Suppose you have two identical agents with shared finances, and three rooms A1, A2, B.
Flip a fair coin.
- If the coin comes up H, put the agents in A1, A2.
- If it comes up T, flip the coin again.
- If it comes up H, put the agents in A1, B.
- If it comes up T, put the agents in A2, B.
(At each point, flip another fair coin to decide the permutation, i.e. which agent goes to which room.)
Now to each agent in either A1 or A2, make the following offer:
Guess whether the first coin-flip came up heads or tails. If you correctly guess heads, you both get $1. If you correctly guess tails, you both get $3. No negative marking.
The agents are told which room they are in, and they know how the game works, but they are not told the results of any coin tosses, or where the other agent is, and they cannot communicate with the other agent.
In terms of resulting winning, if an agent chooses to precommit to always bet heads, its expected earnings are $1, but if it chooses to precommit to always bet tails, its expected earnings are $1.50. So it should bet tails, if it wants to win.
But consider what happens when the agent actually finds itself in A1 or A2 (which are the only cases it is allowed to bet): if it finds itself in A1, it disqualifies the TT scenario, and if it finds itself in A2, it disqualifies the TH scenario. In either case, the probability of heads goes up to 2/3. So then it expects betting heads to provide an expected return of $1.33, and betting tails to provide an expected return of $1. So it bets heads.
(There are no Sleeping Beauty problems here, the probability genuinely does go up to 2/3, because new information -- the label of the room -- is introduced. BTW, I later learned this is basically equivalent to the scenario in Conitzer 2017, except it avoids talking about memory wiping or splitting people in two or anything else like that.)
What's going on? Is this actually a way to beat superrational agents, or am I missing thing? Because clearly tails is the winning strategy, but heads is what EDT tells the agent to bet.