Poker Math Lesson 101: End Game Mathematics


This is the final part of my series on Poker Mathematics. In this section I will discuss some of the most misunderstood concepts in Poker. No Limit Hold 'em tournaments have recently become the format of choice for many casual and inexperienced players. That's why they've recently become my format of choice as well. A very lucrative opportunity exists for any seasoned player who is willing to learn the skills necessary to make the transition from cash game to tournament play.

End Game Mathematics

This article will focus on mathematics as it applies to short stack / large blind play.

In the early stages of a tournament, NLH plays out in much the same way as a NLH cash game. A competent ring game player should have no trouble at all adjusting to the early stages, when blinds are small relative to stack sizes. As the blinds begin to increase in size relative to the size of the stacks in play something interesting happens to the mechanics of the game. Some of the hands that are played by a lot of top players start to lose value and some of the hands that don't play well at all in ring games become quite powerful. I'm going to explain that concept mathematically and then use some sample hands to illustrate the point more clearly.

When the blinds begin to represent a large percentage of the average stack size, the guidelines that dictate the strength of an opening hand are altered drastically. Large blinds and small stacks take away a lot of the value of speculative hands such as 98s or 22. These hands derive most of their value from post- flop playability. They have the potential to flop monster hands; sets, flushes, straights, or full houses. They are also pretty easy to get away from if the flop is an unfavorable one. Much of the power of these speculative hands comes from the fact that they will often win and rarely lose large pots.

A hand like KJ is somewhat different. Many players consider KJ to be a "trap hand" the problem with a hand like KJ is the "negative implied odds" associated with it. Under cash game or low blind conditions, it is unlikely you will win a large pot with KJ, but it is no trick at all to lose a very large pot. The reason is that most of the large pots won in cash games come from both players believing they have the best hand. If you hold KJ and the flop comes K 4 3, you've flopped top pair with a jack kicker. You're going to have to proceed as though you have the best hand until you are given serious evidence to the contrary. The problem is that it is unlikely someone will pay you off with an inferior hand but it is quite likely you would be called by a stronger hand, such as AK. KJ is just a hard hand to play and I recommend that beginners just stay away from it entirely until they become more experienced.

As the blinds at a tournament increase, the "implied value" of speculative hands deteriorates and the fear of "negative implied odds" deteriorates as well. Suited cards, connecting cards, and small pairs lose value as high card hands go up in value. At the final table of a big tournament with blinds representing 10% of the average stack, KT becomes a formidable hand and 98s becomes nearly useless.

In the first round of a major tournament I wouldn't mind calling a small raise from good position with a hand like JTs. At the final table of that tournament, with blinds at 400-800, and me with 6000 chips, it is likely that I'd just throw the hand away. In the first round of a tournament you will rarely see me play a hand like K9o; at the final stages it isn't unusual to see me move all-in with that same hand.

Final Table Action

One area that I feel players, even good players, fail to understand when the blinds escalate is calling for pot odds with live cards. Final table action, with its disproportionate blinds, will force players to make moves like raising and re-raising with nothing just to avoid being blinded out. You have to think of your plays at this stage in terms of "raising to take the pot, but calling for pot odds". I think there are very few players today who truly understand "end game math". I'm going to see if I can equip you with at least a small understanding of it.

Let me give you an example. Let's suppose you are at the final table of a large, "winner takes all" tournament. The stacks started at 2000 chips with blinds of 10 and 20. Now, six hours later, the average stack in play is 35,000 and the blinds are 2,000 and 4,000. In this example there are only 8 players remaining and you are in the big blind and are one of the large stacks with 50,000 in tournament chips. All of the players fold to the dealer button (the short stack), and he goes all-in for his last 8,000 chips. The small blind folds; the action is to you. You hold an abysmal, 9h5d. I think there are a lot of players who would fold in this situation. First of all you have to consider the fact that the short stack on the button is desperate and could be making this move with almost anything. Even if the player is very tight and unlikely to move- in with total garbage, you cannot just fold because you think you are beaten. Let's consider the types of hands that you might be up against and compare those hands to the pot odds. If this player has any clue at all about how to play a short stack at the final table we can assume that he could have made this move with; any pair, any ace, or any two face cards.

The most correct way to dissect this problem is to come up with a range of hand categories and assign each of them a probability, then figure your odds of beating each hand category and compare it to your pot odds. Then make a decision based on the full spectrum of race odd to pot odd EV (estimated value). If our opponent holds a pair smaller than our 5, we are almost 1 to 1 to win a pot that is paying us over 3 to 1. That's a pretty good spot to be in, and an easy call. If our opponent holds one card lower than our five and one card higher than our 9 (A2 for instance) we are still really only a small dog in the hand, about 3 to 2. In that case the pot odds certainly suggest we play. Even if he's made a pair that splits our cards such as 77 our odds are only slightly worse than 2 to 1 against winning the showdown. It's only when our opponent holds a pair higher than our 9 that a fold would be correct. We really don't have any evidence that that's the case. If we balance this problem out by taking into consideration the range of hands we could be up against and our chances of winning the showdown we can see that the pot odds are just to huge for us to fold.

Just to illustrate a point, however, let's assume our opponent has exactly AK. If you knew your opponent had AK, do you think you should fold your hand? Let's do the math; AcKs will beat 9h5d only about 65% of the time. That means the AK is less than a 2 to 1 favorite to win in a showdown. The pot contains 14,000 chips and it will only cost you another 4,000 chips to play for it. The pot, therefore, is paying better than 3 to 1. Even if you knew your opponent held AK exactly, this is a profitable call.

Let's look at another example; during the same orbit of the same tournament, you start the next hand with 44,000 chips after posting the small blind. Everyone folds to you and you raise to14,000 chips with 7d7c. The player in the big blind had 44,000 to start the hand after posting the big blind. He raises all-in. After you met the blind and raise 10,000 more chips, the pot contains 18,000. When the big blind calls the 10,000 chip raise and bets another 30,000 chips it contains 58,000 chips. It will cost you another 30,000 to play for a pot of 58,000. Fold or call? You will still have some chips left, but will be crippled if you call and lose this pot. You can fold now and still have 34,000 chips. Most people would make this decision based on the allocation of prize money paid to various places. The math gets very tricky when you try to figure out how to play in light of complicated pay out structures. But this tournament is a "winner takes all" event. All you need to do is determine the EV (estimated value) of this play and fold or call. Your pot odds are about 2 to 1. I'm going to teach you my technique for calculating EV. I try to keep everything a simple as possible and still remain relatively accurate. I'll round numbers off or even remove them entirely in order to keep the figures simple.

Let's consider the types of hands your opponent might have made this all- in re-raise with. Without having any background information on this person, you really just have to put his hand into a list of possible categories. It is possible that he has a pair of 5's or 9's exactly or that he's drawn two cards under 5 without pairing and then decided to re-raise all- in against one of the chip leaders. Those hands are possible, but unlikely to the point of being irrelevant. The tiny difference the inclusion of these numbers would make when coupled with their utter improbability makes it more practical to just "wash them out". Generally speaking, here are the hand categories. Your opponent could be holding…..

  • 1) unpaired over cards
  • 2) unpaired cards, one over and one under
  • 3) over pair
  • 4) under pair.

Now we just need to compare your odds of beating each hand and compare it to the pot odds to establish your EV in each case. Then it's just a matter of adding the EV's together and dividing them out to reach your conclusion. Because some holdings are more likely than others we will adjust the math accordingly. That sounds very difficult but I have a pretty easy (if rough) method for coming up with the correct figures.

We know that paired hands are much less likely holdings than unpaired hands. You should be dealt one pair in 16 deals. That argues to limit the possibility of your opponent holding a pair substantially, however, the very fact that this player re-raised all-in seems to argue the other way. I weigh the odds of a player starting with a pair against the evidence that he might have on by assigning the odds of our opponent having a pair to about 20% in this case. Because we're assigning pairs a likelihood of 20%, we'll just find our EV's against pairs and count them as 1. Then we'll find our EV against unpaired hands and count them as 4. Then we divide the total EV by 5 to arrive at our eEV (effective estimated value.)

We'll start with pairs. In our example, we held a pocket pair of sevens. There are 7 pairs higher and 5 pairs lower than our sevens. We are roughly a 5 to 1 dog against higher pairs and a 5 to 1 favorite against lower pairs. Let's set up a table to define how this would work out if we had to run against all possible pairs. We'll use the pot odds we have, about 2 to 1 and compare the pot odds to the race odds.

  • 1) 77 vs. 88 (EV=-2)
  • 2) 77 vs. 99 (EV=-2)
  • 3) 77 vs. TT (EV=-2)
  • 4) 77 vs. JJ (EV=-2)
  • 5) 77 vs. QQ (EV=-2)
  • 6) 77 vs. KK (EV=-2)
  • 7) 77 vs. AA (EV=-2)
  • 8) 77 vs. 66 (EV=+2)
  • 9) 77 vs. 55 (EV=+2)
  • 10) 77 vs. 44 (EV=+2)
  • 11) 77 vs. 33 (EV=+2)
  • 12) 77 vs. 22 (EV=+2)

The sum of these E.V.'s is -4. If we divide -4 by twelve we'll get -.33. That's our EV if we run into a random pair. Our estimated value is great when we have the higher pair, but awful when we have the smaller pair. Most pairs will be larger than sevens though, so, if we knew our opponent had a pair we should fold. We don't know that though; let's calculate the EV for unpaired hands.

The two types of unpaired hands we could be up against are two over cards or one over and one under. The chance of our opponent beating us with one over card is, very roughly 3 to 1 against. The odds are actually a bit worse than 3 to 1 against, but I think it's important to keep all of these numbers in your head in a simple form. To begin with, just try to deal with ratios like 3 to 2, 2 to 1, 4 to 1, etc. Trying to memorize and work with ratios like 3.2 to 1 is just not necessary; it won't make enough difference in you results to justify fumbling with cumbersome numbers. Once you have the basics memorized, and feel comfortable working with them you can start getting into the details.

When our opponent only has one card higher than our pair we should win the hand 70% of the time (again, roughly, 2 to 1). If our opponent has two over cards we are essentially flipping a coin (the odds are close to 50/50). Let's see how those two groups average out. It's actually more likely that the player has one over card than two, but it's more likely for a player to make a move like this with two high cards than one. I'd just call it a wash. I'd find the EV of running against two over cards and add it to the EV of running against one over card.

  • 1) 77 vs. 1 live card, one under card (EV=+1.33)
  • 2) 77 vs. two live over cards (EV=+1)

Now we just add those two EV's together and divide by two to give us our effective EV against unpaired hands. Then we add that to our EV of running into a pair. Remember though, we were going to count our odds of running into a pair as only 20% (which is actually very generous). The way we do that is by adding our effective EV against unpaired hands together 4 times, adding our effective EV against a pair only once, and divide the whole thing by 5 to find our "actual effective EV" or "aeEV."

  • 1) eEV (non pairs) +1.165
  • 2) eEV (non pairs) +1.165
  • 3) eEV (non pairs) +1.165
  • 4) eEV (non pairs) +1.165
  • 5) eEV (pairs) -0.33

If we add those groups together and divide by five we get a positive aeEV of +.866. That means you should call the all- in with your 77.

If that seems like a great deal of work to go thru just to solve one problem, just remember that you don't have to memorize every single situation. Use "clumping" techniques. You now know any time the pot is laying 2 to 1 after a player moves in and you hold pocket 7's at a final table, you should call. You should also realize that the size of your pair doesn't make a lot of difference here. With pocket two's this would still be a correct call, just not as profitable in the long run. Most of the information I've covered in this series is not as difficult to comprehend and utilize as it may seem. I'd advise you to read the whole series a few times paying particular attention to the first part and this final part. You should also practice counting outs and figuring pot odds at all times while playing, even when you're not in a hand. I'd even advise putting together your own poker problems to solve. As with any thing else, practice makes perfect. Until next time; Good "Luck"!

Poker Math Series by Dead Money

Poker Math

Poker Math 101: Pot Odds and Counting Outs
Poker Math 201: Drawing Hands vs. Made Hands
Poker Math 301: Starting Hand Odds
Poker Math 401: Application of Knowledge
Poker Math 501: End Game Mathematics

♣ Back to the series of articles on poker hand analysis or poker strategy.