First, there is no positional advantage to acting last. Since the players are kept in a sound-proof booth while any previous players play, order doesn't matter.
Second, there must be a Nash Equilibrium to the game where a strategy to stand with a score of at least x points will be superior to any other strategy. The question is finding x.
What I did was ask myself what would be the strategy if instead of a card numbered 1 to 100, each player got a random number uniformly distributed between 0 and 1 and look for the point x where a perfect logician would be indifferent between standing and switching. With that answer, it is easy to apply the answer to a discrete distribution from 1 to 100.
I'll stop talking at this point and let my readers enjoy the problem. See the links below for the answer and solution.
Answer for a continuous distribution from 0 to 1.
Answer for a discrete distribution from 1 to 100.
For my solution, please click here (PDF).
This question was raised and discussed in my forum at Wizard of Vegas.