His setup is the following: there are two Sleeping Beauties. Two independent fair coins determine the creation of rooms. The first fair coin determines whether a white or black room is created, and if a second fair coin lands heads, a room of the opposite color to the first room is created. So 1⁄4 chance there is one white room, 1⁄4 chance there is one black room, and 1⁄2 chance there is both a white room and a black room. On Monday, one Beauty is always awoken in a white room and the other Beauty is always awoken in a black room, and if their designated room was not created, they stay asleep. They bet under a joint account whose value they are both trying to maximize.
His Dutch book consists of the following bets:
Translating to your example: “wake up in a white room” = “prove Y1“, “wake up in a black room” = “prove Y2”, W = “only Y1 is true”, B = “only Y2 is true”, and WB = “both Y1 and Y2 are true, i.e., X is true”.
The only difference is that the probabilities are Pr(W)=1/4, Pr(B)=1/4, and Pr(WB)=1/2 for Conitzer and Pr(W)=1/4, Pr(B)=1/4, and Pr(WB)=1/4 in your case. So the Dutch book ends up being the same, once you appropriately adjust the stakes for the different probability placed on the “WB” scenario.
The key is once again that the “interpretation of evidential decision theory comes down to requiring that each Beauty, when calculating her expected utility from accepting a bet (or declining it), assumes in this calculation that the other Beauty would do the same”. The justification give by Conitzer is that their epistemic situations (including bets faced) are entirely symmetric.
Nice Dutch book! Vince Conitzer actually gives an almost identical Dutch book in Section 5 of his article on Dutch books against EDT.
His setup is the following: there are two Sleeping Beauties. Two independent fair coins determine the creation of rooms. The first fair coin determines whether a white or black room is created, and if a second fair coin lands heads, a room of the opposite color to the first room is created. So 1⁄4 chance there is one white room, 1⁄4 chance there is one black room, and 1⁄2 chance there is both a white room and a black room. On Monday, one Beauty is always awoken in a white room and the other Beauty is always awoken in a black room, and if their designated room was not created, they stay asleep. They bet under a joint account whose value they are both trying to maximize.
His Dutch book consists of the following bets:
Translating to your example: “wake up in a white room” = “prove Y1“, “wake up in a black room” = “prove Y2”, W = “only Y1 is true”, B = “only Y2 is true”, and WB = “both Y1 and Y2 are true, i.e., X is true”.
The only difference is that the probabilities are Pr(W)=1/4, Pr(B)=1/4, and Pr(WB)=1/2 for Conitzer and Pr(W)=1/4, Pr(B)=1/4, and Pr(WB)=1/4 in your case. So the Dutch book ends up being the same, once you appropriately adjust the stakes for the different probability placed on the “WB” scenario.
The key is once again that the “interpretation of evidential decision theory comes down to requiring that each Beauty, when calculating her expected utility from accepting a bet (or declining it), assumes in this calculation that the other Beauty would do the same”. The justification give by Conitzer is that their epistemic situations (including bets faced) are entirely symmetric.