MarkGritter
06-07-2005, 11:25 PM
I've been working through The Theory of Poker and thought it would be interesting to apply some of the mathematical ideas to triple draw. Chapter 19 lays out a game-theoretic optimum bluffing frequency: Hero should bluff often enough so that the odds of Hero bluffing are the same as the pot odds Villian is getting to call. This has the advantage (or disadvantage) that it doesn't matter if Villian has good judgement about when you are bluffing. You will earn the same amount if Villian always folds, always calls, or makes any combination--- and that amount is greater than simply betting only made hands, and having Villian fold. (Of course, if Villian always calls you should never bluff and if Villian always folds you should always bluff--- but the game-theoretic answer tells you how to earn money even from a clever opponent.)
How does this work out in an example TD 2-7 hand?
Suppose (like in this example (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=2567700&page=&view=&s b=5&o=&vc=1)) Hero is 75322 after the drawing 1 in the second round, and Villian has drawn two with 732xx. Hero bets and Villian calls. Hero draws one and Villian stands pat with a J or T. (Anything stronger he would usually try a raise.)
Hero has a 45% chance of having the winning hand against Villian's distribution of JT732, J9732, J8732, or Tx732. (Villian has a better chance of winning by breaking lower J's.)
If Hero makes a J or a T, betting may not be wise (although I might do so against a thinking opponent, or one who stands pat too easily, since he didn't show strength the previous round.)
If Hero makes a better hand he should bet, since Villian will probably call with far more hands than he will bet here.
But, Hero should bluff some of the time with a bad hand; otherwise Villian can just fold his J or T to bets. The game-theoretic optimium percentage depends on pot size. Here are the results based on hero's 45% chances of a better hand:
Pot size Bluffing percentage
3 bets ------ 11%
4 bets ------ 9%
5 bets ------ 7.5%
6 bets ------ 6.5%
7 bets ------ 5.6%
8 bets ------ 5%
So Hero should be bluffing against an unknown or superior opponent about 5-7% of the time in a typically-sized pot.
The method Sklansky suggests is to pick some set of 'bad cards' as the randomizer for when to bluff. I wonder about this in a lowball game, though, because even an A or K could make you the best hand! To use this system you need to pick 2 or 3 bad cards to bluff on.
Daniel Negreanu's SS2 chapter suggests bluffing in this situation but doesn't provide much in the way of guideline Betting only paired 5's or only paired 7's fits his recommendation to only bluff with big pairs. But note that betting both pairs would be bluffing more often than the game-theoretic optimal, since you would be bluffing on 4-6 bad cards instead of 2-3.
All of the above is just one particular hand--- more often you will have an opportunity to bluff when both of you have been drawing one. In this case your chance of having the best hand may be higher, so you should bluff more often. If Villian has been pat in an earlier draw, his hand is probably better and you should bluff less often.
How does this work out in an example TD 2-7 hand?
Suppose (like in this example (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=2567700&page=&view=&s b=5&o=&vc=1)) Hero is 75322 after the drawing 1 in the second round, and Villian has drawn two with 732xx. Hero bets and Villian calls. Hero draws one and Villian stands pat with a J or T. (Anything stronger he would usually try a raise.)
Hero has a 45% chance of having the winning hand against Villian's distribution of JT732, J9732, J8732, or Tx732. (Villian has a better chance of winning by breaking lower J's.)
If Hero makes a J or a T, betting may not be wise (although I might do so against a thinking opponent, or one who stands pat too easily, since he didn't show strength the previous round.)
If Hero makes a better hand he should bet, since Villian will probably call with far more hands than he will bet here.
But, Hero should bluff some of the time with a bad hand; otherwise Villian can just fold his J or T to bets. The game-theoretic optimium percentage depends on pot size. Here are the results based on hero's 45% chances of a better hand:
Pot size Bluffing percentage
3 bets ------ 11%
4 bets ------ 9%
5 bets ------ 7.5%
6 bets ------ 6.5%
7 bets ------ 5.6%
8 bets ------ 5%
So Hero should be bluffing against an unknown or superior opponent about 5-7% of the time in a typically-sized pot.
The method Sklansky suggests is to pick some set of 'bad cards' as the randomizer for when to bluff. I wonder about this in a lowball game, though, because even an A or K could make you the best hand! To use this system you need to pick 2 or 3 bad cards to bluff on.
Daniel Negreanu's SS2 chapter suggests bluffing in this situation but doesn't provide much in the way of guideline Betting only paired 5's or only paired 7's fits his recommendation to only bluff with big pairs. But note that betting both pairs would be bluffing more often than the game-theoretic optimal, since you would be bluffing on 4-6 bad cards instead of 2-3.
All of the above is just one particular hand--- more often you will have an opportunity to bluff when both of you have been drawing one. In this case your chance of having the best hand may be higher, so you should bluff more often. If Villian has been pat in an earlier draw, his hand is probably better and you should bluff less often.