I guess that does need a lot of explaining.

I would say:

P(green|mostly green bucket) = 1

P(green|mostly red bucket) = 1

P(green) = 1

because P(green) is not the probability that you will get a green marble, it's the probability that someone will get a green marble. From the perspective of the priors, all the marbles are drawn, and no one draw is different from any other. If you don't draw a green marble, you're discarded and the people who did get a green vote. For the purposes of figuring out the priors for a group strategy, your draw being green is not an event.

Of course, you know that you've drawn green. But the only thing you can translate it into that has a prior is "someone got green."

That probably sounds contrived. Maybe it is. But consider a slightly different example:

• Two marbles and two people instead of twenty.
• One marble is green, the other will be red or green based on a coin flip (green on heads, red on tails).

I like this example because it combines the two conflicting intuitions in the same problem. Only a fool would draw a red marble and remain uncertain about the coin flip. But someone who draws a green marble is in a situation similar to the twenty marble scenario.

If you were to plan ahead of time how the greens should vote, you would tell them to assume 50%. But a person holding a green marble might think it's 2/3 in favor of double green.

To avoid embarrassing paradoxes, you can base everything on the four events "heads," "tails," "someone gets green," and "someone gets red." Update as normal.