The first is correct. If you expect all 10 participants to act the same you should not distinguish between the cases when you yourself are the sole decider and when one of the others is the sole decider. Your being you should have no special relevance. Since you are a pre-existing human with a defined identity this is highly counterintuitive, but this problem really is no different from this one: An urn with 10 balls in different colors, someone tosses a coin and draws 1 ball if it comes up head and 9 balls if it comes up tails, and in either case calls out the color of one ball. Suppose that color is red, what is the probability the coin came up tails?
If the coin came up heads there is a 1⁄10 chance of you being drawn, if it came up tails there is a 9⁄10 chance of you being drawn, and a 1⁄9 chance of you being you. Since in your intuition you are always you this probably seems nonsensical. But if you distinguish between yourself and the others a priori you can’t really treat the payoff like that.
The first is correct. If you expect all 10 participants to act the same you should not distinguish between the cases when you yourself are the sole decider and when one of the others is the sole decider. Your being you should have no special relevance. Since you are a pre-existing human with a defined identity this is highly counterintuitive, but this problem really is no different from this one: An urn with 10 balls in different colors, someone tosses a coin and draws 1 ball if it comes up head and 9 balls if it comes up tails, and in either case calls out the color of one ball. Suppose that color is red, what is the probability the coin came up tails?
If the coin came up heads there is a 1⁄10 chance of you being drawn, if it came up tails there is a 9⁄10 chance of you being drawn, and a 1⁄9 chance of you being you. Since in your intuition you are always you this probably seems nonsensical. But if you distinguish between yourself and the others a priori you can’t really treat the payoff like that.