[Alan McIntire]

Speaking of fraction of the pot to bet, I'm wondering what
a roughly optimum strategy should look like. Say the pot
contains 1 bet, each of two players has a stack of 10, and
the rules allow the first player to bet any amount 1 to 10,
and the second player to call or fold.

If the first player is playing a NASH strategy, and bets 1 unit, the second player will have to call half the time
to prevent being robbed by bluffing on everything. The caller will be losing a large fraction of the time since he's calling with half of all hands, but he'll be calling
a lot with relatively weak hands.

On the other hand, if the first player bets 10 units, he
usually won't be called, yet if he is called, the second player will be calling with a lot higher hands, the top 1/11, and there will be a greater than even chance of losing the bet unless you bet only bet 1/22 of the top hands, or something like this.

I'm guessing the optimum strategy for bet sizing should be something like an even money split:

If you have one of the top 1/22 of all hands, bet
10 units once,
bet 9 units 10/9 times,
bet 8 units 10/8 times,
...
bet 2 units 10/2 = 5 times,
bet 1 unit 10 times.

Of course you'd also have to bluff proportionate to the
pot odds you're offering your opponent.

Any ideas or inputs on the optimum way to vary bet size?