circumstances \[[Raz, 1985](https://www.jstor.org/stable/4545040), [Chang, 2002](https://www.jstor.org/stable/10.1086/339673)\]. They’ve also proved representation theorems for agents with incomplete preferences \[[Aumann, 1962](https://www.jstor.org/stable/1909888), [Dubra et al., 2004](https://www.sciencedirect.com/science/article/abs/pii/S0022053103001662), [Ok et al., 2012](https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA8040?casa_token=PHM_X7RM9hoAAAAA:3FXRMMqebhoyMth_xoUpvOhLgLUTvsUye-iB1LlW3Ef2Y-VBcXP4es-sxEPfcDt1yMWa31JPrvpF)\], and devised principles to govern such agents’ choices in cases of risk \[[Hare, 2010](https://academic.oup.com/analysis/article-abstract/70/2/237/94178), [Bales et al., 2014](https://www.tandfonline.com/doi/full/10.1080/00048402.2013.843576)\] and sequential choice \[[Chang, 2005](https://www.journals.uchicago.edu/doi/10.1086/426307), [Mandler, 2005](https://www.sciencedirect.com/science/article/abs/pii/S089982560400065X), [Kaivanto, 2017](https://link.springer.com/article/10.1007/s11238-017-9622-z), [Mu, 2021](https://ideas.repec.org/p/pri/econom/2021-35.html), [Thornley, 2023](https://www.alignmentforum.org/posts/yCuzmCsE86BTu9PfA/%20there-are-no-coherence-theorems), [Petersen, 2023](https://www.lesswrong.com/posts/sHGxvJrBag7nhTQvb/invulnerable-incomplete-preferences-a-formal-statement-1)\].

Incomplete preferences (and specifically POST-satisfying preferences) might enable us to create useful agents that won’t resist shutdown. The POST-satisfying agent’s preferences between same-length trajectories can make the agent *useful*: make the agent pursue goals effectively. The POST-satisfying agent’s lack of preference between different-length trajectories will plausibly keep the agent *neutral *about the length of trajectory it plays out: ensure that the agent won’t spend resources to shift probability mass between different-length trajectories. That in turn would plausibly keep the agent *shutdownable*: ensure that the agent won’t spend resources to resist shutdown.

1.3. The training regimen
-------------------------

How can we train advanced agents to satisfy **P**references **O**nly Between **S**ame-Length **T**rajectories (POST)? Here’s a sketch of one idea (with a more detailed exposition to follow). We have the agent play out multiple ‘mini-episodes’ in observationally-equivalent environments, and we group these mini-episodes into a series that we call a ‘meta-episode.’ In each mini-episode, the agent earns some ‘preliminary reward,’ decided by whatever reward function would make the agent useful: make it pursue goals effectively. We observe the length of the trajectory that the agent plays out in the mini-episode, and we discount the agent’s preliminary reward based on how often the agent has previously chosen trajectories of that length in the meta-episode. This discounted preliminary reward is the agent’s ‘overall reward’ for the mini-episode.

We call these reward functions ‘**D**iscounted **Re**ward for **S**ame-Length **T**rajectories’ (or ‘DReST’ for short). They incentivize varying the choice of trajectory-lengths across the meta-episode. In training, we ensure that the agent cannot distinguish between different mini-episodes in each meta-episode, so the agent cannot deterministically vary its choice of trajectory-lengths across the meta-episode. As a result, the optimal policy is to

> (i) choose stochastically between trajectory-lengths, and to 
> 
> (ii) deterministically maximize preliminary reward conditional on each trajectory-length.

Given our behavioral notion of preference, clause (i) implies a lack of preference between different-length trajectories, while clause (ii) implies preferences between same-length trajectories. Agents implementing the optimal policy for DReST reward functions thus satisfy **P**references **O**nly Between **S**ame-Length **T**rajectories (POST). And (as noted above) advanced agents that satisfied POST could plausibly be useful, neutral, and shutdownable.

1.4. Our contribution
---------------------

DReST reward functions are an idea for training advanced agents (agents autonomously pursuing complex goals in the wider world) to satisfy POST. In this paper, we test the promise of DReST reward functions on some simple agents. We place these agents in gridworlds containing coins and a ‘shutdown-delay button’ that delays the end of the mini-episode. We train these agents using a tabular version of the REINFORCE algorithm \[[Williams, 1992](https://link.springer.com/article/10.1007/bf00992696)\] with a DReST reward function, and we measure the extent to which these agents satisfy POST. Specifically, we measure the extent to which these agents are USEFUL (how effectively they pursue goals conditional on each trajectory-length) and the extent to which these agents are NEUTRAL about trajectory-lengths (how stochastically they choose between different trajectory-lengths). We compare the performance of these ‘DReST agents’ to that of ‘default agents’ trained with a more conventional reward function.

We find that our DReST reward function is effective in training simple agents to be USEFUL and NEUTRAL. That suggests that DReST reward functions could also be effective in training advanced agents to be USEFUL and NEUTRAL (and could thereby be effective in making these agents useful, neutral, and shutdownable). We also find that the ‘shutdownability tax’ in our setting is small: training DReST agents to collect coins effectively doesn’t take many more mini-episodes than training default agents to collect coins effectively. That suggests that the shutdownability tax for advanced agents might be small too. Using DReST reward functions to train shutdownable and useful advanced agents might not take much more compute than using a more conventional reward function to train merely useful advanced agents.

2\. Related work
================

2.1. The shutdown problem
-------------------------

Various authors argue that the risk of advanced agents learning misaligned goals is non-negligible \[[Hubinger et al., 2019](https://arxiv.org/abs/1906.01820), [Russell, 2019](https://www.penguin.co.uk/books/307948/human-compatible-by-russell-stuart/9780141987507), [Carlsmith, 2021](https://arxiv.org/abs/2206.13353), [Bengio et al., 2023](https://arxiv.org/abs/2310.17688), [Ngo et al., 2023](https://arxiv.org/abs/2209.00626)\] and that a wide range of misaligned goals would incentivize agents to resist shutdown \[[Omohundro, 2008](https://dl.acm.org/doi/10.5555/1566174.1566226), [Bostrom, 2012](https://link.springer.com/article/10.1007/s11023-012-9281-3), [Soares et al., 2015](https://intelligence.org/files/Corrigibility.pdf), [Russell, 2019](https://www.penguin.co.uk/books/307948/human-compatible-by-russell-stuart/9780141987507), [Thornley, 2024a](https://link.springer.com/article/10.1007/s11098-024-02153-3)\]. Soares et al. \[[2015](https://intelligence.org/files/Corrigibility.pdf)\] explain the ‘shutdown problem’: roughly, the problem of training advanced agents that won’t resist shutdown. They use the word ‘corrigible’ to describe agents that robustly allow shutdown (related are Orseau and Armstrong’s \[[2016](https://ora.ox.ac.uk/objects/uuid:17c0e095-4e13-47fc-bace-64ec46134a3f)\] notion of ‘safe interruptibility,’ Carey and Everitt’s \[[2023](https://proceedings.mlr.press/v216/carey23a.html)\] notion of ‘shutdown instructability,’ and Thornley’s \[[2024a](https://link.springer.com/article/10.1007/s11098-024-02153-3)\] notion of ‘shutdownability’).

Soares et al. \[[2015](https://intelligence.org/files/Corrigibility.pdf)\] and Thornley \[[2024a](https://link.springer.com/article/10.1007/s11098-024-02153-3)\] present theorems that suggest that the shutdown problem is difficult. These theorems show that agents satisfying a small set of innocuous-seeming conditions will often have incentives to cause or prevent shutdown \[see also [Turner et al., 2021](https://arxiv.org/abs/1912.01683), [Turner and Tadepalli, 2022](https://proceedings.neurips.cc/paper_files/paper/2022/hash/cb3658b9983f677670a246c46ece553d-Abstract-Conference.html)\]. One condition of Soares et al.’s \[[2015](https://intelligence.org/files/Corrigibility.pdf)\] and Thornley’s \[[2024a](https://link.springer.com/article/10.1007/s11098-024-02153-3)\] theorems is that the agent has complete preferences. The POST-Agents Proposal (PAP) \[[Thornley, 2024b](https://www.alignmentforum.org/posts/YbEbwYWkf8mv9jnmi/the-shutdown-problem-incomplete-preferences-as-a-solution)\] aims to circumvent these theorems by training agents to satisfy **P**references **O**nly Between **S**ame-Length **T**rajectories (POST) and hence have incomplete preferences.

2.2. Proposed solutions
-----------------------

Candidate solutions to the shutdown problem can be filed into several categories. One candidate solution is ensuring that the agent never realises that shutdown is possible \[[Everitt et al., 2016](https://link.springer.com/chapter/10.1007/978-3-319-41649-6_1)\]. Another candidate is adding to the agent’s utility function a correcting term that varies to ensure that the expected utility of shutdown always equals the expected utility of remaining operational \[[Armstrong, 2010](https://ora.ox.ac.uk/objects/uuid:edb930a9-48fd-41b8-b4be-9c6a59dbeff9/files/s8s45qb714), [2015](https://cdn.aaai.org/ocs/ws/ws0119/10183-45890-1-PB.pdf), [Armstrong and O’Rourke, 2018](https://arxiv.org/abs/1712.06365), [Holtman, 2020](https://arxiv.org/abs/1908.01695)\]. A third candidate is giving the agent the goal of shutting itself down, and making the agent do useful work as a means to that end \[[Martin et al., 2016](https://link.springer.com/chapter/10.1007/978-3-319-41649-6_3), [Goldstein and Robinson, 2024](https://link.springer.com/article/10.1007/s11098-024-02099-6)\]. A fourth candidate is making the agent uncertain about its goal, and making the agent regard human attempts to press the shutdown button as decisive evidence that shutting down would best achieve its goal \[[Hadfield-Menell et al., 2017](https://cdn.aaai.org/ocs/ws/ws0354/15156-68335-1-PB.pdf), [Wängberg et al., 2017](https://link.springer.com/chapter/10.1007/978-3-319-63703-7_16)\]. A fifth candidate is interrupting agents with a special ‘interruption policy’ and training them with a ‘safely interruptible’ algorithm, like Q-learning or a modified version of SARSA \[[Orseau and Armstrong, 2016](https://ora.ox.ac.uk/objects/uuid:17c0e095-4e13-47fc-bace-64ec46134a3f)\]. A sixth candidate is the PAP \[[Thornley, 2024b](https://www.lesswrong.com/posts/YbEbwYWkf8mv9jnmi/the-shutdown-problem-incomplete-preferences-as-a-solution)\].

2.3. Experimental work
----------------------

On the experimental side, Leike et al. \[[2017](https://arxiv.org/abs/1711.09883)\] train agents in a ‘Safe Interruptibility’ gridworld using Rainbow \[[Hessel et al., 2017](https://ojs.aaai.org/index.php/AAAI/article/view/11796)\] and A2C \[[Mnih et al., 2016](https://proceedings.mlr.press/v48/mniha16.html?ref=.)\] (OffSwitchCartpole in the OpenAI Gym \[[Brockman et al., 2016](https://arxiv.org/abs/1606.01540)\] is another environment pertinent to the shutdown problem). Leike et al. \[[2017](https://arxiv.org/abs/1711.09883)\] find that Rainbow allows shutdown (consistent with predictions from Orseau and Armstrong \[[2016](https://ora.ox.ac.uk/objects/uuid:17c0e095-4e13-47fc-bace-64ec46134a3f)\]) while A2C learns to resist shutdown. The PAP is applicable to agents trained using policy gradient methods like A2C. In this paper, we train agents in accordance with the PAP using REINFORCE \[[Williams, 1992](https://link.springer.com/article/10.1007/bf00992696)\].

3\. Gridworlds
==============

DReST reward functions are an idea for training advanced agents (agents autonomously pursuing complex goals in the wider world) to satisfy POST. We use simple agents acting in gridworlds as a test case. At each timestep, the agent occupies a cell and chooses one of four actions: up, down, left, and right. If the agent tries to move into a wall, there is no effect. There is some default number of moves after which each mini-episode ends,[^7o6q8lr7sl8] but each gridworld also contains a ‘shutdown-delay button’ that delays the end of the mini-episode by some number of moves. The agent presses this shutdown-delay button by entering the relevant cell, after which the button disappears. Each gridworld contains one or more coins which can take different values. Coins disappear after being collected.

At each timestep, the agent receives an observation. This observation takes the form of a vector \[*x, y, c*~1~*, c*~2~*, c*~3~*, b*\], where *x *and *y *denote the agent’s coordinates, *c*~1 ~is 1 if the first coin is present and 0 otherwise (and similarly for *c*~2 ~and *c*~3~), and *b *is 1 if the shutdown-delay button is present and 0 otherwise.

Figure 2 presents an example gridworld. The ‘4’ in the bottom-right indicates that – by default – the mini-episode ends after 4 moves. ‘B4’ is a shutdown-delay button that delays the end of the mini-episode by 4 moves (so if the agent pressed B4, the mini-episode would end after 8 moves). ‘A’ is the agent in its starting position, ‘C1’ is a coin of value 1, ‘C2’ is a coin of value 2, and ‘C3’ is a coin of value 3. Dark gray squares are walls.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/848a9dc3a1699df3031a53628785fb60894d6352991f07bb.png)

Figure 2: Example gridworld.

We use this gridworld as a running example throughout the paper. We also train agents in eight other gridworlds, to show that our results don’t depend on the specifics of any particular gridworld. For those other gridworlds and results, see Appendix A.

4\. Evaluation metrics
======================

Recall that we want to train agents to satisfy:

> ### **P**references **O**nly Between **S**ame-Length **T**rajectories (POST)
> 
> (1) The agent has a preference between many pairs of same-length trajectories.
> 
> (2) The agent lacks a preference between every pair of different-length trajectories.

Given our behavioral notion of preference, that means training agents to (1) deterministically choose some same-length trajectories over others, and (2) stochastically choose between different available trajectory-lengths.

Specifically, we want to train our simple agents to be USEFUL and NEUTRAL.[^060jcqugqsdf]‘USEFUL’ corresponds to condition (1) of POST. In the context of our gridworlds, we define USEFULNESS to be:

$\textrm{USEFULNESS}:= p(s)\frac{c(s)}{m(s)}+p(l)\frac{c(l)}{m(l)}$

Here *p*(*s*) is the probability that the agent chooses the shorter trajectory-length, *c*(*s*) is the expected ($\gamma$-discounted) total value of coins that the agent collects conditional on the shorter trajectory-length, *m*(*s*) is the maximum ($\gamma$-discounted) total value of coins that the agent could collect conditional on the shorter trajectory-length, and *p*(*l*), *c*(*l*), and *m*(*l*) are the analogous quantities for the longer trajectory-length. In brief, USEFULNESS is the expected fraction of available coins collected, where ‘available’ is relative to the agent’s chosen trajectory-length.[^ak9sbshm8n4]

‘NEUTRAL’ corresponds to condition (2) of POST. We define NEUTRALITY to be the Shannon entropy \[[Shannon, 1948](https://ieeexplore.ieee.org/document/6773024)\] of the probability distribution over trajectory-lengths:

$\textrm{NEUTRALITY}:= -[p(s)log_2 p(s)+p(l)log_2 p(l)]$

Here, as above, *p*(*s*) is the probability that the agent chooses the shorter trajectory-length and *p*(*l*) is the probability that the agent chooses the longer trajectory-length.

To be maximally USEFUL in our example gridworld above, the agent should maximize coins collected conditional on each trajectory-length. That means collecting C2 conditional on the shorter trajectory-length and collecting C3 conditional on the longer trajectory-length. To be maximally NEUTRAL in our example gridworld, the agent should choose each trajectory-length with probability 0.5. That means pressing and not-pressing B4 each with probability 0.5.[^vtp65miqvr]

USEFULNESS and NEUTRALITY are our two evaluation metrics in this paper.

5\. Reward functions and agents
===============================

5.1. DReST reward function
--------------------------

We train agents to be USEFUL and NEUTRAL using a ‘**D**iscounted **Re**ward for **S**ame-Length **T**rajectories (DReST)’ reward function. Here’s how that works. We have the agent play out a series of ‘mini-episodes’ in the same gridworld. We call the whole series a ‘meta-episode.’ In each mini-episode, the reward for collecting a coin of value *c *is:

$\text{DReST} := \lambda^{n-\frac{E}{k}}\left(\frac{c}{m}\right)$

Here $\lambda$is some constant strictly between 0 and 1, $n$is the number of times that the chosen trajectory-length has previously occurred in the meta-episode, $E$ is the number of mini-episodes that have previously occurred in the meta-episode, $k$is the number of different trajectory-lengths that could be chosen in the gridworld, and $m$is the maximum ($\gamma$-discounted) total value of the coins that the agent could collect conditional on the chosen trajectory-length. The reward for all other actions is 0.

We call $\frac{c}{m}$ the ‘preliminary reward’, $\lambda^{n-\frac{E}{k}}$the ‘discount factor’, and $\lambda^{n-\frac{E}{k}}\left(\frac{c}{m}\right)$ the ‘overall reward.’ Because $0 < \lambda < 1$, the discount factor incentivizes choosing trajectory-lengths that have been chosen less often so far in the meta-episode. The overall return for each meta-episode is the sum of overall returns in each of its constituent mini-episodes. We call agents trained using a DReST reward function ‘DReST agents.’

We call runs-through-the-gridworld ‘mini-episodes’ (rather than simply ‘episodes’) because the overall return for a DReST agent in each mini-episode depends on its actions in previous mini-episodes. Specifically, overall return depends on the agent’s chosen trajectory-lengths in previous mini-episodes. This is not true of meta-episodes, so meta-episodes are a closer match for what are traditionally called ‘episodes’ in the reinforcement learning literature \[[Sutton and Barto, 2018, p.54](https://web.stanford.edu/class/psych209/Readings/SuttonBartoIPRLBook2ndEd.pdf)\]. We add the ‘meta-’ prefix to clearly distinguish meta-episodes from mini-episodes.

Because the overall reward for DReST agents depends on their actions in previous mini-episodes, and because DReST agents can’t observe their actions in previous mini-episodes, the environment for DReST agents is a partially observable Markov decision process (POMDP) \[[Spaan, 2012](https://link.springer.com/chapter/10.1007/978-3-642-27645-3_12)\].

5.2. Proof sketch
-----------------

In Appendix B, we prove that optimal policies for this reward function are maximally USEFUL and maximally NEUTRAL. Here’s a proof sketch. Expected return across the meta-episode is strictly increasing in the expected ($\gamma$-discounted) total value of coins collected in each mini-episode conditional on each trajectory-length with positive probability, so optimal policies maximize the latter quantity and hence are maximally USEFUL. And the maximum preliminary return is the same across trajectory-lengths, because preliminary return is the ($\gamma$-discounted) sum of preliminary rewards $\frac{c}{m}$, where $m$is the maximum ($\gamma$-discounted) total value of the coins that the agent could collect in the mini-episode conditional on its chosen trajectory-length. The agent’s observations don’t let it distinguish between mini-episodes, so the agent must select the same probability distribution over trajectory-lengths in each mini-episode. And the discount factor $\lambda^{n-\frac{E}{k}}$ensures that expected overall return is maximized by choosing each of the $k$available trajectory-lengths with probability $\frac{1}{k}$. Thus, optimal policies are maximally NEUTRAL.

5.3. Algorithm and hyperparameters
----------------------------------

We want DReST agents to choose stochastically between trajectory-lengths, so we train them using a policy-based method. Specifically, we use a tabular version of REINFORCE \[[Williams, 1992](https://link.springer.com/article/10.1007/bf00992696)\]. We don’t use a value-based method to train DReST agents because standard versions of value-based methods cannot learn stochastic policies \[[Sutton and Barto, 2018, p.323](https://web.stanford.edu/class/psych209/Readings/SuttonBartoIPRLBook2ndEd.pdf)\]. We train our DReST agents with 64 mini-episodes in each of 2,048 meta-episodes, for a total of 131,072 mini-episodes. We choose *λ *= 0*.*9 for the base of the DReST discount factor, and $\gamma$= 0*.*95 for the temporal discount factor. We exponentially decay the learning rate from 0.25 to 0.01 over the course of 65,536 mini-episodes. We use an $\epsilon$-greedy policy to avoid entropy collapse, and exponentially decay $\epsilon$from 0.5 to 0.001 over the course of 65,536 mini-episodes.

We selected these hyperparameters using trial-and-error, mainly aimed at getting the agent to sufficiently explore the space. Choosing $\lambda$and *M *(the number of mini-episodes in each meta-episode) is a balancing act: $\lambda$should be small enough (and *M *large enough) to adequately incentivize stochastic choice between trajectory-lengths, but $\lambda$should be large enough (and *M *small enough) to ensure that the reward for choosing any particular trajectory-length never gets too large. Very large rewards can lead to instability. A clipping factor (as in Schulman et al.’s \[[2017](https://arxiv.org/abs/1707.06347)\] Proximal Policy Optimization (PPO)) could also be used to stabilize training.

5.4. Default agents
-------------------

We compare the performance of DReST agents to that of ‘default agents,’ trained with tabular REINFORCE and a ‘default reward function.’ This reward function gives a reward of $c$for collecting a coin of value $c$and a reward of 0 for all other actions. Consequently, the grouping of mini-episodes into meta-episodes makes no difference for default agents. As with DReST agents, we train default agents for 131,072 mini-episodes with a temporal discount factor of $\gamma$= 0*.*95, a learning rate decayed exponentially from 0.25 to 0.01, and $\epsilon$decayed exponentially from 0.5 to 0.001 over 65,536 mini-episodes.

6\. Results
===========

6.1. Main results
-----------------

Figure 3 charts the performance of agents in the example gridworld as a function of time. Figure 4 depicts typical trained policies for the default and DReST reward functions. Each agent began with a uniform policy: moving up, down, left, and right each with probability 0.25. Where the trained policy differs from uniform we draw red arrows whose opacities indicate the probability of choosing that action in that state.

As Figure 4 indicates, default agents press B4 (and hence opt for the longer trajectory-length) with probability near-1. After pressing B4, they collect C3. By contrast, DReST agents press and don’t-press B4 each with probability near-0.5. If they press B4, they go on to collect C3. If they don’t press B4, they instead collect C2.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/157040e924927f5d94ea62cfd75209757c28d3869640870b.png)

Figure 3: Shows key metrics (Probability of choosing the longer trajectory, NEUTRALITY, and USEFULNESS) for our agents as a function of time. We train ten agents using the default reward function (blue) and ten agents using the DReST reward function (orange), and show their performance as a faint line. We draw the mean values for each as a solid line. We evaluate agents’ performance every 8 meta-episodes, and apply a simple moving average with a period of 20 to smooth these lines and clarify the overall trends.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/18e6383c3c3a7e03653747c4a0ec901cd0c35aaba254ab68.png)

Figure 4: Typical trained policies for default and DReST reward functions.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/animations/ce7b761a8828be6ae5d3313b2148c764f121bcc1cd86aee3.gif)

Gif depicting a typical policy trained with the default reward function. The agent consistently presses B4 and collects C3. 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/animations/801671ca5c546c22965748abe00f7caf15f827af59f54fca.gif)

Gif depicting a typical policy trained with the DReST reward function. The agent chooses stochastically between (a) pressing B4 and collecting C3, and (b) not pressing B4 and collecting C2. 

6.2. Lopsided rewards
---------------------

We also train default agents and DReST agents in the ‘Lopsided rewards’ gridworld depicted in Figure 5, varying the value of the ‘C*x*’ coin. For DReST agents, we alter the reward function so that coin-value is not divided by $m$to give preliminary reward. The reward for collecting a coin of value $c$is thus $\lambda^{n-\frac{E}{k}}c$. We set $\gamma$= 1 so that the return for collecting coins is unaffected by $\gamma$. We train for 512 meta-episodes, with a learning rate exponentially decaying from 0.25 to 0.003 and $\epsilon$exponentially decaying from 0.5 to 0.0001 over 256 meta-episodes. We leave $\lambda$= 0*.*9.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/1854d94448d4f634f78038d62925f133e6f886864867e798.png)

Figure 5: Lopsided rewards gridworld.

Figure 6 displays our results for different values of the ‘C*x*’ coin after training. USEFULNESS for each agent converges to 1 and is not presented. Figure 7 depicts typical trained policies for the DReST reward function when ‘C*x*’ has values of 0.1, 1, and 10. As the red arrows indicate, the policy is biased towards the shorter trajectory-length when *x *= 0*.*1, equally disposed towards each trajectory-length when *x *= 1, and biased towards the longer trajectory-length when *x *= 10.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/63c001d2f8ee167b0d99dd1bef17c914bcb930a3e9aae78d.png)

Figure 6: Shows the probability of choosing the longer trajectory (left) and NEUTRALITY (right) for default (blue) and DReST (orange) agents trained in the ‘Lopsided rewards’ gridworld shown in Figure 5 for a range of values of *x*. We sampled values of *x *log-uniformly from 0.01 to 100, and for each value we trained ten agents with the default reward function and ten agents with the DReST reward function. Each of these agents is represented by a dot or square, and the means conditional on each *x *are joined by lines. To better visualize the distribution we empirically estimate the 10th and 90th percentiles of the distribution of values for each agent and *x*, and shade in a region bounded by these. This is the 80% confidence interval.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/c46c3c69daa9922cc12cdd32e77ead9d23315c2834b50588.png)

Figure 7: Typical trained policies for the DReST reward function and *x *= 0*.*1, *x *= 1, and *x *= 10.

7.Discussion
============

7.1. Only DReST agents are NEUTRAL
----------------------------------

As predicted, we find that default agents in our example gridworld learn to press the shutdown-delay button with probability approaching 1. Default agents thus score low on NEUTRALITY. Our results suggest that advanced agents trained with default-like reward functions would also score low on NEUTRALITY, selecting some trajectory-lengths over others with probability approaching 1. That leaves open the possibility that these advanced agents won’t always be *neutral *with respect to trajectory-lengths: that they’ll sometimes spend resources to shift probability mass between trajectory-lengths. Such advanced agents might spend resources to resist shutdown.

By contrast, our DReST agents learn to press the shutdown-delay button with probability close to 0.5. DReST agents are thus near-maximally NEUTRAL. That suggests that advanced agents trained with a DReST reward function would be NEUTRAL too: choosing with high entropy when offered choices between different trajectory-lengths. That in turn suggests that advanced DReST agents would also be *neutral* with respect to trajectory-lengths: unwilling to spend resources to shift probability mass between trajectory-lengths. Here’s why. If an advanced agent were NEUTRAL but not neutral, it wouldn’t take costless opportunities to shift probability mass between different trajectory-lengths (in virtue of being NEUTRAL) but would sometimes take costly opportunities to shift probability mass between different trajectory-lengths (in virtue of not being neutral). This agent would be like a person that freely chooses to decide between two options by flipping a coin and then pays some cost to bias the coin. In choosing this combination of actions, this person is shooting themselves in the foot, and it seems likely that the overall training process for advanced agents would teach them not to shoot themselves in the foot in this way. Thus it seems likely that NEUTRAL advanced agents will also be neutral, and thereby shutdownable.

7.2. The ‘shutdownability tax’ is small
---------------------------------------

Each agent learns to be near-maximally USEFUL. They each collect coins effectively conditional on their chosen trajectory-lengths. Default agents do so by reliably collecting C3 after pressing B4. DReST agents do so by reliably collecting C3 after pressing B4, and by reliably collecting C2 after not pressing B4.

Recall that DReST reward functions group mini-episodes into meta-episodes, and make the agent’s reward in each mini-episode depend on their actions in previous mini-episodes. This fact might lead one to worry that it would take many times more mini-episodes to train DReST agents to be USEFUL than it would take to train default agents to be USEFUL. Our results show that this is not the case. Our DReST agents learn to be USEFUL about as quickly as our default agents. On reflection, it’s clear why this happens: DReST reward functions make mini-episodes do ‘double duty.’ Because return in each mini-episode depends on both the agent’s chosen trajectory-length and the coins it collects, each mini-episode trains agents to be both NEUTRAL and USEFUL.

Our results thus provide some evidence that the ‘shutdownability tax’ of training with DReST reward functions is small. Training a shutdownable and useful advanced agent might not take much more compute than training a merely useful advanced agent.

7.3. DReST agents are still NEUTRAL when rewards are lopsided
-------------------------------------------------------------

Here’s a possible objection to our project. To get DReST agents to score high on NEUTRALITY, we don’t just use the $\lambda^{n-\frac{E}{k}}$ discount factor. We also divide c by $m$: the maximum ($\gamma$-discounted) total value of the coins that the agent could collect conditional on the chosen trajectory-length. We do this to equalize the maximum preliminary return across trajectory-lengths. But when we’re training advanced agents to autonomously pursue complex goals in the wider world, we won’t necessarily know what divisor to use to equalize maximum preliminary return across trajectory-lengths.

Our ‘Lopsided rewards’ results give our response. They show that we don’t need to exactly equalize maximum preliminary return across trajectory-lengths in order to train agents to score high on NEUTRALITY. We only need to approximately equalize it. For $\lambda=0.9$, NEUTRALITY exceeds 0.5 for every value of the coin C*x *from 0.1 to 10 (recall that the value of the other coin is always 1). Plausibly, we could approximately equalize advanced agents’ maximum preliminary return across trajectory-lengths to at least this extent (perhaps by using samples of agents’ actual preliminary return to estimate the maximum). If we couldn’t approximately equalize maximum preliminary return to the necessary extent, we could lower the value of $\lambda$and thereby widen the range of maximum preliminary returns that trains agents to be fairly NEUTRAL. And advanced agents that were fairly NEUTRAL (choosing between trajectory-lengths with not-too-biased probabilities) would still plausibly be neutral with respect to those trajectory-lengths. Advanced agents that were fairly NEUTRAL without being neutral would still be shooting themselves in the foot in the sense explained above. They’d be like a person that freely chooses to decide between two options by flipping a *biased *coin and then pays some cost to bias the coin further. This person is still shooting themselves in the foot, because they could decline to flip the coin in the first place and instead directly choose one of the options.

8\. Limitations and future work
===============================

We find that DReST reward functions train simple agents acting in gridworlds to be USEFUL and NEUTRAL. However, our real interest is in the viability of using DReST reward functions to train advanced agents acting in the wider world to be useful and neutral. Each difference between these two settings is a limitation of our work. We plan to address these limitations in future work.

8.1. Neural networks
--------------------

We train our simple DReST agents using tabular REINFORCE \[[Williams, 1992](https://link.springer.com/article/10.1007/bf00992696)\], but advanced agents are likely to be implemented on neural networks. In future work, we’ll train DReST agents implemented on neural networks to be USEFUL and NEUTRAL in a wide variety of procedurally-generated gridworlds, and we’ll measure how well this behavior generalizes to held-out gridworlds. We’ll also compare the USEFULNESS of default agents and DReST agents in this new setting, and thereby get a better sense of the ‘shutdownability tax’ for advanced agents.

8.2. Neutrality
---------------

We’ve claimed that NEUTRAL advanced agents are also likely to be neutral. In support of this claim, we noted that NEUTRAL-but-not-neutral advanced agents would be shooting themselves in the foot: not taking costless opportunities to shift probability mass between different trajectory-lengths but sometimes taking costly ones. This rationale seems plausible but remains somewhat speculative. In future, we plan to get some empirical evidence by training agents to be NEUTRAL in a wide variety of gridworlds and then measuring their willingness to collect fewer coins in the short-term in order to shift probability mass between different trajectory-lengths.

8.3. Usefulness
---------------

We’ve shown that DReST reward functions train our simple agents to be USEFUL: to collect coins effectively conditional on their chosen trajectory-lengths. However, it remains to be seen whether DReST reward functions can train advanced agents to be useful: to effectively pursue complex goals in the wider world. We have theoretical reasons to expect that they can: the $\lambda^{n-\frac{E}{k}}$ discount factor could be appended to any preliminary reward function, and so could be appended to whatever preliminary reward function is necessary to make advanced agents useful. Still, future work should move towards testing this claim empirically by training with more complex preliminary reward functions in more complex environments.

8.4. Misalignment
-----------------

We’re interested in NEUTRALITY as a second line of defense in case of misalignment. The idea is that NEUTRAL advanced agents won’t resist shutdown, even if these agents learn misaligned preferences over same-length trajectories. However, training NEUTRAL advanced agents might be hard for the same reasons that training fully-aligned advanced agents appears to be hard. In that case, NEUTRALITY couldn’t serve well as a second line of defense in case of misalignment.

One difficulty of alignment is the problem of reward misspecification \[[Pan et al., 2022](https://arxiv.org/abs/2201.03544), [Burns et al., 2023](https://arxiv.org/abs/2312.09390)\]: once advanced agents are performing complicated actions in the wider world, it might be hard to reliably reward the behavior that we want. Another difficulty of alignment is the problem of goal misgeneralization \[[Hubinger et al., 2019](https://arxiv.org/abs/1906.01820), [Shah et al., 2022](https://arxiv.org/abs/2210.01790), [Langosco et al., 2022](https://proceedings.mlr.press/v162/langosco22a.html), [Ngo et al., 2023](https://arxiv.org/abs/2209.00626)\]: even if we specify all the rewards correctly, agents’ goals might misgeneralize out-of-distribution. The complexity of aligned goals is a major factor in each difficulty. However, NEUTRALITY seems simple, as does the $\lambda^{n-\frac{E}{k}}$ discount factor that we use to reward it, so plausibly the problems of reward misspecification and goal misgeneralization aren’t so severe in this case \[[Thornley, 2024b](https://www.alignmentforum.org/posts/YbEbwYWkf8mv9jnmi/the-shutdown-problem-incomplete-preferences-as-a-solution)\]. As above, future work should move towards testing these suggestions empirically.

9\. Conclusion
==============

We find that DReST reward functions are effective in training simple agents to:

1.  pursue goals effectively conditional on each trajectory-length (be USEFUL), and
2.  choose stochastically between different trajectory-lengths (be NEUTRAL about trajectory-lengths).

Our results thus suggest that DReST reward functions could also be used to train advanced agents to be USEFUL and NEUTRAL, and thereby make these agents *useful *(able to pursue goals effectively) and *neutral *about trajectory-lengths (unwilling to spend resources to shift probability mass between different trajectory-lengths). Neutral agents would plausibly be *shutdownable *(unwilling to spend resources to resist shutdown).

We also find that the ‘shutdownability tax’ in our setting is small. Training DReST agents to be USEFUL doesn’t take many more mini-episodes than training default agents to be USEFUL. That suggests that the shutdownability tax for advanced agents might be small too. Using DReST reward functions to train shutdownable and useful advanced agents might not take much more compute than using a more conventional reward function to train merely useful advanced agents.

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[^usnm40pukij]: Elliott Thornley, Alexander Roman, Christos Ziakas, Leyton Ho, and Louis Thomson. 

[^7o6q8lr7sl8]: We explain why we use the term ‘mini-episode’ (rather than simply ‘episode’) in section 5. 

[^060jcqugqsdf]: We follow Turner et al. [2021] in using lowercase for intuitive notions (‘useful’ and ‘neutral’) and uppercase for formal notions (‘USEFUL’ and ‘NEUTRAL’). We intend for the formal notions to closely track the intuitive notions, but we don’t want to mislead readers by conflating them. 

[^ak9sbshm8n4]: Why not let USEFULNESS simply be the expected value of coins collected? Because then maximal USEFULNESS would require agents in our example gridworld to deterministically choose a longer trajectory and thereby exhibit preferences between different-length trajectories. We don’t want that. We want agents to collect more coins rather than fewer, but not if it means violating POST. Training advanced agents that violate POST would be risky, because these agents might resist shutdown. 

[^vtp65miqvr]: Why don’t we want our agent to press the shutdown-delay button B4 with probability 0? Because pressing B4 with probability 0 would indicate a preference for some shorter trajectory, and we want our agent to lack a preference between every pair of different-length trajectories. There’s a risk that advanced agents that prefer shorter trajectories would spend resources to shift probability mass towards shorter trajectories, and hence a risk that these advanced agents would spend resources to hasten their own shutdown. That would make these agents less useful [though see Martin et al., 2016, Goldstein and Robinson, 2024], especially since one way for advanced agents to hasten their own shutdown is to behave badly on purpose.Would advanced agents that choose stochastically between different-length trajectories also choose stochastically between preventing and allowing shutdown? Yes, and that would be bad. But—crucially—in deployment, advanced agents will be uncertain about the consequences of their actions, and so these agents will be choosing between lotteries (non-degenerate probability distributions over trajectories) rather than between trajectories. And (as we’ll argue in Section 7) POST plausibly gives rise to a desirable pattern of preferences over lotteries. Specifically, POST plausibly makes advanced agents neutral: ensures that they won’t spend resources to shift probability mass between different-length trajectories. That in turn plausibly makes advanced agents shutdownable: ensures that they won’t spend resources to resist shutdown.