Optimal amount of money to put in pot on a draw

[spamuell]

How much is the optimal amount of money for a player to put in the pot on a draw, if (s)he was able to choose the exact amount of money to put in?

For example, if Max is on a flush draw on the turn in a game of hold'em and he is 4.1:1 to make his winning flush and the pot is exactly 4.1 bets in size, does this mean that Max should ideally put in exactly 1 bet to maximise his profit, because any less means that he is not making enough when he is drawing, and any more means that he is losing too much when he doesn't make his draw?

This is from a purely mathematical point of view at the moment, I'm not concerned with pushing people out of the pot or anything like that in this example.

Any help is much appreciated.

[Duke]

If he has 5 opponents then shouldn't he go for a cap, unless the raising makes up a large percentage of his bankroll? There's an existing amount of money in there, and that plus getting paid 5-1 on additional money is a good amount for a 4.1:1 draw.

If he has fewer than 5 opponents for this example, at some point during the raising he'll be about even on the math, so he'll make no money over time on the play, lose money if he goes further, and should only continue to raise if his raise has a chance to win the pot without completing the draw.

~D

[pudley4]

It depends on the number of opponents Max is up against.

Let's go with your assumption that Max is 4.1-1 against making his hand on the river. We'll disregard any river bets - assume if he makes his hand, all his opponents will fold without calling any bets on the river; and if he doesn't make his hand, he'll fold.

If Max has 4 or fewer opponents, he is losing money on every bet he puts into the pot. So he should try to put as few bets into the pot as possible.

If Max has 5 or more opponents, he is making money on every bet that goes into the pot. So he should get as many bets in as possible.

[Louie Landale]

Heads-Up? If he's a 4:1 underdog he does better not to invest any additional amount at all. The optimal amount is $0.

The size of the pot comes into play when the opponent, against your desires, bets. That determines IF you should call or not, but it does NOT determine whether you prefer there to be a bet or not: you do not.

- Louie

[bigpooch]

Anytime there are a total of N players in the pot (including
yourself who I assume is on some sort of draw) and your
equity (which amounts to clean wins + fractions for split
pots) in terms of the pot size is:

<=1/N: 0 is the amount you want to put in
>1/N: the most amount if all N players call

Typically, for a flush draw, you would think only of jamming
the pot on the flop, especially if you had the boss flush
draw and one overcard if there are two or more opponents
that are willing to call all the bets and raises. With only
two opponents, you could often be in trouble because one of
your opponents could drop out or the other player also has
the same flush draw. For three opponents, you have an easy
cap on the flop but always be careful if the board pairs up!
You often see sets in these capped pots.