Let’s see what happens when we adapt this to the canonical instance of “no, really, counterfactuals aren’t conditionals and should have different probabilities”. The cosmic ray problem, where the agent has the choice between two paths, it slightly prefers taking the left path, but its conditional on taking the right path is a tiny slice of probability mass that’s mostly composed of stuff like “I took the suboptimal action because I got hit by a cosmic ray”.
There will be 0 utility for taking left path, −10 utility for taking the right path, and −1000 utility for a cosmic ray hit. The CDT counterfactual says 0 utility for taking left path, −10 utility for taking the right path, while the conditional says 0 utility for left path, −1010 utility for right path (because conditional on taking the right path, you were hit by a cosmic ray).
In order to get the dutch book to go through, we need to get the agent to take the right path, to exploit P(cosmic ray) changing between the decision time and afterwards. So the initial bet could be something like −1 utility now, +12 utility upon taking the right path and not being hit by a cosmic ray. But now since the optimal action is “take the right path along with the bet”, the problem setup has been changed, and we can’t conclude that the agent’s conditional on taking the right path places high probability on getting hit by a cosmic ray (because now the right path is the optimal action), so we can’t money-pump with the “+0.5 utility, −12 utility upon taking a cosmic ray hit” bet.
So this seems to dutch-book Death-in-Damascus, not CDT≠EDT cases in general.
(lightly edited version of my original email reply to above comment; note that Diffractor was originally replying to a version of the Dutch-book which didn’t yet call out the fact that it required an assumption of nonzero probability on actions.)
I agree that this Dutch-book argument won’t touch probability zero actions, but my thinking is that it really should apply in general to actions whose probability is bounded away from zero (in some fairly broad setting). I’m happy to require an epsilon-exploration assumption to get the conclusion.
Your thought experiment raises the issue of how to ensure in general that adding bets to a decision problem doesn’t change the decisions made. One thought I had was to make the bets always smaller than the difference in utilities. Perhaps smaller Dutch-books are in some sense less concerning, but as long as they don’t vanish to infinitesimal, seems legit. A bet that’s desirable at one scale is desirable at another. But scaling down bets may not suffice in general. Perhaps a bet-balancing scheme to ensure that nothing changes the comparative desirability of actions as the decision is made?
For your cosmic ray problem, what about:
You didn’t specify the probability of a cosmic ray. I suppose it should have probability higher than the probability of exploration. Let’s say 1/million for cosmic ray, 1/billion for exploration.
Before the agent makes the decision, it can be given the option to lose .01 util if it goes right, in exchange for +.02 utils if it goes right & cosmic ray. This will be accepted (by either a CDT agent or EDT agent), because it is worth approximately +.01 util conditioned on going right, since cosmic ray is almost certain in that case.
Then, while making the decision, cosmic ray conditioned on going right looks very unlikely in terms of CDT’s causal expectations. We give the agent the option of getting .001 util if it goes right, if it also agrees to lose .02 conditioned on going right & cosmic ray.
CDT agrees to both bets, and so loses money upon going right.
Ah, that’s not a very good money pump. I want it to lose money no matter what. Let’s try again:
Before decision: option to lose 1 millionth of a util in exchange for 2 utils if right&ray.
During decision: option to gain .1 millionth util in exchange for −2 util if right&ray.
That should do it. CDT loses .9 millionth of a util, with nothing gained. And the trick is almost the same as my dutch book for death in damascus. I think this should generalize well.
The amounts of money lost in the Dutch Book get very small, but that’s fine.
(lightly edited restatement of email comment)
Let’s see what happens when we adapt this to the canonical instance of “no, really, counterfactuals aren’t conditionals and should have different probabilities”. The cosmic ray problem, where the agent has the choice between two paths, it slightly prefers taking the left path, but its conditional on taking the right path is a tiny slice of probability mass that’s mostly composed of stuff like “I took the suboptimal action because I got hit by a cosmic ray”.
There will be 0 utility for taking left path, −10 utility for taking the right path, and −1000 utility for a cosmic ray hit. The CDT counterfactual says 0 utility for taking left path, −10 utility for taking the right path, while the conditional says 0 utility for left path, −1010 utility for right path (because conditional on taking the right path, you were hit by a cosmic ray).
In order to get the dutch book to go through, we need to get the agent to take the right path, to exploit P(cosmic ray) changing between the decision time and afterwards. So the initial bet could be something like −1 utility now, +12 utility upon taking the right path and not being hit by a cosmic ray. But now since the optimal action is “take the right path along with the bet”, the problem setup has been changed, and we can’t conclude that the agent’s conditional on taking the right path places high probability on getting hit by a cosmic ray (because now the right path is the optimal action), so we can’t money-pump with the “+0.5 utility, −12 utility upon taking a cosmic ray hit” bet.
So this seems to dutch-book Death-in-Damascus, not CDT≠EDT cases in general.
(lightly edited version of my original email reply to above comment; note that Diffractor was originally replying to a version of the Dutch-book which didn’t yet call out the fact that it required an assumption of nonzero probability on actions.)
I agree that this Dutch-book argument won’t touch probability zero actions, but my thinking is that it really should apply in general to actions whose probability is bounded away from zero (in some fairly broad setting). I’m happy to require an epsilon-exploration assumption to get the conclusion.
Your thought experiment raises the issue of how to ensure in general that adding bets to a decision problem doesn’t change the decisions made. One thought I had was to make the bets always smaller than the difference in utilities. Perhaps smaller Dutch-books are in some sense less concerning, but as long as they don’t vanish to infinitesimal, seems legit. A bet that’s desirable at one scale is desirable at another. But scaling down bets may not suffice in general. Perhaps a bet-balancing scheme to ensure that nothing changes the comparative desirability of actions as the decision is made?
For your cosmic ray problem, what about:
You didn’t specify the probability of a cosmic ray. I suppose it should have probability higher than the probability of exploration. Let’s say 1/million for cosmic ray, 1/billion for exploration.
Before the agent makes the decision, it can be given the option to lose .01 util if it goes right, in exchange for +.02 utils if it goes right & cosmic ray. This will be accepted (by either a CDT agent or EDT agent), because it is worth approximately +.01 util conditioned on going right, since cosmic ray is almost certain in that case.
Then, while making the decision, cosmic ray conditioned on going right looks very unlikely in terms of CDT’s causal expectations. We give the agent the option of getting .001 util if it goes right, if it also agrees to lose .02 conditioned on going right & cosmic ray.
CDT agrees to both bets, and so loses money upon going right.
Ah, that’s not a very good money pump. I want it to lose money no matter what. Let’s try again:
Before decision: option to lose 1 millionth of a util in exchange for 2 utils if right&ray.
During decision: option to gain .1 millionth util in exchange for −2 util if right&ray.
That should do it. CDT loses .9 millionth of a util, with nothing gained. And the trick is almost the same as my dutch book for death in damascus. I think this should generalize well.
The amounts of money lost in the Dutch Book get very small, but that’s fine.