## Aces & Set farming Part 1

### Mathematics, No Limit Texas Holdem

April 26, 2008It’s time for one of the most important things I’ve got to say about no limit Texas holdem. We’ve talked previously about the topic of set farming when discussing the 5/10 rule. Specifically,

Set farming is calling a bet preflop with a small to medium pocket pair (which is unlikely to be best by the river if it doesn’t improve) hoping to hit a set (3 of a kind made with one on board plus your pair). It’s a longshot play where you rarely hit, but when you do you have a hand that’s almost certainly best, and you can comfortably get your stack in.

Set farming is a very central part of correct NL play. In fact, against certain opponents, it is the single most profitable tactic in your arsenal. To understand why, consider this hypothetical hand:

all stacks are 100BB deep

Player 1, under the gun, picks up AhAd and raises to 4BB. Everyone folds around to player 2 in the big blind, who looks down at 4c4d. And calls. At this point, each player decides on their strategy. Player 1 will push his aces - betting roughly the pot on each street, figuring this will deny his opponents the odds to draw. Player 2 will check-fold the flop if he has aces, and try to get his stack in if he hits his set.

Now, this all seems reasonable. Popular poker advice is to push aces hard to prevent a draw. And the usual tactic for set farming is to shut down unless you hit your set, in which case you likewise want to get the money in. So who’s got the best of it here?

Simply ignore the 1/2BB put in by the dead small blind. The pot contains 8BB, all put in by the two remaining players, so their net expectation on the hand cross all streets will be exactly opposite - if player 1 wins 10BB on average, player 2 loses10BB on average and vice versa. Now, there are 4 cases:

- Case 1: Player 2 does not hit his set. In this case, player 1 bets and player 2 folds. Player 1 turns a 4BB profit on the hand.
- Case 2: Player 2 hits a set, and player 1 does not. In this case, all the money goes in. Player 2 wins about 92% of the these hands. Player 2 turn an 84BB profit on average.
- Case 3: Both players hit a set. In this case, all the money goes in. Player 1 wins about 96% of these hands, and turns a 92BB profit on average.

Case 1 happens about 88 % of the time, case 2 happens about 11% of the time, and case 3 happens about 1% of the time.

So who makes money here? Well, player 1’s expectation is as follows:

1’s expectation = .88 * 4BB + .11 * (-84BB) + .01 * (92BB)

1’s expectation = 3.52BB - 9.24BB +.92BB

1’s expectation = -4.8BB

That’s right - the guy with aces LOSES on average 4.8BB every time you play this hand out!!! As we already said, the two players get opposite results, so the guy with the pocket 4s is winning 4.8BB on average.

Now, I hope this shocks you. Player 1 started with the best hand. He played it in a very reasonable way. He was in position. And yet he’s losing money on average. Meanwhile, player 2, who was out of position, and with the worst hand, is turning a profit. Yikes, that sort of turns things around, doesn’t it? There’s more to it though. We picked some “parameters” for this example. The stacks aren’t always going to be 100BB deep and the preflop bet isn’t always going to be 4BB. Let’s make these variables, ESS (effective stack size) and PFB (preflop bet) respectively. Obvious PFB has to be less than or equal to SS. Now let’s re-do the math:

Simply ignore the 1/2BB put in by the dead small blind. The pot contains 2*PFB. Now, there are 4 cases:

- Case 1: Player 2 does not hit his set. In this case, player 1 bets and player 2 folds. Player 1 turns a PFB profit on the hand.
- Case 2: Player 2 hits a set, and player 1 does not. In this case, all the money goes in. Player 2 wins about 92% of the these hands. Player 2 turn an 0.84*ESS profit on average.
- Case 3: Both players hit a set. In this case, all the money goes in. Player 1 wins about 96% of these hands, and turns a 0.92 * ESS profit on average.

Case 1 happens about 88 % of the time, case 2 happens about 11% of the time, and case 3 happens about 1% of the time.

So who makes money here? Well, player 1’s expectation is as follows:

1’s expectation = .88 * PFB + .11 * (-0.84 * ESS) + .01 * (0.92 * ESS)

1’s expectation = .88 * PFB - .0924*ESS +.0092 * ESS

1’s expectation =.88 * PFB - .0832*ESS

So we can see that player 1’s expectation goes up the more money he gets in preflop, and goes down the bigger the stacks are. The opposite is true for player 2. We also see where the “10″ part of the 5/10 rule comes from. If player 1 makes a PFR of 10% or more of the effective stack size with aces, player 2 can’t make money by set farming. So what have we learned? If you have aces, you’d like to get as much money as possible in preflop because your profits go up when you do. We’ve learned aces play better if your stack is a little shorter. We’ve learned that if you get 10%+ of the stacks in preflop with aces, it’s enough that a set farmer isn’t getting the right price.

Now, that’s all well and good. But it’s not that realistic. Often times with aces, there’s no good way to get 10% of the stacks in preflop without losing your customers. This is especially true in deepstack play. If stacks are 500BB deep, it’s highly unlikely that villain will call an preflop open-raise of 50 times the BB. It’s a huge overbet. Now, overbetting with the nuts is safe - you can’t run into a better hand. But it’s not smart because they hands you’d like to play against will be run out of the pot. So player 1 is back to the same old dillema of how to play aces against smaller pairs and come out ahead.

Continued next article…

**This article is part of Project Cash Game No Limit Holdem - You can find more great strategy articles there.**

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