An aside on betting in anthropics
(For a revised version of this sequence, in a more academic style, see here. The revised version was a chapter of my dissertation.)
Previously in sequence: Part 1: Learning from the fact that you exist; Part 2: Telekinesis, reference classes, and other scandals. PDF of the entire sequence here.
This post is the third in a four-part sequence, explaining why I think that one prominent approach to anthropic reasoning (the “Self-Indication Assumption” or “SIA”) is better than another (the “Self-Sampling Assumption” or “SSA”). This part briefly discusses betting in anthropics. In particular: why it’s so gnarly, why I’m not focusing on it, and why I don’t think it’s the only desiderata. If you’re not interested in betting-type arguments, feel free to skip to part 4.
XII. An aside on betting in anthropics
I’ve now covered my main objections to SSA. In part 4, I say more in defense of SIA in particular. Before doing so, though, I want to mention a whole category of argument that I’ve generally avoided in this post: that is, arguments about what sorts of anthropic theories will lead to the right patterns of betting behavior.
I expect to some readers, this will seem a glaring omission. What’s the use of talking about credences, if we’re not talking about betting? What are credences, even, if not “the things you bet with”? Indeed, for some people, the question of “which anthropic theory will get me the most utility when applied” will seem the only question worth asking in this context, and they will have developed their own views about anthropics centrally with this consideration in mind. Why, then, aren’t I putting it front and center?
Basically, because questions about betting in anthropics get gnarly really fast. I’m hoping to write about them at some point, but this series of posts in particular is already super long. That said, I also don’t think that questions about betting are the only desiderata. Let me explain.
Why is betting in anthropics gnarly? At a high-level, it’s because how you should bet, in a given case, isn’t just a function of your credences. It’s also a function of things like whether you’re EDT-ish or CDT-ish, your level of altruism towards copies of yourself/other people in your epistemic position, how that altruism expresses itself (average vs. total, bounded vs. unbounded), and the degree to which you go in for various “act as you would’ve pre-committed to acting from some prior epistemic position” type moves (e.g. “updatelessness”) — either at the level of choices (whether your own choices, or those of some group), or at the level of epistemology itself. Anthropics-ish cases tend to implicate these issues to an unusual degree, and in combination, they end up as a lot of variables to hold in your head at once. Indeed, there is some temptation to moosh them together, egg-ed on by their intertwined implications. But they are, I think, importantly separable.
I’ll give one example to illustrate a bit of the complexity here. You might be initially tempted by the following argument for thirding, rather than halfing, in Sleeping Beauty. “Suppose you’re a halfer. That means that when you wake up, you’ll take (or more specifically, be indifferent to) a bet like: ‘I win $10 if heads, I lose $10 if tails.’ After all, it’s neutral in expectation. But if you take that sort of bet on every waking, then half the time, you’ll end up losing $10 twice: once on Monday, and once on Tuesday. Thus, the EV of a ‘halfer’ policy is negative. But if you’re a thirder, you’ll demand to win $20 if heads, in order to accept a $10 loss on tails. And the EV of this policy is indeed neutral. So, you should be a thirder.”
But this argument doesn’t work if Beauty’s person-moments are EDT-ish (and altruistic towards each other). Suppose you’re a halfer person-moment offered the even-odds bet above on each waking. You reason: “It’s 50% I’m in a heads world, and 50% I’m in a tails. But if I’m a tails-world, there’s also another version of me, who will be making this same choice, and whose decision is extremely correlated with mine. Thus, if I accept, that other version will accept too, and we’ll end up losing twice. Thus, I reject.” That is, in this case, your betting behavior doesn’t align with your credences. Is that surprising? Sort of. But in general, if you’re going to take a bet different numbers of times conditional on outcome vs. another, the relationship between the odds you’ll accept and your true credences gets much more complicated than usual. This is similar to the sense in which, even if I am 50-50 on heads vs. tails, I am not indifferent between a 50% chance of taking the bet “win $10 on heads, lose $10 on tails” conditional on heads vs. a 50% chance of “win $20 on heads, lose $20 on tails” conditionals on tails. Even though both of the bets are at 1:1 odds (and hence both are neutral in expectation pre-coin-flip), I’d be taking the bigger-stakes bet on the condition that I lose. (See Arntzenius (2002) for more.)
Indeed, the EDT-ish thirder, here, actually ends up betting like a fifth-er. That is, if offered a “win twenty if heads, lose ten if tails” bet upon each waking, she reasons: “1/3rd I’m in a heads world and will win $20. But 2/3rds I’m in a tails world, and am about to take or reject this bet twice, thereby losing $20. Thus, I should reject. To accept, the heads payout would need to be $40 instead.” And note that this argument applies both to SIA, and to SSA in the Dorr/Arntzenius “Beauty also wakes up on Heads Tuesday, but hears a bell in that case” version (thanks to Paul Christiano and Katja Grace for discussion). That is, every (non-updateless) altruistic EDT-er is a fifth-er sometimes.
(Or at least, this holds if you use the version of EDT I am most naively attempted by. Paul Christiano has recently argued to me that you should instead use a version of EDT where, instead of updating on your observations in the way I would’ve thought normal, then picking the action with the highest EV, you instead just pick the action that has the highest EV-conditional-on-being-performed-in-response-to-your-observations, and leave the traditional notion of “I have a probability distribution that I update as I move through the world” to the side. I haven’t dug in on this, though: my sense is that the next steps in the dialectic involve asking questions like “why be a Bayesian at all.”)
Note that in these cases, I’ve been assuming that Beauty’s person-moments are altruistic towards each other. But we need not assume this. We could imagine, instead, versions where the person moments will get to spend whatever money they win on themselves, before the next waking (if there is one), with no regard for the future of Beauty-as-a-whole. Indeed, in analogous cases with different people rather than different person-moments (e.g., God’s coin toss), altruism towards the relevant people-in-your-epistemic-position is a lot less of a default. And we’ll also need to start asking questions about what sort of pre-commitments it would’ve make sense to have made, from what sorts of epistemic/cooperative positions, and what the implications of that are or should be. I think questions like this are well worth asking. But I don’t really want to get into them here.
What’s more, I don’t think that they are the only questions. In particular, to me it seems pretty possible to separate the question of how to bet from the question of what to believe. Thus, for example, in the EDT-ish halfer case above, it seems reasonable to me to imagine thinking: “I’m 50% on heads, here, but if it’s tails, then it’s not just me taking this ‘win $10 if heads, lose $10 if tails’ bet; it’s also another copy of me, whose interests I care about. Thus, I will demand $20 if heads instead.” You can reason like that, and then step out of your room and continue to expect to see a heads-up coin with the same confidence you normally do after you flip. Maybe this is in some sense the wrong sort of expectation, but I don’t think your betting behavior, on its own, establishes this.
(One response here is: “you’re mistakenly thinking that you can bet for two, but expect for one. But actually, you’re expecting for both, too. And don’t you care about the accuracy of your copy’s beliefs too? And what is expecting if not a bet? What about the tiny bit of pleasure or pain you’ll experience upon calling it for tails as you step out of the room? Don’t you want that for both copies?”. Maybe, maybe.)
More generally, it doesn’t feel to me like the type of questions I end up asking, when I think about anthropics, are centrally about betting. Suppose I am wondering “is there an X-type multiverse?” or “are there a zillion zillion copies of me somewhere in the universe?”. I feel like I’m just asking a question about what’s true, about what kind of world I’m living in — and I’m trying to use anthropics as a guide in figuring it out. I don’t feel like I’m asking, centrally, “what kinds of scenarios would make my choices now have the highest stakes?”, or “what would a version of myself behind some veil of ignorance have pre-committed to believing/acting-like-I-believe?”, or something like that. Those are (or, might be) important questions too. But sometimes you’re just, as it were, curious about the truth. And more generally, in many cases, you can’t actually decide how to bet until you have some picture of the truth. That is: anthropics, naively construed, purports to offer you some sort of evidence about the actual world (that’s what makes it so presumptuous). Does our place in history suggest that we’ll never make it to the stars? Does the fact that we exist mean that there are probably lots of simulations of us? Can we use earth’s evolutionary history as evidence for the frequency of intelligent life? Naively, one answers such questions first, then decides what to do about it. And I’m inclined to take the naive project on its face.
Indeed, I’ve been a bit surprised by the extent to which some people writing about anthropics seem interested in adjusting (contorting?) their epistemology per se in order to bet a particular way — instead of just, you know, betting that way. This seems especially salient to me in the context of discussions about dynamical inconsistencies between the policy you’d want to adopt ex ante, and your behavior ex post (see, e.g., here). As I discussed in my last post, these cases are common outside of anthropics, too, and “believe whatever you have to in order to do the right thing” doesn’t seem the most immediately attractive solution. Thus, for example, if you arrive in a Newcomb’s case with transparent boxes, and you want to one-box anyway, the thing to do, I suspect, is not to adopt whatever epistemic principles will get you to believe that the one box is opaque. The thing to do is to one-box. (Indeed, part of the attraction of updateless-ish decision theories is that they eliminate the need for epistemic distortion of this kind.) I expect that the thing to say about various “but that anthropic principle results in actions you’d want to pre-commit to not taking” (for example, in cases of “fifth-ing” above) is similar.
Still, I suspect that some will take issue with the idea that we can draw any kind of meaningful line between credences and bets. And some will think it doesn’t matter: you can describe the same behavior in multiple ways. Indeed, one possible response to cases like God’s coin toss is to kind of abandon the notion of “credences” in the context of anthropics (and maybe in general?), and to just act as you would’ve pre-committed to acting from the perspective of some pre-anthropic-update prior, where the right commitment to make will depend on your values (I associate this broad approach with Armstrong’s “Anthropic Decision Theory,” though his particular set-up involves more structure, and I haven’t dug into it in detail). This is pretty similar in spirit to just saying “use some updateless-type decision theory,” but it involves a more explicit punting on/denial of the notion of “probabilities” as even-a-thing-at-all — at least in the context of questions like which person-in-your-epistemic-situation you are. That is, as I understand it, you’re not saying the equivalent of “I’m ~100% that both boxes are full, but I’m one-boxing anyway.” Rather, you’re saying the equivalent of: “all this talk about ‘what do you think is in the boxes’ is really a way of re-describing whether or not you one-box. Or at least, it’s not important/interesting. What’s key is what you do: and I, for one, one-box, because from some epistemic perspective, I would’ve committed to doing so.”
This approach has merits. In particular, it puts the focus directly on action, and it allows you to reason centrally from the perspective of the pre-anthropic-update prior (even if you eventually end up acting like an SIA-er, a Presumptuous Philosopher, and so on), which can feel like a relief. Personally, though, I currently prefer to keep the distinctions between e.g. credences, values, and decision-theories alive and available — partly to stay alert to implications and subtleties one would miss if you mooshed them together and just talked about e.g. “policies,” and partly because, as just discussed, they just seem like different things to me (e.g., “do I believe this dog exists” is not the same as “do I love this dog”). And I also have some worry that the “pre-anthropic update prior” invoked by this approach is going to end up problematic in the same way that the prior becomes problematic for updateless decision theories in general. (E.g., how do you know what pre-commitments to make, if you don’t have credences? Which credences should we use? What if the epistemic perspective from which you’re making/evaluating your pre-commitment implicates anthropic questions, too?)
At some point, I do want to get more clarity about the betting stuff, here — obviously, it’s where rubber ultimately meets road. For now, though, let’s move on.
(Next and last post in sequence: SIA > SSA, part 4: In defense of the presumptuous philosopher.)
Final essay in a four-part series, explaining why I think that one prominent approach to anthropic reasoning (the “Self-Indication Assumption” or “SIA”) is better than another (the “Self-Sampling Assumption” or “SSA”). This part discusses some prominent objections to SIA.
Final essay in a four-part series on expected utility maximization (EUM). Examination of some theorems that aim to justify the subjective probability aspect of expected utility maximization (EUM), namely: Dutch Book theorems; Cox’s Theorem; and the Complete Class Theorem.
Third essay in a four-part series on expected utility maximization (EUM). Examines three axiomatic arguments for acting like an EUM-er: the von Neumann-Morgenstern theorem; an argument based on a very general connection between “separability” and “additivity”; and a related “direct” axiomatization of EUM in Peterson (2017).
This is the second essay in a four-part series on expected utility maximization (EUM). This part focuses on why it can make sense, in cases like “save one life for certain, or 1000 with 1% chance,” to choose the risky option, and hence to “predictably lose.” The answer is unsurprising: sometimes the upside is worth it. I offer three arguments to clarify this with respect to life-saving in particular. Ultimately, though, while EUM imposes consistency constraints on our choices, it does not provide guidance about what’s “worth it” or not — and in some cases, the world doesn’t either. Ultimately, you have to decide.