Actually I find this problem quite perplexing, just like an optical illusion that makes you see different things.
Yes, what I am claiming is that if they observe D1, they should say “yea”. The point is that only player 1 knows whether D1 holds, and no other player can observe D1. Sure, there will be some player i for which Di holds, but you cannot calculate the conditional expectation as above since i is a random variable. The correct calculation in that case is as follows:
Run the game and let i be a player that was selected as a decider. In that case the expected donation conditioned on the fact that Di holds is equal to the expected donation of 550, since i is a decider by definition and thus Di always holds (so the condition is trivial).
Actually I find this problem quite perplexing, just like an optical illusion that makes you see different things.
Yes, what I am claiming is that if they observe D1, they should say “yea”. The point is that only player 1 knows whether D1 holds, and no other player can observe D1. Sure, there will be some player i for which Di holds, but you cannot calculate the conditional expectation as above since i is a random variable. The correct calculation in that case is as follows:
Run the game and let i be a player that was selected as a decider. In that case the expected donation conditioned on the fact that Di holds is equal to the expected donation of 550, since i is a decider by definition and thus Di always holds (so the condition is trivial).