The point of the Allais paradox is less about how humans violate the axiom of independence and more about how our utility functions are nonlinear, especially with respect to infinitesimal risk.
There is an existing Dutch Book for eliminating infinitesimal risk, and it’s called insurance.
Yyyyes and no. Our utility functions are nonlinear, especially with respect to infinitesimal risk, but this is not inherently bad. There’s no reason for our utility to be everywhere linear with wealth: in fact, it would be very strange for someone to equally value “Having $1 million” and “Having $2 million with 50% probability, and having no money at all (and starving on the street) otherwise”.
Insurance does take advantage of this, and it’s weird in that both the insurance salesman and the buyers of insurance end up better off in expected utility, but it’s not a Dutch Book in the usual sense: it doesn’t guarantee either side a profit.
The Allais paradox points out that people are not only averse to risk, but also inconsistent about how they are averse about it. The utility function U(X cents) = X is not risk-averse, and it picks gambles 1A and 2A (in Wikipedia’s notation). The utility function U(X cents) = log X is extremely risk-averse, and it picks gambles 1B and 2B. Picking gambles 1A and 2B, on the other hand, cannot be described by any utility function.
There’s a Dutch book for the Allais paradox in this post reading after “money pump”.
I didn’t mean to imply nonlinear functions are bad. It’s just how humans are.
Picking gambles 1A and 2B, on the other hand, cannot be described by any utility function.
Prospect Theory describes this and even has a post here on lesswrong. My understanding is that humans have both a non-linear utility function as well as a non-linear risk function. This seems like a useful safeguard against imperfect risk estimation.
[Insurance is] not a Dutch Book in the usual sense: it doesn’t guarantee either side a profit.
If you setup your books correctly, then it is guaranteed. A dutch book doesn’t need to work with only one participant, and in fact many dutch books only work with on populations rather than individuals, in the same way insurance only guarantees a profit when properly spread across groups.
Insurance makes a profit in expectation, but an insurance salesman does have some tiny chance of bankruptcy, though I agree that this is not important. What is important, however, is that an insurance buyer is not guaranteed a loss, which is what distinguishes it from other Dutch books for me.
Prospect theory and similar ideas are close to an explanation of why the Allais Paradox occurs. (That is, why humans pick gambles 1A and 2B, even though this is inconsistent.) But, to my knowledge, while utility theory is both a (bad) model of humans and a guide to how decisions should be made, prospect theory is a better model of humans but often describes errors in reasoning.
(That is, I’m sure it prevents people from doing really stupid things in some cases. But for small bets, it’s probably a bad idea; Kahneman suggests teaching yourself out of it by making yourself think ahead to how many such bets you’ll make over a lifetime. This is a frame of mind in which the risk thing is less of a factor.)
The point of the Allais paradox is less about how humans violate the axiom of independence and more about how our utility functions are nonlinear, especially with respect to infinitesimal risk.
There is an existing Dutch Book for eliminating infinitesimal risk, and it’s called insurance.
Yyyyes and no. Our utility functions are nonlinear, especially with respect to infinitesimal risk, but this is not inherently bad. There’s no reason for our utility to be everywhere linear with wealth: in fact, it would be very strange for someone to equally value “Having $1 million” and “Having $2 million with 50% probability, and having no money at all (and starving on the street) otherwise”.
Insurance does take advantage of this, and it’s weird in that both the insurance salesman and the buyers of insurance end up better off in expected utility, but it’s not a Dutch Book in the usual sense: it doesn’t guarantee either side a profit.
The Allais paradox points out that people are not only averse to risk, but also inconsistent about how they are averse about it. The utility function U(X cents) = X is not risk-averse, and it picks gambles 1A and 2A (in Wikipedia’s notation). The utility function U(X cents) = log X is extremely risk-averse, and it picks gambles 1B and 2B. Picking gambles 1A and 2B, on the other hand, cannot be described by any utility function.
There’s a Dutch book for the Allais paradox in this post reading after “money pump”.
I didn’t mean to imply nonlinear functions are bad. It’s just how humans are.
Prospect Theory describes this and even has a post here on lesswrong. My understanding is that humans have both a non-linear utility function as well as a non-linear risk function. This seems like a useful safeguard against imperfect risk estimation.
If you setup your books correctly, then it is guaranteed. A dutch book doesn’t need to work with only one participant, and in fact many dutch books only work with on populations rather than individuals, in the same way insurance only guarantees a profit when properly spread across groups.
Insurance makes a profit in expectation, but an insurance salesman does have some tiny chance of bankruptcy, though I agree that this is not important. What is important, however, is that an insurance buyer is not guaranteed a loss, which is what distinguishes it from other Dutch books for me.
Prospect theory and similar ideas are close to an explanation of why the Allais Paradox occurs. (That is, why humans pick gambles 1A and 2B, even though this is inconsistent.) But, to my knowledge, while utility theory is both a (bad) model of humans and a guide to how decisions should be made, prospect theory is a better model of humans but often describes errors in reasoning.
(That is, I’m sure it prevents people from doing really stupid things in some cases. But for small bets, it’s probably a bad idea; Kahneman suggests teaching yourself out of it by making yourself think ahead to how many such bets you’ll make over a lifetime. This is a frame of mind in which the risk thing is less of a factor.)