The Allais Paradox

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

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

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.

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.

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.

A bird in the hand...

Certainty is a form of utility, too.

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 think defenses of the subject's choices by recourse to nonmonetary values is missing the point. Anything can be rational with a sufficiently weird utility function. The question is, if subjects understood the decision theory behind the problem, would they still make the same choice? After seeing a valid argument that your preferences make you a money pump, you certainly could persist in your original judgment, by insisting that your feelings make your first judgment the right one.

But seriously?---why?

Since people only make a finite number of decisions in their lifetime, couldn't their utility function specify every decision independently? (You could have a utility function that is normal except that it says that everything you hear being called 1A is preferable to 1B, and anything you hear being called 2B is preferable to 2A. If this contradicts your normal utility function, this rule is always more important. Even if 2B leads to death, you still choose 2B.)

The utility function would be impossible to come up with in advance, but it exists.

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.

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?

Since I know myself, I know what I will do after midnight (pay to switch it to A), and so I resign myself to doing it immediately (i.e., leaving the switch at A) so as to save either one cent or two, depending on what happens. I will do this even if I share Don's intuition about certainty. Why pay before midnight to switch it to B if I know that after midnight I will pay to switch it back to A?

*[if the first die comes up 1 to 34]

I think I missed something on the algebraic inconsistency part...

If there is some rational independent utility to certainty, the algebraic claims should be more like this:

• U($24,000) + U(Certainty) > 33/34 U($27,000) + 1/34 U($0)
• 0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0)

This seems consistent so long as U(Certainty) > 1/34 U($27,000).

I'm not committed to the notion there is a rational independent value to certainty, I'm just not seeing how it can be dismissed with quick algebra. Maybe that wasn't your goal. Forgive me if this is my oversight.

This reminds me of the foolish decisions on "deal or no deal". People would fail to follow their own announced utility.

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?

It's rational to take the certain outcome if gambling causes psychological stress. Notwithstanding that stress is intrinsically unpleasant, it increases your risk of peptic ulcers and stroke, which could easily cancel out the expected gain.

If you crunch the numbers differently, you can come to different conclusions. For example, if I choose 1B over 1A, I have a 1 in 34 chance of getting burned. If I choose 2B over 2A, my chance of getting burned is only 1 in 100.

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?

Risk aversion, and the degree to which it is felt, is a personality trait with high variance between individuals and over the lifespan. To ignore it in a utility calculation would be absurd. Maurice Allais should have listened to his homonym Alphonse Allais (no apparent relation), humorist and theoretician of the absurd, who famously remarked "La logique mène à tout à condition d'en sortir". Logic leads to everything, on condition it don't box you in.

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.

I am intuitively certain that I'm being money-pumped all the time. And I'm very, very certain that transaction costs of many forms money-pump people left and right.

As long as it was only one occasion, I wouldn't make the effort to cross the room for two pennies. If I'm playing the game just once, and I feel a one-off payment of 2p tends to zero, I'll play with you, sure. £1 for a lottery ticket crosses the threshold of palpability, even playing once. I can get a newspaper for a pound. Is this irrational? I hope not.

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 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.

My experience of watching game shows such as 'Deal or No Deal' suggests that people do not ascribe a low positive utility to winning nothing or close to nothing - they actively fear it, as if it would make their life worse than before they were selected to appear on the show. It seems this fear is in some sense inversely proportional to the 'socially expected' probability of the bad event - so if the player is aware that very few players win less than £1 on the show, they start getting very uncomfortable if there is a high chance of this happening to them, because winning less than £1 is somehow embarrassing, and winning 1p is somehow significantly worse than winning say 50p. In contrast, on game shows where there's a 'double or nothing' option at the end, it is socially accepted that there's a high chance of winning nothing, so players seem to be much more sanguine about the gamble. I think the psychology of 'face' has a lot to answer for when it comes to such decisions.

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.

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

Constant's response implies that if someone prefers 1A to 1B and 2B to 2A, when confronted with the money pump situation, the person will decide that after all, 1A is preferable to 1B and 2A is preferable to 2B. This is very strange but at least consistent.

"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?

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.

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.

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.

It seems to me that your argument relies on the utility of having a probability p of gaining x being equal to p times the utility of gaining x. It's not clear to me that this should be true.

The trouble with the "money pump" argument is that the choice one makes may well depend on how one got into the situation of having the choice in the first place. For example, let's assume someone prefer 2B over 2A. It could be that if he were offered choice 1 "out of the blue" he would prefer 1A over 1B, yet if it were announced in advance that he would have a 2/3 chance of getting nothing and a 1/3 chance of being offered choice 1, he would decide beforehand that B is the better choice, and he would stick with that choice even if allowed to switch. This may seem odd, but I don't see why it's logically inconsistent.

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

Someone did not read the OP carefully enough.

Hint: re-read the definition of the Axiom of Independence.

Someone isn't thinking carefully enough.

Hint: I did not assert that X is strictly preferred to Y.

Caledonian, Nick T: "Risk aversion" in the standard meaning is when an agent maximizes the expectation value of utility, and utility is a function of money that increases slower than linearly. When an agent doesn't maximize expected utility at all, that's something different.

Do you really want to say that it can be rational to accept a 1/3 chance of participating in a lottery, already knowing that if you got to participate you would change your mind? Risk aversion is (or at least, can be) a matter of taste, this is just a matter of not being stupid.

Dawes gives a very similar 2-gamble example of a money pump on pg 105 of Rational Choice.

Caledonian, Nick T: "Risk aversion" in the standard meaning is when an agent maximizes the expectation value of utility

Oh, I agree.

I just measure utility differently than you do.

Caledonian, if utility is any function defined on amounts of money, then if you are maximizing expected utility, you cannot fall prey to the Allais paradox. 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.

you're violating rationality axioms like the one Eliezer gave in the OP

No. Those axioms are "if => then" statements. I'm violating the "if" part.

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.

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?

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.

Nick is right to object, but removing the phrase "on amounts of money" makes the statement unobjectionable -- and relevant and true.

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

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.

"Nainodelac and Tarleton Nick": This is not about risk aversion. I agree that if it is vital to gain at least $20,000, 1A is a superior choice to 1B. However, in that case, 2A is also a superior choice to 2B. The error is not in preferring 1A, but in simultaneously preferring 1A and 2B.

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 discussion reminded me of the Torture vs. Dust Specks discussion; i.e. in that discussion, many comments, perhaps a majority, amounted to "I feel like choosing Dust Specks, so that's what I choose, and I don't care about anything else." In the same way, there is a perfectly consistent utility function that can prefer A1 to B1 and B2 to B1, namely one that sets utility on "feeling that I have made the right choice", and which does not set utility on money or anything else. Both in this case and in the case of the Torture and Dust Specks, many comments indicate a utility function which places value on the feeling of having made a right choice, without regard for anything else, especially for whether or not the choice was actually right, or for the consequences of the choice.

Not sure if anyone pointed this out, but in a situation where you don't trust the organizer, the proper execution of 1A is a lot easier to verify than the proper execution of 1B, 2A and 2B.

1A minimizes your risk of being fooled by some hidden cleverness or violation of the contract. In 1B, 2A and 2B, if you lose, you have to verify that the random number generator is truly random. This can be extremely costly.

In option 1A, verification consists of checking your bank account and seeing that you gained $24,000. Straightforward and simple. Hardly any risk of being deceived.

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.

It's simple to show that no rational person would actually give money to a Pascal mugger, as the next mugger might threaten 4^^^4 people. I'm not sure whether this solves the problem or just sweeps it under the rug, though.

Well, if Pascal's Mugging doesn't do it, how about the St. Petersburg paradox? ;)

Oh wait... infinite set atheist... never mind.

I'm afraid I don't follow the maths involved, but I'd like to know whether the equations work out differently if you take this premise:

- Since 1A offers a certainty of $24,000, it is deemed to be immediately in your possession. 1B then becomes a 33/34 chance of winning $3,000 and 1/34 chance of losing $24,000.

Can someone tell me how this works out mathematically, and how it then compares to 2B?

The Allais Paradox is indeed quite puzzling. Here are my thoughts:

0. Some commenters simply dismiss Bayesian reasoning. This doesn't solve the problem, it just strips us of any mathematical way to analyze the problem. On the other hand, the fact that the inconsistent choice seems ok does mean that the Bayesian way is missing something. Simply dismissing the inconsistent choice doesn't solve the problem either.

1. If I understand correctly, you argue that situation 1 can be turned into situation 2 by randomization. In other words, if you sell me situation 1, I can sell somebody else (named X) situation 2 by throwing some dies and using your offer. More specifically, I throw a 100-sided die. If it's > 34, X looses. Otherwise, I play X's option with you. However, this can't be reversed. Given only situation 2, I can't sell situation 1, assuming I have only $0 initial capital.

Hence, it seems that assuming invertibility of situations (I can both buy and sell them) and unlimited money buffers for that purpose are important for the demanded consistency.

Nick,

"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."
The Porcine Mugging doesn't bypass the objection. Your estimates of the frequency of simulated people and pigs should be commensurably vast, and it is vastly unlikely that your simulation (out of many with intelligent beings) will be selected for an actual Porcine Mugging that will consume vast resources (enough to simulate vast numbers of humans). These things offset to get you workable calculations.

I would have chosen 1A and 2B, for the following reasons: Any sum of the order of $20,000 would revolutionize my personal circumstances. The likely payoff is enormous. Therefore, I'd pick 1A because I'd get such a sum guaranteed, rather than run the 3% risk (1B) of getting nothing at all. Whereas choice 2 is a gamble either way, so I am led to treat both options as qualitatively the same. But that's a mistake: if the value of getting either nonzero payoff at all is so great, then I should have favored the 34% chance of winning something over the 33% chance, just as I favored the 100% chance over the ~97% chance in choice 1. Interesting.

Surely the answer is dependednat on goal criterion. If the goal is to get 'some' money then the 100% option and the 34% options are better. If your goal is get 'the most' money then the 97% and the 33% options are better. However the goal might be socially construictued. This reminded me of John Nash whom offered one of his sectraries $15 dollars if she shared it equally with a co-worker but $10 if she kept it for her-self. She took the $15 and split it with her co-worker. She chose an option that maximised her social capital but was a weaker one economically.

I agree with Dagon.

This experiment assumes that the subjective probabilities of participants were identical to the stated probabilities. In reality, I feel like people are probably wary of stated probabilities due to experiences with or fears of shysters and conmen. That, is if asked to choose between 1A and 1B, 1B offers the possibility that the `randomising mechanism' that the experimenter is offering is in fact rigged.

Even if the experimenter is completely honest in their statement of their own subjective probabilities, they may simply disagree with that of the participants. Whatever `randomising mechanism' is suggested is, of course, almost certainly completely predictable given sufficient information - a die roll, or similar, predictable using Newtonian mechanics. That, is the experimenter's stated probability is purely a reflection of their own information concerning that mechanism, which may be completely at odds with the participant's knowledge.

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?

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.

"That is, the Axiom of Independence implies dynamic consistency, but not vice versa."
Really? A hyperbolic discounter can conform to the Axiom of Independence at any particular time and be dynamically inconsistent.

I would love to know if the results are different if you repeatedly expose people to the situation rather than communicate it in a formal way. They are likely to observe the outcomes of their strategy and adapt. Perhaps what is being measured is simply the numeracy of the subjects and not their practical inability to determine optimal strategies.

The lottery is another interesting example, what is being bought is the probability of a big win, not a statistically optimal investment. Playing the lottery genuinely increases the chance of you suddenly gaining a life changing amount of money. This is a perfectly rational choice.

Ummm, no. The money pump fails because of the REASON for the preference difference.

The reason is, as some have already stated, that in scenario 1B if you lose you know it's your fault you got nothing. In scenario 2B if you lose, you can rationalise it easily as "Would have lost anyway"

In your money pump scenario, we have a 1/3rd chance of playing 1. If we get to play 1, we know we're playing 1. So your money pump fails, because a standard player would prefer that the switch be on A at all 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.)

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 :)

The problem as stated is hypothetical: there is next to no context, and it is assumed that the utility scales with the monetary reward. Once you confront real people with this offer, the context expands, and the analysis of the hypothetical situation falls short of being an adequate representation of reality, not necessarily because of a fault of the real people.

Many real people use a strategy of "don't gamble with money you cannot afford to lose"; this is overall a pretty successful strategy (and if I was looking to make some money, my mark would be the person who likes to take risks - just make him subsequently better offers until he eventually loses, and if he doesn't, hit him over the head, take the now substantial amount of money and run). To abandon this strategy just because in this one case it looks as if it is somewhat less profitable might not be effective in the long run. (In other circumstances, people on this site talk about self-modification to counter some expected situations as one-boxing vs. dual-boxing; can we consider this strategy such a self-modification?)

Another useful real-life strategy is, "stay away from stuff you don't understand" - $24,000 free and clear is easier to grasp than the other offer, so that strategy favors 1A as well, and doesn't apply to 2A vs. 2B because they're equally hard to understand. The framing of offer two also suggests that the two offers might be compared by multiplying percentage and values, while offer 1 has no such suggestion in branch 1A.

We're looking at a hypothetical situation, analysed for an ideal agent with no past and no future - I'm not surprised the real world is more complex than that.

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"

If you'd ask any person capable of doing the math whether they would want to play 1A or 1B a thousand times you'd probably get a different answer, but not an answer that's more correct.

Also the utility value of money is not directly relative to the amount of money. Imagine that you would need a 1000$ dollars of money to save your dying relative with certainty by paying for his/her treatment. Good enough for explaining 1A > 1B, but doesn't resolve the contradiction with 2B > 2A.

But even a more revealing edit is based exactly onto the certainty. If you would be presented with these two questions, in such a fashion that you would get the money and get to know the result in 1 month after being presented with it. By selecting 1A you would have 0% chance that the plans you make would fail, and with 1B you would have a 1/34 chance that they would fail. Meanwhile regardless of whether you select 2A or 2B you will have to face uncertainty. So you would be frustrated while trying to make plans that are conditionally dependent with you getting the money.

As these conditions are not present in the presentation it's possible to rule these kind of instinctive judgments as flawed, but as it turns out, they're not foolish, on a general level. You could even make a claim that it's costly to perform the calculation that tells you whether the assurance is worth it - but of course instead of saying that you should just figure out how much value this assurance has in each given situation.

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.

I initially chose 1A and 2B, but after reading the analysis of those decisions, I agree that they are inconsistent in a way that implies that one choice was irrational (in the context of this silly little game). So I did some introspection to figure out where I went wrong. Here's what I found:

1) I may have misjudged how small 1/34 is, and this only became apparent when the question was phased as it is in example 2.

2) I think I assumed an implicit costs in these gambles. The first cost is a delay in learning the outcome of these gambles; the second is the implicit need to work to earn this money. I think that these assumptions are reasonable because there is essentially no realistic condition in which I would instantly see the results of a decision that might earn me $27,000; there would probably be a delay of several months (if working) or years (if investing) between making the decision and learning whether I got the money or not. This prolonged uncertainty has a negative utility, since I am unable to make firm plans for the money during that interval. This negative utility would apply to all options except 1A. Furthermore, earning $24,000 would realistically require several months of work on my part. However, a project that had a 1/3 chance of paying out $24,000 might only take a month. The implicit difference in opportunity cost between scenario 1 and scenario 2 has implications for the marginal utility of money in each scenario (making me more risk-averse in scenario 1, which implicitly has a higher opportunity cost).

These implicit costs are not specified in this game, so it is technically "irrational" to incorporate them into my decision-making. However, in any realistic scenario, such costs will exist (regardless of what the salesman says), so it is good that I/we intuitively include them in my/our decision-making.

While Elezier's argument is still correct (that you should multiply to make decisions based on probabilistic knowledge), I see a perfectly rational and utilitarian explanation for choosing 1A and 2B in the stated problem.

The clue lies in Colin Reid's comment: "people do not ascribe a low positive utility to winning nothing or close to nothing - they actively fear it". This fear is explained by Kingreaper: "in scenario 1B if you lose you know it's your fault you got nothing".

That makes the two cases, stated as they are, different. In game 1 the utility of U1($0) has negative value: a sense of guilt (or shame) over having made the bad choice, which doesn't seem possible in game 2 (because game 2 is stated in terms of abstract probabilities, see below).

This makes the inequations compatible:

U($24,000)   >   33/34 U($27,000) + 1/34 U1($0)

e.g. 24 > 33/34 · 27 + 1/34 · -1000

0.34 U($24,000) + 0.66 U2($0)   <   0.33 U($27,000) + 0.67 U2($0)

e.g. 0.34 · 24 + 0.66 · 0 < 0.33 · 27 + 0.67 · 0

Note that stating the game with the "switch" rule turns game 2 into one (let's call it 3) in which the guilt/shame reappears, making U3=U1 -- so a rational player with the described negative U1 would choose A in game 3 and there would be no money pump.

This solution to the paradox is less valid if it is made clear that the subject will be allowed to play the game many times.

Another interesting way to remove this as a possible solution would be to restate case 2 in more concrete terms, to make it clear that you won't get away not knowing that "it was your fault" if you loose:

4A. If a 100-face dice falls on <=34, win $24,000, otherwise win nothing.

4B. If a 100-face dice falls on <=33, win $27,000, otherwise win nothing.

Just to prevent the subject being pattern-matching and not thinking, we should add the phrase "note that if the dice falls on a 34 and you've chosen A, you win 24k, but if you've chosen B, you get nothing".

I believe game 4 is pretty equivalent to game 3 (the one with the switch).

I've checked Allais' document and it suffers the same flaw: it's not an actual experiment in which people are asked to choose A or B and actually allowed to play the game, but a questionnaire asking subjects what they would choose. This is not the same, among other reasons because it doesn't force the experimenter or subject to detail the mechanics of the game (and hence it is not stated whether the subject will be given that sense of shame or even allowed to "chase the rabbit").

It would be interesting to know the result of an actual experiment with this design, possibly with smaller figures to reduce the non-linearity of the utility functions -- since that's not what's being discussed here --, and with subjects filtered against innumeracy (since those are out of hope anyway).

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?

This is an old post, but I guess one resolution is that:

U($24,000) > 33/34 U($27,000) + 1/34 U($0 & Regret that I didn't take the $24000)

Which is consistent with:

0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0)

It's an interesting psychological fact that the regret is triggered in one case, but not the other.

I wonder if this bias is somehow trying to compensate for some other bias. Suppose you think the experimenter is overconfident, i.e., their log-odds are twice as much as they should; so, when they say 100% they do mean 100%, but when they say 97.1% they actually mean 85.2% (and when they say 34% they mean 41.8%, and when they say 33% they mean 41.2%). Now, Option 1B suddenly looks much uglier, doesn't it? (I'm not claiming this happens consciously.)

If flipping the switch before 12:00 pm has no effect on the amount of money one acquires why would one pay anything to do it? why not just flip the switch only once after 12:00 pm and before 12:05PM?

Question: do the rest of you actually find the choice of 1A clearly intuitive?

I think my intuition for examples like this has been safely killed off, so my replacement intuition instead says: "hm, clearly 34*(27-24) > 27, so 1B!" (without actually evaluating 27-24, just noting it's ≥1). Which mainly suggests that I've grown accustomed to calculating expectations out explicitly where they're obvious, not that I'm necessarily good at avoiding real life analogues of the problem.

1A. $24,000, with certainty.

1B. 33/34 chance of winning $27,000, and 1/34 chance of winning nothing.

2A. 34% chance of winning $24,000, and 66% chance of winning nothing.

2B. 33% chance of winning $27,000, and 67% chance of winning nothing.

I would choose 1A over 1B, and 2B over 2A, despite the 9.2% better expected payout of 1B and the small increased risk in 2B. If the option was repeatable several times, I'd choose 1B over 1A as well (but switch back to 1A if I lost too many times).

This does not make me susceptible to a money pump or a Dutch book (you're welcome to try, but note that I don't accept trades with negative expected utility). I simply think that my utility function at this time is such that Utility($24,000)>Utility(97% chance $27,000 + 3% chance $0), yet also Utility(34% chance $24,000 + 66% chance $0)<Utility(33% chance $27,000 + 67% chance $0)

I acknowledge that in one case, I trade expected payout for certainty, and in the other, I trade increased risk (not certainty) for expected payout. I'm not sure I see anything wrong with this, unless you're offended that I am willing to pay for certainty. Certainty is valuable in this world of overconfident people, accidents, and cheaters.

I don't like many of the standard arguments against capital punishment. In particular, I'm tired of the argument "if you just put an innocent person in jail, they might be exonerated later. If you execute an innocent person, and they are exonerated later, it's too late."

Of course, I then point out that people can be exonerated in the time between being convicted and being executed (which can be quite long sometimes), and the response is generally that in the life sentence there's always some chance of being freed due to exoneration while in the capital punishment case, there's a segment of time where there's no chance of being freed.

My response is that a chance X of being freed due to exoneration when sentenced to life in prison is, for some Y, equivalent to having a chance Y of being freed due to exoneration before your execution and zero chance of being freed after being executed. Since there are values of X that are considered acceptable, there are values of Y that must be acceptable too and therefore this argument cannot be used as a basis for an absolutist anti-capital-punishment stance.

I have yet to have anyone understand my response (the few times I've tried it, anyway). But it seems to me that I've stumbled onto something equivalent to the Allais problem. People don't think of "chance X of being freed" and "chance Y of being freed before execution and no chance of being freed after execution" as statements that can ever be equivalent, because they really don't like the certain failure in the last example, even though the two may be mathematically equivalent.

My resolution to this, without changing my intuitions to pick things that I currently perceive as 'simply wrong', would be that I value certainty. A 9/10 chance of winning x dollars is worth much less to me than a 10/10 chance of winning 9x/10 dollars. However, a 2/10 chance of winning x dollars is worth only barely less than a 4/10 chance of winning x/2 dollars, because as far as I can tell the added utility of the lack of worrying increases massively as the more certain option approaches 100%. Now, this becomes less powerful the closer the odds, are, but slower than the dollar difference between the two change. So a 99% chance of x is barely effected by this compared to a 100% chance of .99x, but still by a greater value than .01x, and the more likely option still dominates. I might take a 99% chance of x over a 100% chance of .9x, however, and I would definitely prefer a 99% chance of x over a 100% chance of 0.8x.

EDIT: Upon further consideration, this is wrong. If presented with the actual choice, I would still prefer 1A to 1B, but to maintain consistency I will now choose 2A > 2B.

I don´t really see how me chosing 1A > 1b and 2b >2A is a flaw of mine. First of all, my utility function, which i have inherited from millions of years of evolution, tells me to SOMETIMES take risks IF I CAN AFFORD IT, especially when the increasing stake outweighs the increasing risk.

This is how I see it: If it was my life at stake, I would of course try to raise the odds. But this is extra money. I don´t even starve if i don´t get the money.

If I am not certain I can get the money in case 2, I think that lowering my win-chance with 1/100 is worth to raise the stake with 3000 dollars, which is 3000/24000 = 1/8 of the original stake. When I lower my odds with 1 % I raise the stake with 12,5 %.

Since the outcome is random anyhow, AND not in my favor, and the risk increase is only 1/100, I take my chances.

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

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.

Forgive me if I'm misunderstanding something, but the way I see it, if I choose 1A, it means that I am willing to forgo (i.e. pay) 3000$ for an additional 1/34 ~ 3% chance of getting money. Then if I choose 2B, if means I am unwilling to forgo an additional 3000$ in exchange for an additional 1% chance of getting money. So what I learn from this is that the value I assign an extra percentage chance of getting money is somewhere between 1000$ and 3000$.

So here's why I prefer 1A and 2B after doing the math, and what that math is.

1A = 24000
1B = 26206 (rounded)
2A = 8160
2B = 8910

Now, if you take (iB-iA)/iA, which represents the percent increase in the expected value of iB over iA, you get the same number, as you stated.

(iB-iA)/iA = .0919 (rounded)

This number's reciprocal represents the number of times greater the expected value of iA is than the marginal expected value of iB

iA/(iB-iA) = 10.88 (not rounded)

Now, take this number and divide it by the quantity p(iA wins)-p(iB wins). This represents how much you have to value the first $24000 you receive over the next $3000 to pick iA over iB. Keep in mind that 24/3 = 8, so if $1 = 1 utilon in all cases, you should pick iA only when this quotient is less than 8.

1A/(1B-1A)/[p(1A wins)-p(1B wins)] = 369.92
2A/(2B-2A)/[p(2A wins)-p(2B wins)] = 1088

I have liabilities in excess of my assets of around $15000. That first $15000 is very important to me in a very quantized, thresholdy way, but it is not absolute. I can make the money some other way, but not needing to - having it available to me right now because of this game - represents more utility than a linear mapping of dollars to utility suggests, by a large factor.

The next threshold like this in my life that I can think of is "enough money to buy a house in Los Angeles without taking out a mortgage," of which $3000 is a negligible portion.

I'd say that the utility I assign the first $24000 because of this lies between 370 and 1080 times the utility I assign the next $3000. This is why I take 1A and 2B given that this entire thing is performed only once. Once my debts are paid, all bets (on 1A) are off.

If we're dealing with utilons rather than dollars, or I have repeated opportunity to play (which is necessary for you to "money pump" me) iB is the obvious choice in both cases.

Assuming this is a one off and not a repeated iteration;

I'd take 1A because I'd be *really* upset if I lost out on $27k due to being greedy and not taking the sure $24k. That 1/34 is a small risk but to me it isn't worth taking - the $24k is too important for me to lose out on.

I'd take 2B instead of 2A because the difference in odds is basically negligible so why not go for the extra $3k? I have ~2/3rds chance to walk away with nothing either way.

I don't really see the paradox there. The point is to win, yes? If I play game 1 and pick B and hit that 1/34 chance of loss and walk away with nothing I'll be feeling pretty stupid.

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.  The die comes up 12.  After 12:00PM and before 12:05PM, you pay me a penny to throw the switch to A.

But why would I pay to switch it back to A when I've already won given the conditions of B? And as Doug_S. mentions, you can take my pennies if I'm getting paid out tens of thousands of dollars.

I do see the point in it being difficult to program this type of decision making, though.

Oh, here I come again, I've already commented in similar fashion elsewhere, and several people said the same here: nothing vs. non-nothing as a binary switch may work better if the situation is not repeated to "add up to normality" but only played once. One can argue that repeats may seem as being played once each time, but, being creatures gifted with memory, we can notice a catch of encountering such situations often and modify behaviour.

I would set up an insurance company that pays people to $24,500 to pick 1B and keeps their earning if they win. They get slightly more risk-free money and I profit massively. Isn't that the whole point of insurance?

I think this might just be a rephrasal of what several other commenters have said, but I found this conception somewhat helpful.

Based on intuitive modeling of this scenario and several others like it, I found that I ran into the expected “paradox” in the original statement of the problem, but not in the statement where you roll one dice to determine the 1/3 chance of me being offered the wager, and then the original wager. I suspect that the reason why is something like this:

Loosing 1B is a uniquely bad outcome, worse than its monetary utility would imply, because it means that I blame myself for not getting the 24k on top of receiving $0. (It seems fairly accepted the chance of getting money in a counterfactual scenario may have a higher expected utility than getting $0, but the actual outcome of getting $0 in this scenario is slightly utility-negative.)

Now, it may appear that this same logic should apply to the 1% chance of loosing 2B in a scenario where the counterfactual-me in 2A receives 24000 dollars. However, based on self-examination, I think this is the fundamental root of the seeming paradox: not an issue of value of certainty, but an issue of confusing counterfactual with future scenarios. While in the situation where I loose 1B, switching would be guaranteed to prevent that utility loss in either a counterfactual or future scenario, in the case of 2B, switching would only be guaranteed to prevent utility loss in the counterfactual, while in the future scenario, it probably wouldn’t make a difference in outcome, suggesting an implicit substitution of the future as counterfactual. I think this phenomenon is behind other commenters preference changes if this is an iterated vs one-shot game: by making it an iterated game, you get to make an implicit conversion back to counterfactual comparisons through law of large numbers-type effects.

I only have anecdotal evidence for this substitution existing, but I think the inner shame and visceral reaction of “that’s silly” that I feel when wishing I had made a different strategic choice after seeing the results of randomness in boardgames is likely the same thought process.

I think that this lets you dodge a lot of the utility issues around this problem, because it provides a reason to attach greater negative utility to loosing 1B than 2B without having to do silly things like attach utility to outcomes: if you view how much you regret not switching back through a future paradigm, switching in 1B is literally certain to prevent your negative utility, whereas switching in 2B probably won’t do anything. Note that this technically makes the money pump rational behavior, if you incorperate regret into your utility function: after 12:00, you’d like to maximize money, and have a relatively low regret cost, but after 12:05, the risk of regret is far higher, so you should take 1A.

I’d be really interested to see whether this expeirement played out differently if you were allowed to see the number on the die, or everything but the final outcome was hidden.

It seems that the mistake that people commit is imagining the the second scenario is a choice between 0.34*24000 = 8160 and 0.33*27000 = 8910. Yes, if that was the case, then you could imagine a utility function that is approximately linear in the region 8160 to 8910, but sufficiently concave in the region 24000 to 27000  s.t. the difference between 8160 and 8910 feels greater than between 24000 and 27000... But that's not the actual scenario with which we are presented. We don't actually get to see 8160 or 8910. The slopes of the utility function in the first and second scenarios are identical. 

"Oh, these silly economists are back at it again, asserting that my utility function ought to be linear, lest I'm irrational. Ugh, how annoying! I have to explain again, for the n-th time, that my function actually changes the slope in such a way that my intuitions make sense. So there!" <- No, that's not what they're saying! If you actually think this through carefully enough, you'll realize that there is no monotonically increasing utility function, no matter the shape, that justifies 1A > 1B and 2A < 2B simultaneously. 

"These two equations are algebraically inconsistent". Yes, combining them results into "0 < 0", which is false.

It seems that the axiom of independence doesn't always hold for instrumental goals when you are playing a game.

Suppose you are playing a zero-sum game against Omega who can predict your move - either it has read your source code, or played enough games with you to predict you, including any pseudorandom number generator you have. You can make moves a or b, Omega can make moves c or d, and your payoff matrix is:
   c  d

a 0  4

b 4  1

U(a) = 0, U(b) = 1.

Now suppose we got a fair coin that Omega cannot predict, and can add a 0.5 probability of b to each:

U(0.5 a + 0.5 b) = min(0.5*0  + 0.5*4, 0.5*4 + 0.5*1) = 2

U(0.5 b + 0.5 b) = U(b) = 1

The preferences are reversed. However, the money-pumping doesn't work:

I have my policy switch at b. You offer me to throw a fair coin and switch it to a if the coin comes up heads, for a cost of 0.1 utility. I say yes. You throw a coin, it comes up heads, you switch to a and offer me to switch it back to b for 0.1 utility. I say no thanks.

You could say that the mistake here was measuring utilities of policies. Outcomes have utility and policies only have expected utility. VNM axioms need not hold for policies. But money is not a terminal goal! Getting money is just a policy in the game of competing for scarce resources.

I wonder if there is a way to tell if someone's preferences over policies is irrational, without knowing the game or outcomes.