Are you claiming that it’s rational to precommit to saying “nay”, but upon observing D1 (assuming no precommitment happened) it becomes rational to say “yea”?
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).
Are you claiming that it’s rational to precommit to saying “nay”, but upon observing D1 (assuming no precommitment happened) it becomes rational to say “yea”?
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).