I think I found a way to do slightly better than .69 for 2 and .31 for 3 when cooperating with yourself:
Play 1 with probability .12, 2 with .60, and 3 with .28. If you played the same number, repeat. If you played different numbers, the higher one can play a 3 and the lower one can play a 2 (or whichever cooperating pattern you choose), and they can start alternating from there.
If my calculations are correct, this only loses 2.10 points on average from the maximum possible, which seems to be the best you can do. The goal of the randomization is to pick a different number than your opponent so that you can start cooperating, so the reason this does better is that by picking out of 3 numbers, you have a higher change of not picking the same number.
I think I found a way to do slightly better than .69 for 2 and .31 for 3 when cooperating with yourself:
Play 1 with probability .12, 2 with .60, and 3 with .28. If you played the same number, repeat. If you played different numbers, the higher one can play a 3 and the lower one can play a 2 (or whichever cooperating pattern you choose), and they can start alternating from there.
If my calculations are correct, this only loses 2.10 points on average from the maximum possible, which seems to be the best you can do. The goal of the randomization is to pick a different number than your opponent so that you can start cooperating, so the reason this does better is that by picking out of 3 numbers, you have a higher change of not picking the same number.