It hadn't occured to me to think of this in a game theoretical framework. It is a (very) finate game and so we could just write down the decision nodes (the possible outcomes) which are fixed regardless of either player's play. Howver I think getting to a subgame perfect Nash equilibrium would be hard.
(A SPNE is a pair of strategies for a player and bank such that neither player can change his strategy without possibly doing worse in each subgame. The best way to thoink about it is that the strategy pair is optimal at all the last decision node. So you know what each player will do last. Knowing this, you can plot your second to last move. But now you know how each player's last two moves will work. So now you can plot your third to last move... etc.)
SPNE are of questionable value because they may not be 'optimal' in some sense. Moreover, applying game theory to this would be a mess, I think.
Let's take a really dumbed down version of the game. In my version there are only two suicases and three moves.
1. You pick a case
2. I make you an offer
3. You chose
This is actually the last possible move of the regular game, so looking at it is necessary to analysing the regular game anyways.
Now the first move is completely stochastic and has no strategy implications. In fact, we could replace it with a coin flip or whatever. So the bank here has the first move. It's goal would be to offer the least amount of money that is less than it's certainty equivalent for the negative (relative to the bank) lottery it's about to play and it believes I will take. Since there will be no more moves, the bank can assume that my strategy will be to take the offer if it is greater than my certainty equivalent for the positive (relative to me) lotteries.
If the bank knew my utility function, it could do this perfectly. It would offer me 1 cent over my certainty equivalent and I'd take it (if I were behaving rationally). Then, it would have 'won' the difference.
The bank doesn't know my utility function, however. In my simplified game, the bank knows nothing about my utility function. It's offer has to be optimal against all of my strategies (but we already know that my strategy is determined by my utility). So the bank would have to do some crazy integration over the weighted population of utility functions. If it knew nothing about the population of utility functions, it would have to assume a uniform distribution. In that case it's only optimal strategy (and this is terrible for the bank) would be (I think) to offer a penny below it's certainty equivalent.
Now, you argue, in the real game the bank does have some knowedge of my utility function. If we get to the last 2 suicases, it's made a few (8 or so?) offers already. So it has some data points about my preferences over past lotteries. While it technically might be possible to get some information, I think the problem is really, really hard. Maybe I'll bounce it off of a few friends that are better than me at this stuff...
