Yeah, you’re correct—I shouldn’t have conflated “outcomes” (things utilities are non-derivatively assigned to) with “objects of preference.” Thanks for this.
fbacus
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As Richard Kennaway noted, it seems considerations about time are muddling things here. If we wanted to be super proper, then preferences should have as objects maximally specific ways the world could be, including the whole history and future of the universe, down to the last detail. Decision theory involving anything more coarse-grained than that is just a useful approximation—e.g. I might have a decision problem with only two outcomes being “You get $10” and “You lose $5,” but we would just be pretending these are the only two ways the world can end up for practical purposes, which is a permissible simplification since in my actual circumstances my desire to have more money over less is independent of other considerations. In particular the objects of preference are not states of the world at a time, since it is not a rational requirement that just because I prefer that one state instead of another obtain at I must also prefer likewise for . So, our agent who is perfectly content with their eternal A-B-C cycle is just one who likes this strange history of the world they can bring about, who nevertheless has potentially acyclic preferences. Indeed, this agent is still rational on orthodox decision theory even if they’re paying for every such cycle. Similarly, in your case of the agent who goes through to get to , we would want to model the objects of preference as things like “Stay in the state indefinitely” and “Be in a -state for a little while and then be in a -state,” and we would not want , , and themselves to be the objects of preference.
So what does this say about the money-pump argument for acyclicality, which seems to treat preferences as over states-at-a-time? I think the sorts of considerations you raise do show that such arguments don’t go through unless we make certain (perhaps realistic) assumptions about the structure of agents’ preferences, assuming we view the force of the money-pump arguments to come from the possibility of actual exploitation (some proponents don’t care so much about actual exploitability, but think the hypothetical exploitability is indicative of an underlying rational inconsistency). For example, if I have the preferences , the argument assumes that there is some slightly worse version of that I still prefer to , that my preferences are coarse-grained enough that we can speak of my “trading” one of these states of affairs for another, that they are coarse-grained enough that my past trades don’t influence whether I now prefer to trade for the slightly worse version of , and so on. If the relevant assumptions are met, then the money-pump argument shows that my cyclic preferences leave me with an outcome that I really will disprefer to having just stuck with , as opposed to ending up in a state I will later pay to transition out of (which is no fault of my rationality). You’re right to point out that these really are substantive assumptions.
So, in sum: I think (i) the formalism for decision theory is alright in not taking into account the possibility of transitions between outcomes, and only taking into account preferences between outcomes and which outcomes you can affect in a given decision problem; and (ii) formalism aside, your observations do show limitations to/assumptions behind the money-pump arguments insofar as those arguments are supposed to be merely pragmatic. I do not know the extent to which I disagree with you, but I hope this clarifies things.
Could that domain not just be really small, such that the ratio of outcomes you’d accept the bet at get closer and closer to 1? It seems like the premise that the discounting rate stays constant over a large interval (so we get the extreme effects from exponential discounting) is doing the work in your argument, but I don’t see how it’s substantiated.
Thanks for the reply!
So, in my experience it’s common for decision theorists in philosophy to take preferences to be over possible worlds and probability distributions over such (the specification of which includes the past and future), and when coarse-graining they take outcomes to be sets of possible worlds. (What most philosophers do is, of course, irrelevant to the matter of how it’s best to do things, but I just want to separate “my proposal” from what I (perhaps mistakenly) take to be common.) As you say, no agent remotely close to actual agents will have preferences where details down to the location of every particle in 10,000 BC make a difference, which is why we coarse-grain. But this fact doesn’t mean that maximally-specific worlds or sets of such shouldn’t be what are modeled as outcomes, as opposed to outcomes being ways the world can be at specific times (which I take to be your understanding of things; correct me if I’m wrong). Rather, it just means a lot of these possible worlds will end up getting the same utility because the agent is indifferent between them, and so we’ll be able to treat them as the same outcome.
I worry I might be misunderstanding your point in the last few paragraphs, as I don’t understand how having maximally-specific worlds as outcomes is counter to our commonsense thinking of decision-making. It is true I don’t have a felt preference between “I get $100 and the universe was in state S five billion years ago” and “I get $100 and the universe was in state T five billion years ago,” let alone anything more specific. But that is just to say I am indifferent—I weakly prefer either outcome to the other, and this fact is as intuitively accessible to me as the fact that I prefer getting $100 to getting $10. As far as I can tell, the only consequence of making things too fine-grained is that you include irrelevant details to which the agent is indifferent and you make the model needlessly unwieldy—not that it invokes a concept of preference divorced from actual decision-making.
Every decision problem can be modeled as being between probability distributions over such histories, or over sets of such histories if we want to coarse-grain and ignore minute details to which the agent is indifferent. Now, of course, a real human never has precise probabilities associated with the most minute detail—but the same is true even if the outcomes are states-at-times, and even non-maximally-specific states-at-times. Idealizing that matter away seems to be something any model has to do, and I’m not seeing what problems are introduced specifically by taking outcomes to be histories instead of states-at-times.
But having outcomes be sets of maximally-specific world-histories doesn’t prevent us from being able to model sequential decisions. All we need to do so is to assume that the agent can make choices at different times which make a difference as to whether a more-or-less preferred world-history will obtain. For example, say I have a time-dependence preference of preferring to wear a green shirt on weekends and a red shirt on weekdays, so I’m willing to pay $1 to trade shirts every Saturday and Monday. This would not make me a money-pump, rather I’m just someone who’s willing to pay to keep their favorite shirt at the moment on. But allowing time-dependent preferences like this doesn’t prevent money-pump and diachronic Dutch book arguments from ever going through, for I still prefer wearing a red shirt next Monday to wearing a green shirt then, if everything else about the world-history is kept the same. If I in addition prefer wearing a yellow shirt on a weekday to wearing a red one, but then also prefer a green shirt on a weekday to a yellow one, then my preferences are properly cyclic and I will be a money-pump. For let’s say I have a ticket that entitles me to a red shirt next Monday; then suppose I trade that ticket for a yellow shirt ticket, then that one for a green shirt ticket, then I pay $1 to trade that for my original red shirt ticket. Assuming my preferences merely concern (i) what color shirt I wear on what day and (ii) my having more money than less, I will end up in a state that I think is worse than just having made no trade at all. This is so even though my preferences are not merely over shirt-colors and money, but rather coarse-grained sets of world-histories.
All that is needed for the pragmatic money-pump argument to go through is that my preferences are coarse-grained enough that we can speak of my making sequential choices as to which world-histories will obtain, such that facts about what choices I have made in these very cases do not influence the world-history in a way that makes a difference to my preferences. This seems pretty safe in the case of humans, and a result showing transitivity to be rationally required under such assumptions still seems important. Even in the case of a realistic agent who really likes paying to go through the A-B-C-A cycle, we could imagine alternate decision problems where they get to choose whether they get to go through that cycle during a future time-interval, and genuinely exploit them if those preferences are cyclic.
I worry that what I described above in the shirt-color example falls under what you mean by “or which only vary with time in some specified manner, rather than being totally unconstrained as your proposal would have it.” A world-history is a specification of the world-state at each time, similar in kind to “I wear a red shirt next Monday.” As I said before, the model would allow for weird fine-grained preferences over world-histories, but realistic uses of it will have as outcomes larger sets of world-histories like the set of histories where I wear a red shirt next Monday and have $1000, and so on. This is similar to how standard probability theory in principle allows one to talk about probabilities of propositions so specific that no one could ever think or care about them, but for practical purposes we just don’t include those details when we actually apply it.
I’m concerned that I may be losing track of where our disagreement is. So, just to recap: I took your arguments in the OP to be addressed if we understand outcomes (the things to which utilities are assigned) to be world-histories or sets of world-histories, so we can understand the difference between an agent who’s content with paying a dollar for every A-B-C-A cycle and an agent for whom that would be indicative of irrationality. You alleged that if we understand outcomes in this way, and not merely as possible states of the world with no reference to time, then (i) we cannot model sequential decisions, (ii) such a model would have an unrealistic portrayal of decision-making, and (iii) there will be too few constraints on preferences to get anything interesting. I reply: (i) sequential decisions can still be modeled as choices between world-histories, and indeed in an intuitive way if we make the realistic assumption that the agent’s choices in this very sequence by themselves do not make a difference to whether a world-history is preferred; (ii) the decisions do seem to be intuitively modeled if we coarse-grain the sets of world-histories appropriately; and (iii) even if money-pump arguments only go through in the case of agents who have a certain additional structure to their preferences, this additional structure is satisfied by realistic agents and so the restricted results will be important. If I am wrong about what is at issue, feel free to just correct me and ignore anything I said under false assumptions.