A Little Bit of Game Theory

By: The Monster Stack (Guest Blogger)

Hello,

I do not profess to be an expert in game theory - far from it. In fact, I am still struggling to understand its basic concepts. However, a situation came up recently while watching a friend play Pot Limit Omaha that was a classic example of game theory in action. When applied correctly to poker, game theory can be a tremendous asset and pros such as Chris Ferguson do it masterfully.

The game is $5-$10 Pot Limit Omaha and my friend, a very aggressive player we'll call John, was sitting in front of about $740 when the following situation arose. He was holding 8h 9d Jh Jd in the small blind, a very good hand for Omaha, particularly when you flop a big draw. A middle position player raised to $30, the button called as did my friend. The big blind folded. With $100 in the pot, the flop came down 6d 7h 2h. John was first to act and led out with a bet of half the size of the pot, $50. The pre-flop raiser raised $100 more. The button folded over to John, who was at this point, even against a set, a coin flip to win the pot. He smooth-called the bet bringing the pot to $400. The turn was the As keeping John's outs the same but still giving him a 37% chance at winning the pot, assuming all his outs were live (that is, if his opponent didn't have something like 2s 2c Kh Qh, which would still give John a 20% chance at taking down the pot).

John checked and his opponent bet the pot and John called. When the river was a blank for John he pushed in his remaining $160 and to my great surprise his opponent folded and John took down a huge pot without a showdown. His play left me with many questions. Why did he not press his big draw on the flop? With a flush draw, straight draw and an over pair he had to know he was in good shape no matter what. Why did he not push his last $160 in when his opponent bet the pot on the turn? It's not as though he was planning on folding after putting nearly $600 into the pot.

He explained it as follows. Once he sees this flop he is going to see the next two cards no matter what. In Omaha, or even Hold 'Em for that matter, you are going to play a draw this big for all your chips if you have to. So, if he pushes the flop and his opponent has a set or two pair John is going to have to hit one of his many outs to win the pot. On the turn, if he comes over-the-top for his last $160 he is going to get called as his opponent is getting the right price to call with just about any reasonable hand. The only chance he has to win on the turn is if his opponent was on a complete and total steal, but even then, with so much in the pot and one card to come, he might have to call and hope for miracle. Since John's money is going in anyway he wants to have the greatest chance of winning the pot.

If he hits one of his outs on the river he can push in, and if he gets called, he will win the pot that way. But, if he misses the river, as he did, and pushes there is the small chance that his opponent too was on a big draw and missed and can't call. Throughout the hand John gave himself the best chance to win the pot. Leaving a small amount of bluffing chips behind is ideal since betting the missed draw on the river in this manner only has to work a small percentage of the time to be profitable. As a matter of fact, it has to work less than 14% of the time to be profitable.

So here is how the hand played and the options John had on each street. Once we examine it in this way we can see that each decision gave John the best possible chance of winning the pot. It is important to keep in mind that John is playing this hand from out of position. Although this could work in position, you actually control more of this situation from out of position.

Flop:
A) John goes all-in. Opponent has a big hand. Opponent calls. John must hit to win.
B) John calls and see the turn. John hits. Bets. Wins by opponent folding or wins when opponent calls.
The best option here B is to call and see what develops.

Turn:
A) John hits. Wins by betting whether or not opponent calls or folds.
B) John misses. Goes all-in. Same result as on the flop. John must hit.
C) John check-calls. John hits the river. John bets and wins if his opponent calls or folds.

If John misses the turn the best option is C. If he hits, the best option is A.

River:
A) John hits. Wins by betting whether or not opponent calls or folds.
B) John misses. Goes all-in and wins the pot if opponents folds. Wins the pot if opponent calls and John has the best hand. Loses if opponent calls and has him beat.

Notice that losing on the river by betting and getting called is the same result as getting all-in on the flop or turn and then missing his outs. The difference is that John gave himself some extra chances to win along the way. Remember that this is a perfect example. I have seen hands that he has played in this same manner and lost, however this is to be expected. The difference is that each time he gave himself the greatest chance to win and combined, this proposition is a winning one. Before committing chips to a pot ask yourself if it is the best way to win the hand. Sometimes a passive approach works best. Other times aggression is vital. Knowing your opponents' tendencies is key. An aggressive opponent might call you down here simply because he can't stand the thought of all that money being out there rather than in his stack. Don't be surprised when he rolls over a weak hand on the river that just narrowly beats yours. Your opponent is getting the right price to call you even if he has one pair. He too only has to be right less than 12% of the time for this call to be profitable. But remember, you gave yourself the best chance to win the hand every step of the way. Do that every time and you will be a winner.

One more thing. Watch out for this tactic when it is being used against you. John is an aggressive player and this passive mode of play is out of character for him. When you see an aggressive player playing a hand like this, perhaps he is just giving himself the best chance if winning the hand. Picking off the bluff a small percentage of the time will be a profitable play.

See you on the felt,
The Monster Stack