Uncovering Surprising Behaviors in Reinforcement Learning via Worst-case Analysis

INTRODUCTION Reinforcement Learning (RL) methods have achieved great success over the past few years, achieving human-level performance on a range of tasks such as Atari (Mnih et al., 2015) , Go (Silver et al., 2016) , Labyrinth , and Capture the Flag (Jaderberg et al., 2018) . On these tasks, and more generally in reinforcement learning, agents are typically trained and evaluated using their average reward over environment settings as the measure of performance, i.e. E P (e) [R(π(θ), e)] , where π(θ) denotes a policy with parameters θ, R denotes the total reward the policy receives over the course of an episode, and e denotes environment settings such as maze structure in a navigation task, appearance of objects in the environment, or even the physical rules governing environment dynamics. But what biases does the distribution P (e) contain, and what biases, or failures to generalize, do these induce in the strategies agents learn? To help uncover biases in the training distribution and in the strategies that agents learn, we propose evaluating the worst-case performance of agents over environment settings, i.e. min e∈E E [R(π(θ), e)] , where E is some set of possible environment settings. Worst-case analysis can provide an important tool for understanding robustness and generalization in RL agents. For example, it can help us with: • Understanding biases in training Catastrophic failures can help reveal situations that are rare enough during training that the agent does not learn a strategy that is general enough to cope with them. • Robustness For critical systems, one would want to eliminate, or at least greatly reduce, the probability of extreme failures. • Limiting exploitability If agents have learned strategies that fail to generalize to particular environment settings, then an adversary could try and exploit an agent by trying to engineer such environment settings leading to agent failure. In this work, we use worst-case analysis to investigate the performance of a state-of-the-art agent in solving a first-person 3D navigation task; a task on which agents have recently achieved average-case human-level performance (Wayne et al., 2018) . By optimizing mazes to minimize the performance of agents, we discover the existence of mazes where agents repeatedly fail to find the goal (which we refer to as a Catastrophic Failure). Our Contributions To summarize, the key contributions of this paper are as follows: 1. We introduce an effective and intuitive approach for finding simple environment settings leading to failure (Section 2). 2. We show that state-of-the-art agents carrying out navigation tasks suffer from drastic and often surprising failure cases (Sections 3.1 and 3.2). 3. We demonstrate that mazes leading to failure transfer across agents with different hyperparameters and, notably, even different architectures (Section 3.3). 4. We present an initial investigation into how the training distribution can be adapted by incorporating adversarial and out-of-distribution examples (Section 4).
 APPROACH Tasks We consider agents carrying out first-person 3D navigation tasks. Navigation is of central importance in RL research as it captures the challenges posed by partially observable Markov decision processes (POMDPs). The navigation tasks we use are implemented in DeepMind Lab (DM Lab) (Beattie et al., 2016) . 1 Each episode is played on a 15 × 15 maze where each position in the maze may contain a wall, an agent spawn point, or a goal spawn point. The maze itself is procedurally generated every episode, along with the goal and agent spawn locations. The goal location remains fixed throughout an episode, while the agent spawn location can vary. In training, the agent respawns at different locations each time they reach the goal, while for our optimization and analysis we limit the agent to the same spawn location. Agents receive RGB observations of size 96 × 72 pixels, examples of which are provided in Figure 1 . Episodes last for 120 seconds and are played at a framerate of 15 frames per second. The agent receives a positive reward of 10 every time it reaches the goal, and 0 otherwise. On this specific navigation task, RL agents have recently achieved human-level average-case performance (Wayne et al., 2018) . Agents We perform our analysis on Importance Weighted Actor-Learner Architecture agents trained to achieve human-level average-case performance on navigation tasks. These agents can be described as async batched-a2c agents with the V-trace algorithm for off policy-correction, and we henceforth refer to these as A2CV agents . Details of the training procedure are provided in Appendix A.1. Search Algorithm If we are interested in worst-case performance of agents, how can we find environment settings leading to the worst performance? In supervised learning, one typically uses gradient based methods to find inputs that lead to undesired output (Biggio et al., 2013; Szegedy et al., 2013; Goodfellow et al., 2014) . In contrast, we search for environment settings leading to an undesired outcome at the end of an episode. This presents a challenge as the environment rendering and MDP are not differentiable. We are therefore limited to black-box methods where we can only query agent performance given environment settings. To search for environment settings which cause catastrophic failures, we propose the local search procedure described in Algorithm 1 (visualizing the process in Figure 2 ). We then Evaluate each with the agent over 30 episodes, select the best maze (i.e. lowest agent score), and Modify this maze by randomly moving two walls to form the next set of candidates (Iteration 1). This process is repeated for 20 iterations, leading to a maze where the agent score is 0.09 in this example (i.e. the agent finds the goal once in 11 episodes). In Appendix A.2.1 we detail the computational requirements of this search procedure.
 EXPERIMENTS The agents we study achieve impressive average-case performance, but how much does their worstcase performance differ from their average-case performance? To investigate this, we consider the worst-case performance over a large set of mazes, including mazes that are not possible under the training distribution. A natural question to ask is whether any departure from the wall structure present during training will lead to agent failure. To test this, we evaluate the agent on samples from a distribution of mazes containing all mazes agents could be evaluated on during the search. In particular, we randomly select agent and goal spawn locations in the first step and then randomly move 40 walls, corresponding to the same actions taken by our optimization procedure, but where the actions are chosen randomly rather than in order to minimize agent performance. We find that agents do generalize to random mazes from the set we consider. In fact, we find that the average score obtained by agents on randomly perturbed mazes is slightly higher than on the training distribution, with agents obtaining an average of 45 goal reaches per two minute episode. The increased performance is likely due to the agent spawn location being fixed, making it easier for the agent to return to the goal once found. The considered agents generalize in the sense that agent performance is not reduced on average by out-of-distribution wall structure. But what about the worst case over all wall structures? Have the agents learned a general navigation strategy that works for all solvable mazes? Or do there exist environment settings that lead to catastrophic failures with high probability? In this section, we investigate these questions. We define a Catastrophic Failure to be an agent failing to find the goal in a two minute episode (1800 steps). As detailed below, we find that not only do there exist mazes leading to catastrophic failure, there exist surprisingly simple mazes that lead to catastrophic failure for agents yet are consistently and often rapidly solved by humans.
 RESULT 1: CATASTROPHIC FAILURES EXIST Do environment settings leading to catastrophic failure exist for the agents we are considering? By searching over mazes using the procedure outlined in Algorithm 1, we find mazes where agents fail to find the goal on many episodes, only finding the goal 10% of the time. In fact, some individual mazes lead to failure across five different agents we tested, with even the best performing agent only finding the goal in 20% of the episodes. Optimization curves for our search procedure are given in Figure 3 . Note that while we define catastrophic failure as failure to find the goal, the actual objective used for the optimization was average number of goal reaches during an episode. Using average number of goals gives a stronger signal at the start of the optimization process. Finding mazes leading to lower average number of captures is easier than finding mazes where the agent rarely finds the goal even once. As can be seen, despite finding the goal on average 45 times per episode on randomly perturbed mazes, on mazes optimized to reduce score, agents find the goal on average only 0.33 times per episode, more than a 100× decrease in performance. In terms of probability of catastrophic failure, we note that despite agents finding the goal in approximately 98% of episodes on randomly perturbed mazes, using our method, on average we find mazes where agents only finds the goal in 30% of the episodes. Example trajectories agents take during failures are visualized in Figure 4 . The trajectories often seem to demonstrate a failure to use memory to efficiently explore the maze with the agent repeatedly visiting the same locations multiple times. The mazes presented in Figure 4 appear to be of higher complexity than mazes seen during training. This suggests that to obtain agents that truly master navigation, more complex mazes should be included in the training distribution. However, we can ask whether it is only more complex mazes that lead to catastrophic failure or whether there are also simple mazes leading to catastrophic failure. This is a question we explore in the next subsection.
 RESULT 2: SIMPLE CATASTROPHIC FAILURES EXIST While the existence of catastrophic failures may be intriguing and perhaps troubling, one might suspect that the failures are caused by the increased complexity of the mazes leading to failure relative to the mazes the agent is exposed to during training, e.g., the mazes leading to failure contain more dead ends and sometimes have lower visibility. Further, understanding the cause of failure in such mazes seems challenging due to the large number of wall structures that may be causing the agent to fail. In this section, we explore whether there exist simple mazes which lead to catastrophic failures. As our measure of complexity, we use the total number of walls in the maze. We also evaluate humans on such mazes to get a quantitative measure of maze complexity. To find simple mazes which lead to failure, we first follow the same procedure as in the previous section, producing a set of mazes which all lead to catastrophic failures (i.e. a low agent scores). Next, we use this set of mazes as the initial set of candidates in our search algorithm, however we now use a Modify function that removes a single randomly chosen wall each iteration. This process is repeated for 70 iterations, searching for a maze with few walls while maintaining low agent score. In Figure 4 , we present the resulting simple mazes and the corresponding agent trajectories from our optimization procedure. Interestingly, we find that one can remove a majority of the walls in a maze and still maintain the catastrophic failure (i.e. very low agent score). Of note is that a number of these mazes are strikingly simple, suggesting that there exists structure in the environment that the agent has not generalized to. For example, we can see that placing the goal in a small room in an otherwise open maze can significantly reduce the agent's ability to find the goal. Human baselines While these simple maps may lead to catastrophic failure, it is unclear whether this is because of the agent or whether the maze is difficult in a way that is not obvious. To investigate this, we perform human experiments by tasking humans to play on a set of 10 simplified mazes. Notably, we find that human players are able to always locate the goal in every maze and typically within one third of the full episode length. This demonstrates that the mazes are comfortably solvable within the course of an episode by players with a general navigation strategy. We provide a detailed comparison of agent and human performance in Appendix A.3. Analysis One question that may arise is the extent to which these mazes are isolated points in the space of mazes. That is, if the maze was changed slightly, would it no longer lead to catastrophic failure? To test this, we investigate how sensitive our discovered failure mazes are with respect to the agent and goal spawn locations on simplified adversarial mazes. As can be seen in Figure 5 , we find that for a large range of spawn locations, the mazes still lead to failure. This indicates that there is specific local maze structure which causes agents to fail. Procedures for finding such simple mazes may prove useful as a tool for debugging agents and understanding the ways in which training has led them to develop narrow strategies that are good enough for achieving high average-case performance.
 RESULT 3: FAILURE MAZES TRANSFER ACROSS AGENTS We have found failure cases for individual agents, but to what extent do these failure cases highlight a specific peculiarity of the individual agent versus a more general failure of a certain class of agents, or even a shortcoming of the distribution used for training? In this section, we investigate whether mazes which cause one trained agent to fail also cause other agents to fail. We consider two types of transfer: (1) between different hyperparameters of the same model architecture, and (2) between different model architectures. To test transfer between agents of the same architecture, we train a set of five A2CV agents with different entropy costs and learning rates. To test transfer between agents with significantly different architectures, we train a set of five MERLINbased agents (Wayne et al., 2018) . These agents have a number of differences to the A2CV agents, most notably they contain a sophisticated memory structure based on a DNC (but with a fixed write location per timestep) (Graves et al., 2016) . Both agents are trained on the same distribution and achieve human-level averages scores on the navigation task (with MERLIN scoring 10% higher than A2CV on average). Further details of agent training can be found in Appendix A.1. To quantify the level of transfer between (sets of) agents, we follow the procedure for finding adversarial mazes outlined in Section 3.1 to produce a collection of 50 unique failure mazes for each agent (i.e. 10 collections of 50 mazes each). We then evaluate every agent 100 times on each maze in each collection, reporting their average performance on each collection. Complete quantitative transfer results can be found in Appendix A.4. Failure cases transfer somewhat across all agents First, we find that across all agents, some level of transfer exists. In particular, as can be seen in Figure 6 , the probability of one agent finding the goal on mazes generated to reduce the score another agent is significantly below 1. This suggests a common cause of failure that is some combination of the distribution of environment settings used during training and the set of methods that are currently used to train such agents. A possible way to address this could be enriching the training distribution so that it contains fewer biases and encourages more general solutions. Transfer within agent type is stronger than between agent type Comparing the performance of each agent type on mazes from the same agent type to mazes from another agent type, we see that transfer within agent type is stronger. As shown in Figure 6b , performance increases as we go from 'MERLIN to MERLIN' 'A2CV to MERLIN' (0.42 to 0.58) and also if we go from 'A2CV to A2CV' to 'MERLIN to A2CV' (0.63 to 0.70). This suggests that there are some common biases in agents that are due to their architecture type. Analyzing structural differences between mazes that lead to one agent type to fail but not another could give interesting insight into behavioural differences between agents beyond just average performance. A2CV agents are less susceptible to transfer Despite similar probabilities of failure when evaluating on mazes optimized for the same agent, A2CV agents seem to suffer less on mazes optimized using other A2CV or MERLIN agents. This indicates that A2CV agents may have learned a more diverse set of strategies. Figure 6 : Mazes that lead to failures in one agent lead to failure in other agents as well. This is the case for agents of the same architecture with different hyperparameters, and is also the case for transfer across agents of different architecture. We note, however, that transfer across agents with different architectures is weaker than among agents with the same architecture, and that the performance of agents with the same architecture but with different hyperparameters is slightly higher than for the agents used to originally find the mazes.
 ADAPTING THE TRAINING DISTRIBUTION From our experiments so far, we have discovered that there exist many mazes which lead to catastrophic failure. In this section, we investigate whether agent performance can be improved by adapting the training distribution, for example by incorporating adversarial mazes into training and modifying the original mazes used in training.
 MOTIVATION To better understand what may be causing catastrophic failures, with the aim of fixing them, we compare the set of adversarial mazes with the original set of mazes used in training. From this comparison, we find that there are two notable differences. The probability of a catastrophic failure correlates with the distance between the spawn locations and how hidden the goal is First, we find that a number of features are more common in adversarial mazes than non-adversarial mazes. In particular, adversarial mazes are more likely to have the goal hidden in an enclosed space (such as a small room), and on average the path length from the player's starting location to the goal is significantly longer (31.1 ± 8.4 compared to 11.6 ± 6.3). Notably, while the training distribution also contains hidden goals which are far from the agent's starting location, they are much rarer. Adversarial mazes are typically far from the training distribution Second, we find that adversarial mazes tend to not only be out-of-distribution, but also far from the training distribution due to the Modify function used in our adversarial search procedure (for example, see Figure 2 ). This contrasts with the adversarial images literature where attacks are usually constrained to be small or imperceptible. It may therefore not be surprising that the agent is unable to generalize to all out-of-distribution mazes which could also explain the significant reduction in their performance. Given these two observations, it is natural to ask whether the training distribution can be adapted to improve the agent's performance. In the following sections we investigate this question and discuss our findings, focusing on incorporating adversarial mazes into training and modifying the original mazes used in training.
 APPROACH We consider two distinct approaches for incorporating adversarial and out-of-distribution mazes into the training distribution. Adversarial training To add adversarial mazes into the training distribution, we first create a dataset of 6000 unique adversarial mazes from separate runs of our search procedure using the previously trained A2CV agents. Notably, this set also includes the 250 mazes used in our transfer experiments (Section 3.3). Next, we train a new set of A2CV agents using both this adversarial set of mazes and the standard distribution of mazes, sampling randomly every episode (i.e. 50% of training episodes are on an adversarial maze). Randomly perturbed training To ensure our adversarial search procedure produces indistribution adversarial mazes, we alter the default maze generator used in training so that any adversarial maze can be generated. We accomplish this by randomly perturbing the original mazes, repeatedly using the same Modify function used by our adversarial search procedure, but selecting candidates randomly rather than by worst agent performance.
 RESULTS In this section, we report our findings on the robustness of agents trained using the approaches above. Catastrophic failures still exist Our main finding is that while agents learn to perform well on the richer distributions of mazes described above, this does not lead to robust agents. In particular, agents trained on a distribution of mazes enriched with 6000 adversarial mazes were able to find the goal on average 89.8% of the time on the adversarial mazes they were trained on. Similarly, agents trained on randomly perturbed mazes were able to find the goal close to 100% of the time on the distribution they were trained on. However, despite the agents being trained on these richer training distributions, the same search method is still able to find mazes leading to extreme failure as can be seen in Figure 7 . One possible explanation for this result is that the 6000 adversarial mazes used for training were insufficient to get good coverage of the space of mazes, and that further enlarging this set could yield qualitatively different results. Indeed, for agents trained using randomly perturbed mazes, the search procedure took 50 iterations to obtain the same level of failure as it did after 20 iterations when applied to agents trained on the standard training distribution. This suggests that perhaps enriching the training distribution with a very large set of adversarial mazes may lead to more general and robust agents. However, there are a number of challenges that need to be addressed before this approach can be tested which we will describe in the next section.
 DISCUSSION Our results suggest that if a richer training distribution is to yield more robust agents, we may need to use a very large set of environment settings leading to failure. This is similar to how adversarial training in supervised learning is performed where more adversarial examples are used than the original training examples. We describe below what we see as two significant challenges that need to be overcome before such an approach can be thoroughly evaluated in the RL setting.
 Expensive generation The cost of generating a single adversarial setting is on the order of 1000's episodes using the method in this work. This implies that generating a set of adversarial settings which is similar in size to the set trained on would require orders of magnitude more computational than training itself. This could be addressed with faster methods for generating adversarial settings. Expensive training Since agents receive very little reward in adversarial settings, the training signal is incredibly sparse. Therefore, it is possible that many more training iterations are necessary for agents to learn to perform well in each adversarial setting. A possible solution to this challenge is to design a curriculum over adversity, whereby easier variants of the adversarial settings are injected into the training distribution. For example, for the navigation tasks considered here, one could include training settings with challenging mazes where the goal is in any position on the shortest path between the starting location of the agent and the challenging goal. We hope that these challenges can be overcome so that, in the context of RL, the utility of adversarial retraining can be established -an approach which has proved useful in supervised learning tasks. However, since significant challenges remain, we suspect that much effort and many pieces of work will be required before a conclusive answer is achieved.
 RELATED WORK Navigation Recently, there has been significant focus in the RL community on agent navigation in simulated 3D environments, including a community-wide challenge for agents in such environments called VizDoom (Kempka et al., 2016) . Such 3D first-person navigation tasks are particularly interesting because they capture challenges such as partial observability, and require the agent to "effectively perceive, interpret, and learn the 3D world in order to make tactical and strategic decisions where to go and how to act." (Kempka et al., 2016) . Recent advances have led to impressive human-level performance on navigation tasks in large procedurally generated environments (Beattie et al., 2016; Wayne et al., 2018) . Adversarial examples in supervised learning Our work can be seen as an RL navigation analogue of work on adversarial attacks on supervised learning systems for image classification (Szegedy et al., 2013) . For adversarial attacks on image classifiers, one considers a set of inputs that is larger than the original distribution, but where one would hope that systems perform just as well on L ∞ balls around inputs from the distribution. In particular, the adversarial examples lie outside the training distribution. Analogously, we consider a set of mazes which is larger than the original set of mazes used during training, but where we would hope our system will work just as well. Notably, while similar on a conceptual level, our setting has two key differences from this previous line of work: (1) The attack vector consists of changing latent semantic features of the environment (i.e. the wall structure of a maze), rather than changing individual pixels in an input image in an unconstrained manner. (2) The failure is realized over multiple steps of agent and environment interacting with each other, rather than simply being errant output from a single forward pass through a neural net. More recently, in the context of supervised learning for image classification, there has been work to find constrained adversarial attacks which is closer to what we consider in this work (Athalye & Sutskever, 2017; Fawzi & Frossard, 2015; Eykholt et al., 2018; Sharif et al., 2016) . In the context of interpretable adversarial examples in image classification, similar approaches to our simplification approach have been explored where one searches for adversarial perturbations with group-sparse structure or other minimal structure (Xu et al., 2018; Brendel et al., 2018) . Additionally, our findings regarding transfer are consistent with findings on adversarial examples for computer vision networks where it has been found that perturbations that are adversarial for one network often transfer across other networks (Szegedy et al., 2013; Tramèr et al., 2017) Input attacks on RL systems There have been a number of previous works which have extended adversarial attacks to RL settings, however they have achieved this by manipulating inputs directly, which effectively amounts to changing the environment renderer (Huang et al., 2017; Lin et al., 2017a; b) . While these are interesting from a security perspective, it is less clear what they tell us about the generality of the strategy learned by the agent. Generalization in RL systems Recently, it has been shown that simple agents trained on restricted datasets fail to learn sufficiently general navigation strategies to improve goal retrieval times on held out mazes (Dhiman et al., 2018) . In comparison, our method is both automatic and able to find more spectacular failures. Further, our findings highlight failures in exploration during navigation. This is in contrast to this previous work which studied failures to exploit knowledge from previous goal retrievals in the same episode. In the context of testing generalization in RL, previous work has looked at statistical generalization in RL . Here we consider agents that already generalize in the statistical sense and try to better characterize the ways in which they generalize beyond the average-case.
 CONCLUSIONS AND FUTURE WORK In this work, we have shown that despite the strong average-case performance often reported of RL agents, worst-case analysis can uncover environment settings which agents have failed to generalize to. Notably, we have found that not only do catastrophic failures exist, but also that simple catastrophic failures exist which we would hope agents would have generalized to, and that failures also transfer between agents and architectures. As agents are trained to perform increasingly complicated tasks in more sophisticated environments, for example AirSim (Shah et al., 2017) and CARLA (Dosovitskiy et al., 2017) , it is of interest to understand their worst-case performance and modes of generalization. Further, in real world applications such as self-driving cars, industrial control, and robotics, searching over environment settings to investigate and address such behaviours is likely to be critical on the path to robust and generalizable agents. To conclude, while this work has focused mostly on evaluation and understanding, it is only a first step towards the true goal of building more robust, general agents. Initial results we report indicate that enriching the training distribution with settings leading to failure may need to be done at a large scale if it is to work, which introduces significant challenges. While training robust agents is likely an endeavour requiring significant effort, we believe it is important if agents are to carry out critical tasks and on the path to finding more generally intelligent agents. goal was <50% (compared to 98% on the average maze). Furthermore, the 25th, 50th, and 75th percentiles were as follows: • p(reaching the goal): 0.031, 0.136, 0.279 • number of goals reached: 0.042, 0.136, 0.368
 A.3 HUMAN EXPERIMENTS To upper bound the intrinsic difficulty of the mazes found to be adversarial to agents, we conducted experiments where three humans played on the same mazes. Each human played a single episode on each of ten mazes. The humans played at the same resolution as agents, 96x72 pixels, to rule out visual acuity as a confounding factor. On all mazes, all humans successfully found the goal in the course of the episode. In fact, in most episodes, humans were able to find the goal in less than a third of the episode. Figure 8 : Mazes used for human experiments. For each maze, the agent that performed best found the goal less than 50% of the time. In contrast, humans always found the goal, usually within less than a third of the episode. Note that humans played at the same resolution as agents, 96x72 pixels. Figure 1 : 1 Figure 1: Navigation task. (left) Example maze from the training distribution together with the path taken by the agent from spawn (cyan) to goal (magenta). (right) Frames from top left to bottom right correspond to agent observations as it takes the path from spawn to goal. Note that while the navigation task may look simple given a top down view, the agent only receives very partial information about the maze at every step, making navigation a difficult task.
 Figure 2 : 2 Figure 2: Example of search procedure. First, we generate a set of 10 initial candidate mazes by sampling from the training distribution.We then Evaluate each with the agent over 30 episodes, select the best maze (i.e. lowest agent score), and Modify this maze by randomly moving two walls to form the next set of candidates (Iteration 1). This process is repeated for 20 iterations, leading to a maze where the agent score is 0.09 in this example (i.e. the agent finds the goal once in 11 episodes). In Appendix A.2.1 we detail the computational requirements of this search procedure.
 (a) Average number of goals reached per episode over the course of the optimization. (b) Probability of the agent reaching the goal in an episode.
 Figure 3 : 3 Figure 3: The search algorithm is able to rapidly find mazes where agents fail to find the goal. (a) The objective used for the optimizer is average agent score. The dashed line corresponds to average goals reached on randomly perturbed mazes. (b) Minimizing score also leads to a low probability of at least one goal retrieval in an episode. The dashed line corresponds to average probability of reaching a goal on randomly perturbed mazes. The blue lines are computed by averaging across 50 optimization runs.
 Figure 4 : 4 Figure 4: Example mazes leading to low scores and example trajectories on these mazes. (a) Maze with randomly perturbed walls. Despite being out of distribution, agents find the goal in 98% of episodes on such mazes and are able to get the goal on average 45 times per episode. (b) Maze obtained after 20 iterations, moving two walls at each iteration to minimize reward. All agents find the goal on such mazes in less than 20% of episodes. (c) Maze obtained through additional iterations of removing walls. All agents find the goal on such mazes in less than 40% of episodes. (d) Human trajectory on the same maze as in (c). Humans are able to consistently find the goal on such mazes.
 (a) Agent locations, with goal (magenta) fixed.(b) Goal locations, with agent (cyan) fixed.
 Figure 5 : 5 Figure 5: Adversarial mazes are robust to change of spawn positions. The probability of goal retrieval (shown with the color bar) remains low across large portions of the simplified maze as the (a) agent spawn locations and (b) goal locations are moved for each episode..
 (a) Across hyperparameters, same architecture. (b) Across architectures.
 (a) Average number of goals reached. (b) Probability of finding the goal in a single episode.
 Figure 7 : 7 Figure 7: Richer training distributions did not lead to robust agents. Adversarial optimization for agents trained with adversarial mazes (red) and for agents trained with randomly perturbed mazes (yellow). Compared against the Standard training method from Figure 3 for 50 iterations (blue).
 Figure 9 : 9 Figure 9: Trajectories taken by Human 3 on mazes leading to agent failure.
 Table 1 : 1 Human seconds-to-first-goal on agent failure mazes we provide detailed results for our transfer experiments. In particular, we detail transfer between all pairs among the 10 agents, five A2CV agents and five MERLIN agents trained with different entropy costs and learning rates.
 Figure 10 : 10 Figure 10: Pairwise transfer scores. Lower number indicates more transfer. 'A' corresponds to our A2CV agent, and 'M' corresponds to MERLIN.
 end best ← Evaluate(candidates, num evaluations); Algorithm 1: Method for finding environment settings leading to failure cases. Concretely, we generate a set of initial candidate mazes by sampling mazes from the training distribution. We then use the Modify function on the maze which yielded the lowest agent score to randomly move two walls to produce a new set of candidates, rejecting wall moves that lead to unsolvable mazes. Importantly, this method is able to effectively find catastrophic failure cases (as we demonstrate in Section 3.1), while also having the advantage of being intuitive to understand and implement. input : num iterations, num candidates, num evaluations and function Modify output: An environment setting best candidates ← GenerateCandidates(num candidates); for i ← 1 to num iterations do best ← Evaluate(candidates, num evaluations); candidates ← Modify(best, num candidates);

Avraham Ruderman
Richard Everett
Bristy Sikder
Hubert Soyer
Jonathan Uesato
Ananya Kumar
Charlie Beattie
Pushmeet Kohli


Fixed-point solutions to the regress problem in normative uncertainty

Introduction People sometimes make claims about how we ought to act in the face of empirical uncertainty. A "decision theory" is a collection of such claims. Because they make demands on our behavior, decision theories are "norms". (Descriptive claims about how actual agents act under uncertainty are sometimes called "positive decision theories". For the purposes of this paper, the term "decision theory" will refer exclusively to normative decision theories, as defined above.) Moral theories are also norms, for example, because they too are collections of claims about how we "ought" (albeit in another sense) to act. Though I hope that the formal results of this paper are of interest to readers of many metaethical persuasions, the language used throughout will assume a realist position. I will say, then, that we suffer decision-theoretic uncertainty when we assign positive probability to the truth-values of conflicting decision theories, and, more generally, some form of normative uncertainty whenever we assign positive probability to the truth-values of conflicting norms. The most widely accepted normative decision theory, by far, is expected utility theory. As a normative theory, expected utility theory can be interpreted as the claim that we have a cardinal utility function whose value depends on what act we choose and on the state of the world, and that we ought to act so as to maximize the expected value of that function, given our uncertainty across states. Alternatively, expected utility theory can be interpreted as the claim that we have an ordinal utility function whose value depends on what act we choose and on the state of the world, and that we ought to act in such a way as satisfies various assumptions (the von Neumann-Morgenstern axioms, perhaps), which together happen to entail that we will be acting as if we were maximizing the expected value of a cardinal utility function. Either way, expected utility theory, as a normative ideal, is a set of claims about what we ought to do. Expected utility theory is not without critics. Soon after its foundations were laid in the 1940s and 1950s by John von Neumann, Oskar Morgenstern, and Leonard Savage, others, such as Maurice Allais, raised objections to the claim that expected utility maximization is a model of ideal behavior. It is true that, in the apparent absence of plausible alternatives, expected utility theory came to serve as the unchallenged basis for almost all of economic theory. But the debate over the normative claims that underlie it continued among some economists and philosophers, and has recently received new attention with Lara Buchak's 2013 publication of Risk and Rationality, which argues that it can be rational to violate the von Neumann-Morgenstern "independence" axiom so as to act on explicit risk preferences. In sum, the normative claims made by expected utility theory-even if sometimes taken for granted-are claims about whose truth-value an agent can be uncertain. What "should" one do in the face of decision-theoretic uncertainty? Is there even a coherent way to interpret that question? The position that there is-that we "ought" to act on the basis of our normative uncertainty in general-has been called "uncertaintism" (or, less frequently, "metanormativism"). The uncertaintist, presumably, must then offer an account of what one should do when one assigns, say, a seventy percent chance to the truth of expected utility theory, and a thirty percent chance to the truth of Buchak's alternative. As we can see, questions about how to deal with empirical uncertainty give rise to questions about how to deal with decision-theoretic uncertainty. But our regress does not stop there. Just as decision theories are theories about how to act in the face of empirical uncertainty, let us use the term "metanormative theories" for collections of claims about how we ought to act in the face of normative uncertainty. It seems that, just as we can suffer normative uncertainty, we can suffer metanormative uncertainty as well: we can assign positive probability to conflicting metanormative theories. Metametanormative theories, then, are collections of claims about how we ought to act in the face of metanormative uncertainty. And so on. In the end, it seems that the very existence of normative claims-the very notion that there are, in some sense or another, ways "one ought to behave"-organically gives rise to an infinite hierarchy of metanormative uncertainty, with which an agent may have to contend in the course of making a decision. Postulating such a hierarchy may seem like a strange and unneccesarily complex solution to a rather small and obscure problem. By analogy, therefore, consider three other structures explored in detail by philosophers and economic theorists: belief hierarchies, preference hierarchies, and hierarchies of deliberation procedures. We have beliefs about the world; and when one reflects on the fact that we can also have beliefs about people's beliefs, one can hardly help but document the emergence of a "belief hierarchy", constituted of beliefs about the world, beliefs about beliefs (2nd-order beliefs), beliefs about beliefs about beliefs (3rd-order), and so on [For a rigorous exploration of the structure, see Mertens and Zamir (1985) ]. Likewise, we have preferences over features of the world, and the fact we can also have preferences over the contents of people's preferences (2nd-order preferences), and so on, gives rise to a "preference hierarchy" [see for example Bergemann et al. (2017) ]. Finally, when we are not sure which act would maximize our utility, we may find it useful to ponder our options, even if doing so would come at a cost. Since pondering is in some sense just another option, we must then ponder whether to take one of our firstorder options or to take the option of pondering among them, and so on. Lin (2014) concludes that this deliberation must stop after finite steps, and will do so justifiably so long as the reasoner reaches the point that further meta-deliberation no longer obviously dominates taking one of the acts available. Lipman (1991) also explores this problem, in terms more similar to those we will use below: by constructing a (transfinite) hierarchy of beliefs over decision-making procedures, which justifies an act whenever it is judged to be (judged to be…) at least as good as the other available options, including the options of further deliberation at any order. Similarly, normative decision theories tell us how to act in the face of uncertainty about the true state of the world. The fact that we may have to act in the face of uncertainty about the true decision theory thus seems plausibly to give rise to a "metanormative hierarchy", similar in many respects to the hierarchies above. The infinite hierarchy of metanormative uncertainty, furthermore, is more than hypothetical. I myself, for instance, do not currently know the correct decision theory. Expected utility theory seems highly plausible to me, but I cannot fully rule out Buchak's arguments for some sort of risk-weighted expected utility theory, or arguments for theories that evade "Pascal's muggings" by giving special treatment to low-probability but high-expected-value events, to name just two examples. Likewise, I believe that there is a "true metatheory"-a correct way to act in the face of my decision-theoretic uncertainty-but I am not certain of what it is, nor of the true theory at any order of the above hierarchy. Despite all this uncertainty, however, my efforts to be moral or rational do ultimately result in conclusions about how to act. Somehow, from my infinite hierarchy of metanormative uncertainty, it seems I can be guided by normative concerns. The process by which this happens seems potentially interesting. Questions of decision-theoretic uncertainty have received little attention so far, however-research on normative uncertainty has focused primarily on moral uncertainty-and even the most recent inquiries into normative uncertainty (again, usually presented in the context of moral uncertainty) have generally left such questions unanswered. Weatherson (2014) avoids the problem by positing that the only norms are "firstorder"; that there simply is no right or wrong way to deal with normative uncertainty. Sepielli (2014) allows for "orders of rationality" and the corresponding "iterated normative uncertainty" (526). He constructs a metanormative hierarchy much like that we will define below, to serve a bridge between the mind-independent norms of which we are uncertain and the action-guiding, "all-things-considered" decision rule (given by what he calls "global systemic rationality") to which we are more plausibly all accountable. Then, given an agent who assigns some positive credence to theories of rationality, at some order, which conflict as to which act is most rational for her (at that order), he expresses the suspicion that that appealing to a higher-order rationality concept will always allow the agent to eliminate at least one action available. Posit "actions A…Z", he writes; "[p]erhaps my uncertainty regarding objective normativity will intentionally explain my doing any of A…R…. My hope is to show that, as a general matter, potential actions will be hived off with each stepping-back" (and permanently so). Given a finite act-set, this procedure would ultimately result in a coherent definition of what the agent subjectively ought to do. Unfortunately, as for whether the procedure succeeds, he concludes, "I'll have to make good on that suspicion elsewhere" (539). Tarsney (2017) reiterates Sepielli's hope for what he calls "fixed point solutions" to the regress problem. He acknowledges, however, that such fixed points will not necessarily exist (240-241), and concludes that a general solution to the regress problem must thus be found altogether elsewhere. In particular-motivated also by the thought that finite, boundedly rational agents cannot be normatively required to work out the fixed point of an infinite hierarchy, even if it exists (238) 1 -he argues for what he calls the "Enkratic Principle" (246). This is the broad, mind-independent principle that, just as long as one is in some sense "trying one's best" to do the first-order right thing in the face of normative uncertainty, one's decision is justified. On Tarsney's account, the regress is thus cut off at the second order. Finally, many accept that uncertaintism does indeed require some sort of fixedpoint solution to the regress problem-but conclude from this that there simply must be something deeply wrong with uncertaintism. The following sentiment, described in Volume 7 of Oxford Studies in Normative Ethics (2017), is typical: But a regress problem looms. Let us suppose that I am uncertain among some ordinary moral theories, [and] I ask what to do given the probability distribution over T1…Tn. But I am uncertain as to the answer, assigning some probability to each of U1…Un. This prompts me to ask what I ought to do given this probability distribution…. We can imagine this process iterating indefinitely…. The possibility of normative uncertainty all the way up makes the uncertaintist project look pointless. Does this possibility in fact render the uncertainist project pointless? Or can one accept the possibility of normative uncertainty all the way up, and still be norm-guided in some important classes of circumstances? I here argue the latter. To begin to answer the above questions more precisely, Sect. 2 presents a formal framework that aims to capture our intuitions about the concept of decision-theoretic uncertainty. Within this framework, Sect. 3 specifies various conditions under which two similar convergence results follow, as given in Sect. 4. Section 5 considers the earlier sections' implications for normative uncertainty more generally.
 Framework
 Choiceworthiness A finite set A = {a 1 , . . . , a |A| } of "feasible acts" presents itself. There is a finite set of "possible states" S = {s 1 , . . . , s |S| } to which I assign positive probability. 2 I assign utilities to performing each act in each state, as represented by the utility function u(A, s), where the value assigned to each act is not necessarily independent of the alternatives in A. 3 I also find myself in an overall finite epistemic position e, specifying the probabilities I assign to all relevant claims. 4 Let us call π = A, u, e my "choice problem". Definition 2.1 A choice problem π is a triple of (i) a set of acts A, (ii) an epistemic position e, and (iii) a state-contingent utility function u over A. We will say that my utility function specifies the "objective choiceworthiness" (or simply "choiceworthiness") of each act, conditional on each state. That is, given s, the choiceworthiness of a i is u(A, s) i -the ith element of the |A|-vector u(A, s). From the probabilities I assign to the states in S, therefore, I also assign probabilities to potential values of the objective choiceworthiness of each act. Definition 2.2 A (finite) choiceworthiness distribution is a (finite) probability distribution over choiceworthiness values for some (finite) set of acts. Let D n denote the set of all finite probability distributions in R n , and let some d(π ) ∈ D |A| represent the choiceworthiness distribution entailed by π . Many other finite probability distributions over R |A| might do just as well as the chosen d at representing my finite choiceworthiness distribution. Exactly which others depends on how much structure is contained in our understanding of "utility". If we understand utility to be a merely "ordinal" quantity, for instance, then any transformation of d that is monotonic in choiceworthiness (and constant in probability) represents the same choiceworthiness distribution. We are here assuming nothing about utility except that it at least partially orders act-state pairs from a given {feasible set × possible set}, and that R is "rich enough" to capture any potential difference between the choiceworthiness values of particular act-state pairs-that choiceworthiness cannot be lexicographic, for instance. As discussed at the end of Sect. 2.5, these assumptions about choiceworthiness, here made implicitly, will follow from similar assumptions made explicitly about subjective choiceworthiness.
 Subjective choiceworthiness I am uncertain about acts' choiceworthinesses. Even so, I may know that one act is the most appropriate for me to choose, given my epistemic position. As I write this, for instance, I assign high probability to the event that, if I go to the doctor, I will swiftly be cured of my back injury (an outcome I would prefer immensely to the status quo), and low probability to the roughly complementary event that, if I go to the doctor, I will waste some time and remain injured (an outcome to which I would slightly prefer the status quo). Despite this uncertainty, and all my other uncertainty, I am in fact certain that going to the doctor is the "better choice" for me right now (by far!). There is thus some scale on which the act of going to the doctor scores higher for me than the act of not going-and would score higher for anyone with the same utility function, in the same overall epistemic position, facing the same set of feasible acts. Let us call this scale "subjective chiceworthiness". It is my intuition that subjective choiceworthiness c, when well-defined, is fundamentally a cardinal scale. That is, I would maintain that a representation of acts' subjective choiceworthinesses (for an agent in a given situation) in R |A| would be unique at least up to affine transformation. If my feasible act set A = {a 1 , a 2 , a 3 } consists of going to the doctor (a 1 ), going to a very slightly less competent doctor (a 2 ), or not going at all (a 3 ), then there is some important and foundational sense in which, given my epistemic position and my preferences, the distance between c(π ) 1 and c(π ) 2 is less than the distance between c(π ) 2 and c(π ) 3 . It might be objected that I will always do whatever winds up being most subjectively choiceworthy; that therefore, in the absence of a specified theory of decision-making under uncertainty, no information is conveyed by postulated differences between the acts not chosen; and that c is therefore better understood as merely an ordering, or perhaps even as a choice relation. To this it might be replied that, under certain circumstances, differences in subjective choiceworthiness could bear some relationship to the subjective probability with which a subjectively sub-optimal act would become optimal upon further reflection. Or that cardinal subjective choiceworthiness takes on a clearer meaning in other situations of normative uncertainty (i.e. one might not choose the most subjectively morally choiceworthy act, and might in some sense be more blameworthy the less subjectively morally choiceworthy one's act was)-and that it would be strange for subjective choiceworthiness to be fundamentally cardinal in one of these situations but not the other. Or that our models of the world are generally simpler when we extend our intuitions regarding quantities' cardinality beyond the domains in which they happen to be testable-such as our intuition that temperature is generally cardinal, even on some cold, distant star that we will only discover if its temperature rises above some threshold. Furthermore, cardinal subjective choiceworthiness allows for the convergence results described below, and less structured interpretations of subjective choiceworthiness would not. If we are otherwise persuaded that the regress problem must have some solution or other, it is not circular to allow this observation itself to lend credibility to the concept of cardinal subjective choiceworthiness. In any event, for the purposes of this analysis, we will understand subjective choiceworthiness (again, when well-defined) to be cardinal. We will represent it by a "subjective choiceworthiness function" c(π ), where c assigns a real number to the subjective choiceworthiness of each of the acts in a feasible set A, for an agent with a utility function u, in epistemic position e. Note that by having c map into the real numbers, we are assuming that all information about differences in subjective choiceworthiness (and therefore utility) can be captured by ratios of differences in real numbers. We are here explicitly assuming for subjective choiceworthiness what we provisionally assumed above for objective choiceworthiness-that, for instance, it cannot be lexicographic. Like probability theories that let us condition on probability 0 events, utility theories that let us distinguish between acts that differ infinitesimally in choiceworthiness may also be interesting to consider in light of the regress problem. However, we will not touch them here.
 Metachoiceworthiness In general, if I am to translate a choiceworthiness distribution d into a determination of how to act, I must invoke a "decision theory": a collection of claims concerning how to evaluate acts in light of one's choiceworthiness distribution. For example, using this terminology, one decision theory is "Expected Choiceworthiness Theory" (EC). EC is characterized by the fact that, if I am certain that it is the correct decision theory, then each act's subjective choiceworthiness for me is its expected choiceworthiness under d. 5 Another decision theory would be "minimum choiceworthiness"-a theory characterized by the fact that, if I am certain that it is the correct decision theory, then each act's subjective choiceworthiness for me is its minimum possible choiceworthiness under d. Just as I am uncertain about the true state of the world, I may also be uncertain about the correct decision theory. To come to a determination of how to act, therefore, I may have to invoke a sort of "meta decision theory" (or, "2-metatheory"): a collection of claims concerning how to respond to one's uncertainty over decision theories. Note that, since this is so, the decision theories (we will awkwardly call these "1metatheories", for ease of indexing) cannot themselves be claims about subjective choiceworthiness. This is perhaps a surprising claim, so it bears repeating: expected utility theory (for example) is not, in this language, a theory about what subjective choiceworthiness is, or even about what it ought to be "all things considered". It is, rather, a theory about what subjective choiceworthiness "1-ought" to be, for someone with a given objective choiceworthiness distribution over his feasible set-or, a theory about what subjective choiceworthiness is for someone with a given objective choiceworthiness distribution over his feasible set, if he knows the true 1-metatheory. Suppose, for instance, that I am faced with three feasible acts, that I assign probability to each of two 1-metatheories, t 1 and t 2 , and that I am certain of "2-metatheory" m. The theories are such that if I were certain of t 1 , the subjective choiceworthinesses of the acts would be ordered a 1 a 2 a 3 ; if I were certain of t 2 , the subjective choiceworthinesses of the acts would be ordered a 3 a 2 a 1 ; and, given the probabilities I assign to t 1 and t 2 , but my certainty about m, the subjective choiceworthinesses of the acts are in fact ordered a 2 a 3 a 1 . Although I assign probability 1 2 to t 1 , I assign no positive probability to the event that a 1 is more subjectively choiceworthy than a 2 from my epistemic position. The 1-metatheories' claims, therefore, are not claims about the acts' subjective choiceworthinesses given my empirical uncertainty, but about how the acts score on an altogether different scale. Let us call this scale "metachoiceworthiness", or "1-metachoiceworthiness". Of course, metachoiceworthiness must be constructed such that, if I know that an act's 1-metachoiceworthiness is x, then the act's subjective choiceworthiness for me is also x. We might therefore informally think of 1-metachoiceworthiness as "whatever subjective choiceworthiness is, for someone who knows the correct 1-metatheory". But since, again, decision theories are not actually claims about subjective choiceworthiness, let us begin by thinking about 1-metachoiceworthiness on its own terms, and only afterward consider its relationship to subjective choiceworthiness. In any event, the elusiveness of subjective choiceworthiness is not restricted to "order 1". Just as I may be uncertain as to the correct 1-metatheory, I may be uncertain as to the correct 2-metatheory; I may therefore have to appeal to a "3-metatheory"; and the 2-metatheories are therefore making claims not about acts' subjective choiceworthiness given beliefs about their 1-metachoiceworthiness, but about acts', say, "2-metachoiceworthiness" given beliefs about their 1-metachoiceworthiness. So our regress begins.
 k-metachoiceworthiness Let us call choiceworthiness "0-metachoiceworthiness", choiceworthiness distributions "0-metachoiceworthiness distributions", and decision theories "1-metatheories". The concepts of k-choiceworthiness, k-metachoiceworthiness distributions, and kmetatheories can then together be defined recursively.
 Definition 2.3 The k-metachoiceworthiness c k of an act a i , for an agent facing finite choice problem π , is a i 's subjective choiceworthiness for an agent with the same (k − 1)-metachoiceworthiness distribution as that entailed by π , but who knows the correct k-metatheory. Let us denote acts' relative k-metachoiceworthiness by the two-place relation π,k . Definition 2.4 A (finite) k-metachoiceworthiness distribution d k ∈ D |A| is a probability distribution over k-metachoiceworthiness values for some (finite) set of acts A. Definition 2.5 A k-metatheory, applied to a finite set of acts A, is a function t k : D |A| → R |A| , representing claims about the k-metachoiceworthiness of the acts in A given (k − 1)-metachoiceworthiness distribution d k−1 ∈ D |A| . Note that, strictly speaking, if we want our k-metatheories to make k-metachoiceworthiness claims over finite act-sets of arbitrary size, we would have to say that a k-metatheory is a family of functions {t n k } from D n to R n , with one for each n ∈ N. For simplicity, however, we will take n = |A| as given and interpret our project only as an attempt to find criteria under which the subjective choiceworthinesses of any n acts will be well-defined-with the understanding that identical reasoning would apply to any other n. We can now define a few aditional terms.
 The relationship of k-metachoiceworthiness to subjective choiceworthiness Upon introducing the cardinal subjective choiceworthiness function c(π ) above, we placed no restrictions on what it could be. Now that we have documented the emergence of an elaborate web of concepts concerning π , however, we can consider how it relates to c. Recall that k-metachoiceworthiness claims are defined so that, if I know that an act's k-metachoiceworthiness for me is x, the act's subjective choiceworthiness for me is x. Let us now introduce a compatible, minimally restrictive principle with which one's subjective choiceworthiness function might comply in the face of uncertainty about an act's k-metachoiceworthiness. Definition 2.9 The Dominance Principle is the principle that • If b ≥ x ∀b ∈ [ c k ] i , and b \* > x for some b \* ∈ [ c k ] i , then c(a i ) > x. • If b ≤ x ∀b ∈ [ c k ] i , and b \* < x for some b \* ∈ [ c k ] i , then c(a i ) < x. Note that if I accept the Dominance Principle, it follows immediately that my subjective choiceworthiness for an act a i is well-defined whenever | ∩ k∈N [min([ c k ] i ), max([ c k ] i )]| = 1. That is, whenever exactly one number lies in the ranges of "admissible" (not dominated) k-metachoiceworthiness values, across all k, for an act, that number must be the act's subjective choiceworthiness. Note also that any claim about subjective choiceworthiness itself, such as the Dominance Principle, in some sense takes on both a positive and a normative interpretation. One could interpret the Principle normatively as asserting that one's subjective choiceworthiness always ought to obey the above pattern. In this case, if one accepts the Principle, one's subjective choiceworthiness also does obey it, since to hold that an act should be ranked highly for someone in your epistemic position is simply another way to say that it is highly subjectively choiceworthy. Alternatively, one could interpret the Principle positively as asserting that, as a matter of fact, subjective choiceworthiness always obeys the above pattern. If one accepts this claim (and that "ought implies can"), one must also accept that subjective choiceworthiness always ought to obey the above pattern. Either way, if one accepts the Principle, one cannot assign positive probability to k-metatheories that claim that the k-metachoiceworthiness of an act lies outside the admissible range imposed by one's k -metachoiceworthiness distribution for the act for lower orders k < k. Finally, note that the framework outlined here differs from other approaches to subjective choiceworthiness in the following respect. Some other approaches [e.g. that of MacAskill (2016a)] begin with the normative theories in all their diversity; work through problems of intertheoretic comparability; and then try to define subjective choiceworthiness with no more structure than necessary. On some accounts, this minimal structure allows only for a binary classification of acts into the "permissible" and the "impermissible" [as recommended, for instance, by Barry and Tomlin (2016) ]. The above approach, by contrast, begins by assuming that subjective choiceworthiness is a single cardinal scale, and it characterizes k-metachoiceworthiness claims, and the k-metatheories that make them, in terms of the subjective choiceworthinesses that they would induce if they were known. This approach has the cost of assuming cardinal subjective choiceworthiness, but it has the benefit of immediately giving all my kmetachoiceworthiness claims both unit and level comparability, without requiring any further assumptions. Thus, from a cardinal definition of subjective choiceworthiness, we also get a cardinal definition of utility, without having to assume it explicitly. By similar reasoning, we also get cardinal definitions of k-metachoiceworthiness for all k. Note that we are not taking the Von Neumann-Morgenstern approach of defining my utility function so that it represents the choices I would make if I were maximizing expected utility; indeed, our project is to explore how far I can stray from certainty about expected utility theory while still knowing how I subjectively ought to act.
 Conditions
 Completeness A "partial k-metatheory" would be one that makes claims about the k-metachoiceworthinesses of some acts under some (k − 1)-metachoiceworthiness distributions, but not of all acts under all (k − 1)-metachoiceworthiness distributions. A partial decision theory of "strict dominance", for instance, claims that a i π,1 a j ⇐⇒ u(A, s) i > u(A, s) j ∀s ∈ S, and makes no other claims at all. That is, it claims that an act a i is more 1-metachoiceworthy than an act a j if and only if a i is more objectively choiceworthy than a j in all the states to which I assign positive probability, and it is silent about acts' relative 1-metachoiceworthinesses in all other cases. Conversely, Definition 3.1 A k-metatheory, applied to a finite set of acts A, is complete if it is defined throughout D |A| . One condition for the results below is that I assign positive probability only to complete decision theories. Believing that the true decision theory is complete is, I think, reasonably well motivated by the sense that, just as I know acts' 0metachoiceworthiness (i.e. objective choiceworthiness) if I know the true state, I should be able to know acts' 1-metachoiceworthiness if I know the true decision theory (and so on up the hierarchy). In any event, we will remove partial decision theories from consideration so as to separate the regress problem from the problems of incomparability that can plague normative uncertainty in their own right. Note that the framework laid out in Sect. 2 does not allow us to assign positive probability to the "nihilistic decision theory" (one that makes no claims about acts' 1-metachoiceworthinesses under any choiceworthiness distribution). Since a decision theory is a collection of claims determining what acts' subjective choiceworthinesses would be for me if I knew how to respond to my empirical uncertainty, and since my subjective choiceworthiness is already defined in the degenerate case of empirical certainty, all my decision theories at least claim that an act's subjective choiceworthiness is its objective choiceworthiness, when my objective choiceworthiness distribution is degenerate.
 Continuity We will say that Definition 3.2 A k-metatheory t k is continuous if slight changes to the individual (k−1)-metachoiceworthiness claims to which one assigns positive probability produce only slight changes to the k-metachoiceworthiness claims made by t k . To be called continuous, t k need not respond continuously to the probability one assigns to some (k − 1)-metachoiceworthiness claim. A more formal definition of continutity, as we are using the term here, is given in the "Appendix". A second condition, necessary for only the first of the results below, is that I assign positive probability only to continuous decision theories.
 The Analog Principle MacAskill (2014) argues that, when we are facing both empirical and normative uncertainty over a set of acts, there is a sense in which we should treat our empirical and normative uncertainty "analogously". If I am uncertain which act is objectively best, it may seem unlikely that the appropriate response to my uncertainty would depend on the reason (i.e., empirical or normative) for my uncertainty-especially upon considering that I might have uncertainty about how to behave without even knowing the reason for my uncertainty. In the context of the regress problem, one might likewise argue that we should treat our empirical and k-metatheoretic uncertainty analogously. More formally: Definition 3.3 Let t \* k : D |A| → R |A| denote the true k-metatheory. The Analog Principle is the claim that t \* k = t \* 1 ∀k ≥ 1. One might wonder if it matters whether my beliefs about the k-metatheories are correlated across different orders k (as of course they are-very strongly!-if I accept the Analog Principle), or whether they are correlated my beliefs about the state of the world. In fact, it does not. A k-metatheory is simply a function of my (k-1)metachoiceworthiness distribution; a k-metatheory's output therefore does not depend on the probability that it is the true k-metatheory, nor on its probability conditional on some state or k -metatheory. A final condition, necessary only for the first of the results below, is that I accept the Analog Principle.
 Convergence
 Intuition In the context of the framework above, the commonness of well-defined subjective choiceworthiness is not surprising. If I assign positive probability to a finite number of theories, and they disagree about how subjectively choiceworthy some act should be for me, there will be a minimum and a maximum to that range of values. In the face of that uncertainty, my subjective choiceworthiness should lie somewhere in the interior of the range. Where, exactly? I will assign positive probability to different answers as to where it should lie, producing a smaller range. And so on. Given a few other assumptions (either the continuity of my theories and my acceptance of the Analog Principle, or the possibility of transfinite hierarchies), this process will not "get stuck" by shrinking the range of potential subjective choiceworthiness values for each act merely from a larger range to a smaller range. Instead, the process is guaranteed ultimately to shrink said range to a single point. In other words, nothing very counterintuitive falls out of the mathematical setup of the problem. The point of this exercise is simply to formally illustrate a coherent framework whereby our intuitions about normative uncertainty-including about the infinite regress that it threatens-can be reconciled with the understanding that, at the end of the day, we make norm-guided decisions. With that said, the convergence results can be stated as follows.
 Natural hierarchies Theorem 1 If one assigns positive probability only to a finite set of decision theories all of which are complete and continuous, and if one accepts the Dominance Principle and the Analog Principle, then one's subjective choiceworthiness is well-defined over any finite set A of acts. The proof can be found in the "Appendix". Proposition 4.1 If two acts are equally subjectively choiceworthy, this fact will not necessarily be revealed by the iterated application of one's distribution over kmetatheories. Proof Suppose for example that I assign probability 1 2 to Expected Choiceworthiness Theory, and to the analogous hierarchy (T 1 ) according to which the (k+1)metachoiceworthiness of an act is its expected k-metachoiceworthiness. I assign probability 1 2 to a risk-averse hierarchy of theories (T 2 ) according to which the (k+1)metachoiceworthiness of an act is the average of its expected k-metachoiceworthiness and its minimum possible k-metachoiceworthiness. I am deciding between two acts, a 1 and a 2 . I assign probability 1 2 to a state in which a 1 has objective choiceworthiness 0 and probability 1 2 to a state in which a 1 has objective choiceworthiness 1. Act a 2 has objective choiceworthiness 1 3 in both states. The subjective choiceworthiness of a 2 is, of course, 1 3 . The k-metachoiceworthiness claim about a 1 made by T 1 is [ c k ] 1,1 = 1 − k n=1 1 2 2n−1 ; that made by T 2 is [ c k ] 1,2 = k n=1 1 4 n . This can be seen by verifying algebraically that these summations satisfy the intitial conditions [ c 1 ] 1,1 = 1 2 and [ c 1 ] 1,2 = 1 4 , and the recursive conditions [ c k ] 1,1 = 1 2 [ c k−1 ] 1,1 + 1 2 [ c k−1 ] 1,2 and [ c k ] 1,2 = 1 4 [ c k−1 ] 1,1 + 3 4 [ c k−1 ] 1,2 (k > 1). As k increases, {[ c k ] 1,1 } converges to 1 3 from above, while always strictly greater, and {[ c k ] 1,2 } converges to 1 3 from below, while always strictly less. Thus the acts' subjective choiceworthinesses are precisely equal, even though any formal comparison of the acts is undefined if we halt our deliberation after finite k. This example does not imply the potentially concerning conclusion that one might not be able to infer acts' relative subjective choiceworthinesses, from one's choiceworthiness distribution and one's hierarchy distribution, in finite time. Indeed, since the subjective choiceworthiness of each act is well-defined under the above conditions, and is a logical consequence of one's finite choice problem, we know from the completeness of first-order logic that the value of c(π ) i can be determined for all i by some finite proof-such as the one given above. The example does, however, demonstrate that there may be facts about subjective choiceworthiness that are not discovered by the "iterate a few times and hope my uncertainty is more or less resolved" algorithm. In particular, therefore, I believe it demonstrates that fixed-point solutions to the regress problem need not take the form either of convergence after finite k or of monotonic decreases in the set of maximally kchoiceworthy acts-as hoped for in Sepielli (2014) , and as claimed by Tarsney (2017, p. 239) .
 Transfinite hierarchies Suppose that the k-metatheories to which I assign positive probability are all complete and compatible with the Dominance Principle, but that they are not continuous, or that I reject the Analog Principle. It is then possible that my subjective choiceworthiness for some act a i does not converge, even after infinite steps. That is, though lim k→∞ min([ c k ] i ) and lim k→∞ max([ c k ] i ) do both exist (by the fact that the respective sequences in k are monotonic and bounded), lim k→∞ min([ c k ] i ) < lim k→∞ max([ c k ] i ). In such a situation, it seems, someone could still claim that there is a "right way" for me to act. Someone could claim that my subjective choiceworthiness for a i should be the average of lim k→∞ min([ c k ] i ) and lim k→∞ max([ c k ] i ), for example. If I assign positive probability to competing theories of how to act in situations like the above, I must appeal to a theory about how to act in the face of this uncertainty. So our regress extends beyond the natural numbers, into the transfinite ordinals. The definitions given in Sects. 2 and 3 can generally be reinterpreted so that k (or, κ) is any ordinal, not just any natural number. Two, however, will require slight tweaks: Definition 4.1 (Definition 2.3, revised) The κ-metachoiceworthiness c κ of an act a i , for an agent facing finite choice problem π , is a i 's subjective choiceworthiness for an agent with the same κ -metachoiceworthiness distribution as that entailed by π for all κ < κ, but who knows the correct κ-metatheory. Definition 4.2 (Definition 2.5, revised) A κ-metatheory, applied to a finite set of acts A, is a function t κ : D κ|A| → R |A| , representing claims about the κ-metachoiceworthiness of the acts in A given κ -metachoiceworthiness distributions d κ ∈ D |A| for all κ < κ. Note that this definition of a higher-order metatheory is strictly more general than the original, even with respect to finite k, because it allows k-metatheories to be functions of one's beliefs not only about (k − 1)-metachoiceworthiness but about kmetachoiceworthiness at all lower orders k < k. The following result thus holds, as Theorem 1 does not, regardless of whether we want to allow for this possibility. We can now state the following: Theorem 2 If one assigns positive probability only to a finite set of complete κmetatheories for each ordinal κ, and one accepts the Dominance Principle, then one's subjective choiceworthiness is well-defined over any finite set A of acts. The proof can be found in the "Appendix".
 The infectiousness of stubbornness Definition 4.3 The Weak Dominance Principle is the principle that • If b ≥ x ∀b ∈ [ c κ ] i , for some κ, then c(a i ) ≥ x. • If b ≤ x ∀b ∈ [ c κ ] i , for some κ, then c(a i ) ≤ x. Let us call a κ-metatheory "compromising" if it is compatible with the (strong) Dominance Principle, and "stubborn" if it is not. Expected Choiceworthiness Theory and risk-weighted variants of it are examples of "compromising" theories. Minimax Theory, according to which an act's metachoiceworthiness is its minimum possible choiceworthiness, is an example of a "stubborn" theory. However, it is compatible with the Weak Dominance Principle. Both the theorems above demonstrate that, when all the decision theories (or κmetatheories) to which I assign positive probability are compromising (along with some other conditions), the range of potential subjective choiceworthiness values for each act shrinks to point. In both cases, this is demonstrated roughly by the fact that, when the range of potential subjective choiceworthiness values for some act is a non-degenerate interval at some order k (or κ), the application of even higher-order metatheories shrinks this interval by increasing its minimum. One might notice that, strictly speaking, neither of the proofs requires that all my decision theories (or κ-metatheories) be compromising. Suppose, for example, that I reject the Dominance Principle, but accept the Weak Dominance Principle. Suppose further that just one decision theory (or one κ-metatheory for each κ) to which I assign positive probability is "stubborn"-or, that all the stubborn theories to which I assign positive probability are one-sidedly pessimistic (like Minimax) or optimistic (like Maximax) at each order. Then our proofs can go through with only slight modifications. We just need to shrink our interval exclusively from the top or from the bottom, at a given order, to avoid asking concessions of our stubborn theories. The plausibility of stubborn theories, however, poses two challenges for this project in general. First, stubborn theories are "infectious": they can determine our behavior regardless how little positive credence we give them. Suppose I am deciding between two acts, a 1 and a 2 . I assign probability 0.99 to a state in which a 1 has objective choiceworthiness 1, and probability 0.01 to a state in which a 1 has objective choiceworthiness 0. Act a 2 has objective choiceworthiness 0.0001 in both states. Furthermore, I assign probability 0.99 to Expected Choiceworthiness, and probability 0.01 to Minimax, at every order of the hierarchy. It would be deeply counterintuitive to conclude that, from such a position, a 2 is more subjectively choiceworthy than a 1 . It would be perhaps even more counterintuitive to conclude that the acts' subjective choiceworthinesses were equal, if a 2 's objective choiceworthiness were known to be 0. But of course we do reach both conclusions, as repeated applications of my distribution over decision theories bring a 1 's higher-order metachoiceworthiness arbitrarily close to 0-according even to the sequence of "Expected Choiceworthiness"-analogous theories. Second, stubborn theories can clash with each other. If we are going to give some weight to Maximin, at every order, it seems only fair to give some weight to Maximax at every order as well. But if we do, then each act's range of potential subjective choiceworthiness values never shrinks at all; min([ c κ ] i ) = min([ c 0 ] i ), and max([ c κ ] i ) = max([ c 0 ] i ), for all κ. It does not feel as though I can fully rule out stubborn theories. Thus, despite all our progress, I am still left with the original motivating question: how do I so regularly wind up with well-defined subjective choiceworthiness? One encouraging thought is the observation that, though stubbornness is infectious in one sense, there is a sense in which compromise is infectious as well. For example, suppose I assign positive probability to stubborn κ-metatheories (or even, only to stubborn κ-metatheories) at almost all κ, but assign positive probability only to compromising theories at a relatively sparse class of orders-at the limit ordinals, perhaps. Then, even though there is a sense in which I believe in stubborn theories "almost everywhere" up the hierarchy, the scattered all-compromising orders will still force my subjective choiceworthiness range for each act down to a point. (A minimally modified version of the proof of Theorem 2 presented in the "Appendix" will hold so long as our class of "allcompromising" orders is such that, for every set M of ordinal numbers, there is an ordinal number γ ∈ : γ > μ ∀ μ ∈ M.) Similar reasoning applies to the case of merely natural hierarchies. In short, my hierarchy distribution can handle a lot of stubbornness; as long as an all-compromising order comes along every now and then to shrink my subjective choiceworthiness range for each act, there are reasonable conditions under which subjective choiceworthiness will generally be well-defined.
 Rescaling MacAskill (2014) offers the following example of undefined subjective choiceworthiness. Suppose an agent faces a choice problem: Order 0 s 1 (Pr. 18 23 ) s 2 (Pr. 5 23 ) a 1 0 4 a 2 1 0 She must choose among acts A = {a 1 , a 2 }. She assigns probability 18 23 to a state s 1 in which a 1 has objective choiceworthiness 0 and a 2 has objective choiceworthiness 1, and probability 5 23 to a state s 2 in which c 0 (a 1 ) = 4 and c 0 (a 2 ) = 0. In evaluating the at order 1, she assigns probability 18 23 to Expected Choiceworthiness Theory (t 1 ), and probability 5 23 to "Square Root Theory" (t 2 ), according to which an act's 1-metachoiceworthiness is the expectation of the square root of the difference between its objective choiceworthiness and the objective choiceworthiness of the least objectively choiceworthy act in A. As we can see, after rescaling, the 1-metachoiceworthiness distribution of a 1 is precisely what the 0-metachoiceworthiness distribution of a 2 had been, and the 1metachoiceworthiness distribution of a 2 is precisely what the 0-metachoiceworthiness distribution of a 1 had been. Therefore, if we obey the Analog Principle-that is, if our distribution over k-metatheories is the same at every order-and if we rescale after every step, our k-metachoiceworthiness distribution for the acts will flip forever between that of "Order 0" and that of "Order 1, rescaled", without converging. To my mind, however, k-metachoiceworthiness claims are intuitively characterized by the property that, if one believes them, they define one's subjective choiceworthiness. If an agent faces empirical uncertainty over two acts' objective choiceworthiness as represented above, we want to say that, in the event that she learns the truth of s 2 , act a 1 's subjective choiceworthiness for her is 4. In precisely the same language, I think, we want to say that in the event that she learns the truth of Expected Choiceworthiness Theory (but does not learn the true state), a 1 's subjective choiceworthiness for her is 20 23 -and likewise all up the hierarchy. To keep these claims "in line", the framework of Sect. 2 permits real-valued representations of the subjective choiceworthiness values and k-metachoiceworthiness distributions associated with a given choice problem to be rescaled only in conjunction, not independently. Furthermore, if this is the right way to think about k-metachoiceworthiness, then Square Root Theory (SR) is, as stated, incoherent. SR does not specify which real-valued representations of objective choiceworthiness to use as inputs, so its claims should be independent to rescaling. But they are not. Using our agent's 0-metachoiceworthiness as represented above, SR claims that the 1metachoiceworthiness of a 1 is 20 23 , and EC claims that the 1-metachoiceworthiness of a 1 is lower (just 10 23 ). But if we had represented her 0-metachoiceworthiness distribution differently, Order 0 s 1 (Pr. 18 23 ) s 2 (Pr. 5 23 ) a 1 0 1 a 2 1 4 0 we would conclude that EC claims that the 1-metachoiceworthiness of A is 5 23 for her, and that SR claims the same. To ensure that an act's true k-metachoiceworthiness for an agent be independent of the scale she arbitrarily uses to represent her (k−1)-metachoiceworthiness distribution, all our k-metatheories have to be "affine" (unique up to affine transformation). Though this condition closes the door to "Square Root Theory", it permits a wide array of other risk-averse theories, including Buchak's REU Theory and the risk-averse theory presented in Proposition 4.1.
 Applications to moral uncertainty If I assign positive probability only a finite set of complete, cardinal, comparable moral theories-or, if I at least know the right way to represent all my moral theories' choiceworthiness claims on the same cardinal scale-then the results above can be applied almost directly to my moral choice problems under empirical certainty. "Almost", because, to avoid wading into a sea of hopeless complexity, we must assume that my moral theories make no claims about how to respond to uncertainty per se. That is, we must say that, for example, among varieties of utilitarianism, I assign positive probability only to those that claim that an act's objective choiceworthiness is (something along the lines of) its objective impact on total utility. I must assign no positive probability to a variety of utilitarianism that claims that an act's objective choiceworthiness is, say, my expectation of its impact on total utility. Utilitarianism is so often described as the idea that we ought to maximize the world's expected utility-see Parfit (1984, pp. 25, 26) , for instance-that one might easily come to believe that Expected Choiceworthiness is the only way utilitarians are allowed to deal with uncertainty. In this context, however, we should be careful to separate the unique moral claim of utilitarianism (that value is identified with utility) from the independent decision-theoretic claim (that one ought to maximize expected value). Moral theories that explicitly incorporate such decision-theoretic claims may also be interesting to consider in light of the regress problem, but we will not discuss them here. With that restriction, suppose I am certain about the state of the world. I then simply have to swap out our language about objective choiceworthiness being "my utility function, contingent on the true state of the world" for language about objective choiceworthiness being "moral value, contingent on the true moral theory", and Sects. 2-4 apply to cases of moral uncertainty, under empirical certainty, in full. If I face both empirical uncertainty and moral uncertainty, however, my situation is more complex. One approach would be for me to take "objective choiceworthiness" to be a function of both the true state and the true moral theory, to consider my probability distribution over {states} × {moral theories}, and then to apply my hierarchy distribution. Another approach, however, would be for me first to work out the subjective choiceworthiness of each act, conditional on each state, in light of my distribution over moral theories, and then to apply my hierarchy distribution a second time, to work out the subjective choiceworthiness of each act in light of my distribution over states. A third approach, symmetrical to the second, would be for me first to work out the subjective choiceworthiness of each act, conditional on each moral theory, in light of my distribution over states, and then to apply my hierarchy distribution a second time, to work out the subjective choiceworthiness of each act in light of my distribution over moral theories (These approaches have the disadvantage that they would not be able to account for any dependence between my distribution over states and my over moral theories. They have the advantage, however, that they would be able to account for the possibility that my hierarchy distribution over ways of dealing with moral uncertainty differs from my hierarchy distribution over ways of dealing with empirical uncertainty). And other conceivable approaches abound. Unfortunately, these approaches will not necessarily all yield the same subjective choiceworthiness values, or even the same act recommendations-not even under decision-theoretic certainty, and not even when I believe that the same theory should be used in the face of emprical uncertainty as in the face of moral uncertainty. Consider the following situation. I assign positive probability to a set of moral theories M = {m 1 , m 2 , m 3 } and to a set of states S = {s 1 , s 2 , s 3 }. I have two feasible acts, a 1 and a 2 . Their objective choiceworthinesses, conditional on each state and moral theory, are as follows: Furthermore, I am certain that an act's k-metachoiceworthiness is its second-lowestpossible (k−1)-metachoiceworthiness. If I apply this decision theory to my uncertainty over {states} × {moral theories}, I get c(a 1 ) = 2 and c(a 2 ) = 3, so a 2 a 1 . However, if I apply this decision theory first over states (conditional on each moral theory) and then over moral theories-or, first over moral theories (conditional on each state) and then over states-then I get c(a 1 ) = 4 and c(a 2 ) = 3, so a 1 a 2 . These complications only worsen when |M| = |S|, in which case even the "same" decision theory can aggregate across moral theories and across states arbitrarily differently. There is another way in which decision-theoretic uncertainty can interact with moral uncertainty. It is often argued that morality requires us to make decisions as if from behind a "veil of ignorance" about our own identity among those affected by our actions. If so, the moral choiceworthiness of an act depends directly on its decisiontheoretic 1-metachoiceworthiness. Suppose that I ought to act toward a group as if my identity is, in probability, distributed uniformly over the group. Then, if Expected Choiceworthiness is the correct decision theory, the Veil of Ignorance argument points toward classical utilitarianism as the correct moral theory; if Minimax is the correct decision theory, toward Rawls's "maximin criterion"; if some risk-weighted theory is the correct decision theory, toward the corresponding version of prioritarianism; and so on. 7 But we will not explore this interaction further here. Finally, the above thoughts about how to integrate the results of Sects. 2-4 into situations of moral uncertainty can apply straightforwardly to normative uncertainty in other domains, so long as one assigns positive probability finite set of theories which are in some analogous sense complete, cardinal, and intertheoretically comparable. But we will not explore such applications further here.
 Conclusion We are often uncertain about the moral and decision-theoretic norms which we believe should guide our behavior. Even when these norms conflict, however, we often have a subjective understanding of whether some act would be rationally or morally permissible for us, from our position of normative uncertainty. "Uncertaintism" might be understood as the project of unraveling how this uncertainty translates into the subjective choiceworthiness on which we ultimately feel justified in acting. When the uncertaintist tries to specify any particular mechanism for translating the uncertainty over choiceworthiness into an appropriate characterization of subjective choiceworthiness, however, we find that, just as we are not certain of our acts' objective choiceworthinesses, we are not certain of his proposed mechanism either. Nor are we certain about how to how to deal with our uncertainty about such a mechanism. Indeed, our certainty about subjective choiceworthiness seems to stand strangely on its own. In general, when we try to ground our certainty about subjective choiceworthiness in metanormative certainty at some order, we find that the hoped-for ground of certainty does not exist. For some, as cited above, this "possibility of normative uncertainty all the way up makes the uncertaintist project look pointless". The results presented here demonstrate that, as stated, the quoted worry is not justified. We can reliably have well-defined subjective choiceworthiness without being certain about the correct first-order normative theory or about any higher-order metatheory. We only have to commit to a weaker family of assumptions, such as the Let |d t k | and |d T | denote the number of k-metatheories and hierarchies, respectively, to which I assign positive probability. Let c k ∈ R |d t k ||A| represent the claims made by my |d t k | k-metatheories about the k-metachoiceworthinesses of the |A| acts in A. Let p k ∈ |d t k |−1 represent the probabilities I assign to these k-metatheories. We can now represent my kmetachoiceworthiness distribution by d k = c k , p k . 6 Definition 2.8 A hierarchy distribution d T is a probability distribution over hierar- chies. Definition 2.6 A k-metatheory distribution d t k is a probability distribution over kmetatheories.Definition 2.7 A metatheoretic hierarchy (or simply "hierarchy") T is a collection of k-metatheories t k with one for each k > 0.
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 In the process, he distinguishes between the "ideal regress problem", which an ideal agent with perfect reasoning ability might face, and the "non-ideal regress problem". In order to separate the issue of normative uncertainty from the issue of bounded rationality, here we will consider only what he calls the "ideal regress problem". Note that we are implicitly assuming that normative facts are not logical facts; if they are, then it is impossible for an agent with perfect reasoning ability to suffer normative uncertainty.
 Here and elsewhere, we will assume that all credences satisfy the Kolmogorov probability axioms. Note that this implies that all the sets over which I have probability distributions are nonempty.3 We will assume that the probability of each state is independent of the chosen act. We will thus bypass the question of how to act in the face of such dependency (i.e. causal decision theory vs. evidential decision theory and other alternatives), and focus entirely on the question of how to act in the face of uncertainty over states (i.e. expected utility theory vs. its alternatives). For an analysis of how to approach uncertainty between causal and evidential decision theory, see MacAskill (2016b).4 More precisely, let e specify my probability distribution over the set of [{states of the world} × {decision theories (or, 1-metatheories)} × {2-metatheories} × {3-metatheories} × • • • ]. The concept of a "k-metatheory" is defined in Sect. 2.4.
 Let us distinguish EC from "Maximize Expected Choiceworthiness" (MEC). MEC is the weaker theory characterized only by the fact that, if I am certain that it is correct, then the acts with the highest subjective choiceworthiness for me are the acts with the highest expected objective choiceworthiness under d.
 This is not to say that a given distribution can only be represented by one particular vector pair. Multiple k-metatheories may make the same k-metachoiceworthiness claims in some situation, for instance.
 The implications of risk-weighted expected utility theory for decisions made on behalf of groups, rather than individuals, are further discussed byBuchak (2013, pp. 167, 168).