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This means that the model can and will implicitly sacrifice next-token prediction accuracy for long horizon prediction accuracy.

Are you claiming this would happen even given infinite capacity?

I think that janus isn't claiming this and I also think it isn't true. I think it's all about capacity constraints. The claim as I understand it is that there are some intermediate computations that are optimized both for predicting the next token and for predicting the 20th token and that therefore have to prioritize between these different predictions.

Here's a simple toy model that illustrates the difference between 2 and 3 (that doesn't talk about attention layers, etc.).

Say you have a bunch of triplets . Your want to train a model that predicts  from  and  from .

Your model consists of three components: . It makes predictions as follows:


(Why have such a model? Why not have two completely separate models, one for predicting  and one for predicting ? Because it might be more efficient to use a single  both for predicting  and for predicting , given that both predictions presumably require "interpreting" .)

So, intuitively, it first builds an "inner representation" (embedding) of . Then it sequentially makes predictions based on that inner representation.

Now you train  and  to minimize the prediction loss on the  parts of the triplets. Simultaneously you train  to minimize prediction loss on the full  triplets. For example, you update  and  with the gradients

and you update  and  with the gradients

.
(The  here is the "true" , not one generated by the model itself.)

This training pressures  to be myopic in the second and third sense described in the post. In fact, even if we were to train  with the  predicted by  rather than the true  is pressured to be myopic.

  • Type 3 myopia: Training doesn't pressure  to output something that makes the  follow an easier-to-predict (computationally or information-theoretically) distribution. For example, imagine that on the training data  implies , while under  follows some distribution that depends in complicated ways on . Then  will not try to predict  more often.
  • Type 2 myopia:  won't try to provide useful information to  in its output, even if it could. For example, imagine that the s are strings representing real numbers. Imagine that  is always a natural number, that  is the -th Fibonacci number and  is the -th Fibonacci number. Imagine further that the model representing  is large enough to compute the -th Fibonacci number, while the model representing  is not. Then one way in which one might think one could achieve low predictive loss would be for  to output the -th Fibonacci number and then encode, for example, the -th Fibonacci number in the decimal digits. (E.g., .) And then  computes the -th Fibonacci number from the -th decimal. But the above training will not give rise to this strategy, because  gets the true  as input, not the one produced by . Further, even if we were to change this, there would still be pressure against this strategy because  () is not optimized to give useful information to . (The gradient used to update  doesn't consider the loss on predicting .) If it ever follows the policy of encoding information in the decimal digits, it will quickly learn to remove that information to get higher prediction accuracy on .

Of course,  still won't be pressured to be type-1-myopic. If predicting  requires predicting , then  will be trained to predict ("plan") .

(Obviously, $g_2$ is pressured to be myopic in this simple model.)

Now what about ? Well,  is optimized both to enable predicting  from  and predicting  from . Therefore, if resources are relevantly constrained in some way (e.g., the model computing  is small, or the output of  is forced to be small),  will sometimes sacrifice performance on one to improve performance on the other. So, adapting a paragraph from the post: The trained model for  (and thus in some sense the overall model) can and will sacrifice accuracy on  to achieve better accuracy on . In particular, we should expect trained models to find an efficient tradeoff between accuracy on  and accuracy on . When  is relatively easy to predict,  will spend most of its computation budget on predicting .

So,  is not "Type 2" myopic. Or perhaps put differently: The calculations going into predicting  aren't optimized purely for predicting .

However,  is still "Type 3" myopic. Because the prediction made by  isn't fed (in training) as an input to  or the loss, there's no pressure towards making  influence the output of  in a way that has anything to do with . (In contrast to the myopia of , this really does hinge on not using  in training. If  mattered in training, then there would be pressure for  to trick  into performing calculations that are useful for predicting . Unless you use stop-gradients...)

* This comes with all the usual caveats of course. In principle, the inductive bias may favor a situationally aware model that is extremely non-myopic in some sense.

Very interesting post! Unfortunately, I found this a bit hard to understand because the linked papers don’t talk about EDT versus CDT or scenarios where these two come apart and because both papers are (at least in part) about sequential decision problems, which complicates things. (CDT versus EDT can mostly be considered in the case of a single decision and there are various complications in multi-decision scenarios, like updatelessness.)

Here’s an attempt at trying to describe the relation of the two papers to CDT and EDT, including prior work on these topics. Please correct me if I’m misunderstanding anything! The writing is not very polished -- sorry!

Ignoring all the sequential stuff, my understanding is that the first paper basically does this: First, we train a model to predict utilities after observing actions, i.e., make predictions conditional on actions. So in particular, we get a function a ---> E[utility | a] that maps an observed action by the agent onto a prediction of future reward/utility. Then if we use some procedure to find the action a that maximizes E[utility | a], it seems that we have an EDT agent. I think this is essentially the case of an “EDT overseer” who rewards based on actions (rather than outcomes) in “Approval-directed agency and the decision theory of Newcomb-like problems”. Also see the discussion of Obstacle 1 in "Two Major Obstacles for Logical Inductor Decision Theory".

Now what could go wrong with this? I think in some sense the problem is generally that it's unclear how the predictive model works, or where it comes from. The second paper (the DeepMind one) basically points out one issue with this. Other issues are known to this community. I’ll start with an issue that has been known to this community: the 5 and 10 problem / the problem of counterfactuals. If the agent always (reliably) chooses the action a that maximizes E[utility | a], then the predictive model’s counterfactual predictions (i.e., predictions for all other actions) could be nonsensical without being strictly speaking wrong. So for example, in 5 and 10, you choose between a five dollar bill and a ten dollar bill. (There’s no catch and you should clearly just take the ten dollar bill.) The model predicts that if you take the five dollar bill, you will get five dollars, and (spuriously / intuitively falsely) that if you take the ten dollar bill, you get nothing. Because you are maximizing expected utility according to this particular predictive model, you take the five dollars. So the crazy prediction for what happens if you take the ten dollars is never falsified.

In non-Newcomb-like scenarios, a simple, extremely standard solution to this problem is to train the predictive model (the thing that gives a ---> E[utility | a]) while the agent follows some policy that randomizes over all actions (perhaps one that takes actions with probabilities in proportion to the model's predictions E[utility | a]). My understanding is that this is how the first paper avoids these issues and gives good results. Unfortunately, in Newcomb-like problems these approaches tend to lead to pretty CDT-ish behavior, as shown in "Reinforcement Learning in Newcomblike Environments".

Anyway, the second paper (the DeepMind one) points out another issue related to where the E[utility | action] model comes from. Roughly, the story — which I think is very well described in Section 2 — seems to be the following: the E[utility | action] model is trained on the actions of an expert who knows whether X=1,2 and acts on that fact by choosing A=X; then the E[utility | action] model won't work for a non-expert agent, i.e., one who doesn’t observe X. I view this as a distributional shift issue — you train a model (the a ---> E[utility | a] one) in a setting where A=X, and then you apply it in a setting where sometimes A and X are uncorrelated.

It’s also similar to the Smoking Lesion/medical Newcomb-like problems! Consider the following medical Newcomb-like problem: First we learn the fact that sick people go to the doctor and healthy people don’t go to the doctor. Then without looking at how healthy I am, I don’t go to the doctor so as to gain evidence that I am healthy. Arguably what goes wrong here is also that I’m using a rule for prediction out of distribution on someone who doesn’t look at whether they’re sick. I think it relates to one of the least challenging versions of medical Newcomb-like problems and it’s handled comfortably by the so-called tickle defense.

Interlude: The paper talks about how this relates to hallucination in LLMs. So what’s that about? IIUC, the idea is that when generating text, LLMs incorrectly update based on the text they generate themselves. For example, imagine that you want an LLM to generate ten tokens. Then after generating the first nine tokens, it will predict the tenth token from its learned distribution . But this distribution was trained on fully human- not LLM-written text. So (in my way of thinking),  might do poorly (i.e., not give a human-like continuation of ), because it was trained on seeing nine tokens created by a human and having to predict a continuation by a human rather than nine tokens by itself/an LLM and having to predict a continuation by a human. For example, we might imagine that if  are words that only a human expert confident in a particular claim C would say, then the LLM will predict continuations that confidently defend claim C, even if the LLM doesn’t know anything about C. I'm not sure I really buy this explanation of hallucination. I think the claim would need more evidence than the authors provide. But it's definitely a very interesting point.

Now, back to the original toy model. Again, I would view this as a distribution shift problem. If we make some assumptions, though, we can infer/guess a model (i.e. function a ---> E[utility | a]) that predicts the utility obtained by a non-expert, i.e., an agent who doesn't observe X. Specifically, let’s assume that we are told the conditional distributions P(utility | X=1, A=0) and P(utility | X=0, A=1) (which we never see in training if the agent in training always knows and acts on X). Let’s also assume that we know that the difference between the training distribution and the new setting is that in the new setting the agent chooses A independently of X. Then in the new model we just need to make X and A independent and change nothing else. Formally you use the new distribution P’(X,U|A) = P(X)P(U|A,X), where the Ps on the right-hand side are just the old distribution, instead of P(X,U|A) = P(X|A)P(U|A,X).

It turns out that if we put the original distribution into a causal graph with edges X->A and A->U and X->U and then make a do-intervention on A (a la Pearl), then we get this exact distribution, i.e., P(X,U|do(A)) = P’(X,U|A). (Intuitively, removing the inference from A to X is exactly what the do(A) does if A's parent is X.) So in particular maximizing E[U | do(A)] gives the same result as maximizing E’[U|A]. Anyway, the paper uses the do operator to construct the new predictor, rather than the above argument. They seem to claim that the causal structure (or reasoning about causality) is necessary to construct the new predictor, with which I disagree.

Is this really CDT? I’m not sure… In the above type of case, this doesn’t come apart from EDT. If we buy that their scenario is a bit like a Smoking Lesion, then one could argue that part of the point of CDT is to solve this type of scenario. (In some sense my response is as in most versions of the Smoking Lesion: Because of the tickle defense, EDT applied properly gets this right anyway, so there’s actually nothing to fix here.) In my view it’s basically just about using the do-calculus to concisely specify the scenario P’ (based on P plus a particular causal graph for P). It seems that one can do these things without being committed to using do(A) in a scenario where there’s some non-causal dependence between A and U (that doesn't disappear outside of training), perhaps via some common cause Y. In any case, the paper doesn’t tell us how to distinguish between U <- Y -> A and A -> Y -> U — all causal relationships are assumed. So while nominally they construct their predictor as E[U | do(A)], it’s a bit unclear how wedded they are to CDT.

Anyway, with a (maybe-causalist) E[U | do(A)] in hand, we can of course build a (maybe-)CDT agent by choosing a to maximize E[U | do(A)]. But I think the paper doesn’t say anything about where to get the causal model from that gives us E[U | do(A)]. They pretty much assume that the model is provided.

I think the “counterfactual teaching” stuff doesn’t really say anything about CDT versus EDT, either. IIUC the basic idea is this. Imagine you want to train an LLM and you want to prevent the issue above. Then intuitively — in my distribution shift view — what we need to do is just train the LLM to make a good prediction  upon observing  that were generated by itself (rather than humans). The simplest, most obvious way to do this is to let the LLM generate some tokens , then get a probabilistic prediction about the next token from the LLM and then ask a human to give a next token . The loss of the LLM is just the, e.g., log loss of its prediction against the  provided by the human. One slightly tricky point here is that we only train the LLM to make good predictions on . We don’t want to train it to output  that make  easier to predict. So we need to be careful to choose the right gradient. I think that’s basically all they’re doing, though. It doesn’t seem like there’s anything causalist here.

So, in conclusion: While very interesting, I don't think these papers tell us anything new about how to build an EDT or a CDT agent.

>I'm not sure I understand the variant you proposed. How is that different than the Othman and Sandholm MAX rule?

Sorry if I was cryptic! Yes, it's basically the same as using the MAX decision rule and (importantly) a quasi-strictly proper scoring rule (in their terminology, which is basically the same up to notation as a strictly proper decision scoring rule in the terminology of the decision scoring rules paper). (We changed the terminology for our paper because "quasi-strictly proper scoring rule w.r.t. the max decision rule" is a mouthful. :-P) Does that help?

>much safer than having it effectively chosen for them by their specification of a utility function

So, as I tried to explain before, one convenient thing about using proper decision scoring rules is that you do not need to specify your utility function. You just need to give rewards ex post. So one advantage of using proper decision scoring rules is that you need less of your utility function not more! But on to the main point...

>I think, from an alignment perspective, having a human choose their action while being aware of the distribution over outcomes it induces is much safer than having it effectively chosen for them by their specification of a utility function. This is especially true because probability distributions are large objects. A human choosing between them isn't pushing in any particular direction that can make it likely to overlook negative outcomes, while choosing based on the utility function they specify leads to exactly that. This is all modulo ELK, of course.

Let's grant for now that from an alignment perspective the property you describe is desirable. My counterargument is that proper decision scoring rules (or the max decision rule with a scoring rule that is quasi-strictly proper w.r.t. the max scoring rule) and zero-sum conditional prediction both have this property. Therefore, having the property cannot yield an argument to favor one over the other.

Maybe put differently: I still don't know what property it is that you think favors zero-sum conditional prediction over proper decision scoring rules. I don't think it can be not wanting to specify your utility function / not wanting the agent to pick agents based on their model of your utility function / wanting to instead choose yourself based on reported distributions, because both methods can be used in this way. Also, note that in both methods the predictors in practice have incentives that are determined by (their beliefs about) the human's values. For example, in zero-sum conditional prediction, each predictor is incentivized to run computations to evaluate actions that it thinks could potentially be optimal w.r.t. human values, and not incentivized to think about actions that it confidently thinks are suboptimal. So for example, if I have the choice between eating chocolate ice cream, eating strawberry ice cream and eating mud, then the predictor will reason that I won't choose to eat mud and that therefore its prediction about mud won't be evaluated. Therefore, it will probably not think much about how what it will be like if I eat mud (though it has to think about it a little to make sure that the other predictor can't gain by recommending mud eating).

On whether the property is desirable [ETA: I here mean the property: [human chooses based on reported distribution] but not compared to [explicitly specifying a utility function]]: Perhaps my objection is just what you mean by ELK. In any case, I think my views depend a bit on how we imagine lots of different aspect of the overall alignment scheme. One important question, I think, is how exactly we imagine the human to "look at" the distributions for example. But my worry is that (similar to RLHF) letting the human evaluate distributions rather than outcomes increases the predictors' incentives to deceive the human. The incentive is to find actions whose distribution looks good (in whatever format you represent the distribution) in relation to the other distributions, not which distributions are good. Given that the distributions are so large (and less importantly because humans have lots of systematic, exploitable irrationalities related to risk), I would think that human judgment of single outcomes/point distributions is much better than human judgment of full distributions.

>the biggest distinction is that this post's proposal does not require specifying the decision maker's utility function in order to reward one of the predictors and shape their behavior into maximizing it.

Hmm... Johannes made a similar argument in personal conversation yesterday. I'm not sure how convinced I am by this argument.

So first, here's one variant of the proper decision scoring rules setup where we also don't need to specify the decision maker's utility function: Ask the predictor for her full conditional probability distribution for each action. Then take the action that is best according to your utility function and the predictor's conditional probability distribution. Then score the predictor according to a strictly proper decision scoring rule. (If you think of strictly proper decision scoring rules as taking only a predicted expected utility as input, you have to first calculate the expected utility of the reported distribution, and then score that expected utility against the utility you actually obtained.) (Note that if the expert has no idea what your utility function is, they are now strictly incentivized to report fully honestly about all actions! The same is true in your setup as well, I think, but in what I describe here a single predictor suffices.) In this setup you also don't need to specify your utility function.

One important difference, I suppose, is that in all the existing methods (like proper decision scoring rules) the decision maker needs to at some point assess her utility in a single outcome -- the one obtained after choosing the recommended action -- and reward the expert in proportion to that. In your approach one never needs to do this. However, in your approach one instead needs to look at a bunch of probability distributions and assess which one of these is best. Isn't this much harder? (If you're doing expected utility maximization -- doesn't your approach entail assigning probabilities to all hypothetical outcomes?) In realistic settings, these outcome distributions are huge objects!

The following is based on an in-person discussion with Johannes Treutlein (the second author of the OP).

>But is there some concrete advantage of zero-sum conditional prediction over the above method?

So, here's a very concrete and clear (though perhaps not very important) advantage of the proposed method over the method I proposed. The method I proposed only works if you want to maximize expected utility relative to the predictor's beliefs. The zero-sum competition model enables optimal choice under a much broader set of possible preferences over outcome distributions.

Let's say that you have some arbitrary (potentially wacky discontinuous) function V that maps a distributions over outcomes onto a real value representing how much you like the distribution over outcomes. Then you can do zero-sum competition as normal and select the action for which V is highest (as usual with "optimism bias", i.e., if the two predictors make different predictions for an action a, then take the maximum of the Vs of the two actions). This should still be incentive compatible and result in taking the action that is best in terms of V applied to the predictors' belief.

(Of course, one could have even crazier preferences. For example, one's preferences could just be a function that takes as input a set of distributions and selects one distribution as its favorite. But I think if this preference function is intransitive, doesn't satisfy independence of irrelevant alternatives and the like, it's not so clear whether the proposed approach still works. For example, you might be able to slightly misreport some option that will not be taken anyway in such a way as to ensure that the decision maker ends up taking a different action. I don't think this is ever strictly incentivized. But it's not strictly disincentivized to do this.)

Interestingly, if V is a strictly convex function over outcome distributions (why would it be? I don't know!), then you can strictly incentivize a single predictor to report the best action and honestly report the full distribution over outcomes for that action! Simply use the scoring rule , where  is the reported distribution for the recommended action,  is the true distribution of the recommended action and  is a subderivative of . Because a proper scoring rule is used, the expert will be incentivized to report  and thus gets a score of , where  is the distribution of the recommended action. So it will recommend the action  whose associate distribution maximizes . It's easy to show that if  -- the function saying how much you like different distribution -- is not strictly convex, then you can't construct such a scoring rule. If I recall correctly, these facts are also pointed out in one of the papers by Chen et al. on this topic.

I don't find this very important, because I find expected utility maximization w.r.t. the predictors' beliefs much more plausible than anything else. But if nothing else, this difference further shows that the proposed method is fundamentally different and more capable in some ways than other methods (like the one I proposed in my comment).

Nice post!

Miscellaneous comments and questions, some of which I made on earlier versions of this post. Many of these are bibliographic, relating the post in more detail to prior work, or alternative approaches.

In my view, the proposal is basically to use a futarchy / conditional prediction market design like that the one proposed by Hanson, with I think two important details:
- The markets aren't subsidized. This ensures that the game is zero-sum for the predictors -- they don't prefer one action to be taken over another. In the scoring rules setting, subsidizing would mean scoring relative to some initial prediction $p_0$ provided by the market. Because the initial prediction might differ in how bad it is for different actions, the predictors might prefer a particular action to be taken. Conversely, the predictors might have no incentive to correct an overly optimistic prediction for one of the actions if doing so causes that action not to be taken. The examples in Section 3.2 of the Othman and Sandholm paper show these things.
- The second is "optimism bias" (a good thing in this context): "If the predictors disagree about the probabilities conditional on any action, the decision maker acts as though they believe the more optimistic one." (This is as opposed to taking the market average, which I assume is what Hanson had in mind with his futarchy proposal.) If you don't have optimism bias, then you get failure modes like the ones pointed out in Obstacle 1 of Scott Garrabrant's post "Two Major Obstacles for Logical Inductor Decision Theory": One predictor/trader could claim that the optimal action will lead to disaster and thus cause the optimal action to never be taken and her prediction to never be tested. This optimism bias is reminiscent of some other ideas. For example some ideas for solving the 5-and-10 problem are based on first searching for proofs of high utility. Decision auctions also work based on this optimism. (Decision auctions work like this: Auction off the right to make the decision on my behalf to the highest bidder. The highest bidder has to pay their bid (or maybe the second-highest bid) and gets paid in proportion to the utility I obtain.) Maybe getting too far afield here, but the UCB term in bandit algorithms also works this way in some sense: if you're still quite unsure how good an action is, pretend that it is very good (as good as some upper bound of some confidence interval).


My work on decision scoring rules describes the best you can get out of a single predictor. Basically you can incentivize a single predictor to tell you what the best action is and what the expected utility of that action is, but nothing more (aside from some degenerate cases).

Your result shows that if you have two predictors with the same information, then you can get slightly more: you can incentivize them to tell you what the best action is and what the full distribution over outcomes will be if you take the action.

You also get some other stuff (as you describe starting from the sentence, "Additionally, there is a bound on how inaccurate..."). But these other things seem much less important. (You also say: "while it does not guarantee that the predictions conditional on the actions not taken will be accurate, crucially there is no incentive to lie about them." But the same is true of decision scoring rules for example.)

Here's one thing that is a bit unclear to me, though.

If you have two predictors that have the same information, there's other, more obvious stuff you can do. For example, here's one:
- Ask Predictor 1 for a recommendation for what to do.
- Ask Predictor 2 for a prediction over outcomes conditional on Predictor 1's recommendation.
- Take the action recommended by Predictor 1.
- Observe an outcome o with a utility u(o).
- Pay Predictor 1 in proportion to u(o).
- Pay Predictor 2 according to a proper scoring rule.

In essence, this is just splitting the task into two: There's the issue of making the best possible choice and there's the issue of predicting what will happen. We assign Predictor 1 to the first and Predictor 2 to the second problem. For each of these problems separately, we know what to do (use proper (decision) scoring rules). So we can solve the overall problem.

So this mechanism also gets you an honest prediction and an honest recommendation for what to do. In fact, one advantage of this approach is that honesty is maintained even if the Predictors 1 and 2 have _different_ information/beliefs! (You don't get any information aggregation with this (though see below). But your approach doesn't have any information aggregation either.)

As per the decision scoring rules paper, you could additionally ask Predictor 1 for an estimate of the expected utility you will obtain. You can also let the Predictor 2 look at Predictor 1's prediction (or perhaps even score Predictor 2 relative to Predictor 1's prediction). (This way you'd get some information aggregation.) (You can also let Predictor 1 look at Predictor 2's predictions if Predictor 2 starts out by making conditional predictions before Predictor 1 gives a recommendation. This gets more tricky because now Predictor 2 will want to mislead Predictor 1.)

I think your proposal for what to do instead of the above is very interesting and I'm glad that we now know that this method exists that that it works. It seems fundamentally different and it seems plausible that this insight will be very useful. But is there some concrete advantage of zero-sum conditional prediction over the above method?

Minor bibliographical note: A related academic paper is Arif Ahmed's unpublished paper, "Sequential Choice and the Agent's Perspective". (This is from memory -- I read that paper a few years ago.)

Nice post!

What would happen in your GPT-N fusion reactor story if you ask it a broader question about whether it is a good idea to share the plans? 

Perhaps relatedly:

>Ok, but can’t we have an AI tell us what questions we need to ask? That’s trainable, right? And we can apply the iterative design loop to make AIs suggest better questions?

I don't get what your response to this is. Of course, there is the verifiability issue (which I buy). But it seems that the verifiability issue alone is sufficient for failure. If you ask, "Can this design be turned into a bomb?" and the AI says, "No, it's safe for such and such reasons", then if you can't evaluate these reasons, it doesn't help you that you have asked the right question.

Sounds interesting! Are you going to post the reading list somewhere once it is completed?

(Sorry for self-promotion in the below!)

I have a mechanism design paper that might be of interest: Caspar Oesterheld and Vincent Conitzer: Decision Scoring Rules. WINE 2020. Extended version. Talk at CMID.

Here's a pitch in the language of incentivizing AI systems -- the paper is written in CS-econ style. Imagine you have an AI system that does two things at the same time:
1) It makes predictions about the world.
2) It takes actions that influence the world. (In the paper, we specifically imagine that the agent makes recommendations to a principal who then takes the recommended action.) Note that if the predictions are seen by humanity, they themselves influence the world. So even a pure oracle AI might satisfy 2, as has been discussed before (see end of this comment).
We want to design a reward system for this agent such the agent maximizes its reward by making accurate predictions and taking actions that maximize our, the principals', utility.

The challenge is that if we reward the accuracy of the agent's predictions, we may set an incentive on the agent to make the world more predictable, which will generally not be aligned without mazimizing our utility.

So how can we properly incentivize the agent? The paper provides a full and very simple characterization of such incentive schemes, which we call proper decision scoring rules:

We show that proper decision scoring rules cannot give the [agent] strict incentives to report any properties of the outcome distribution [...] other than its expected utility. Intuitively, rewarding the [agent] for getting anything else about the distribution right will make him [take] actions whose outcome is easy to predict as opposed to actions with high expected utility [for the principal]. Hence, the [agent's] reward can depend only on the reported expected utility for the recommended action. [...] we then obtain four characterizations of proper decision scoring rules, two of which are analogous to existing results on proper affine scoring [...]. One of the [...] characterizations [...] has an especially intuitive interpretation in economic contexts: the principal offers shares in her project to the [agent] at some pricing schedule. The price schedule does not depend on the action chosen. Thus, given the chosen action, the [agent] is incentivized to buy shares up to the point where the price of a share exceeds the expected value of the share, thereby revealing the principal's expected utility. Moreover, once the [agent] has some positive share in the principal's utility, it will be (strictly) incentivized to [take] an optimal action.

Also see Johannes Treutlein's post on "Training goals for large language models", which also discusses some of the above results among other things that seem like they might be a good fit for the reading group, e.g., Armstrong and O'Rourke's work.

My motivation for working on this was to address issues of decision making under logical uncertainty. For this I drew inspiration from the fact that Garrabrant et al.'s work on logical induction is also inspired by market design ideas (specifically prediction markets).

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