I think you're missing a term in your second calculation. And why are anthropism and copies of you necessary for this puzzle. I suspect the answer will indicate something I'm completely missing about this series.

Take this for straight-up probability:

I have two jars of marbles, one with 18 green and 2 red, the other with 18 red and two green. Pick one jar at random, then look at one marble from that jar at random.

If you pick green, what's the chance that your jar is mostly green? I say 90%, by fairly straightforward application of bayes' rule.

I offer a wager: you get $1 per green and lose $3 per red marble in the jar you chose.

After seeing a green marble, I think your EV is $5.60. After seeing a red marble, I think your EV is $0 (you decline the bet). If you are forced to make the wager before seeing anything, conditional on drawing green, I think your EV is $2.80. I calculate it thus: 50% to get mostly-green jar, and 90% of that will you see green and take the bet, which is worth +$1*18 - $3*2 in this case. 50% to get mostly-red, 10% of which will you draw green, worth +1*2 - $3*18. 0.5 * 0.9 * (1 * 18 - 3 * 2) + 0.5 * 0.1 * (1 * 2 - 3 * 18) = 2.80, which is consistent: half the time you pick green, with EV of 5.60.

I think you left out the probability that you'll get green and take the bet in each of your 0.5 probabilities for the conditional strategy. Multiply a 0.9 to the first term and 0.1 into the second, and everything gets consistent.