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Comment by cubefox | Hacker News Reader

> Even weirder, if we would have played this game repeatedly, the second urn is clearly preferable.

I don't think that's the case?

Edit: I get it now, under repetition the probability distribution of the second urn could be inferred, and the gambler could use this information to bet on the more frequent color. However, the Ellsberg paradox only says we prefer known distributions to unknown ones, and under repetition the distributions of both urns would be known. So it doesn't say people would prefer betting on the first urn in the repetition case.

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It is the case if the balls are being put back. As you play more games, you would figure out the proportion of red and green balls in the second urn. Therefore, you would figure out which is the better colour.

Repeated games under epistemic uncertainty actually allows you to get an edge by extracting information and reducing that uncertainty, in comparison to a game which is guaranteed to have no edge for you.

Yeah, but that would change an unknown distribution to a known distribution. Then the Ellsberg paradox does no longer apply.

Yes, but why do we intuitively avoid epistemic uncertainty, even though it is something which can only _give_ us an edge here?

To make clear: playing this game with an urn which has an equal amount of red and green balls, is the worst case scenario when you get to pick the colour that wins. The second urn can only be the same or better, whether or not you can figure out any information about the balls in it.

It can only give us an edge in the repetition case. And the Ellsberg paradox doesn't apply to the repetition case. Ellsberg doesn't say anything about people preferring the first urn in the repetition case. In fact, I see no evidence that they do.

In a one time game it’s a 50% known bet vs. an unknown bet. 50% is not bad odds?

Because the real consequence of losing is not $0, so there is (potentially unacceptable) cost to gain knowledge which gives you enough edge to win?

Reducing the uncertainty does not give you an edge vs having certainty a priori though.

For instance, you are not better off discovering that the urn has a 50:50 ratio than you would be if you knew that 50:50 ratio ahead of time.

If you're saying that you could discover a better ratio, then the thing that gives you the edge is the better ratio itself; if you knew the better ratio ahead of time, the epistemic uncertainty would only harm you.

I think the second urn is preferable in the repeated case because for really unbalanced ball setups there is a high correlation between the color that gets drawn each time. You can just guess that the next ball will be the color that was most common among previous drawings.

With the 30/30 red/green urn, on the other hand, you don't really have a much greater edge than 50%, at least in the early phases. In the later phases, perhaps you could count balls to get an edge, unless the balls are put back in the urn after being drawn.

When the game is repeated, does the second urn always have the same distribution of balls?

Can you add any more? Cause, the other theory and Wikipedia click hole are pretty convincing. But, I'm not a maths wiz.

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