Ignoring Small Chances

On September 9, 1713, so the story goes, Nicholas Bernoulli proposed the following problem in the theory of games of chance, after 1768 known as the St Petersburg paradox …:

Peter tosses a coin and continues to do so until it should land heads when it comes to the ground. He agrees to give Paul one ducat if he gets heads on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled.

Nicholas Bernoulli … suggested that more than five tosses of heads are [seen as] morally impossible [and so ignored]. This proposition is experimentally tested through the elicitation of subjects‘ willingness-to-pay for various truncated versions of the Petersburg gamble that differ in the maximum payoff. … All gambles that involved probability levels smaller than 1/16 and maximum payoffs greater than 16 Euro elicited the same distribution of valuations. … The payoffs were as described …. but in Euros rather than in ducats. … The more senior students seemed to have a higher willingness-to-pay. … Offers increase significantly with income. (more)

This isn’t plausibly explained by risk aversion, nor by a general neglect of possibilities with a <5% chance. I suspect this is more about analysis complexity, i.e., about limiting the number of possibilities we’ll consider at any one time. I also suspect this bodes ill for existential risk mitigation.

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  • J Storrs Hall

    This isn’t explained very clearly, but if I understand it right the payoff is rather obviously linear in cutoff (series length). Could none of the subjects do induction?

    • Dan Weber

      What would you pay for a lottery ticket with an expected value of infinity, even if the variance is huge?

      Really should have included the link to the modern English explanation: http://en.wikipedia.org/wiki/St_Petersburg_Paradox

      • Paul

        I agree entirely with Dan about this inclusion… in fact may I suggest scrolling down: http://en.wikipedia.org/wiki/St._Petersburg_paradox#Finite_St._Petersburg_lotteries

        Dog of Justice also mentions this later down…the true expected value of St. Petersburg “Paradox” is actually a quite reasonable $10 or so, because the tail probabilities of earning zillions are CORRECTLY ignored (set to 0%, because they would literally never be paid).

      • If you calculate the expected value of the most likely 99.9999% of the outcomes it is only $10. The most likely 99.9% and it is only $5.

        To get an expected value that is reasonable by reppeatedly betting, you would have to include the value of your time. At any reasonable value of your time the expected value becomes very low.

  • Khoth

    The paper appears to be arguing that the effect can be explained by neglect of small probabilities and by diminishing marginal utility of money. What there is making you say otherwise?

  • Jamie_NYC

    Robin, this problem is well known. What is at work here is logarithmic (or some other, non-linear) utility function. From the base case, large loss (initial payment) has higher absolute utility than large gain (total eventual pay-off) not because of loss aversion, but because of non-linear utility function.

    • Jamie_NYC

      Addendum: the utility function must be asymptotic, not just non-linear, since the expected payoff is infinite.

    • If wealth is logarithmic, shouldn’t the wealthier have disdained gambling since the gain is that much less utility for them? Yet:

      > Offers increase significantly with income.

      More evidence that the rich are able to be less risk averse…

    • The level of risk aversion required to explain this behavior is extreme, and completely inconsistent with lots of other risk-taking behavior by the same sort of subjects.

      • Drewfus

        I agree with Robin. Manipulating the shape of these unobservable utility functions can’t explain away something like this. More likely the cognitive load of imagining very low probability events outways the benefits of considering them (in some sort of rational manner).

        There is a case for considering humans as not only utility maximizers, but also as embodiments of the principle of least effort.


      • If change in risk aversion doesn’t explain why the poorer participants were even more irrational than the richer participants, then what is going on? The probabilities are the same for both group, are they not, so shouldn’t they neglect them equally…?

  • Dog of Justice

    Uh, isn’t the dominant factor here counterparty risk? I.e. if you flip 50+ heads in a row, the guy offering the bet will only be able to afford to pay you as if you flipped ~20, so even without logarithmic utility this bet isn’t worth more than 20.

    • Subjects are acting as if the experimenter can’t pay more than 16 Euros! And yet they showed they could pay what was promised by displaying a big bag of money.

    • Paul

      I do appreciate the “display a big bag of money” step, but I still feel that this is insufficient to change people’s core belief that they wont get paid the big money! (Possibly I have not overcome my confirmation bias with respect to this belief, though!)

      Displaying a big bag of money to me would just make me recheck my math, thinking “what the heck do they know that I don’t?” Surely, no one would offer a bet of negative expected gain to them…so something suspicious must be going on!

      I hope I am not being stubborn on this point…also I admit have not read the entire article (it is quite long…talk about info processing).

  • Alexander Kruel

    Interesting! This is related to a comment I wrote earlier over at Accelerating Future:

    I don’t think that these discussions will lead anywhere because they miss the underlying reason for most of the superficial disagreement about risks from AI.

    There are a few people who disagree about the possibility of AGI in general, for more or less weird reasons. Forget those. The problem becomes more obvious when you turn your attention to people like mathematician and climate activist John Baez, computer science professor Stan Franklin, Douglas Hofstadter or organisations like GiveWell (I personally contacted many more experts about this topic). The disagreement all comes down to a general averseness to options that have a low probability of being factual, even given that the stakes are high.

    Should we really concentrate our efforts on such vague possibilities as risks from AI? Technically, from the standpoint of maximizing expected utility, given the absence of other existential risks, the answer might very well be yes. But even though we believe to understand this technical viewpoint of rationality very well in principle, it does also lead to problems such as Pascal’s Mugging. But it doesn’t need a true Pascal’s Mugging scenario to make people feel deeply uncomfortable with what Bayes’ Theorem, the expected utility formula, and Solomonoff induction seem to suggest one should do.

    There seems to be a fundamental problem with the formalized version of rationality. The problem might be human nature itself, that some people are unable to accept what they should do if they want to maximize their expected utility. Or we are missing something else and our theories are flawed. Either way, to solve this problem we need to research those issues and thereby increase the confidence in the very methods used to decide what to do about risks from AI, or to increase the confidence in risks from AI directly, enough to make it look like a sensible option, a concrete and discernable problem that needs to be solved.

    If you view the criticism the SIAI does encounter in the light of the above then what most people mean when they doubt the reputation of people who claim that risks from AI need to be taken seriously, or who say that AGI might be far off, what those people mean is that risks from AI are too vague to be taken into account at this point, that nobody knows enough to make predictions about the topic right now.

    Many people perceive the whole world to be at stake, either due to climate change, war or engineered pathogens. Telling them about something like risks from AI, even though nobody seems to have any idea about the nature of intelligence, let alone general intelligence or the possibility of recursive self-improvement, seems like just another problem, one that is too vague to outweigh all the other risks. Most people feel like having a gun pointed to their heads, telling them about superhuman monsters that might turn them into paperclips then needs some really good arguments to outweigh the combined risk of all other problems.

    It is true that a lot of people already work to mitigate climate change while major existential risks are completely ignored. But that argument seems to fail if people perceive the existential risk in question to be under a certain threshold. There seems to be a point where things become vague enough that they get discounted completely.

    Summary: The problem isn’t that people doubt the possibility of artificial general intelligence. But most people would sooner question their grasp of “rationality” than give five dollars to a charity that tries to mitigate risks from AI because their calculations claim it was “rational”.

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