Diachronic Dutch Books and Sleeping Beauty

One version of the Sleeping Beauty problem (Elga 2000) is this: Beauty is the subject of an experiment that begins when she is put to sleep on Sunday night. She is awakened on Monday morning. That afternoon she is informed that it is Monday. She is put back to sleep on Monday night. The experimenters then flip a fair coin. If it comes up heads, Beauty is not awakened on Tuesday. If it comes up tails, her memories of Monday are erased and she is awakened on Tuesday morning. Her experiences on Tuesday morning (if she has any) are identical to her experiences on Monday morning. Beauty is perfectly rational and knows in advance (with certainty) that the experiment will go exactly as planned. Thus, on Sunday her credence in HEADS (the proposition that the coin toss comes up heads) is 1/2. But what is her credence in HEADS on Monday morning? Of those philosophers who have explicitly addressed this question, the more numerous group, "thirders," claim that the correct answer is 1/3, though a smaller group of "halfers" endorse 1/2. Hitchcock (2004) attempts to settle the matter by advancing a diachronic Dutch Book argument (hereafter DDB) for 1/3. Specifically, he advances a DDB against 1/2 and then shows that no similar DDB can be used against the thirder. Bradley and Leitgeb (2006) (hereafter B&L) argue that Hitchcock's DDB fails. We demonstrate the following: (a) B&L's criticism of Hitchcock is unconvincing; (b) nevertheless, there are serious reasons to worry about the success of Hitchcock's argument; (c) however, it is possible to construct a new DDB for 1/3 about which such worries cannot be raised.