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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