I don't understand how the offer gets rescinded after "accept".  Is this just a game of "player says higher or lower about her threshold until banker and player agree to flip or pay?".  Is there an assumption that it's not anchored or path-dependent (that is, if player would accept 100k, but after banker offers 500k, rescinds it, and offers 110k, she rejects it?

In real-world games, I always wonder how much arbitrage goes on.  In the big-buyin poker and sports-betting world, it's common for players to lay off some of their action, and if someone I knew was about to play this game, I'd absolutely offer to guarantee some amount if she refused offers below a higher amount and split the difference with me.  For instance, I'd guarantee $500K if she rejects offers below $990K and gives me 90% of any payout over $500k.  I'd base this on a risk pool of people who buy shares, so none of us face the full $500K risk.  

It is interesting, though, when you play these kinds of games at parties, to see how the results differ when you talk about absolute values of money, vs "a coin flip for a year's paid vacation vs offers of number of weeks off", or "flip for 2 months' pay vs percent-bonus offers".

 

Quick note: might be easier to replace your utility function as $u\left(x\right)=1-e^{-\lambda x}$ for some parameter $\lambda >0$ (which is equivalent to the one you have, after rescaling and shifting). Utility functions should be convex but this is very convex, being bounded above.

Utility functions are discussed a lot here; I think it's worth poking around a bit.