I just thought of another argument. Imagine that before being faced with counterfactual mugging, the agent can make a side bet on Omega’s coin. Let’s say the agent who doesn’t care about counterfactual selves chooses to bet X dollars on heads, so the income is X in case of heads and -X in case of tails. Then the agent who cares about counterfactual selves can bet X-5050 on heads (or if that’s negative, bet 5050-X on tails). Since this agent agrees to pay Omega, the income will be 10000+X-5050=4950+X in case of heads, and 5050-X-100=4950-X in case of tails. So in both cases the caring agent gets 4950 dollars more than the non-caring agent. And the opposite is impossible: no matter how the two agents bet, the caring agent always gets more in at least one of the cases.
“Imagine that before being faced with counterfactual mugging, the agent can make a side bet on Omega’s coin”—I don’t know if that works. Part of counterfactual mugging is that you aren’t told before the problem that you might be mugged, otherwise you could just pre-commit.
I just thought of another argument. Imagine that before being faced with counterfactual mugging, the agent can make a side bet on Omega’s coin. Let’s say the agent who doesn’t care about counterfactual selves chooses to bet X dollars on heads, so the income is X in case of heads and -X in case of tails. Then the agent who cares about counterfactual selves can bet X-5050 on heads (or if that’s negative, bet 5050-X on tails). Since this agent agrees to pay Omega, the income will be 10000+X-5050=4950+X in case of heads, and 5050-X-100=4950-X in case of tails. So in both cases the caring agent gets 4950 dollars more than the non-caring agent. And the opposite is impossible: no matter how the two agents bet, the caring agent always gets more in at least one of the cases.
“Imagine that before being faced with counterfactual mugging, the agent can make a side bet on Omega’s coin”—I don’t know if that works. Part of counterfactual mugging is that you aren’t told before the problem that you might be mugged, otherwise you could just pre-commit.