Among my general family of weird niche opinions is that I don’t believe that the common assumption made in decision theory that preferences should be transitive is a valid one. This isn’t a post arguing in favour of my opinion. This is instead a post arguing against one of the most common arguments that preferences should be transitive: The dutch book/money pump argument. As such I think this post should be valid regardless of whether transitive preferences are good or bad.
Transitivity of preference is this: Suppose you prefer A over B and B over C, then you should also prefer A over C.
What “Prefer” mostly means in this context is “If I were to give you a choice between two options, which would you pick?”
(if you disagree with my above definitions, do say. In general please nitpick this post as much as you like. I really do want to understand people’s beliefs on this subject)
The argument goes like this:
Suppose you prefer A to B, B to C and C to A.
Given that these preferences are strict, there must be some (possibly very small) amount of money \(\epsilon\) such that you prefer A to B + \(\epsilon\), B to C + \(\epsilon\) and C to A + \(\epsilon\) (aside: I don’t entirely buy this but don’t have any strong objections to it and will leave it as mostly unquestioned).
Now suppose you have C. I offer you B in exchange for you giving me C + \(\epsilon\). Now you have B. I offer you A in exchange for B + \(\epsilon\). Now you have A. I offer you C in exchange for A + \(\epsilon\). Now you’re back where you started with C but you’ve paid me \(3 \epsilon\) for the privilege.
(Again, if you feel like I have oversimplified this argument or expressed it badly, please say).
I don’t think this argument holds water, for a number of reasons.
The first is that it does not actually model the sort of preference we started with: The question “Given A or B, neither of which you currently have, which would you prefer to acquire?” is fundamentally a different question to “If you already have one of A or B, would you swap it for the other?”.
In particular, you should expect the latter to exhibit a status quo bias: A greater reluctance to switch from B to A than to pick A over B when presented with both options.
And a sufficiently large status quo bias could potentially wipe out intransitivities found in the original preference relation, so it may be that the original preference relation is intransitive while this new sort of “Would you switch?” relation is not.
You can argue that these two relations should be the same, and that a status quo bias is intrinsically irrational, but I’m not sure that’s valid. A status quo has a lot going for it: You know how things work, you’re experienced with working within it, etc. Things can and should be improved, but in general the burden of proof is higher for comparing something with the status quo than when comparing two new things, and I think that’s quite reasonable.
Lets assume for the sake of the argument that we replace our notion of preference with this new one: I prefer A to B if when I have B and someone offers me A for it I will choose to switch.
The next question is then: How is this preference actually being elicited? I think when people talk about this sort of preference process they imagine some sort of perfect copy of you is asked for its preferences and then wiped out of existence, so you are left with no memory of the question. This might be thought of as a “static preference”: It is your preference as captured at some snapshot point in time.
But that’s not what’s happening in the dutch book scenario: In this scenario you do have memory. This is a set of dynamic preferences being elicited by a series of questions where you know what has happened before. So it could be that your static preferences are intransitive but you will never exhibit intransitive dynamic preferences because you have a rule that if you notice you’re being dutch booked you stop trading in the resource.
This can be quite rational. I’ve previously argued that any agent for whom solving NP hard problems isn’t free will engage in this sort of path dependence where the exhibited set of preferences depends on the order in which you ask the questions (not even which questions you ask! Just the order).
Finally, supposing that for whatever reason we’ve decided to actually exhibit intransitivity even in this dynamic preferences case. Then, yes, we do get dutch booked and can be used as a money pump.
This is an argument against intransitive preferences in the same way that the possibility of being attacked is an argument for never going outside.
When you argue for something, you have to argue not only that it has benefits but also that the costs outweigh the benefits, and it’s not clear to me that they do.
This is both because insisting on transitive preferences in the first place is potentially very high cost (it requires essentially global knowledge and solving NP hard problems), and also because it’s unclear what the real cost of money pumps is to an agent: Money pumps are a problem if the agent eliciting your opinions is intrinsically adversarial, but most cases for preference elicitation are going to be neutral or cooperative. The universe doesn’t hate you, and your fellow agents are probably more or less out to cooperate (life is an iterated prisoners dilemma, not a one-shot one), so the actual instances of deliberate money pumping in the wild are relatively low. If you’re playing the investment markets you should probably remove your intransitive dynamic preferences as you notice them, but it’s not really clear that they cause much harm in broader contexts and I don’t think the argument that they do has been convincingly made (I’m not sure I’ve even seen it made unconvincingly).
In conclusion, I think dutch book arguments prove much less than they are usually claimed to, and I don’t think they should be taken as having much significance for how actual agents should reason.