I think this is a strong argument that EDT agents shouldn’t do bayesian updates on empirical observations. I thought that it might still be ok to change your mind on the basis of logical arguments and reasoning (not empirical data or observations). But I think a very similar argument bites against that.
Example:
There’s 2 mathematical propositions, each of which you think have an independent 50% probability of being true: Y1 and Y2.
The proposition X = “Y1 and Y2”.
Presumably you assign 25% to X being true.
Let’s say you try to prove Y1 to be true, and succeed. You don’t have time to prove Y2. Naively, you’d know expect to assign 50% to X being true, or be willing to be on 1:1 odds.
However, let’s say that there are many copies of you across the universe, and equally many of them tried to prove statement Y1 and Y2. For simplicity, let’s say everyone who tried to prove a true statement succeeded, and no one had time to attempt to prove more than one statement.
Given an opportunity to bet on X being true, and thinking about your odds, you reason:
If X is true, then Y2 will be true (in addition to Y1).
So if X is true, and I bet on X, then everyone will bet on X and everyone will win. (Assuming that someone who proved Y2 is relevantly in the same position as me, so that my choosing to bet provides strong evidence that they will bet.)
If X is false, then Y2 will be fase.
So if X is false, and everyone in my position bets on X, in expectation just 1⁄2 people will lose. (The ones who proved Y1.)
So the stakes are twice as high if X is true than false.
Since I assign X a 50% chance (or 1:1) of being true, I will bet on (1:1) * (2:1) = (2:1) odds that X is true. I.e., from this perspective, the EV calculation becomes:
EV(bet on 2:1 odds) = 50% [that X is true] * 2 [people who win if X is true] * 1 [payout if X is true] + 50% [that X is false] * 1 [people who lose if X is false] * (-2) [payout if X is false] = 0.
This strategy will lose money in expectation.
From the ex-ante perspective, there are 4 equiprobable worlds where (Y1,Y2) have different truth values. In 1 of them, neither is true; in 2 of them, exactly 1 is true; and in 1 one of them, both are true. From the ex-ante perspective, there’s 2 people who prove their statement true when X is false; and 2 people who prove their statement true when X is true. If they all bet at 2:1 odds that X is true, they’ll lose money in expectation.
One difference from the empirical case in the post above is that you need to perceive yourself as correlated with people who proved a different statement than you did.
Edit 2026-04-20:
I significantly simplified the example above.
I want to flag that I think the argument against logical updates is somewhat weaker than the argument against empirical updates. In particular, this even more unappealing argument doesn’t apply to the logical case as far as I can tell. (And the toy example above — disjunction of logical statements where you’ve proven one — is more rare than unreliable empirical evidence of logical facts, as in the calculator example above.)
Edit 2026-05-05: Dutch book variant:
Before proving, the bookie offers:
If X is true → you pay $1
If [the statement you’re about to prove] is true and X is false → you get $1.1
Then if you fail to prove your statement, revealing it to be false, no pay-out happens.
If you do prove your statement, the bookie offers:
If X is true → you get $0.7.
If [the statement you just proved] is true and X is false → you pay $1.2.
(2nd offer is accepted because the EDT agent perceives the stakes to be twice as high if X is true.)
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.
It seems to me that the only relevant difference here is the set of agents that you perceive yourself as being correlated with, and that the update being from empirical observation vs. logical reasoning is unimportant.
Replace “Y1” and “Y2″ with empirical statements which are probabilistically independent according to my prior, say, “the Hubble constant is x” and “the physical baryon density is y” and replace “prove a statement” with “conduct a definitive cosmological experiment”, and everything seems to go through the same. (If the thought is that we want all agents in a large world to verify the same statements, that’s dealt with by having the empirical statements be about global cosmological parameters.)
As noted in this comment, if you took yourself to control both agents that saw “X is true” and “X is false” in the original Calculator Bet, then you bet at 99:1 odds. And in your example, if you took yourself to only control ‘agents who proved Y1’ (rather than ‘agents who proved a statement’), then you bet at 1:1 odds.
I think this is a strong argument that EDT agents shouldn’t do bayesian updates on empirical observations. I thought that it might still be ok to change your mind on the basis of logical arguments and reasoning (not empirical data or observations). But I think a very similar argument bites against that.
Example:
There’s 2 mathematical propositions, each of which you think have an independent 50% probability of being true: Y1 and Y2.
The proposition X = “Y1 and Y2”.
Presumably you assign 25% to X being true.
Let’s say you try to prove Y1 to be true, and succeed. You don’t have time to prove Y2. Naively, you’d know expect to assign 50% to X being true, or be willing to be on 1:1 odds.
However, let’s say that there are many copies of you across the universe, and equally many of them tried to prove statement Y1 and Y2. For simplicity, let’s say everyone who tried to prove a true statement succeeded, and no one had time to attempt to prove more than one statement.
Given an opportunity to bet on X being true, and thinking about your odds, you reason:
If X is true, then Y2 will be true (in addition to Y1).
So if X is true, and I bet on X, then everyone will bet on X and everyone will win. (Assuming that someone who proved Y2 is relevantly in the same position as me, so that my choosing to bet provides strong evidence that they will bet.)
If X is false, then Y2 will be fase.
So if X is false, and everyone in my position bets on X, in expectation just 1⁄2 people will lose. (The ones who proved Y1.)
So the stakes are twice as high if X is true than false.
Since I assign X a 50% chance (or 1:1) of being true, I will bet on (1:1) * (2:1) = (2:1) odds that X is true. I.e., from this perspective, the EV calculation becomes:
EV(bet on 2:1 odds) = 50% [that X is true] * 2 [people who win if X is true] * 1 [payout if X is true] + 50% [that X is false] * 1 [people who lose if X is false] * (-2) [payout if X is false] = 0.
This strategy will lose money in expectation.
From the ex-ante perspective, there are 4 equiprobable worlds where (Y1,Y2) have different truth values. In 1 of them, neither is true; in 2 of them, exactly 1 is true; and in 1 one of them, both are true. From the ex-ante perspective, there’s 2 people who prove their statement true when X is false; and 2 people who prove their statement true when X is true. If they all bet at 2:1 odds that X is true, they’ll lose money in expectation.
One difference from the empirical case in the post above is that you need to perceive yourself as correlated with people who proved a different statement than you did.
Edit 2026-04-20:
I significantly simplified the example above.
I want to flag that I think the argument against logical updates is somewhat weaker than the argument against empirical updates. In particular, this even more unappealing argument doesn’t apply to the logical case as far as I can tell. (And the toy example above — disjunction of logical statements where you’ve proven one — is more rare than unreliable empirical evidence of logical facts, as in the calculator example above.)
Edit 2026-05-05: Dutch book variant:
Before proving, the bookie offers:
If X is true → you pay $1
If [the statement you’re about to prove] is true and X is false → you get $1.1
Then if you fail to prove your statement, revealing it to be false, no pay-out happens.
If you do prove your statement, the bookie offers:
If X is true → you get $0.7.
If [the statement you just proved] is true and X is false → you pay $1.2.
(2nd offer is accepted because the EDT agent perceives the stakes to be twice as high if X is true.)
Overall payout:
[your statement] is false → $0
X is true → -$0.3
[your statement] is true and X is false → -$0.1
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.
It seems to me that the only relevant difference here is the set of agents that you perceive yourself as being correlated with, and that the update being from empirical observation vs. logical reasoning is unimportant.
Replace “Y1” and “Y2″ with empirical statements which are probabilistically independent according to my prior, say, “the Hubble constant is x” and “the physical baryon density is y” and replace “prove a statement” with “conduct a definitive cosmological experiment”, and everything seems to go through the same. (If the thought is that we want all agents in a large world to verify the same statements, that’s dealt with by having the empirical statements be about global cosmological parameters.)
As noted in this comment, if you took yourself to control both agents that saw “X is true” and “X is false” in the original Calculator Bet, then you bet at 99:1 odds. And in your example, if you took yourself to only control ‘agents who proved Y1’ (rather than ‘agents who proved a statement’), then you bet at 1:1 odds.