CardSharp

Last article we looked at the set farming strategy, and saw that it was an effective way to take money from an opponent with a big pair if both players adopted certain strategies. We also looked at two possible “defenses” the guy with aces could employ – raising more or having a smaller stack. However, both of these are infeasible in many circumstances. You don’t have much control of your stack size, and raising the needed 10% of the effective stacks is often not feasible in deepstack games. Clearly there’s only one option left for the aces: they can’t always pay off the set for a full stack. In fact, against the strategy I described for the set farmer (check-fold postflop if no set), there’s no reason the aces should ever pay off the set. Any time the guy bets, or even calls, you know he’s got you beat (unless you also hit a set). This creates an odd strategy for the guy with aces – bet out every time, and if your opponent folds, fine. If he gives you any action, check-fold the rest of the streets. This strategy beats the set farmer out of almost 4BB/hand on average with the setup from the last article.

That strategy is optimal against the set farmer. However, there are some pretty substantial problems with it. First, it makes no use of the made hand strength of the the aces – player 1 could adopt this strategy with any two cards whatsoever. That ought to strike you as wasteful. Second, it fails to extract value out of weaker made hands that would pay the aces off. That’s a huge problem – there’s a lot of money to be made with aces when your opponent has AK and the flop comes K high. Keeping the pot small or giving it away outright against a hand you’re over a 9:1 favorite against is not an attractive solution. Perhaps worst of all, it makes you horribly bluff vulnerable – anyone with any two cards can bet and take the pot from you. So what we’ve got here is a strategy that’s the best possible when you hold aces against one possible opponent holding and line of play, and that’s easily beaten if your opponent adopts a different line. As you can probably guess if you read Cardsharp regularly, what we have here is a strategy rho. This rho, which I’ll describe from the perspective of an AA vs. set farmer confrontation has 4 strategies in the head:

1. player 1(with aces) always tries to get the stacks in
2. player 2 (with the smaller pocket pair) set farms – he never bets or calls postflop without a set
3. player 1 folds to any action
4. player 2 bluffs at every opportunity

As you can see, 2 beats 1, 3 beats 2, 4 beats 3, and of course 1 beats 4 – it’s a standard rho. Now, previously we’ve been discussing how to use game theory to balance strategies from the head of the rho. And we know that the way to do it is to select between the strategies at a rate that makes your opponent’s decision not matter.  So let’s do that, but I want to do it in a fairly general way that applies not just to set farming and aces, but all made hand vs. draw confrontations.

First, let’s consider the general course of events in a made hand vs. draw situation.  To begin with, there’s some money in the pot  – usually a small amount relative to the stacks.  On the street of play where the made hand vs. draw situation arises, some amount of money goes in.  Then the draw hits, or at least appears to hit. (it may not be totally clear what is being drawn at). Then the made hand has to decide whether to pay off the likely draw if he bets.  This scenario is incredibly important to NL holdem – I would say well over half the hands have a trajectory that looks something like this.  Now, back to the game theory.  From the perspective of the player with the made hand (we’ll call him player 1), he wants to make the decisions of the guy with the draw (player 2) not matter.  Which decision by player 2?  Why, the decision whether or not to play the draw or not.  If he can accomplish this, player 1 can make it unprofitable to draw against him.  To start, let’s make a simplifying assumption: the amount of money already in the pot is trivial compared to the stack sizes.  In other words, let’s ignore pot odds for a second, and look only at implied odds.  In order to make player 2’s decision not matter, we need to look at 3 variables: the size of the bet from the made hand (BET), the probability the draw will hit (P(draw)) and the effective stack size remaining (ESS).  We then get the following rule:

The Payoff Rule

The pot is small, you on a made hand, and you believe your opponent is on a draw.  The made hand has made a bet (BET) that charges the draw to see the next card.  The draw has probability P(draw) of hitting.  There is ESS money behind at the start of the street.  On the next card, the draw hits.  If BET > P(draw) * ESS, you should aways pay off the draw, up to and including your entire stack.  If BET < P(draw)*ESS, you should try to limit the percentage of your remaining stack that goes in on future streets based on the fraction of how much smaller BET is than P(draw) * ESS.  In other words, apply pot control.  If you can’t accomplish that, you need to fold some percentage of the time on future streets such that you pay off no more than that amount on average.

The payoff rule is not an absolute thing – the more money that was in the pot to start, the more willing you should be to pay off.  If your opponent may have a made hand you beat instead of a draw, you should pay of more.  If your opponent may have a made hand that beats you, you should pay off less.

An example is in order, so let’s go back to the set farming from the last article.  Player 1 bet 4BB preflop and was called.  That qualifies as a small pot.    Player knows his opponent is on a draw, because he has the nuts.  The most likely draw his opponent has is a set draw (P(draw)=0.12 or 12%).  The remaining stacks were 100BB at the start of the street.  So BET=4, P(draw)=0.12 and ESS=100.  P(draw)*ESS = 12, so the bet was 1/3 the size player 1 would need to have made to commit against the draw.  That means player 1 wants to limit how much additional money he puts in the pot to no more than 33BB.  Since player 1 is in position, he can probably do that even if player 2 has a set.  However, if player 2 starts betting very aggressively, the aces ought to fold some percentage of the time to keep the average payoff around 33BB when the set hits.

I said at the start of the previous article that this was one of the most important things I have to say about NL.  The reason is simply that a very high percentage of the mistakes I see NL players at all levels make are violations of the payoff rule. Yet the amazing thing is that I’ve never seen the payoff rule in print before.  It’s something all good NL players know from experience, but I believe I’m the first one to put it in print and name it.

There’s slightly more to implementing the payoff rule in practice than what I’ve said here – we still need to talk about how to achieve pot control in practice, and how to decide which times to fold.  But the general principle isn’t too hard to understand – if there was a draw, and it didn’t cost much for your opponent to draw, and the draw hits, you need to take steps to avoid paying off too much.  I’ll come back to these topics shortly.