Function of pot odds, math related question

[lotus776]

I've stumbled over a theory element of poker that I haven't read about yet and thought I'd post it to see what others who may have realized the same thing say.

When playing No-Limit, the idea of pot odds governs the amount which the bettor bets, not considering all psychological reasons for betting. Primarily pot odds are given to the caller based on how much the bettor decides to bet. For example:

pre-flop the pot contains $100 ([25+25]+50 from the blinds, no callers for simplicity). After the flop the small Blind bets 50, the pot is now $150 and the caller must pay $50 to see the next card. The caller is being offered $150 to $50 or 3 to 1 pot odds.

What I've noticed is that as the bet amount which the bettor places increases (obviously the pot odds for the caller decrease) the pot odds approach 1 to 1, but never quite reach it. This conclusion lead me to believe that there is a simple function that can define that movement of this curve on a graph. For Example:

-if you bet 1/4 of the pot you offer the caller 5 to 1 odds
-if you bet 5/6 of the pot you offer the caller 2.5 to 1
-if you bet 3/2 of the pot you offer the caller 1.667 to 1

this continues on and on for infinity, the bettor is never able to give the caller 1 to 1 odds, but the number approaches an asymptote; all of this leads me to believe that the function looks something like:

f(x)=(x+p)/x, where p is the pot. But I'm not entirely sure.

Has anyone thought of this before? This is really just theory as no one could lay down a bet of infinity times the pot, but for my personal satisfaction I'd like to know if anyone knows this measureable function or if they can help me determine it.

thanks a lot
-Brent

[PaultheS]

(I replied to this post in the Science, Math, and Religion forum but I just noticed you posted here too so I'll cross-post my reply.)

You just want a function for pot odds?

f(x) = (1 + x) / x, where x is the fraction of the pot you bet (1/2, 1, 3/2, 2, etc.).

e.g. if you bet 1/2 pot, f(1/2) = (1 + 1/2) / (1/2) = 3. So the player is getting 3:1 on his call (he's calling 1/2 pot to win 3 1/2 pots).

The curve is f(x) = (1/x) + 1, which approaches 1 as x goes to infinity and infinity as x goes to 0 (from the positive side), as you stated.

Hope that helps.

[AaronBrown]

This is a very important concept in Poker.

To remove the effect of information, assume both players have displayed their pocket cards. Only the river remains to be dealt. You have a pair of Aces, your opponent has a four flush in hearts. You will win unless he gets one of the nine remaining hearts out of the 44 remaining cards in the deck. There is $44 in the pot, and it is your bet in no-limit.

If you check, he has infinite pot odds and will (of course) not fold. You have 35/44 chance of winning the pot, for $35 equity.

If you make a large bet, say $44, he will fold and you will get $44 equity. His pot odds are 2 to 1, but his odds of winning are only 35 to 9. If he were to call, he would be paying $44 but only getting pot equity of 9/44 * $132 = $27.

If you bet $198/13 or about $15, he will be indifferent between folding and calling. He will pay $15 for 9/44 chance to win $74.

Now let’s change the situation a bit. Suppose the hands are not exposed. You think he’s going for a flush, and you think he thinks you have a pair of Aces. In fact you’ve got three Aces, and therefore a chance for a full house. Or maybe you’re holding one of the hearts, so his chances are 8/44 instead of 9/44. If you bet more than $15, you think he’ll fold, but that he’ll call $15. It’s to your advantage to make this bet, because it gets more money in the pot, and your odds of winning are actually better than 35/44. But you have to do the calculation, from his point of view, in order to know the maximum amount you can bet.