those figures hold for you if you do not have other information. but do they necessarily hold for the other players? that is one implicit assumption there already. maybe the other players have some kind of method (econ speak: technology, maybe?) that allows them to get the white ball with 50% probability rather than 33%, like some weird stephen chow movie like that one where he is a trickster with weird gadgets.

or maybe the other players are stephen chow and chow yun fatt - gambling king, got special eyesight or sixth sense?

in order to know these numbers, then we need to assume complete information about each players abilities. usually we do not, so we assume that everyone has the same ability.
if so, then your payoff in the second period would be (0.5)*(0.33), which would be less than the 0.33 you would get if you went in the first period, and vice versa. so you could then say that if that implicit assumption that all players face the same probabilities is not valid, then if the other players have some way of increasing their probability of getting white above the standard level of 33%, you will want to go first, or if they have some impaired ability, you will want to go second (if say they only got 10% chance of getting in the first round).
similarly, if it is you who has a technological advantage or disadvantage, then the outcome is obvious from extrapolating the above logic/example. you can also consider what if you have to pay for the additional technology, then mention in general about how to set up your choice function. like what is your expected payoff if you pay for the technology and go first or go second... et cetera.
got a lot of nice ways to spin this one la. can smoke one! but in the end, still dependent on the original result.