Why should I care if I together with other people are jointly getting Dutch booked, if I myself am not? If my neighbour loses money but I do not, I do not care that “we” lost money, if his affairs have no connection with mine.
First off, I am quite sympathetic, and by no means would argue that the Dutch Book for the common prior assumption is as convincing as other Dutch Book Arguments.
However, it’s still intriguing.
If you and your neighbor are game-theoretic partners who sometimes cooperate in prisoner’s-dilemma like situations, then you might consider this kind of joint Dutch Book concerning. A coalition which does not manage to jointly coordinate to act as one agent is a weaker coalition.
If the collision is able to negotiate bets internally, then for any instance of the coalition getting Dutch booked, they can just agree to run that bet without the bookie, and split the bookies cut.
If everyone else changes to my prior, that’s great. But if I change from my prior to their prior, I am just (from the point of view of someone with my prior, which obviously includes myself) making myself vulnerable to be beaten in ordinary betting by other agents that have my prior.
Let’s say your prior is P and mine is Q. I take your argument to be that P always prefers bets made according to P (bets made according to Q are at best just as good). But this is only true if P thinks P knows better than Q.
It’s perfectly possible for P to think Q knows better. For example, P might think Q just knows all the facts. Then it must be that P doesn’t know Q (or else P would also know all the facts.) But given the opportunity to learn Q, P would prefer to do so; whereupon, the updated P would be equal to Q.
Similar things can happen in less extreme circumstances, where Q is merely expected to know some things that P doesn’t. P could still prefer to switch entirely over to Q’s beliefs, because they have a higher expected value. It’s also possible that P trusts Q only to an extent, so P moves closer to Q but does not move all the way. This can even be true in the Aumann agreement setting: P and Q can both move to a new distribution R, because P has some new information for Q, but Q also has some new information for P. (In general, R need not even be a ‘compromise’ between P and Q; it could be something totally different.)
So it isn’t crazy at all for rational agents to prefer each other’s beliefs.
A weaker form of the common prior assumption could assert that this is always the case: two rational agents need not have the same priors, but upon learning each other’s priors, would then come to agree. (Either P updates to Q, or Q updates to P, or P and Q together update to some R.)
Why should I care if I together with other people are jointly getting Dutch booked, if I myself am not? If my neighbour loses money but I do not, I do not care that “we” lost money, if his affairs have no connection with mine.
First off, I am quite sympathetic, and by no means would argue that the Dutch Book for the common prior assumption is as convincing as other Dutch Book Arguments.
However, it’s still intriguing.
If you and your neighbor are game-theoretic partners who sometimes cooperate in prisoner’s-dilemma like situations, then you might consider this kind of joint Dutch Book concerning. A coalition which does not manage to jointly coordinate to act as one agent is a weaker coalition.
If the collision is able to negotiate bets internally, then for any instance of the coalition getting Dutch booked, they can just agree to run that bet without the bookie, and split the bookies cut.
If everyone else changes to my prior, that’s great. But if I change from my prior to their prior, I am just (from the point of view of someone with my prior, which obviously includes myself) making myself vulnerable to be beaten in ordinary betting by other agents that have my prior.
This isn’t always true!
Let’s say your prior is P and mine is Q. I take your argument to be that P always prefers bets made according to P (bets made according to Q are at best just as good). But this is only true if P thinks P knows better than Q.
It’s perfectly possible for P to think Q knows better. For example, P might think Q just knows all the facts. Then it must be that P doesn’t know Q (or else P would also know all the facts.) But given the opportunity to learn Q, P would prefer to do so; whereupon, the updated P would be equal to Q.
Similar things can happen in less extreme circumstances, where Q is merely expected to know some things that P doesn’t. P could still prefer to switch entirely over to Q’s beliefs, because they have a higher expected value. It’s also possible that P trusts Q only to an extent, so P moves closer to Q but does not move all the way. This can even be true in the Aumann agreement setting: P and Q can both move to a new distribution R, because P has some new information for Q, but Q also has some new information for P. (In general, R need not even be a ‘compromise’ between P and Q; it could be something totally different.)
So it isn’t crazy at all for rational agents to prefer each other’s beliefs.
A weaker form of the common prior assumption could assert that this is always the case: two rational agents need not have the same priors, but upon learning each other’s priors, would then come to agree. (Either P updates to Q, or Q updates to P, or P and Q together update to some R.)