Mortons Theorem of Poker

Betting and Folding in Poker
Playing The Poker Game

We all have heard many times the term “Poker takes a day to learn and a life time to master”. There is a lot of truth to this statement. It takes many hours of experience to gain the ability to evaluate what the correct tactics and strategies are, but experience alone does not bring this ability. It also requires spending time studying and training.

Online technology has changed the game of poker in a number of ways. Not only has Texas Hold ‘em, seven-card stud, and Omaha poker moved to online, but also the ability to find any thing you want to know about the subject has evolved to a greater level. Most tactics for beginners or intermediate poker players are pretty much clear or cut-and-dried. Most top poker article writers and poker book authors agree that playing a well rounded poker game making rational decisions, thinking positive, paying attention, playing good starting hands, playing the percentages, mixing up your play and a combination with bankroll management will make you a winning player!

But, is this always true? Not necessarily. Some instances of course yes in low-limit stakes, but on the other stakes there are players who practice these exact tactics and complain at the end of the day because they fall short of their goal many a times. Why is this? Let’s take a step back and find out where they could be possibly failing.

Thinning the field is a reference in poker where some big hands (KK, QQ, AK) make more money when the whole table of opponents isn’t chasing them to the river trying to catch a possible miracle card. When you have many players trying to draw to flushes, straights and two pair, the win expectation can be significantly reduced. Many quality hands earn more profits on average when targeted by fewer competitors than when targeted by many competitors.

This sounds reasonable, does it not? But, I wanted some game mathematical theory to back this up, not mere conjecture.

Several years ago, after a lot of browsing through as much as poker literature, hand evaluations, poker tips and basic every day to day life experiences on and off the poker tables, I found something compatible with what I was looking for – some theory to relate to some simple aspects of the game: a well-written article by an author named Andy Morton on recpoker.com. If you do not know Andy Morton, he is pretty well known in southern California for his poker playing and blogging on newsgroups all over the Internet. On this newsgroup, he has made some excellent points on the discussion of thinning the field in multi way pots, which is also referred to as Mortons Theorem of Poker.

Mortons Poker Theorem
Morton’s Theory Of Poker

Let me quote him: “What I’m going to tell you is that if you bet the best hand with more cards to come against two or more opponents, you will often make more money if some of them fold, even if they are folding correctly, and would be making a mistake to call your bet. Put another way, you want your opponents to fold correctly, because their mistaken chasing you will cost you money in the long run”.

For example:

You’re holding Ace of diamonds and King of clubs in limit Hold ‘em, the flop (first three community cards) is King of spades-Nine of Hearts-Three of hearts (Ks9h3h), now you have flopped top pair with the best kicker possible.

After the betting on the flop is complete, you have two opponents remaining. Player A is holding the top flush draw (Ace of hearts and Ten of Hearts), giving him nine winning cards (Kh-Qh-Jh-8h-7h-6h-5h-4h-2h).

The second player, player B, holds Queen of clubs and Nine of clubs, this player has flopped second pair with four outs (9s-9d-Qs-Qd). You exclude the Queen of hearts, because even though it makes two pair against you and gives him a winning hand but doing this he loses to player A with making a flush.

Here comes the turn card (Six of Diamonds) and both drawing players, A and B miss totally!

Player A with the nut flush draw is most definitely calling for a single big bet. Player B must make a decision whether to call or fold his hand.

Player B expectations are the following:

  • Player B folding = 0
  • Player B calling = 4/46 * (P+2) – 42/46 * (1)

*P=pot size

Setting these two expectations equal to each other and solving for P lets us determine the pot size at which he is indifferent to calling or folding:

  • E (player B>folding) = E(player B>calling) => P’ B = 8.5 Big bets

What this means in simpler terms, when player B decides to fold, he does not lose any thing. But if he calls, he will win 4/46 of the times minus the one big bet the remainder of the time. Basically the pot should be larger than 8.6 times the big blind amount to call the bet to get a positive expectation.

Your expectations are the following:

E (you>B folds) = 37/46 * (P+2)

E (you>B calls) = 33/46 * (P+3)

*P=pot size

E (you>B calls) = E (you>B folds) => P’ you = 6.25 Big bets.

What this means in simpler terms is when the pot is smaller than 6.25 big bets, you profit when player B is chasing, but when the pot is larger than this, your expectation is higher when B folds.

There are two things you have to remember:

If B calls incorrectly, you’re not the only one that benefits, player A will stack his chips when he makes his flush. In fact, player A is benefiting more from B’s call – since you are also losing expectation due to B’s call.

However if player B folds, your expectation heads up against a flush draw climbs and you have a more positive expectation than if player B called.

So you ask, “What is the breaking point in which you want player B to call or fold”, it is depicted in the picture bellow:

    • The “X” dictates the region in big bets you want him to fold correctly if the pot turns a blank
The math of Poker
The Math of Poker

Why is this blog post a diamond in the rough? This goes against every thing we have been taught by other respected authors like David Sklansky who wrote about the “Fundamental Theorem of Poker”.

To sum up, Sklansky himself had stipulated that his theorem should be applied to heads-up situations more. Morton’s Theorem is based on the economic principle of implicit collusion. This principle, applied to poker, obviously would occur when the pot would be split among a number of players. In this case, all players flat call through the river, in hopes that all hands except the short stack split the pot in the end.

I promise: More on Poker and Poker Theorems to come. The math and the beauty of it all is fascinating and fun.

Developing Poker Software

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