Diachronic Dutch Books

You have a Dutch Book (DB) against you at t provided that, given your credences at t, you would assent to each of a set of bets such that you're guaranteed to lose on balance if you assent to them all.

This morning, I was thinking about cases where people are offering diachronic DB argument.

Suppose you rationally change your mind about p, adjusting your credence between today and tomorrow, say from 1/4 to 3/4, in the light of new evidence, all duly according to Bayes. My initial thought was that there is then a diachronic DB against you in the following sense: there is a pair of bets such that if one is offered today and another tomorrow, you will accept both and be guaranteed to lose overall. (For instance, today, you will accept the deal that you will pay three dollars if p and get a dollar if not p, and tomorrow you will accept the deal that you will pay three dollars if not p and get a dollar if p. But then you're going to lose two dollars whether or not p is true.)

But that was careless of me. A Dutch Book would do better to be defined as a set of bets that you're individually rational in accepting and that are sure to lose you money given the information you have. But you don't have a guarantee that you will change your mind about p from 1/4 to 3/4 in this case. (This is at the heart of the diachronic DB argument that has been given for the Reflection Principle.)

Is there anything to be learned from my case above, other than to be more careful in thinking about DBs? Maybe. Consider your situation during the second bet, the one tomorrow. You accept that bet. In accepting the bet, you bring it about that by your present lights you are bringing it about that you have played a game that you are sure to have overall lost. So one lesson of this is that it is not irrational to bring it about that you have played a game that you are sure to have lost. Moreover, this case suggests that there is a crucial temporal or causal directionality to DB-based arguments. DB arguments have been used to argue that you should now adopt any credences you know for sure you will rationally have (with some provisos). But one had better not use DBs to argue that you should now adopt any credences you know for sure you rationally had: that way lies stasis.