I had encountered this idea some years ago in the rationalist canon, but I had never been able to think of examples where it really happened. Imagine my joy when I thought I had finally run across it when, after dinner one night, my (irrational) toddler demonstrated intransitive preferences when eating M&M's and trading with me. She prefers green over orange, orange over brown, and brown over green. Here it is!, I thought. A money pump! But exactly how could I benefit from this?
As it turned out, there was no way to money-pump her, and might not be a way to ever meaningfully money-pump anybody. But to illustrate the point I'll give you a hypothetical example of how it could work. This hypothetical example alters her real behavior considerably - to see what I changed and why it could not work in the real world, skip ahead to "Where Are the Real Money Pumps?" in bold below.
Say we both start out with 12 green, 12 orange, and 12 brown, and assume the following preference rules (individual exchange rates):
I have transitive preferences: I like green twice as much as orange, which I like twice as much as brown. (1 green = 2 orange = 4 brown)
She has intransitive preferences: she likes green twice as much as orange, which she likes twice as much as brown, which she likes twice as much as green, etc.) In fact, I would argue that she has a subtype of intransitive preferences, cyclic preferences: with intransitive preferences you can also just say that there is no value on something and it can't be used in trade. (Though irrational by Von Neumann-Morgenstern and other standards, this is in fact how normal human beings behave; when someone will put a price on anything, that person is called a psychopath.)
It helps to see the specifics of how you can take advantage of someone with cyclic preferences, but it's dry and boring, so I'll include it in a "supplement" at the end if you're interested and don't just want to take my word for it. Suffice it to say after at most 7 trades, I would have all the M&M's except for two brown ones.
THEN WHERE ARE THESE MONEY PUMPS? I.E., WHERE ARE THE SYSTEMS OF CYCLIC PREFERENCES THAT ARE ECONOMICALLY TAKEN ADVANTAGE OF?
I went looking for cases in the real world where people get money-pumped, and found:
a) none, and
b) I'm not the only one who has noticed this gap.
In fact money-pumping seems to be an entirely theoretical risk, predicted deductively from rationality models. So what's going on? Likely some combination of:
- People are irrational and their preferences are a mess, but they aren't neatly cyclical like this. In fact we should expect that most intransitive preferences are just that - they have no relation to any other preferences - therefore precluding a cyclic system of intransitive preferences. That is, sets of cyclic preferences are a subset of intransitive preferences, but because of the nature of intransitive preferences (basically they're a mess with little relation to each other or even consistency on short time scales), cyclic preferences are very rare or nonexistent in the real world.
- Humans have many heuristics that can cause reasoning errors, but that also must have been on net beneficial to our ancestors, and (sometimes) they only make sense in the context of their environment. For example, we like to punish wrongdoers, to the point where we will spend more resources to do it, even when the damage has already been done. This seems irrational, unless you realize that there are not many limited-round-games in life, and if there was someone doing bad things in our tribe fifty thousand years ago, it made sense to invest those punishment-resources now to deter further wrongdoing later. A more germane example is the endowment effect, which is a heuristic that clearly has the effect of keeping people from getting taken advantage of in markets with asymmetric information. I expect that with cyclic preferences, there is a meta-preference that usually acts like a circuit-breaker for this sort of thing, for example just thinking about the goods in dollar figures. Of course, you could rightly object, if someone can do that, they don't really have cyclic preferences, since the dollar is like a color of M&M that doesn't admit of intransitive valuation - and you're right! Which is why people don't actually get money-pumped! In this same way, I expect that even my toddler would notice her overall number of M&M's is shrinking and mine is growing and at some point say she's not playing this stupid game anymore.
- There actually are some cyclic preference-sets revealed by people engaging in repetitive behavior that makes their life consistently worse. This includes compulsive gambling, junk food, substance abuse, and staying in abusive relationships (or seeking out new ones.) It's interesting that these aren't about trading with currency, or at least don't centrally involve explicit trade or currency exchange, which are relatively new things in terms of evolutionary psychology, and something where along with learning them explicitly, we have developed learned defenses against being taken advantage of (as noted above.) Even in those cases where there appear to be cyclic preferences, these are better understood as predictably shifting preferences (due to things like future discounting), but this is a semantic distinction since they have the same outcome.
SO WHERE DO WE SEE IRRATIONAL BEHAVIOR DUE TO INTRANSITIVE PREFERENCES?
Clashes between the system of transitive preference systems - speaking broadly, the economy - and intransitive preferences are somewhat rare, but they occur, even when they are not cyclic. You don't think of people repeating abusive relationships as part of the economy, but that's a great example of intransitive preferences. Gambling and substance use part of the economy, but at the fringes (it's interesting that societies usually have prohibitions or regulations about trading the same sorts of things, things which involve strongly affective parts of our cognition and behavior like gambling, sex, drugs, and firearms.)
Some psychopaths recognize the intransitive nature of most humans' valuation of other human life (it's "priceless"), and take hostages who they will kill unless they are given money or some other objective. In those cases, many humans magically overcome our intransitiveness and kill the hostage-takers, or allow the hostage-takers to kill their victims in order to avoid negotiating in the future.
A fortunately more common but unfortunately probably more intractable problem is healthcare: about 70% of healthcare dollars in the U.S. spent by someone are spent in the last 6 months of that person's life. We could spend more and more getting additional minutes at the end of life. Unless we're going to ruin ourselves in this way, there has to be a rule, regardless of whether we're in a public or private healthcare system. This is something we don't like thinking about.
Finally, whenever people try to create a system of transitive preferences outside of the mother-system (the economy), the gravity of the currency economy inevitably connects to it and sucks it in, whether we're talking about Ithaca-dollars, or the charade of "no currency" at Burning Man.
SUPPLEMENT - AN EXAMPLE OF HOW CYCLICAL PREFERENCE MONEY-PUMP WOULD WORK, IF IT EVER ACTUALLY HAPPENED
Starting out we have:
Me: 12 green, 12 orange, 12 brown
Her: 12 green, 12 orange, 12 brown
Round 1. I offer her 6 of my orange for all 12 of her brown. Now we have:
Me: 12 green, 6 orange, 24 brown
Her: 12 green, 18 orange, 0 brown
Round 2. I offer her 6 of my brown for all 12 of her green. Now we have:
Me: 24 green, 6 orange, 18 brown
Her: 0 green, 18 orange, 6 brown
Round 3. I offer her 3 of my orange for all of her brown. Now we have:
Me: 24 green, 3 orange, 24 brown
Her: 0 green, 21 orange, 0 brown
Round 4. I offer her 11 of my green for all 21 of her orange. (Give her a good exchange rate and round up. She's irrational, I'll get it back!) Now we have:
Me: 13 green, 24 orange, 24 brown
Her: 11 green, 0 orange, 0 brown
Round 5. I offer her 6 of my brown for all 11 of her green (rounding up again.) Now we have:
Me: 24 green, 24 orange, 18 brown
Her: 0 green, 0 orange, 6 brown
Round 6. I offer her 3 of my green for all 6 of her brown. Now we have:
Me: 21 green, 24 orange, 24 brown
Her: 3 green, 0 orange, 0 brown
Round 7. I offer her 2 of my brown for all 3 of her green (rounding up.) Now we have:
Me: 24 green, 24 orange, 22 brown
Her: 0 green, 0 orange, 2 brown
...you get the picture.
As I was calculating this out I actually found it quite hard to think about the irrational player's decisions. There is value exchange symmetry in rational trading, which is to say, it doesn't matter if I am getting higher-value units or lower-value units. Whereas I would be tempted to say to the irrational player in round 7 above, "Look, why are we going through all this? Why don't you just give me those last 2 brown because they're worth less than themselves!" (Also more than themselves. But I want them, so I wouldn't say that.)
Originally I couldn't see how to benefit from this, even hypothetically - I thought "I could keep the trade going indefinitely but not accumulate anything". The errors were a) I actually had no preference for one color over another, and b) her preference for the "better" one in any pair was arbitrarily small (e.g., she just barely liked orange better than brown), and you can't subdivide M&M's (so with rounding either you could never arrive at having almost entirely fleeced the other party, or it would take too long. On the other hand, if you benefit from the trade itself and have no preferences, and you CAN usefully subdivide, you could still benefit. But I wasn't charging M&M commissions.
There is a total wealth (by my measurement, in units of "browns") of 168 in the game, with each side (by my measurement) starting with 84 brown-units. At the end of each round, with my trades, the value I hold is 84, 126, 126, 124, 162, 156, 166. You actually can't even talk about the other player's total value because what unit do you use to measure it? If we held differing, but rational, valuations - as people do in the real world - then say my daughter values browns twice as much as orange, orange twice as much as green - we'd quickly both wind up with her holding all the brown and me holding all the green. And that would be fine. In fact there have been cases in history where people became worried about the problems that could arise when preferences were different - Isaac Newton noted that because the English and Chinese relative valuations for gold and silver were different, that in a simple system eventually one would end up with all the silver and the other with all the gold, and trade would grind to a halt. But of course the system isn't that simple, and in any event as long as preferences are rational - not circular - it wouldn't matter.