When cloning is involved, you could argue that the right way to decide
whether to take a bet is to look a the outcome on a "per instance" basis.
Let us suppose that the answer to question 3 ("you are not a clone.
what is the probability that the coin landed heads?") is 1/3, as Eliezer
has it. He may be willing to give 2-1 odds in making a bet that the
coin will land tails.
If he did, he would put up $200 and the experimenter would put up $100.
If the coin lands tails, he'd win $100 and not be cloned. If the coin
landed heads, he'd lose $200, but on a per instance basis that averages
out to a loss of $100 per person. So the potential gains and losses are
equal on this basis.
When you get cloned, all your property gets divided by two, at least on
average. It may not be a step to take lightly. But the result of the
cloning is that your productivity is doubled. You can handle twice the
liability afterwards as you could before. This changes how you should
evaluate bets.
In effect, cloning makes it a non-zero-sum game. The changes introduced
by cloning in potential future earnings cause the gains or losses of the
bet to be evaluated differently. The result is that it may be rational
to take a bet which gives an expected profit to the other player.
Hal