For $24,000, you can have my two cents. ;)

Actually, that makes me think of another explanation besides overreaction to small probabilities: if a person takes 1B and loses, they know they would have won if they'd chosen differently. If they take 2B and lose, they can tell themselves (and others) they probably would have lost anyway.

People don't maximize expectations. Expectation-maximizing organisms -- if they ever existed -- died out long before rigid spines made of vertebrae came on the scene. The reason is simple, expectation maximization is not robust (outliers in the environment can cause large behavioral changes). This is as true now as it was before evolution invented intelligence and introspection.

If people's behavior doesn't agree with the axiom system, the fault may not be with them, perhaps they know something the mathematician doesn't.

Finally, the 'money pump' argument fails because you are changing the rules of the game. The original question was, I assume, asking whether you would play the game once, whereas you would presumably iterate the money pump until the pennies turn into millions. The problem, though, is if you asked people to make the original choices a million times, they would, correctly, maximize expectations. Because when you are talking about a million tries, expectations are the appropriate framework. When you are talking about 1 try, they are not.

If I knew the offer wouldn't be repeated, I might take 1A because I'd really rather not have to explain to people how I lost $24,000 on a gamble.

It took me 30 minutes of sitting down and doing math before I could finally accept that 1A+2B was an irrational preference. I finally realized that a lot of it came down to: with a 66% vs 67% chance of losing, I could take the riskier option and not feel as bad, because I could sweep it under the rug with "oh, I probably would have lost anyways."

Once I ran a scenario where I'd KNOW whether it was that 1% that I controlled, or the 66% that I didn't control, that comfort evaporated.

I learned a lot about myself by working through this exercise, so thank you very much :)

Eliezer, I see from this example that the Axiom of Independence is related to the notion of dynamic consistency. But, the logical implication goes only one way. That is, the Axiom of Independence implies dynamic consistency, but not vice versa. If we were to replace the Axiom of Independence with some sort of Axiom of Dynamic Consistency, we would no longer be able to derive expected utility theory. (Similarly with dutch book/money pump arguments, there are many ways to avoid them besides being an expected utility maximizer.)

I'm afraid that the Axiom of Independence cannot really be justified as a basic principle of rationality. Von Neumann and Morgenstern probably came up with it because it was mathematically necessary to derive Expected Utility Theory, then they and others tried to justify it afterward because Expected Utility turned out to be such an elegant and useful idea. Has anyone seen Independence proposed as a principle of rationality prior to the invention of Expected Utility Theory?

The idea is that $ amount equals your utility, while in reality the history of how you got this amount also matters (regret, emotions, etc.).

There's no paradox here - as your utility expressed in $ just doesn't match utility of the subjects. As for money pump - you just have a win win situation - you earn money, and the subjects earn good feelings.

Yes, philosophers, and others, do often too easily accept the advice of strong intuitions, forgetting that strong intuitions often conflict in non-obvious ways.

Nainodelac, if you prefer 1A to 1B and 2A to 2B, as you should if you need exactly $24,000 to save your life, that is a perfectly consistent preference pattern.

How trustworthy is the randomizer?

I'd pick B in both situations if it seemed likely that the offer were trustworthy. But in many cases, I'd give some chance of foul play, and it's FAR easier for an opponent to weasel out of paying if there's an apparently-random part of the wager. Someone says "I'll pay you $24k", it's reasonably clear. They say "I'll pay you $27k unless these dice roll snake eyes" and I'm going to expect much worse odds than 35/36 that I'll actually get paid.

So for 1A > 1B, this may be based on expectation of cheating. For 2A < 2B, both choices are roughly equally amenable to cheating, so you may as well maximize your expectation.

It seems likely that this kind of thinking is unconscious in most people, and therefore gets applied in situations where it's not relevant (like where you CAN actually trust the probabilities). But it's not automatically irrational.

Your description is not a money pump. A money pump occurs when you prefer A > B and B > C and C > A. Then someone can trade you in a round robin taking a little out for themselves each cycle. I don't feel like typing in an illustration, so see Robyn Dawes, Rational Choice in an Uncertain World.

There is a significant difference between single and iterative situations. For a single play I would prefer 1A to 1B and 2B to 2A. If it were repeated, especially open-endedly, I would prefer 1B to 1A for its slightly greater expected payoff. This is analogous, I think, to the iterated versus one-time prisoner's dilemma, see Axelrod's Evolution of Cooperation for an interesting discussion of how they differ.

The large sums of money make a big difference here. If it were for dollars, rather than thousands of dollars, I'd do what utility theory told me to do, and if that meant I missed out on $27 due to a very unlucky chance then so be it. But I don't think I could bring myself to do the same for life-changing amounts like those set out above; I would kick myself so hard if I took the very slightly riskier bet and didn't get the money.

When I made the (predictable, wrong) choice, I wasn't using probability at all. I was using intuitive rules of thumb like: "don't gamble", "treat small differences in probability as unimportant", and "if you have to gamble against similar odds, go for the larger win".

How do you find time to use authentic probability math for all your chance-taking decisions?

The Allais "Paradox" and Scam Vulnerability by Karl Hammer is a much needed update for anyone who reads the OP.

Risk and cost of capital introduce very strange twists on expected utility.

Assume that living has a greater expected utility to me than any monetary value. If I need a $20,000 operation within the next 3 hours to live, I have no other funding, and you make me offer 1, it is completely rational and unbiased to take option 1A. It is the difference between a 100% of living and a 97% chance of living.

If I have $1,000,000,000 in the bank and command of legal or otherwise armed forces, I may just have you killed - for I would not tolerate such frivolous philosophizing.

I wonder how the results would change if the experiment changes so that the outcomes of 2B are, "You have a 33% chance of receiving $27k, a 66% chance of not getting anything, and a 1% chance of having someone laugh in your face for not picking 2A"

When we speak of an inherent utility of certainty, what do we mean by certainty? An actual probability of unity, or, more reasonably, something which is merely very much certain, like probability .999? If the latter, then there should exist a function expressing the "utility bonus for certainty" as a function of how certain we are. It's not immediately obvious to me how such a function should behave. If probability 0.9999 is very much more preferable to probability 0.8999 than probability 0.5 is preferable to probability 0.4, then is 0.5 very much more preferable to 0.4 than 0.2 is to 0.1?

My intuitions match the stated naive intuitions, but I reject your assertion that the pair of preferences are inconsistent with Bayesian probability theory.

You really underestimate the utility of certainty. "Nainodelac and Tarleton Nick"'s example in these comments about the operation is a perfect counter.

With a 33% vs. 34% chance, the impact on your life is about the same, so you just do the straightforward probability calculation for expected value and take the maximum.

But when offered 100% of some positive outcome, vs. a probability of nothing, it seems perfectly rational to prefer the guarantee. Maximizing expected dollar winnings is not necessarily the same as maximizing utility. And you're right, the issue isn't decreasing returns. But the issue is the cost of risk.

Your money pump doesn't convince me either. I'd be happy to pay the two cents, both times, and not regret the cost at the end, just as I don't regret paying for insurance even if I happen not to get sick.

A bird in the hand...

Certainty is a form of utility, too.

I know this was posted 4 years ago, but I had a thought. If I was offered a certainty of $24,000 vs a 33/34 chance of $27,000, my preference would depend on whether this was a once-off. If this was a once-off, my primary concern would be securing the money and being able to put food on the table tonight. Option 1 will put food on the table with 100% certainty, while Option 2 will not.

If, however, the option was to be offered many times, I would optimise for greatest return - Option 2. If I miss out this month, I'll just scrape for food until next month, when chance are I'll get the money.

I think I just answered my own question. If my goal can be reached with $24,000, then Option 1 is the best one because it reaches the goal in one guaranteed fell swoop. However, if my goal is to make lots of money, then Option 2 is the way to go, because it makes the most over time.

That make sense to anyone?

Please correct me if any of my assumptions are innacurate, and I apologize if this comment comes off as completely tautological.

Expected utility is explicity defined as the statistic

U(x)})

where X is the set of all possible outcomes associated with a particular gamble, p(x) is the proportion of times that outcome x occurs within the gamble, and U(x) is the utility of outcome x, a function that must be strictly increasing with respect to the monetary value of outcome x.

To reduce ambiguity:

• 1A, 1B, 2A, and 2B are instances of gambles.

• For 1B, the possible outcomes are $27000 and $0.

• For 1B, the expected utility is p($27000) * U($27000) + p($0) * U($0) = 33/34 * U($27000) + 1/34 * U($0).

If you choose 1A over 1B and 2B over 2A, what can we conclude?

• that you are not using the rule "maximize expected utility" to make your decisions. Thus you do not fit the definition, as given by the Axiom of Independence, of consistent decision making.

If you choose 1A over 1B and 2B over 2A, what can we not conclude?

• that your decision rule changes arbitrarily. You could, for example, always follow the rule, "Maximize minimum net utility. In the case of a tie, maximize expected utility." In this case, you would choose 1A and 2B.

• that you would be wrong or stupid for using a different decision rule when you only get to play one time, than the rule you would use when you get to play 100 times.

How do I alleviate feeling pleased at myself for having read the statement of the paradox - that people preferred 1A>1B but 2B>2A - and immediately going "WHAT?" and boggling at the screen and pulling confused faces for about thirty seconds, so flabbergasted I had to reread that this choice pattern was common?

(Personally I'm really strongly biased these days toward a bird in the hand and would have chosen 1A and 2A every time. I occasionally do bits of sysadmin for dodgy dot-coms that friends are working for. There are people who offer equity; I take an hourly fee. "No, no, that's fine, I am but humble roadie." This may not always be the best life strategy, but it seems to work for me at present.)

Agree with Denis. It seems rather objectionable to describle such behaviour as irrational. Humans may well not trust the experimenter to present the facts of the situation to them accurately. If the experimenter's dice are loaded, choosing 1A and 2B could well be perfectly rational.

I hate to discuss this again, but...

Is Michael Vassar's variant Pascal's Mugging (with the pigs), bypassing as it does Robin's objection, the reductio of expected value? If you don't care about pigs, substitute something else really really bad that doesn't require creating 3^^^3 humans.

Is Pascal's Mugging the reductio ad absurdum of expected value?

No. I thought it might be! But Robin gave an excellent reason of why we should genuinely penalize the probability by a proportional amount, dragging the expected value back down to negligibility.

(This may be the first time that I have presented an FAI question that stumped me, and it was solved by an economist. Which is actually a very encouraging sign.)

This may be related to the phenomenon of overconfident probability estimates. I would not be surprised to find that people who claim a 97% certainty have a real 90% probability of being right. Maybe someone who hears there's 1 chance in 34 of winning nothing interprets that as coming from an overconfident estimator whereas the 34% and 33% probabilities are taken at face value.

On the other hand, the overconfidence detector seems to stop working when faced with asserted certainty.

Upon rereading the thread and all of its comments, I suspect the person I originally quoted meant something along the lines of "preferring 1A to 1B but 2B to 2A is irrational", which seems more defensible.

There is nothing irrational about preferring 1A and 2B by themselves, it's choosing the first option in the first scenario and the second in the second that's dodgy.

You can define a utility function on gambles that is not the expected value of a utility function on amounts of money, but then that function is not expected utility, and you're outside of normal models of risk aversion, and you're violating rationality axioms like the one Eliezer gave in the OP.

Having a utility function determined by anything other than amounts of money is irrational? WTF?

Choosing 1A and 2B is irrational regardless of your level of risk aversion.

No, only if the utility of avoiding risk is worth less than the money at risk. Duh.

"Nainodelac and Tarleton Nick", why are you using my (reversed) name?

steven: not if you're nonlinearly risk averse. As many have suggested, what if you take a large one-time utility hit for taking any risk, but you're not averse beyond that?

tcpkac: no one is assuming away risk aversion. Choosing 1A and 2B is irrational regardless of your level of risk aversion.

I confess, the money pump thing sometimes strikes me as ... well... contrived. Yes, in theory, if one's preferences violate various rules of rationality (acyclicity being the easiest), one could conceivably be money-pumped. But, uh, it never actually happens in the real world. Our preferences, once they violate idealized axioms, lead to messes in highly unrealistic situations. Big deal.

James D. Miller has a proposal for Lottery Tickets that Usuallly Pay Off.

Robin, were you thinking of a certain colleague of yours when you mentioned accepting intuition too readily?

Let's say you prefer 1A over 1B, and 2B over 2A, and you would pay a single penny to indulge each preference. The switch starts in state A. Before 12:00PM, you pay me a penny to throw the switch to B.

I don't understand why I would pay you a penny to throw the switch gefore 12:00?

Would I pay $24k to play a game where I had a 33/34 probability of winning an extra $3k? Let's consult our good friend the Kelly Criterion.

We have a bet that pays 1/8:1 with a 33/34 probability of winning, so Kelly suggests staking ~73.5% of my bankroll on the bet. This means I'd have to have an extra ~$8.7k I'm willing to gamble with in order to choose 1b. If I'm risk-averse and prefer a fractional Kelly scheme, I'd need to start with ~$20k for a three-fourths Kelly bet and ~$41k for a one-half Kelly bet. Since I don't have that kind of money lying around, I choose 1a.

In case 2, we come across the interesting question of how to analyze the costs and benefits of trading 2a for 2b. In other words, if I had a voucher to play 2a, when would I be willing to trade it for a voucher to play 2b? Unfortunately, I'm not experienced with such analyses. Qualitatively, it appears that if money is tight then one would prefer 2a for the greater chance of winning, while someone with a bigger bankroll would want the better returns on 2b. So, there's some amount of wealth where you begin to prefer 2b over 2a. I don't find it obvious that this should be the same as the boundary between 1a and 1b.

This is a problem because the 2s are equal to a one-third chance of playing the 1s. That is, 2A is equivalent to playing gamble 1A with 34% probability, and 2B is equivalent to playing 1B with 34% probability.

Equivalence is tricky business. If we look at the winnings distribution over several trials, the 1s look very different from the 2s and it's not just a matter of scale. The distributions corresponding to the 2s are much more diffuse.

Surely, the certainty of having $24,000 should count for something. You can feel the difference, right? The solid reassurance?

A certain bet has zero volatility. Since much of the theory of gambling has to do with managing volatility, I'd say certainty counts for a lot.