The fallacy is in the assumption that the player's valuation is linear in money. It isn't. Given the choice between 40,000 guaranteed, or a 50/50 choice between 100,000 and 0, almost everyone would pick the 40,000 guaranteed because the valuation tails off above a certain point.
And that is why the game works.
If it was a repeated game, not a single-play, then it wouldn't work this way, because the averaging strategy would work. But the answer above doesn't quite fit into classical game theory. Can anyone provide a better framework?