I suggest explicitly stepping outside of an expected-utility framework here.
EV seems fine. You just need to treat it as the multi-stage decision problem it is, and solve the MDP/POMDP. One of the points of my Kelly coin-flip exercises is that the longer the horizon, and the closer you are to the median path, the more Kelly-like optimal decisions look, but the optimal choices looks very unKelly-like as you approach boundaries like the winnings cap (you ‘coast in’, betting much less than the naive Kelly calculation would suggest, to ‘lock in’ hitting the cap) or you are far behind when you start to run out of turns (since you won’t lose much if you go bankrupt and the opportunity cost decreases the closer you get to the end of the game, the more greedy +EV maximization is optimal so you can extract as much as possible, so you engage in wild ‘overbetting’ from the KC perspective, which is unaware the game is about to end).
Ah, yeah, this looks pretty close to what I was looking for.
OK, so if I’m understanding correctly, the basic idea is EV maximization with a cap on total possible winnings? (Which makes sense—there’s only ever so much money to win.)
So is the claim that this approaches Kelly in the limit of simultaneously increasing cap and horizon?
Yes, for some classes of games in some sense… MDP/POMDPs are a very general setting so I don’t expect any helpful simple exact answers (although to my surprise there were for this specific game), so I just have qualitative observations that it seems like when you have quasi-investment-like games like the coin-flip game, the longer they run and the higher the cap is, the more the exact optimal policy looks like the Kelly policy because the less you worry about bankruptcy & the glide-in period gets relatively smaller.
I suspect that if the winnings were not end-loaded and you could earn utility in each period, it might look somewhat less Kelly, but I have not tried that in the coin-flip game.
EV seems fine. You just need to treat it as the multi-stage decision problem it is, and solve the MDP/POMDP. One of the points of my Kelly coin-flip exercises is that the longer the horizon, and the closer you are to the median path, the more Kelly-like optimal decisions look, but the optimal choices looks very unKelly-like as you approach boundaries like the winnings cap (you ‘coast in’, betting much less than the naive Kelly calculation would suggest, to ‘lock in’ hitting the cap) or you are far behind when you start to run out of turns (since you won’t lose much if you go bankrupt and the opportunity cost decreases the closer you get to the end of the game, the more greedy +EV maximization is optimal so you can extract as much as possible, so you engage in wild ‘overbetting’ from the KC perspective, which is unaware the game is about to end).
Ah, yeah, this looks pretty close to what I was looking for.
OK, so if I’m understanding correctly, the basic idea is EV maximization with a cap on total possible winnings? (Which makes sense—there’s only ever so much money to win.)
So is the claim that this approaches Kelly in the limit of simultaneously increasing cap and horizon?
Yes, for some classes of games in some sense… MDP/POMDPs are a very general setting so I don’t expect any helpful simple exact answers (although to my surprise there were for this specific game), so I just have qualitative observations that it seems like when you have quasi-investment-like games like the coin-flip game, the longer they run and the higher the cap is, the more the exact optimal policy looks like the Kelly policy because the less you worry about bankruptcy & the glide-in period gets relatively smaller.
I suspect that if the winnings were not end-loaded and you could earn utility in each period, it might look somewhat less Kelly, but I have not tried that in the coin-flip game.