Is this game solveable?

[poker-penguin]

Headsup match.

I have 20 red chips, you have 20 blue chips.

We both choose, unseen by the other and simultaneously, how many chips to gamble. These chips are placed into a container, shaken up, and one is selected at random by the referee. The player whose chip was selected wins all the chips in the pot.

His opponent's chips are replaced with chips of his colour, and we repeat the wager.

So, what's the optimal strategy?

My envelope is a horrible mess right now, involving the probability of your opponent playing a certain number of chips.

There's a part of me that says this is a trick question because the marginal return of betting more than 1 chip decreases - you're adding 1 to the prize pool but not increasing your share by 1/x but instead, 1/(x+1) so obviously the best strategy is to keep betting one chip and hope to suckout.