Just for fun: an alternative treatment that eliminates indexical probabilities.
Instead of one absent-minded driver, we have some essentially identical people who are pretty good at decision theory problems, one per intersection. They reason the same way, all know that they reason the same way, etc. A random 1:1 mapping of people to intersections is chosen, and they are individually instructed to decide on a probability for whether the car should continue through the intersection or turn, should it make it there. They all receive the same utility payoff depending upon where the car ends up as in the original problems.
There are no philosophical questions here about whether it makes sense to assign probability to being a particular forgetful observer. It’s just a straightforward game theory application to objective, external probabilities. However, the mathematical derivation is exactly the same as in the single absent-minded driver. They all know that whatever reasoning they use to assign a probability to continuing, by symmetry so will the others and it reduces to finding an optimal common p.
In this view it is clearer that the “action” utility function simply isn’t a useful concept.
Just for fun: an alternative treatment that eliminates indexical probabilities.
Instead of one absent-minded driver, we have some essentially identical people who are pretty good at decision theory problems, one per intersection. They reason the same way, all know that they reason the same way, etc. A random 1:1 mapping of people to intersections is chosen, and they are individually instructed to decide on a probability for whether the car should continue through the intersection or turn, should it make it there. They all receive the same utility payoff depending upon where the car ends up as in the original problems.
There are no philosophical questions here about whether it makes sense to assign probability to being a particular forgetful observer. It’s just a straightforward game theory application to objective, external probabilities. However, the mathematical derivation is exactly the same as in the single absent-minded driver. They all know that whatever reasoning they use to assign a probability to continuing, by symmetry so will the others and it reduces to finding an optimal common p.
In this view it is clearer that the “action” utility function simply isn’t a useful concept.