Money pumping: the axiomatic approach

by Stuart_Armstrong5 min read5th Nov 200993 comments


VNM Theorem
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This post gets somewhat technical and mathematical, but the point can be summarised as:

  • You are vulnerable to money pumps only to the extent to which you deviate from the von Neumann-Morgenstern axioms of expected utility.

In other words, using alternate decision theories is bad for your wealth.

But what is a money pump? Intuitively it is a series of trades that I propose to you, that end up bringing you back to where you started. All the trades must be indifferent or advantageous to you, so that you will accept them. And if even one of those trades is advantageous, then this is a money pump: I can charge you a tiny amount for that trade, making free money out of you. You are now strictly poorer than if you had not accepted the tradesat all.

A strict money pump happens when every deal is advantageous to you, not simply indifferent. In most situations, there is no difference between a money pump and a strict money pump: I can offer you a tiny trinket at each indifferent deal to make it advantageous, and get these back later. There are odd preference systems out there, though, so the distinction is needed.

The condition "bringing you back to where you started" needs to be examined some more. Thus define:

A strong money pump is a money pump which returns us both to exactly the same situations as when we started: in possession of the same assets and lotteries, with none of them having come due in the meantime.

A weak money pump is a money pump that returns us to the same situation that would have happened if we had never traded at all. Lotteries may have come due in the course of the trades.

What is the difference? Quite simply with a strong money pump, since we return to exactly the same setup, I can then money pump you again, and again, charging you a penny at each round, and draining you of cash until you run out completely or your preferences change. Remember that I charge you that penny only for a deal that is strictly to your advantage, so even with that charge, you are coming out ahead on every deal. It's just at the end of the loop that you're losing out.

For a weak money pump, since we don't return exactly to the same setup, I cannot just money pump you again instantly. I can get cash from you only once for each setup. It might never happen again; or it may happen regularly. But it's not as reusable as a strong money pump.

Now if your preferences are inconsistent, I can certainly money pump you. In his post Zut Allais (French puns remain the property and responsibility of their authors) Eliezer presents a fun version of this, one where subjects prefer A to B and prefer B to A, depending on how they are phrased. Thus I will assume that your preferences are consistent - that you will only invert your preferences under objectively different conditions. I will also assume that you follow the completeness and continuity axioms of the von Neumann-Morgenstern formulation (completeness assumes you are actually capable of deciding between options, while continuity is a technical assumption). Note that the axioms on the Wikipedia article seemed to be incorrect, based on sources such as this book; I have corrected them by replacing >'s with ≥ where appropriate. A lot of the complexities of this post revolve around the difference between these two symbols.

The third von Neumann-Morgenstern axiom is transitivity: that if you like A at least as much as B (designated A ≥ B) and B at least as much as C, then you also like A at least as much as C. Strict transitivity is the weaker statement replacing ≥ with > in the above. This is precisely where the strong money pumps come in:

If your preferences are consistent, complete and continuous, then you are immune to a strong money pump if and only if your preferences are transitive. You are immune to a strict strong money pump if and only if your preferences are strictly transitive.

Proof: A strong money pump is equivalent with a sequence of preferences A ≥ B ≥ ... ≥ S > T ≥ ... ≥ Z ≥ A. Such a sequence can only exist if your preferences are not transitive (note the strict inequality in the middle).

A strict strong money pump is equivalent with a sequence of preferences A > B > ... > Z > A. Such a sequence can only exist if your preferences are not strictly transitive.

The fourth von Neumann-Morgenstern axiom is independence: that if A ≥ B, then for all C, pA + (1-p)C ≥ pB + (1-p)C for all 0 ≤ p ≤ 1. It means essentially that all lotteries can be considered in isolation from each other. This is where the weak money pumps come in:

If your preferences are consistent, complete, continuous and transitive, then you are immune to a weak money pump if and only if your preferences are independent. You are immune to a strict weak money pump if and only if your preferences are not strictly dependent.

The rest of the post is a proof of this, and can be skipped for those unkeen on mathematics.

First of all, we need to explain "strictly dependent" (there is no canonical definition of the term). Given all the other axioms, we can replace independence with the following equivalent axiom:

  • (Independence II) For all A, B, C, D and 0 ≤ p ≤ 1, if pA + (1-p)C > pB + (1-p)D, then A > B or C > D.

The four standard axioms together imply the expected utility hypothesis, and Independence II is a simple consequence of that. Conversely, if we take C and D to be the same lottery, then the axiom states that pA + (1-p)C > pB + (1-p)C implies A > B (since C < C is inconsistent). The contrapositive of this statement is that if A ≤ B, then pA + (1-p)C ≤ pB + (1-p)C. This is just standard independence once more. Thus we can equivalently replace Independence with Independence II.

A decision theory is dependent if it is not independent. Dependency means that there exists p, A, B, C and D such that pA + (1-p)C > pB + (1-p)D while A ≤ B and C ≤ D. A decision theory is strictly dependent if we replace all ≤ in that expression with <. We're now ready to prove the result.

Proof: Assume you have the lottery L, and that there will be a draw to determine one of the random elements. This means L can be written as pA + (1-p)B, where it will end up in as A or B after the draw. Then if I were to trade you L for M = pC + (1-p)D, with M > L, independence implies that A > C or B > D.

Consequently, if you start with lottery L and I trade it for M > L, then for at least one outcome after the random draw, you are left with a lottery strictly better than what would have had if we had not traded. This result continues to be true no matter how often the draws happen, and hence I will not be able to trade you back to your initial situation with an advantageous or indifferent deal. Thus independence implies that you cannot be weakly money pumped with certainty.

Conversely, if your preferences are dependent, then there are p, A, B, C and D such that pA + (1-p)C > pB + (1-p)D and yet A ≤ B and C ≤ D. Then I can weakly money pump you, building on Eliezer's example. Assume you are in possession of pB + (1-p)D, with the first draw being to determine which of B or D you have. Then I can propose a binding contract: I will trade A for B and C for D after that first draw, replacing your current lottery with pA + (1-p)C. This is advantageous to you, so you will accept it. Then, after the first draw, I will propose to trade back B for A or D for C, another advantageous trade that you will accept. Congratulations! You have just been (weakly) money pumped.

In the strict dependency situation, the proof works out exactly the same way, with ≥ replacing > where appropriate, and proving that you cannot be strictly weakly money pumped if your preferences are consistent, transitive, complete, and not strictly dependent.

Note: Of course, if you do not follow independence, you truly do not follow independence. You can play present lotteries off against future lotteries; you can look ahead and see that I will attempt to money-pump you, and compensate for that. You can therefore behave as if you were following independence in these situations, even if you do not. This sort of "arbitraged independence" will be the subject of a future post.

Addendum: Following a discussion with MendelShmiedekamp, I realised that the continuity axiom is not needed for any of the results, as long as one uses independence II instead of independence. Anything that violates independence also violates independence II, so the "if you violate independence, you can be weakly money-pumped" result goes straight through to the non-continuous case.


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This is arbitrage, plain and simple. Almost all of the literature about arbitrage can be applied here. Making use of arbitrage opportunities makes them go away. I think LW frequently tries to reinvent the wheel.

3Stuart_Armstrong12yMy next post is going to be on arbitrage, in fact :-) However, the results here are still useful, as they illustrate exactly what types of money pumps/arbitrages can occur from precisely which violations of the axioms.
1RobinZ12yPlease elaborate on the similarity - it is not immediately obvious upon looking up the definition of "arbitrage". (That LW frequently attempts to reinvent wheels is perfectly plausible, of course.)
0wedrifid12yAnd it is even still plausible if you note that this particular post doesn't seem intended to be original research.
1Stuart_Armstrong12yEr, it is original research. I might be reiterating known results, but if so, I'm unaware of them. Stuart, proud re-inventor of the wheel since... the invention of the wheel.

It might be helpful (for new LW readers) to make careful distinctions between money and utility - diminishing marginal utility of money is important for applying this.

1Stuart_Armstrong12yMy next post will return to that issue (it'll be the last of this series).

Of note, you don't explain why discontinuous preferences necessarilly cause vulnerability to money pumping.

I'm concerned about this largely because the von Neumann-Morgenstern continuity axiom is problematic for constructing a functional utility theory from "fun theory".

5Stuart_Armstrong12yYou really, really, really don't want to be touching continuity without knowing exactly what you're doing. See the hyperreals [] for an example of the sort of thing that happens in this case. Also look at non-measurable functions [] to see the fun in store. But most of the time, when people deny continuity, it's not on theoretical grounds but because they have a particular non-continuous preference theory in mind. That's perfectly fine. But generally, the non-continous theory can be approximated arbitrarily well by a continuous version that looks exactly the same in virtually all circumstances.
0MendelSchmiedekamp12yThis has been helpful. I'm much more familiar with the mathematics than the economics. Presently, I'm more worried about the mathematical chicanery involved in approximating a consistent continuous utility function out of things.
0Stuart_Armstrong12yContinuity is no longer needed for these results...
0MendelSchmiedekamp12yIf we're using the Independence II as an axiom, you should be a little more precise, when you introduced it above, you referred to the base four axioms, including continuity. Now, I only noticed consistency needed to convert between the two Independence formulations, which would make your statement correct. But on the face of things, it looks like you are trying to show a money pump theorem under discontinuous preferences by calling upon the continuity axiom.
-1Stuart_Armstrong12yMathematically: Independence + other 3 axioms => Independence II Independence II => Independence Hence: ~Independence => ~Independence II My theorem implies: ~Independence II => You can be money pumped Hence: ~Independence => You can be money pumped
1MendelSchmiedekamp12yNote, Independence II does not imply Independence, without using at least the consistency axiom.
1Stuart_Armstrong12yThe contrapositive of independence II is: For all A, B, C, D and p, if A ≤ B and C ≤ D, then pA + (1-p)C ≤ pB + (1-p)D. If we now take C and D to be the same lottery, we get independence, as long as C ≤ C. Now, given completeness, C ≤ C is always true (because at least one of C=C, CC must be true, and thus we can always get C ≤ C, -- switching C with C if needed!). So we don't need consistency, we need a weak form of completeness, in which every lottery can be at least compared with itself.
1RobinZ12yTransitivity and Continuity are unnecessary, however.
0MendelSchmiedekamp12yThat is my reading of it too. I know Stuart is putting forward analytic results here, I was concerned that this one was not correctly represented.
1Douglas_Knight12yThe continuity hypothesis really is an unimportant "technical assumption." The only kind of thing it rules out are lexicographical preferences, like if you maximize X, but use Y as a tie-breaker. Specifically, it follows from independence that if AP; the only thing the continuity axiom requires is that at P there is no preference between B and the mixture; there is no tie-breaker. (Without the continuity axiom, it may well be that P is 0 or 1.) This is still true if you only have preferences involving p a rational number: the above is a Dedekind cut. If you restrict p to some smaller set that isn't dense, it's probably bad, but then I'd say you aren't taking probability seriously.
0MendelSchmiedekamp12yCorrect, by definition, if you have a dense set (which by default we treat the probability space as) and we map it into another space than either that space is also dense, in which case the converging sequences will have limits or it will not be dense (in which case continuity fails). In the former case, continuity reduces to point-wise continuity. Note, setting the limit to "no preference" does not resolve the discontinuity. But by intermediate value, there will exist at least one such point in any continuous approximation of the discontinuous function.
0Douglas_Knight12yWhat is the discontinuous function? the function that assigns a preference to a dilemma? (particularly, mixed dilemmas parameterized by probabilities) With discrete range, that can never be continuous. I think you are complaining about the name "continuity axiom"; I am not the right target of that complaint! I don't know why it's called that, but I suspect you have jumped from the name to false beliefs about the axiom system. There is another continuous function, which is the assignment of utilities to lotteries. But I think this is continuous (to the extent that it can be defined) without invoking the continuity axiom. It is more the inverse map, from utilities to indifference-classes of lotteries, that risks not being continuous. I would complain more that this map is not well-defined, but there may be a way of arranging something like indifference-classes to have a finer topology than the order topology (eg, the left-limit topology, or the discrete topology).
0MendelSchmiedekamp12yI was talking about utility functions, but I can see your point about generalizing the result to the mapping from arbitrary dilemmas to preferences. Realize though, that preference space isn't discrete. You can describe it as the function from a mixed dilemma to the joint relation space for < and =. Which you can treat as a somewhat more complex version of the ordinals (certainly you can construct a map to a dense version of the ordinals if you have at least 2 dilemmas and dense probability space). That gives you a notion of the preference space where a calculus concept of continuity does apply (as the continuity axiom is a variation on the intermediate value theorem for this space which implies typical continuity). From this perspective, the point I'm making about continuous approximations should make more sense.
0Douglas_Knight12yThe function whose continuity is at issue is the function from real numbers to lotteries that mixes A and B. C is being used to build open sets in the space of lotteries of the form of all lotteries better (or worse) than C, whose preimage in the real numbers must be open, rather than half-open.
0MendelSchmiedekamp12yWe are talking about the same thing here just at different levels of generality. The function you describe is the same as the one I'm describing, except on a much narrower domain (only a single binary lottery between A and B). Then you project the range to just a question about C. In the specific function you are talking about, you must hold that this is true for all A, B, and C to get continuity. In the function I describe, the A, B, and C are generalized out, so the continuity property is equivalent to the continuity of the function.
0Douglas_Knight12ySo what did you mean by
0MendelSchmiedekamp12yI meant that setting the limit to no preference for a given C doesn't equate to a globally continuous function. But that when you adjust your preferences function to approximate the discontinuous function by a continuous one, the result will contain (at least one) no preference point between any two A < B. Now perhaps there is a result which says that if you take the limit as you set all discontinuous C to no preference, that the resulting function is complete, consistent, transitive, and continuous, but I wouldn't take that to be automatic. Consider, for example, a step discontinuity, where an entire swatch of pA + (1-p)B are stuck on the same set of < and = mappings and then there is a sharp jump to a very large set of < and = mappings at a critical p'. If you map the ordinals to the real line, this is analogous to a y-coordinate jump. To remove this discontinuity you would need to do more than split the preferences at p' around no preference, because all this does is add a single point to the mix. To fully resolve it, you need to add an entire continuous curve, which means a process of selecting new A, B, and C, and showing that the transfinite limit always converges to a valid result.
0Stuart_Armstrong12yThanks, that's very useful to know. Do you have a link to the proof?
0Johnicholas12yCan you elaborate? Maybe there is another solution to your problem than abandoning continuity.
0MendelSchmiedekamp12yI'm very busy at the moment, but the short version is that one of my good candidates for a utility component function, c, has, c(A) < c(B) < c(pA + (1-p)B) for a subset of possible outcomes A and B, and choices of p. This is only a piece of the puzzle, but if continuity in the von Neumann-Morgenstern sense falls out of it, I'll be surprised. Some other bounds are possible I suspect.
0Stuart_Armstrong12yIndependence fails here. We have B > A, yet there is a p such that (pA + (1-p)B) > B = (pB + (1-p)B). This violates independence for C = B. As this is an existence result ("for a subset of possible A, B and p..."), it doesn't say anything about continuity.
0MendelSchmiedekamp12ySorry I left this out. It's a huge simplification, but treat the set of p as a discrete subset set in the standard topology.
0Stuart_Armstrong12yAnd that is discontinuous; but you can model it by a narrow spike around the value of p, making it continuous.
0Stuart_Armstrong12yHum, this seems to imply that the set of p is a finite set... Still doesn't change anything about the independence violation, though.
1MendelSchmiedekamp12yBut does doesn't the money pump result for non-independence rely on continuity? Perhaps I missed something there. (Of note, this is what happens when I try to pull out a few details which are easy to relate and don't send entirely the wrong intuition - can't vouch for accuracy, but at least it seems we can talk about it.)
0Stuart_Armstrong12yActually, I realised you didn't need continuity at all. See the addendum; if you violate independence, you can be weakly money-pumped even without continuity (though the converse may be false).
0Technologos12yPerhaps I'm confused, but I thought that the inequality you described simply refers to a utility function with convex preferences (i.e. diminishing returns). I agree in general that discontinuity does not by itself entail the ability to be money-pumped--this should be trivially true from utility functions over strictly complementary goods.
[-][anonymous]12y 2

I wrote a critical response to this and posted it to my drafts:

Summary: you're ignoring the fact that utilities can change; it's easy to come up with an example of a case where both receiving an item A tomorrow and receiving an item B tomorrow are preferable to an equal chance of receiving either tomorrow. You're also ignoring the fact we're not mathematical oracles, which irritates me but doesn't assist my point.

Once I feel more on top of myself, I'll probably revise it and post it.

1Stuart_Armstrong12yI'm not ignoring these facts as in "I don't know them". I'm ignoring them as in "simplifying them away" so as to get rigorous results. Rigorous results that can then be built upon in more general cases. The big question is, why do we prefer A and B to the chance of either? There are some objective reasons for this that do not violate any of the axioms: take A being a huge cake and B being a video game, and you lack both a fridge and a games consol, and these can only be bought today. So you'd have to buy them both, and your choices are AuF or BuGC tomorrow versus ((A+B)/2) u F u GC, where u denotes union. Other situations can be modelled similarly. In fact, whether you are money pumpable or not really comes down to how you model the situation: Maybe uncertainty makes you nervous, and you lose happiness over this. Then either I'm weakly money pumping you if I act on these preferences, or I'm objectively granting you the removal of your worry as a service. Most people at the time feel that I'm granting them a service, but afterwards they feel I money pumped them. Especially if I repeat it. Which is a note of caution on the subtlety of blindly applying the results of my post directly to the real world. But if your utilities are simply changing for no valid reason, then you are completly money pumpable. First assume humans are perfect spheres. Then assume they're mathematical oracles. Get results in this case. Add the complexities of reality afterwards.
0[anonymous]12yDoes this mean we should start treating certain types of money pumping as payment for a service rather than something rational agents always avoid? When Less Wrongers say that expected utility is the sole fundamental decision-making method used by practical rational agents (as opposed to ones that require impossible computation ability), are they blindly applying the results of your post directly to the real world, or is there more to it? AIXI, Bayes, Solomonoff, CEV. I'm ready for my complexities now.
0Stuart_Armstrong12yThis is part of the process of adding the complexities. Now you know precisely in what way's you can be exploited when you abbandon which axiom. Yet people do abandon these axioms, and are only moderately exploited. Thus we can assume that real decision theories, when itterated in the presence of people trying to money pump you, tends approximately to the expected utility hypothesis. This dramatically reduces the number of reasonable/probable decision theories out there.
3ChrisHibbert12yThe name of the service is "insurance". This is a business in which customers repeatedly make bets that they wish they hadn't made in retrospect, but it still makes sense to make the bet ex ante.
1bgrah44912yPlease forgive the nitpicking but as an actuary, I do try to make this point whenever I feel it's helpful to do so: Insurance is not betting. Insurance is removing variation and chance from your life, not introducing variation and chance to your life. A bet introduces risk where there was none before. Insurance removes risk when it already exists. End of nitpicking.
0JamesAndrix12yThat's exactly the same as hedging bets.
0bgrah44912yWhich is why hedging is understood by people who hedge as insurance (unlike the bet they are trying to hedge).
0Technologos12yA good point to remember, and I'd say the most useful way to think of it. The problematic word seems to be "bet," and while I agree that most bets do increase variation, I feel like Chris/Stuart take bet to mean "an amount of money that pays returns when one outcome happens and not when another does." This adequately captures both traditional bets (bets that some thing will happen because one believes the probability of it happening is higher than one's betting partner believes it is) and insurance or hedging bets.
0ChrisHibbert12yAgreed. I work on prediction markets, so I see it all as bets, and am used to thinking that both participants in a purely financial trade can gain from it, even though many people on the outside of the deal see it as zero sum. Sometimes you increase your variance because you think it's worth increasing your expected return, other times you reduce your variation.
1Stuart_Armstrong12yActually, for most insurances, it makes no sense to do the bet at any point. Aggregating [] the risk over your lifetime, you're better off not paying the insurance (this doesn't apply to insurance for major disasters).
0[anonymous]12yIs my post not an example of someone abandoning the axiom of independence and not being exploited? I assume you mean "tends to something which is approximately the expected utility hypothesis".

"A weak money pump is a money pump that returns us to the same situation that would have happened if we had never traded at all."

Weak money pumps occur all the time in real life. People pay big bucks to let other people manage their money without knowing that they might have been better off doing it themselves. The pump is disguised by normal expected gains (in a well functioning economy), which the stock broker or hedgefund manger can claim was his doing.

Strong money pumps I would think almost as a rule don't exist in individuals (since people wise up fast) but might occur institutionally in the form of regulatory loopholes. Any lawyers on the blog?

5[anonymous]12yI really don't see how this is a money pump. The broker isn't exploiting the client's bizarre preferences, he is exploiting their ignorance. Money pumps are about preferences not beliefs. Losing money due to ignorance/incorrect beliefs is just bad trading, not being money pumped.
0LauraABJ12yFair point. A better example would be that said money manager convinces the client at each trade that it is in his interest based on his preferences, when really these preferences lead back to where the client would have been if he hadn't traded at all (minus the manager's fee of course), but the client is unaware that his choices did him no good, because of marginal gains anyway. Unless you mean to say the pumped party must be aware that he is back where he started in order to be considered a pump.
0Stuart_Armstrong12yYou can see this in several ways - either the client is being inconsistent, as he makes different decisions based on how the the choices are presented; or he is being cheated by the money manager who lies about the value of the trades in question; or the client simply has non-independent preferences, which the money manager is exploiting (strict weak money pump). I'm not sure either of the explanations is better than the other in this set-up; you'd have to experiment with the situation, change some variables, and see what comes up.
0pwno12yWouldn't most regrettable decision be considered a weak money pump?

I'm missing something. Suppose that my preferences are strictly transitive, but that they violate the other axioms and that there are lots of trades which I view as incomparable (none of AB holds), and that I won't make an incomparable trade. Why would this leave me vulnerable to being money pumped?

2Stuart_Armstrong12yIt wouldn't (at least not to a strong money pump). But decreeing that things are incomparable is often rather dumb; if your house was flooding, would you grab certain things, or would you just refuse to choose anything, because the choices are "incomparable"?
2soreff12yThanks, I was wondering if all of the axioms were crucial, or mostly the transitivity one. Perhaps "incomparable" is the wrong approximation. Perhaps a better way to view it is that I view transactions as having frictional costs (if nothing else, the cost of working out to sufficient precision what my actual preferences are). There are a lot of (A, B) pairs such that, if I had A and was offered B in exchange, I would turn down the offer, and the same if I had B and was offered A.. Very roughly, assume that I treat each exchange transaction as having some probability of going wrong in some way (e.g. failing in such a way that I wind up with neither object), so the new object's utility has to be say 10% higher than the old object's utility to offset the transaction risk. Would this model leave me vulnerable to being money pumped?
2pengvado12yYour model is safe from being money pumped by another agent. The disadvantage is that you'll pass up some certain gains, which is equivalent (modulo loss aversion []) to taking on some certain losses. But if you really do think that there is a nonnegligible probability that any given exchange will go bad, then you don't have to violate any of the preference axioms, all your caution is in the probability estimate.
0Stuart_Armstrong12yIn a certain sense, it does (as long as your 10% beliefs are inaccurate). If you have a lottery A that gives you negative value, I can trade it for a lottery B that is slightly more negative (you 10% chance of getting neither will make you accept this deal). And then itterate. Or, what about me selling you (in a single transaction) an insurance, good for a hundred trades, that guarantees you against the 10% loss chance? Generally, inaccurate beliefs leave you open to some sort of arbitrage, even if it's not technically a money pump as described above.

Fun investment fact: the two trades that over 40 years turned 1'000 USD into >1'000'000 USD

1'000 USD in Gold on Jan 1970 for 34.94 USD / oz (USD 1'000.00)

1st Trade Sell Gold in Jan 1980 at 675.30 USD / oz (USD 19'327.41) Buy Dow on April 18 1980 at 763.40 (USD 19'327.41)

2nd Trade Sell Dow on Jan 14 2000 at 11'722.98 (USD 296'797.14) Buy Gold on Nov 11 2000 at 264.10 USD / oz (USD 296'797.14)

Portfolio value today: ~1'187'188.57 USD


0[anonymous]12yTechnically, you started and ended not with 1'000 USD and 1'000'000 USD, but 28.62 ounces of gold and 1124 ounces of gold, which is not quite as impressive-sounding. Still, four trades could have gotten precisely what you said.

This claim is wrong for two reasons.

  1. The usual money pump argument deals only with the difficulty of making a choice when your preferences are cyclic. The VM axioms have nothing to do with it.

  2. Even in the presence of cyclic choices, there could be a maximal element among the convex combinations of the choices, see Peter Fishburn on SSB Utility, or Skew Symmetric Bilinear Utility.

0Stuart_Armstrong12yCyclic choices are what I termed a strict strong money pump. The VM axiom of transitivity forbids this. I don't really see the relevance of the maximal element. Nowhere did I assume that there were no maximal elements.
2mjwebster12y@Stuart, you have misunderstood. There may be a maximal element among the convex combinations of cyclic preferences, when the VM axioms fail to hold. SSB utility axioms have to hold in this case. This maximal is a good candidate for choice from the cyclic preferences. So the claim that a violation of VM axioms leads to a money pump is false, even in the presence of cyclic preferences. Read Fishburn's Nonlinear Preference and Utility Theory (1988) or the very recent Essays in Honor of Fishburn, edited by S. Brams et al. You should probably start your discussion from Merrill Flood's 1952 article on preference cycles, available from Rand. []
0Stuart_Armstrong12yIf I understand this correctly, you're not saying that you can't be money pumped with cyclic preferences; you're saying that if you start with the maximal, or choose to go to the maximal, then you can no longer be money pumped. Is this what you are saying?
0mjwebster12yPage 42-44, Non Linear Preference, "The money-pump concept also reveals a narrow perspective on how choice might be based on preferences, and perhaps a lack of imagination in dealing with cyclic patterns. Although there is no transparent way to make a sensible choice from {p,q,r} when p>q>r>p, nothing prevents a person from considering preferences over the set of convex combinations of p, q, and r. And, if there is a combination in the set, then that persons has an ex ante maximally preferred alternative. As first shown in Kreweras (1961), this indeed can be the case, and we shall consider it later as part of the SSB theory." Here is a modern paper addressing some of these issues: []
0Stuart_Armstrong12yIndeed; but I can still money pump you for cyclic preferences if I'm the only trader around, and I only offer you the pure lotteries p, q and r rather than convex combinations. And if you never change your preferences after you realise what I'm doing... Restrictive conditions, to be sure, but mathematically you can't escape. The fact that you generally do escape implies that something else is going on than your simple preferences.

Is this anything like the Red paper clip trade story?

4JamesAndrix12yLikewise, I was wondering. If I know I'm not a money pump, and every trade I make is beneficial to me, can I model the rest of the market as a single big agent that is a money pump? Can I reverse this process to build a model of the world as an agent, and then find a sequence of beneficial trades?
2Jack12yFor sure you can. This is just what traveling merchants did/do. You buy X from people with Y>X and sell it to people with X>Y while at the same time buying Y from the second group and selling it back to the first. Of course, you're not actually hurting anyone, indeed, everyone in this scenario benefits.
0Stuart_Armstrong12yAs you asking "can I money pump the world"?
2RobinZ12yIf I recall, the red paper clip story involved several points at which the man was able to manage a grossly unfair trade. There's a similarity in the sense that the man made the money by profitable trades, but he wasn't using any theory along the lines of Armstrong's.
2Alicorn12yWhich trades seemed to you "grossly" unfair - keeping in mind that the traders got to participate in an interesting process, as well as getting whatever they traded for?
2gwern12y [] A fish-shaped pen for such a doorknob? (For that matter, who would want a red paperclip instead of a perfectly good pen?) From Boy Scouts, I remember that camp stoves were expensive. Even the smallest cheapest ones, for backpackers, are no less than 60$ new: [] And from the picture, he traded for one of the big ones. A snowmobile for some beer & a neon sign? IIRC, snowmobiles, even used ones, cost hundreds & thousands of dollars; whereas a keg of beer is <$100 and a Budweiser neon sign probably not much more. A cube van. I looked in the Blue Book; that was probably worth well upwards of 2000 dollars. The comments on the blog estimate the year's rent at 5 or 7k. The recording contract was 30 hours in the studio & 50 hours of post-production. I suspect someone was ripped off here but I can't tell whom. The movie is a straight-to-DVD film I've never heard of, and which still hasn't come out. The role is described as credited & speaking, but who knows how much one actually gets. I'd say the town was the loser in that deal, unless they value the publicity that highly. -------------------------------------------------------------------------------- So yeah, I'd say on multiple deals the traders were financially exploited, and this says more about their willingness to participate in a well-advertised stunt than it does about whether the deals were actually fair.
4Alicorn12yPerhaps my intuitions are shaped by the feeling that it wouldn't have been outrageous for any of these traders - assuming they didn't want and had little to no use for the things they traded away - to give them away for free. (I've participated in Freecycle - that's where I got my perfectly-functional digital camera.) Trading them for a thing of nonzero value doesn't seem to me like it can be less smart, unless the motivations are explicitly mercenary - and even then, if the thing is hard to sell in a timely manner, it might be worth expediting by taking a weird offer now rather than a "fair" one later.
3gwern12yGiving stuff away is one thing, and well understood. If Warren Buffet decides to give away a few dozen billion dollars, he's a hero. But if some guy comes up to him and trades him a red pen for a few dozen billion dollars, I suspect no one would react the same way. And then there is the issue of truth in advertising. His website says "My name is Kyle MacDonald and I traded one red paperclip for a house. I started with one red paperclip on July 12 2005 and 14 trades later, on July 12, 2006 I traded with the Town of Kipling Saskatchewan for a house located at 503 Main Street." He doesn't say, 'I started with one red paperclip, and by a combination of exploiting my friends & acquaintances and relentless self-promotion, I got a local government to give me a house.' (And don't forget that he's flogging a book as well.) The implication is that he did it by sheer skill and laudably increasing economic efficiency, when he has done so little more than Madoff did.
4Alicorn12yGiving stuff away is an understood phenomenon only in limited contexts. If Warren Buffet decides to give away a few dozen billion dollars to a charity that fights AIDS in Africa, or teaches illiterate adults to read and write, or to a scholarship fund for deserving underprivileged teens to attend college, or whatever - that we understand. If he gives five hundred dollars to a guy he meets on the Internet because the guy could really use five hundred dollars, that's incomprehensible to most people. This precise distinction is why I brought up Freecycle, which is entirely about giving your belongings to people you meet on the Internet who could really use them. But neither Freecycling, nor giving up a swell fish pen for one red paperclip, seem bizarre to me.
2gwern12yIf neither freecycling nor a bad trade like a pen for a paperclip seem bizarre to you, then why do you object to Buffet giving $500 away on the Internet? If you do not personally object, and are merely describing how you think other people would react, then what would you find bizarre? So far we've established that you don't find: 1. a fair trade bizarre 2. an unfair trade bizarre 3. a non-trade (gift/charity) bizarre either Which would seem to suggest you find no voluntary exchange bizarre, and so I don't think you're in a mental position to have any opinion one way or the other about the red paperclip scheme. (As for actually giving away $500 online: happens all the time. I'm sure you read webcomics or blogs that are funded in part by readers donating. Not to mention little things like Wikipedia or all the online microlending sites like Kiva [].)
2Alicorn12yI find no paradigmatic voluntary exchange bizarre. There are sometimes factors that bizarre-ify exchanges of any of the three types you divide up. I don't think that these factors were at work in the red paperclip "scheme", but could be mistaken - I haven't done extensive red paperclip related research. I know people sometimes give away substantial amounts to content producers they want to support, which looks like a hybrid of gift and either fair or unfair trade (depending on the content). I don't find it bizarre. I'm sure many people would, but I'm willing to be proven wrong.
0Zachary_Kurtz12yBut what do you mean by "fair trade" here. It's not like any party didn't know what the deal was. This just means there's something else at play which was determining the value of the goods to both participants. What is "unfair" about this type of exchange?
0gwern12yWell, if you're going to assume that any deal which seems unfair to a mere outsider will have 'something else at play' rendering the deal actually fair, then there's nothing I can say about that. Such assertions are like that of the Austrians - no mere evidence can falsify it. (After all, who knows what evil lurks in the hearts of men? Er, I mean, differing preferences.)
0Zachary_Kurtz12yWell the Austrians are right in that the traditional assumptions of rationality (and therefore fairness) don't make much sense. Although I think the behavioral economists are doing a decent job of backing up these type of assertions: the point that "fairness" is subjective. Can someone engage in a trade they don't believe is fair? Yes of course. But if both parties say a trade is fair, who is an outside observer (with all the market data in the world) to tell them otherwise?
0Eliezer Yudkowsky12y"Fairness is not agreement, fairness is symmetry. []"
1RobinZ12yThat's what I meant, actually - the only reason for many of the trades is to buy participation in the media exercise.
0Zachary_Kurtz12yOk I'm going to re-read this article again. I thought I saw a similarity in there. But I tend to agree with Alicorn below me. If both parties agree to a free trade, can it really be characterized as "unfair" just because the normal market values don't seem to line up? It's all about personal expected utility after all, not what the market thinks.
0RobinZ12yI really don't know what I should say that hasn't been said. My point isn't that Kyle MacDonald hoodwinked any of the people whom he traded with - my point is that he wasn't using money pumps. The story has negligible relevance to Stuart Armstrong's post.
0Zachary_Kurtz12yfair enough, I was getting off track.

Conversely, if your preferences are dependent, then there are p, A, B, C and D such that pA + (1-p)C > pB + (1-p)D and yet A ≤ B and C ≤ D. Then I can weakly money pump you, building on Eliezer's example. Assume you are in possession of pB + (1-p)D, with the first draw being to determine which of B or D you have. Then I can propose a binding contract: I will trade A for B and C for D after that first draw, replacing your current lottery with pA + (1-p)C. This is advantageous to you, so you will accept it. Then, after the first draw, I will propose to t

... (read more)

All the trades must be indifferent or advantageous to you, so that you will accept them. And if even one of those trades is advantageous, then this is a money pump: I can charge you a tiny amount for that trade, making free money out of you. You are now strictly poorer than if you had not accepted the tradesat all.

I think this is part I don't get.

Lets say we're making a trade I would be indifferent about under normal circumstances (say a red paper clip for a pen). If you then try to turn around and try to "pump" me for a transaction cost of $... (read more)

2RobinZ12yThe alteration is to the advantageous trade in the network. Example: Say you are indifferent to trading your paperclip for my penny, indifferent to trading your pen for my paperclip, and happy to trade your penny for my pen. Under this situation, there is some small increment to the price of the pen which will leave you still happy about the trade - say, two cents. In that case, I can make you pay two cents for my pen, swap it for a paperclip, then swap the paperclip for a penny.
0Zachary_Kurtz12yOK this makes a lot more sense now. So you're engaging arbitrage by taking advantage of my transitive preferences? I could see this scenario being possible, but not if you described all the trades to me beforehand (I assume limiting information is a prerequisite for the pump to work)
0RobinZ12yPerhaps not this specific set of trades, but money-pumps have been demonstrated experimentally []. It's obviously wrong, but there's no theoretical reason why someone might not be susceptible to it.

Great post!

There's a defense of bad preferences (not yet demonstrated in this article, but probably will be in the comments) along the lines of "You couldn't really money pump me; I'd catch on after a round or two and quit talking to you."

I think this fails, since there's no reason that only one agent will try to turn you into a money pump once you are found to be vulnerable. If there's a group of smart agents, they could engage you in a series of transactions over the course of years, taking your money all the while.

2Stuart_Armstrong12yThis brings up the fact that money pumps force you to act more like an expected utility maximiser, as long as you are aware of them and want to avoid them.
0whpearson12yA more nuanced approach is to have some form of reflectivity that looks at whether you are losing money in general and tries to figure out if you are being money pumped or not (by a single person or group).