The Ellsberg Paradox

I'm reading Iconoclast, by Gregory Berns, the distinguished chair of neuroeconomics at Emory University. He's a professor of psychiatry and economics, which makes for an interesting combination. The book is about the way successful and creative people think and act, with a special focus on fear and how it affects behavior. Early in the book, Berns describes something called The Ellsberg Paradox. Berns uses it as an example of people's fear of the unknown.

There are two large urns placed in front of you. The urns are completely opaque, so you cannot see their contents. The urn on the left contains ten black marbles and ten white ones. The urn on the right contains twenty marbles, but you do not know the proportion of black to white. Now, the game is to draw a black marble from one of the urns. If you are successful, you win $100. You only have one chance, so which urn will you draw from? Keep the answer in mind.

Let's play again. Now, the game is to draw a white marble. Again, you only have one chance, so which urn will it be?

Most people when confronted with these choices choose the urn on the left -- the one with the known proportions of black and white marbles. And therein lies the paradox. If you choose the left-hand urn when trying to pull a black marble, that means you think your chances are better for that urn. But because there are only two colors in both urns, the odds of pulling a white must be complementary to the odds of pulling a black. Logically, if you thought the left-hand urn was the better choice for a black marble, the right-hand urn should be the better choice for a white marble. The fact that most people avoid the right-hand urn altogether suggests that people have an inherent fear of the unknown (also called the ambiguity aversion).

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• [monstrinho_do_biscoito]

  hang on, if i need both a black and a white to maximise my money, then it’s in my interest to go for the left urn with the equal ratio.

• [retchdog]

  The statement is reasonable. The paradox as stated is resolved (in a very weak way) if we suppose that the player assumes the urn on the right has 10 of each marble as well.

  It’s a weak resolution because if you repeat this experiment, players use the “left, left” strategy much more than half the time.

  This statement is an improvement over its incarnation as the “boxer vs wrestler” problem: two equally-skilled boxers will get 2:1 odds on either, but if the best boxer in the world and the best wrestler in the world are in a match, would you take 2:1 payoff on either?

  (The idea being that the uncertainty about two totally different styles is a different kind of uncertainty. Of course, nowadays everyone knows that the wrestler would obliterate the boxer.)

• [nprnncbl]

  I agree with those saying this is not a paradox, and further I don’t think choosing the first urn in both cases is fallacious reasoning. This can be viewed as a game theoretic max-min problem: what’s the best I can do in the worst case? That is, I want to maximize my expected payoff based on my choice of urn, where the distribution of balls in the unknown urn is as unfavorable as possible.

  In the first case, the worst case is that all the balls in the second urn are white, yielding a payoff of zero for that urn; but the first urn has an expected payoff of $50, so it is preferred.

  In the second draw, we have gained no additional information about the second urn, and so we again make a worst-case assumption, only this time, the worst case is that all of the balls are black, the first urn is again preferable.

  The fallacy lies in assuming that we think about the second urn probabilistically: not only is this not what we do, it is not the only way to analyze the problem logically.

  Thinking probabilistically, #18 Airshowfan averaged over possible distributions of the balls in the hidden urn, but made an assumption about this distribution, that it was uniform. This is often a reasonable assumption, in that it is the distribution over the number of balls in the urn which has maximum entropy, although as it turns out, whatever the distribution of balls in the urn, the probability of drawing white vs. black are, of course, complementary. And in the case of a uniform distribution, we end up again with an expected gain of $50 for either color. So there is no prior reason to prefer the second urn.

  So (Bayesian) probability (with the maximum entropy principle) yields no preference; but game theory (max-min principle) yields a preference for the first urn.

  If we change the number of balls in the first urn so the odds are not even for that urn, then thinking probabilistically will lead us to choose urn 1 for the favored color, and urn 2 (which still has even odds) for the disfavored color. If n>10 for one of the balls, then this strategy has an expected yield of (n/20)*100 + 50 dollars. But the game theoretic prediction remains unchanged, and we will choose the first urn both times.

  Things can change if we know we are going to be making multiple draws, and there is substantial research into the exploration-exploitation* tradeoff, from both Bayesian and min-max perspectives. (*Question for Portuguese speakers: how would you translate that, since both words are exploração?)

  In this case, if we know in advance we are making the two draws, then the worst case distribution for the second urn is evenly split, and we have no preference. If we choose the second urn, the ball we draw does not change that assessment. So max-min no longer predicts a preference for the first urn.

  Probabilistically, if the first urn is balanced, then we have seen that the second urn has the same expected payoff, so we have no reason to prefer it base on what we know. But if we draw from the second urn, we gain information about it, even from a single draw. Drawing a white ball yields more support to there being a preponderence of what balls, and less to there being a preponderence of black. So, probabilistically, we would choose from the second urn first, and then if we chose a white ball, choose from the second urn again, otherwise switch to the first urn for the second ball. (Note that I’m assuming that all choices are made with replacement.)

  So if we know in advance we are making the two draws, the predictive capability of the two theories is reversed: max-min gives no preference, while Bayesian probability yields a non-trivial strategy.

• [airshowfan]

  KyleTexas is right. Not only about how you have more info if you didn’t replace the first marble you drew (but the puzzle seems to imply that you do replace it, since it does everything it can to make things 50-50 and does not seem to have an “A-ha, but I didn’t say you replaced the marble!” trick answer), but also about “It’s not really interesting to say people are adverse to risk ‘all things being equal’. What’s interesting is how *unequal* odds will a person accept in exchange for more certainty?”. That’s kinda the point I was trying to make: If the odds are equal, OF COURSE anyone will pick more knowledge. A much more interesting question is, indeed, whether you’re willing to pass up better odds in cases when you have less-complete knowledge.

  And in response to #25′s “I KNOW I have a 50% chance on the left. I have no way of knowing my chances on the left. My chances swing as low as 5% all the way up to 95%, but I don’t KNOW”; You can “know” your odds in that weird Schroedinger’s Cat kind of way, by adding up all possible odds and assigning them equal probability. You don’t know your odds, but if you average all your possible odds (which really is not that different from averaging all the possible marble picks from the urn where you do know the odds), you get your odds given the information you have (and it turns out that they’re 50-50). Since you have no information about how the person who set up the game set it up, just assume any outcome is equally likely.

  All you have to do is divine from what you know of me whether I’m more likely to put the poison into my own goblet, or my enemy’s. ;)

• [qlfwyyd]

  Thanks to everyone who posted comments. I was incredibly confused – and frustrated – by the initial post. But I think some of these further explanations have cleared it up. It would have been much clearer to me if in the initial post they explained that logically the urn on the right has the same chance of drawing a black or white ball as the urn on the left using AIRSHOWNFAN’s explanation.

• [Falcon_Seven]

  This whole ‘exercise’ focuses on the ‘probability’ that you will chose one scenario over another based on the information supplied. There is no ‘paradox’, probabilistic or otherwise, that represents a negative consequence, so you will choose whatever ‘gamble’ you deem is most likely to ‘payoff’, given the information you are to base your decision on. You are either rewarded or you are not, so this is essentially a ‘guessing game’.

  However, if the ‘paradox’ were framed thusly, “‘Box A’ and ‘Box B’, both of which are capable of delivering either unbelievably excruciating pain or unbelievable pleasure -however, it is impossible to tell which box will deliver what at any given time until you stick your hand into it- yet there is a small but detectable difference in ‘Box B’ that it could deliver unbelievable pleasure, at twice the intensity of ‘Box A’ and at a slightly higher rate than ‘Box A’.”, what would you choose if this were the only information on which you were to base your decision?

  This exercise is similar to an Einsteinian ‘thought experiment’ in an effort to explain choice. The only problem with it is that it does not represent the ‘real world’ when it comes to risk because there is no downside to your choice.

• [gregberns]

  Let’s be clear, I never said that it is irrational to stick with known probabilities, but it is a paradox from a mathematical point of view (as I show in 36). But keep in mind that we are talking about subjective probabilities, which are different than objective probabilities. Subjective probabilities are revealed by a person’s choices, and in this example, many people flip them back and forth. Why?

  Lots of discussion here that is essentially rationalization of this behavior, and I think many have touched on the reason I believe: the human brain evolved in a world that imposed higher costs on ambiguity. When researchers at Caltech did fMRI on the brains of people faced with similar dilemmas, they observed strong responses in parts of the brain associated with fear and arousal — what most feal as gut-level responses.

  The point of the book (and this example)is that if you want to do truly innovative things, it takes a lot of effort by the prefrontal cortex to override these deep seated responses. Most people don’t.

• [mwitthoft]

  Anyway, thanks for making it clear that I don’t want to buy the book.

• [lvvvop]

  I don’t know what is in the urn to the right, but it’s a game, so I risk nothing either way: I can’t risk what I don’t have.
  But what if it’s not a game, what if my survival depends on which urn I choose and which colour I specify? The point I’m trying to make is that theoretically we have time to assign probabilities and then make a decision that gives us the best outcome, how often though, do we, when confronted with a situation that requires us to act quickly do we immediately choose what we know as opposed to what we don’t? Maybe always, and those who know that make a lot of money at our expense, and deep down we know that too: But how much are you willing to risk before you are no longer willing to risk anything?
  Think of Monty Hall: two thirds of the time I’ll win if I switch, but I’ll have only one chance to play, what if chose right on my first turn? And there lies the dark heart of probability: I only have one chance, therefore I need to know more than what happens in the long run to make an informed decision.

• [Dewi Morgan]

  Rechecking my reasons for picking the right urn, I think it was healthy skepticism. The “unknown” urn is the mystery. “What’s in the box?” – it’s well worth losing the potential for the win, to find what the “gimmick” is: you can’t do that without checking the mystery urn. You’d walk away and *never know* if the urn held a puma. That’s not worth a 50% chance of a win.

• [Ugly Canuck]

  Yes. Instead of marbles black & white use boxes of snapping turtles and fluffy puppies.

• [mystical limpet]

  “You have no way of knowing what’s in the right jar, so drawing the black ball might be impossible (0/20). However, it’s just as likely that drawing a black ball is garaunteed (20/20).”

  A lot of this assumes that the 2nd urn contains a truly random sample of marbles. But the original question doesn’t say that, it just says you don’t know the proportion of black to white marbles. For all I know the person giving me the test deliberately filled the 2nd urn with white marbles, hoping I would be drawn to the unknown.

  It’s like in Family Guy, when Peter is offered the prize of either a boat (known) or the mystery box (unknown). He chooses the mystery box, since it could be anything, including a boat.

  Anyhow, if the point of the exercise is that successful people (whatever that means) are more likely to take risks, I guess that’s true. But they’re only successful because their risk paid off. It would be just as true to say that unsuccessful people take more risks.

• [KyleTexas]

  The Wikipedia article posted by SCHORSH @ #15 is really good:

  http://en.wikipedia.org/wiki/Ellsberg_paradox

  Seems to me, especially from reading this discussion, that the answer is almost an Occam’s Razor-type thing. Basically, when people are presented with two statistically equal “gambles”, but one is dead simple to figure out, and the other requires a bit of mathematical figuring to understand, people overwhelmingly pick the simpler one.

  Wow, people don’t like to think. Unless they think there’s something in it for them, or some pain that can be avoided. If not, they’d just as soon not be bothered.

  I guess according to the “Ellsberg Paradox”, if you were a dictator you could pretty much invade any other country for any reason, and the populace wouldn’t really question you as long as you kept them insulated from the cost of the invasion. Of course, in order to do that, you’d need to make sure the people doing the actual fighting were only a small percentage of the population, which would mean you’d need some mechanism to keep them conscripted into the military indefinitely so that nobody among the broader population would be compelled to serve (i.e., consider whether the balls in the jar add up). It might also be a good idea to enact a monetary policy of incredibly low interest rates, facilitating unprecedented access to credit. That way people could enrich themselves by selling each other houses without actually having to produce anything — sort of like asking people to choose a white ball from two jars, one of which has 19 white balls, and the other has an unknown number of white balls. On the liberal blogs on a place called the internet, some geeks might debate the merits of which jar you should choose, but that’s cool. Just so long as nobody asks, “Why the fuck are you offering me $100 to pick marbles out of jars?”

• [certron]

  There is another trick at work here… being paralyzed into non-action by the uncertainty between the available choices.

  Fear is the mind-killer.

  (It doesn’t hurt that I’m currently listening to the Adam Freeland song with a similar title.)

• [Nylund]

  Think of this: You have two coins, one you know is fair, and the other you have no idea if its fair, two heads, or two tails. If I want to get a heads, which coin would you choose to flip? well, you might think, “I know one has a 50-50 chance of getting heads. The other may have a zero percent chance, a 100%, or a 50% chance. If you assume all are just as likely, your choice of coins does not matter. The weighted average of the 2nd coin, 1/3(50%) + 1/3(100%) +1/3 (0%) = 50%.

  Now it may not be right to assume that each is just as likely, but without any more information, its about as logical an assumption as you can make.

  So, if each coin is just as likely, it doesn’t matter which you choose, so regardless of whether or not I wanted heads or tails, I’d be indifferent to either coin.

  But, if I knew beforehand that I was going to do this twice, I’d of course pick the “unknown” coin in an attempt to gain more information about it. I could then eliminate one of my 3 terms from my weighted average stated above.

  So not knowing that there will be a second choice affected my first choice. The “unknown” that screws me up in this problem is not the “unknown” contents of the urn, but the “unknown” fact that I will be asked the question twice in a row.

• [arbitraryaardvark]

  Instead of marbles, package contained Bobcat.
  Would not play again.

• [mdh]

  #53 – the one thing i do know is they’ll give me 20 bucks for being right, and I don’t know what happens to the 20 bucks if I’m wrong.

  That’s at least evidence of the potential for their own self-interest right there, so I assume there is at least some chance they’re gaming me.

  And I loved the Princess Bride ref.

• [CharlesSpongeworth]

  Maybe I’m missing the point, but isn’t it just an Optimist/Pessimist game? I thought that the 50:50 odds of getting a black marble weren’t that good, so I thought I’d take a gamble with the urn on the right, which might be better or might be worse. I would choose it again for the white marble as I’m still feeling lucky.

  Does this mean there’s something wrong with my head?

• [ryuubu]

  I chose the right one both times…

• [bardfinn]

  Even the Wikipedia article is badly worded:

  “The balls are well mixed so that each individual ball is as likely to be drawn as any other.”

  Which makes a claim of probability of draw, which goes directly to proportion – because of that sentence, the yellow and black balls are in equal proportion to the red balls. Either the sentence is false or you now know the exact proportion.

  What that sentence /ought/ to read as, is “The balls are well mixed so that there are no abnormal groupings of one colour.”

• [Anonymous]

  @kyletexas,

  Brilliant! Funny and to the point about the silliness of this experiment.

  As to the “point” of the book, about innovation and so-called “gut-level” versus intellectual or innovative decision-making — this has a lot more to do with learning styles etc. For the majority, “gut-level” is social behaviour — for some, math and logic are the fundamental elements, and social is hard.

  Innovation happens only when someone is out-of-touch with the world around them. This is practically the definition of the word.

• [gregberns]

  Full disclosure: I am the author of the original passage which is quoted from my new book. I’m glad that it has provoked such a debate.
  As many here have noted, all things being equal, they prefer the urn where probabilities are known. If you think your chance of picking a black marble is greater in the left urn then your subjective probabilities are:
  P(B)L > P(B)R
  but the probability of picking a white marble is complimentary to picking a black marble:
  P(B)L = 1 – P(W)L
  P(B)R = 1 – P(W)R
  if you plug these back into the original ordinal relationship, you get:
  P(W)R> P(W)L
  which is why it’s “paradoxical” to stick with the left urn.
  But my book is about the neural circuitry that underlies both innovative thinking and why the brain is not really evolved to be truly innovative (because it operates under a strict energy budget). Uncertainty is only part of it. Perception is another big part. Check out the excerpts on Amazon!

• [Divers Hands]

  One of the larger problems I find with some of the logic here is that it misses the real point of how possible probabilities function when looking at the thought experiment proposed by Schroedinger’s famed Cat. Shroedinger was not trying to point out that all possibilities exist prior to a choice, but how ridiculous that prospect is when applied to the real world. For example, prior to choosing which urn one chooses from, the mathematical reasoning is that the probabilites are equal. However, the minute one makes the choice to choose from the unknown urn, the probabilities cease being equal, as whatever the actual quantities of marbles in the urn are is now the actual odds one is dealing with. Hence Schroedinger’s use of a cat: no one really beleives that the cat is actually both alive and dead in the box. It is one or the other. Only in the realm of theory is the cat both alive and dead.

  Regardless, I still fail to see how any part of this paradox (either the poorly worded one from the post or the more logically consitent one at the Wikipedia link) demonstrate anything pertaining to humanity’s fear of the unknown. as the postings here seem to demonstrate, it really seems to seperate us into how inclined we are to place our faith in a theoretical universe or an actual one.

• [robwiss]

  I’m not sure I agree with the author’s premise. Making a less than optimal or irrational choice due to not fully understanding a problem doesn’t seem like an example of fear of the unknown to me. Rather, it seems like straightforward evidence that average people are awful at probability theory. You don’t need to do an experiment to understand that. All you need to do is look at the class average for intro statistics and probability classes.

  A more interesting experiment would be to see what odds the urn with the known odds would need to have in order to get people to prefer the unknown urn.

  I’d guess not much below 50%, maybe 40%.

• [Teej]

  Nonsense.

• [Anonymous]

  No matter which way you bet, the expected value of the outcome is $100 if you don’t know how many black and yellow balls there are.

  More importantly, if an adversary was loading the urn, he or she could make the expected value no less than $100 if A&D or B&C are chosen, but could affect the expected value if A&C or B&D are chosen. I think people are wary of situations where an adversary may know in advance which decision they are likely to make. This seems well founded, because if there were even a slight statistical likelihood of humans choosing A&C more than B&D, it would benefit the urn owner to put more black balls in the urn. People might not know *why* or even *if* they were more statistically inclined to choose A&C over the other choices, but the smart people would pick a safe choice over one that might allow an adversary to get ahead.

  There have been a few studies showing that when playing these kinds of games, people play to minimize other people’s winnings as well as to maximize their own.

• [neurolux]

  How would I choose which urn to draw from? I guess I’d flip a coin.

• [airshowfan]

  @#49: There are other words for “exploitation” in Portuguese, but they are not English cognates for “exploration”; they are cognates for “abuse” and “advantage” and “use”.

  @#50, paragraph 2: That’s what prompted me to make the Princess Bride reference: Either you can deduce something about the person who set up the game, or you cannot. If you can, then use that information, and offset the odds from 50-50 by some amount. If you cannot, then I don’t see any course of action more reasonable than assuming that every outcome is equally likely.

  As far as I can tell, there is no real difference between saying “this urn contains marbles, half are white and half are black, so I have a 50% chance of picking out a black one” and saying “this urn contains some black marbles and some white marbles, I have no idea what the proportion is, so given my knowledge it is fair to expect a 50% chance of picking out a black one”.

  If you disagree with my statement (that there is no real difference), you might claim “But there could be NO black marbles in there” or “the person setting up this game could want you to lose and have done everything in their power to make that happen” or some other possibility. Yes, that is true, but the opposite of each of those possibilities is about as probable, if you have no knowledge about it. I mean, as an illustrative example, take the urn you DO know the ratio for (it’s half and half). Maybe a minor hand injury some time ago leads you to be more likely to draw from the left side of the box, where the concentration of white marbles is a tad higher. Or maybe you like to draw from near the bottom, where the concentration of white marbles is a tad higher. Or maybe your skin cells have very minor photosensitivity which will draw your fingertips towards white marbles. All kinds of things COULD make it more likely for you to draw a white marble from the half-and-half urn. But since you have zero knowledge of these factors, you assume that each of those factors is about as likely to cause a white-marble preference as a black-marble preference, and you say “Since I cannot know, I’ll assume every outcome (picking any one marble) is equally likely”.

  Why is it ok to make that assumption with respect to taking any marble from the half-and-half urn, but not with respect to the distribution of marble colors on the other urn?

• [dainel]

  Keep the answer in mind.

  Did everyone picked the left urn? Am I the only one who picked the right urn? Both times?

• [Ugly Canuck]

  A bird in hand is worth two in the bush.
  Does “fear” = “wariness”? Or is “fear” perhaps too strong a word?

• [JArmstrong]

  “The urn on the right contains twenty marbles, but you do not know the proportion of black to white.”

  Nor do I know whether all twenty marbles are blue or green or candy-striped. It wasn’t until the third sentence of the third paragraph that I knew there were only black and white marbles in the second urn; I don’t know if this would have affected my decision, but the set-up could have been crafted differently.

• [dragonfrog]

  If you’re doing the second test after the first, then you may well have info to point you in the right direction.

  If you picked from the left urn and didn’t replace the marble, then the ratio will be 10-9 one way or the other. If you won the first time, stick with the left urn; if you lost the first time, switch to th right urn.

  Just sayin, is all…

• [bpratt]

  “Logically, if you thought the left-hand urn was the better choice for a black marble, the right-hand urn should be the better choice for a white marble.”

  Well, no. The left hand urn is a better choice in both cases since you have more information about it. I think reading that book would be pretty annoying if it’s based on that kind of spurious reasoning…

• [Wordguy]

  @ #3: Right. You know you have a 50/50 chance with the urn on the left. For the urn on the right you know that you either have the same 50/50 chance, or a worse chance for one of the colors. Hopefully the whole book isn’t like that.

• [chip]

  This is an unfair test of logic. The “You only have one chance” in the first paragraph pushes you toward (what ends up being) the illogical choice. By the time you find out you get another shot at it, it’s too late.

  If I had known from the beginning that I would be playing twice, I’d have chosen from the right urn both times. If the proportion of black to white in that urn is not 50/50, then I would have a better chance on one of the two tries. If the proportion was 50/50, then it would have been a wash.

• [Tommy]

  That example is just horrible. You’re not making any assumption about the ratio of marbles in the urn on the right. How would you?

• [TJIC]

  @2 posted by JArmstron:

  > Nor do I know whether all twenty marbles are blue or green or candy-striped.

  Well said.

  Based on the environments we evolved in, the left urn makes a hell of a lot more sense.

  Would you rather forage in a field that has both berry bushes AND apple trees, or in another field that you’ve never been to?

  There are some interesting quirks in human nature, but there are huge numbers of “illogical” behaviors that, when analyzed, either (a) were stated so badly by the researchers that there’s no obvious illogic in people’s reactions, or (b) actually make sense – e.g. loss aversion, because declining marginal utility arises from the real world we live in.

• [PaulR]

  If the options was to lose $100, the choice would be to go for the right urn.

  It’s a fairly well-known phenomenon called prospect theory. We’re willing to take a chance on a loss, but not so much when there’s a gain. Bruce Schneier has discussed this a few times on his blog.
  http://www.schneier.com/blog/archives/2008/04/risk_preference.html

• [jmzero]

  The second choice is no different than the first, since you have no more information about the unknown jar than you did when you started.

  I think it’s interesting that most people would choose the “known” jar – and I certainly would – but the second choice is really no different.

• [Dewi Morgan]

  My personal picks were: first round, pick the urn with the unknown proportions, in order to gain information, even if I gained no money. It wouldn’t be much information, but it would be “there is at least one ball of the colour I extracted”. I did not (because I hadn’t read onwards) expect to be able to use the information for anything, but I did it anyway, because information is always useful.

  Second round, as others have pointed out, my answer was “same urn if I lost last time, otherwise, change urns”.

  I disagree that it’s aversion to uncertainty: If that were the case, then everyone would select as I did. I think it’s a far less esoteric aversion: the aversion to methematical complexity.

  The odds of drawing from the 50% urn are *easy*. The odds of drawing from the other urn would take some time to think about. Nobody wants to take that time, when there is a reward waiting to be had. Sure, intuitively the odds appear to be 50-50 too, but just a “more complicated” 50-50, with potential for pitfalls. Avoid the complexity, go for the simple.

  The only reason I chose different was that I decided NOT to bother figuring out the odds. Instead, I decided that since, knowing my luck, I’d lose a 50-50 split nine times out of ten, I’d go with the other one, because then I’d at least gain something from the experience even if the pot were rigged.

• [HarveyBoing]

  If you choose the left-hand urn when trying to pull a black marble, that means you think your chances are better for that urn.

  The “paradox” is an interesting observation. But it’s not a true paradox. In particular, the above quote is logically unjustified. It makes an unsubstantiated claim with an obvious counter-example: if you choose the left-hand urn when trying to pull a black marble, that means that you acknowledge your changes might be better for the other urn, but you prefer to stick with a known “decent chance”.

  Change the puzzle so that the known ratio is 19 white marbles and 1 black marble, and now I am more likely to choose the other urn because it would be difficult for that urn to have a worse chance of choosing black than the left-hand urn (assuming we’re assured it has only black and white marbles). Even if the right-hand urn does have fewer black marbles than the left-hand one (i.e. none), I’ve lost very little by taking the gamble, because I had very little chance of drawing a black marble from the left-hand urn anyway.

  It’s the fact that the left-hand urn is “good enough” that leads to the identical choice in both cases, not that it is known to be a superior choice as compared to the right-hand urn.

  There’s no paradox, though I agree that it’s an interesting way to present a concept of human behavior with respect to the unknown. The reluctance to choose the right-hand urn certainly is related to the tendency to avoid the unknown, but there’s no paradox in always preferring the known to the unknown in an example like this.

  The example doesn’t even demonstrate that the unknown is always less-preferred. It only demonstrates that it is when all else is equal. In other situations, the same person might well prefer the unknown.

• [maoinhibitor]

  You don’t know the proportion of black to white marbles in the urn on the right when you are asked to draw a black marble.

  You don’t know the proportion of black to white marbles in the urn on the right when you are asked to draw a white marble.

  Making your first choice does not change the state of the marbles in the urn on the right. You still have absolutely no idea about the proportion over there.

  Is this a poor summary of the Ellsberg Paradox? Or is this just a good illustration of the weakness of binary thinking?

• [Daemon]

  That example definately doesn’t support the concept that we have a fear of the unknown… at most it shows a preferance for the known, provided the known is sufficiently non-lame.

• [Anonymous]

  I always tell people as an illustration of my understanding:
  When it comes to the lottery, one ticket is infinitely better than none; but two or two hundred is really no better. An over-simplification, to be sure, but a useful one at that.

• [randomcat]

  #3 is spot-on. Not having any information about the right urn, the left urn is the only sensible choice. Especially considering the possibility that someone with $100 at stake may have had access to the right urn. Rational pursuit of self-interest is not the same thing as fear.

• [doug117]

  Agree with #3 et al. Spurious thinking.

  “Logically, if you thought the left-hand urn was the better choice for a black marble, the right-hand urn should be the better choice for a white marble.”

  That is the statement that is goofy.

  Ambiguity aversion is perhaps fear, but nevertheless quite a useful thing.

• [Falcon_Seven]

  Maybe we should ask Schrodinger’s cat and see how it feels about its chances.

• [Schorsch]

  As usual, why buy this book with the tortured, illogical description of the paradox, when Wikipedia does it better: http://en.wikipedia.org/wiki/Ellsberg_paradox

  The Wikipedia description makes sense. Check it out.

• [asuffield]

  Well yes, if there is no difference at all in the comparative value of the two choices (which there isn’t, the probabilities are identical) then I’ll pick the option about which I have more information. But only if there is no difference.

  That’s not a paradox. It’s as good a way to choose between two equivalent options as any other.

• [airshowfan]

  Shroedinger was not trying to point out that all possibilities exist prior to a choice, but how ridiculous that prospect is when applied to the real world.

  Okay then. Every marble is actually half-black and half-white, until you pull it out at which point you force it to choose… ;)

• [Spikeles]

  Or… you could just… “accidentally” knock them both over and see which one the marble comes out of. Oops.. sorry.. so clumsy.. can i have my $100 now?

• [airshowfan]

  See, but the thing is, no urn is the OBVIOUS choice. With both urns, you have a 50-50 chance of drawing a white marble or a black marble (assuming all marbles in both urns are white or black).

  There are twenty-one possible configurations for the right urn. 1: All twenty are black. 2: Nineteen black and one white. 3: Eighteen black and two white. And so on. You have no idea which of those 21 configurations you’re dipping your hand into. So if you assume each of the 21 configurations is equally likely (and it is the discomfort with making this assumption that will keep people from going with the right urn if they have a choice), the odds of drawing out a black marble is (1/21)*(20/20) plus (1/21)*(19/20) plus (1/21)*(18/20)… which adds up to half. Just like the left urn. So the odds of the right urn are statistically the same as the odds of the left urn, in a weird Schrodinger’s Cat kind of way.

  The point of the “paradox” is that, all other things being equal (i.e. two urns both with 50-50 odds), you’d rather take a gamble in a system with fewer unknowns – or, more precisely, in a system where you have to make fewer assumptions in order to estimate your odds.

• [highlyverbal]

  “Logically, if you thought the left-hand urn was the better choice for a black marble…”

  But I didn’t think the left-hand one was BETTER! The fact that I picked it only shows I thought it was equivalent -or- better. Sheesh.

• [SeamusAndrewMurphy]

  So, what I’m getting is that you only have one chance to make a decision, but you’ll have two.

  That’s perhaps not the most empirical methodology.

  It seems this tells nothing about fear of the unknown, but something of using your thinking cap to determine odds on known info versus unknown. That’s fear?

  Being risk averse is not the same as fearing the future. It’s logical thinking.

  Using this example, Las Vegas odds makers must exist in a near panic state.

  As an aside, everyone knows the axiom that 50% of all doctors graduate in the bottom half of their class. Well so do economist/psychiatry profs.

• [juanpa]

  #13:

  “Logically, if you thought the left-hand urn was the better choice for a black marble, the right-hand urn should be the better choice for a white marble.”

  If you thought the left-hand urn was better for a black marble then you also thought the probability on the right-hand urn for the black marble where less than 0.5.

  Given the information available here I should said this would not be warranted (It should be also 0.5 for the right-hand urn). But, if you thought that for whatever reason, then yes, you should also conclude that the chances for the white-marble on the right-hand urn are greater than 0.5.

• [KyleTexas]

  GREGBERNS @ #36 thanks for joining the discussion.

  I think the point you’re missing is what I’d like to term the “Cider In Your Ear” angle, from the quote from Sky Masterson’s advice in the musical Guys and Dolls:

  “One of these days in your travels, a guy is going to show you a brand-new deck of cards on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand-new deck of cards and squirt cider in your ear. But, son, do not accept this bet, because as sure as you stand there, you’re going to wind up with an ear full of cider.”

  What your saying is that if test subject believes urn #1 is the best one to go for when choosing white marbles, it would therefore be *irrational* for same subject to choose urn #2 for black marbles.

  But this isn’t really true, because the math shows that statistically, it doesn’t matter which urn you pick, given a truly random distribution of the marbles in the unknown urn an *otherwise*fair*experiment*. Your saying if you pick one urn for one color marble, the rational thing is to pick the other urn for the other color. But it doesn’t *really* matter which urn you pick first. But that’s not really the position the test subject is operating from, either:

  We’re conditioned as human beings to expect a catch. We’re conditioned to think there’s something you know that we don’t — that you, the gamble offerer, know how to make that jack of spades jump out of the deck and squirt cider in our ears. Given this fact, the natural inclination is to go with the option with more certainty, because that option appears to contain fewer unknown variables, thereby reducing your “cider-to-ear-squirting” advantage.

  The problem with your experiment is your setting it up in sociological terms: somebody is offering you something of real value in your life — in this case $100 — under circumstances which are somewhat fishy. So the natural inclination is to respond in a psychologically appropriate way. But then you’re putting on your mathematical hat, crunching the numbers and saying “Look at the irrational humans!”

  Of course they’re behaving (mathematically) irrationally, because you’re giving them a (psychologically) irrational test: Who in the world is ever going to come up to you and offer you $100 *real* US dollars to pick marbles?

  This isn’t a math problem. And it’s not a human nature problem, either. Strictly speaking, it’s a *humans* (plural) nature problem. Because the decision-making of the test subject isn’t the result of their own internal silliness; it’s the result of you, the test-giver, presenting a weirdly contrived situation. What, you’re going to give me $100 for doing nothing but pick a marble? That’s ridiculous!

  The rational thing to do is to respond from a position of distrust, which means to always gravitate toward the option which provides more certainty. No P(B)L = 1 – P(W)L proof can account for that.

• [forgeweld]

  Thanks for the direction to the coherent explanation at Wikipedia, Schorsch.

• [highlyverbal]

  The tie-breaker for me on the urns was on the value of information! If I had a chance to draw another without replacement, as sometimes happens in my life, then knowing the rest of the marbles has value.

  I prefer information, I am not adverse to ambiguity.

• [juanpa]

  Actually I think the difference is that in the case of the left-hand urn, in order to assign probabilities you can use the frequencies.

  In the right-hand urn you have to resort to some other heuristic in order to assign probabilities, in this case the appropiate is the indiference principle (or symmetry principle).

  But i’d say this principle is not obvious to people not exposed to probability thinking, where as frequency ==> probability is perhaps a basic brain function.

• [nigel1965]

  “Logically, if you thought the left-hand urn was the better choice for a black marble, the right-hand urn should be the better choice for a white marble.”

  I’m sorry, but how is that logical? I KNOW I have a 50% chance on the left. I have no way of knowing my chances on the left. My chances swing as low as 5% all the way up to 95%, but I don’t KNOW.

• [KyleTexas]

  Actually, the correct answer for which urn you should use for the second pick depends on the *color* of the marble you withdrew with your first pick.

  If you pull out a black marble from the 1st (known proportion urn), you should pick from that urn again, because you now know it has 10 white marbles to 9 black, giving you a .5263 probability of pulling out a white marble with your second pick. If, on the other hand, you pull out a white marble on the first try, you should switch urns (where you can assume random probability distribution of marbles) b/c you only have a .4737 probability of picking white again from the known urn.

  The same rule holds true when drawing marbles from the second urn, although the value of the data point from your first pick decreases. Basically, if you pull out a black marble on your first pick, you should still stay with that urn, but you only have a .5013 probability of picking out a white one (again, assuming random distribution of marbles)

  If you put the marbles *back* after the first draw, and you pull from the “known-urn” first, it really doesn’t matter which urn you pick from second.

  However, if you put the marbles *back* and you pull from the unknown urn first, then you should *switch* urns if you pull a black marble first, and *stay* with the unknown urn if you pull a white marble first. Because you’re more likely than not to pick the dominant color on your first draw.

  But agreed, overall this is a pretty dumb thought experiment, at least as far as proving the point that people are *afraid* of the unknown. *Maybe* you could stretch it to say people tend to favor options about which they have the most information, but I think that just implies they’re rational. I mean, the goal is to win $100 bucks, right?

  People naturally pick the known urn because of issues stated by #2 and others — e.g., it’s unclear whether some of the marbles in the unknown urn might be “candy-striped.” Now, once the Puzzle Daemon explains “No, there are only black and white marbles,” you might think about it more and realize it doesn’t really matter which urn you pick on your first go. But again, that’s just another way of making your decision based on more information–in this case the information that there are absolutely *no* other variables at play than the unknown proportion of the marbles in urn #2. In that case, what the heck, I’ll pick either urn.

  But I only feel that way because now I have more information about *both* urns. Before, I only had reliable information about one urn. If I knew there were 9 black marbles and 11 white marbles, I wouldn’t pick from the known-proportion urn when drawing for a black marble. I’d take my chance with the unknown

  Now, here’s the interesting question: If I knew there were 499 black marbles and 501 white marbles in the known urn, and an unknown distribution in the second urn, which would I pick? I might actually pick the first urn, because of nagging questions like “are any of the marbles candy-striped?” Getting dinged by the probability of .001 might be an acceptable “insurance premium” against the fact that the Puzzle Daemon is pulling a fast one on me that I haven’t been able to figure out.

  It’s not really interesting to say people are adverse to risk “all things being equal.” What’s interesting is how *unequal* odds will a person accept in exchange for more certainty?

• [mdh]

  @#2 – my sentiments exactly. Just because I don’t know the mix of black and white marbles on the right-side does not preclude there being zero of either color in that one, and just 20 blue marbles instead.

  Plus, if I draw correctly from the left my first time, that increases my chances slightly for the second draw from the left.

  At least from the way they phrased it in the set-up, that’s the smart bet.

  #16 – I’m ‘afraid’ your right.

• [Bugs]

  #3 is incomplete, the article is correct. Arguably badly worded, but correct.

  You have no way of knowing what’s in the right jar, so drawing the black ball might be impossible (0/20). However, it’s just as likely that drawing a black ball is garaunteed (20/20). Therefore to calculate the “expected” return, you take the average of all the probabilities: impossible, certain and every value in between. This gives you exactly equal odds, the same as the left jar.

  Most people (including me for a few minutes) spotted the chance of success being impossible but not the equal chance of success being certain. This proves the article’s point that we’re pessimistic about unknowns. It’s also further proof that we are, as a species, rubbish at dealing with probability.

  The article is badly worded because it doesn’t tell you what happens to the jars between your choices.

  If you’re given a new, randomly filled right jar that’s fine.

  However, if they’re reset to their starting state, the colour of the first marble drawn from the right jar gives you information you can use to adjust your odds. For example, if you drew a black marble on the first try you know there can’t be more than 19 white balls in the jar. Now you know there are somewhere between 19 and 0 white balls, but still somewhere between 20 and 0 black balls. The calculation now tells us to expect slightly more black balls than white (10.5:9.5) so now we’re better off going to the left basket. The inverse is true too, obviously.

  Another way of looking at it:

  Forget the left urn (10/20) now, just consider the right one. As an example, let’s say the black:white ratio is 17:3. (so 17 black balls out of 20 i.e. 17/20)

  First you need to draw a black ball. Obviously, your chance of success is 17/20.

  The ball you pick is put back. Now you need to pick a white ball. Obviously your chance of success is 3/20.

  Those two probabilities, whatever the actual numbers are, will always be in proportion such that they average out to be 10/20. If you’re trying to draw alternate coloured balls you can expect to be lucky half the time. If the jar is randomly refilled between each draw, this still works because the black:white ratio will average out to be even. With no information, the most rational course is to act as if it’s 10/20.