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Sports & Games

Game Theory and Math

Daniel Negreanu

Lesson time 11:10 min

Daniel discusses Game Theory Optimal poker and provides tips on how to calculate pot odds and fold frequency at game speed.

Daniel Negreanu
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So the latest trend or fad in poker, and the latest evolution of it, is understanding game theory optimal play. So what exactly does game theory optimal play mean? Well, essentially, what it means is you create a strategy that is unexploitable. So let's use a real life example. You all know rock, paper, scissors, right? You throw a rock, you throw paper, you throw scissors. Well, now, what would be the game theory optimal percentage to throw each one? I'll let you think about that for a second. Well, obviously, throwing one third, one third, one third would be the game theory optimal approach. The problem with that is while you won't lose, you also won't win if you stick to that-- unless you get lucky, of course. But that's not an exploitative strategy. Now, what if you find-- you notice that your opponent throws rock a lot? You start to notice a pattern of he's throwing rock. Let's say he's throwing rock every time. Well, what should you do? So obviously, you throw paper. The question is, with poker, how often should you throw paper? Do you throw paper 100% of the time? That seems logical. The problem with that is if you start throwing paper 100% of time, he notices and says, I caught on to something you're doing, so I'm going to adjust as well. So what you want to do is instead of doing 33%, 33%, 33% across the board, if he's throwing rock 100% of the time, you up your paper to maybe 40% or 45% of the time. So you're going to exploit the weakness in his strategy, but you're going to do it while he doesn't even realize you're doing it. Because every time you deviate from a game theory structure, you become exploitable as well. Another way to look at it, really, is game theory is a defensive strategy. It ensures that I'm protected. There's no way that you're going to be able to exploit me. Now, exploitative play, which is something that I've been doing for 20 years before I fully had a understanding of game theory, is always looking for the mistakes in my opponent, and going on the aggressive, and trying to take advantage of them. Now, of course, a byproduct of that is, well, if they're being perceptive, they can start exploiting me because I'm willing to give up on the defensive side to play aggressively. So it's very important to have a mix of both, starting with a baseline of game theory and then exploiting from there. So there's a debate in poker, which is which is the better approach. Should we be focusing on playing strictly game theory optimal, or should we focus on playing strictly exploitative? And I think the right answer is somewhere in the middle. So you take what you can from game theory and then adjust to your opponents. Because if you're not doing that, you're not going to maximize profit. So for example, if you have Player A, who plays game theory in a game full of recreational players, Player A is going to win. But if you're a Player B, and you're using some of that game theory, but you're also exploiting...

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Daniel is very engaging, that's point number one. I found this to be very thorough but never hard to follow. I suspect repeated viewings will be rewarded. I have read a few poker books and therefore some of these concepts were not new but illustrated in an engaging, entertaining manner and I though the videos used to illustrate points and conecpts were clear and well chosen.


A fellow student

It might be a stupid question but I'm a new player. Do you use fold frequency at all stages or just when deciding to call before the flop?

Rick N.

Hi I am some what confused by the different explanation between the video and the hand book. The video talks about pot odds and counting outs using the outs count as 9 hearts x 4 which says 36% = 2:1. In the work book the explanation is described using pot odds and counting outs and using the graph which shows 9 outs as needing 4:1 one to continue. What is the right answer.

Cameron P.

The section on "Fold Frequency" is absolutely wrong as presented here and betrays Daniels lack of depth in understanding of game theory. The "Fold Frequency" he gives, is better characterized as "if you fold this much or more you opponent can make money by bluffing with any two cards" or as it is better known "Minimum Defense Frequency" or MDF. It is /absolutely incorrect/ to say you "should"fold this much. To give an example: if there is 100 in the pot and I bet 100, if my opponent folds 50% of the time, I win money, even if when they call I /never/ win. Therefore we know that if we are facing a bet of 100 into 100, we should call /at least/ 50% of the time or our opponent can bet any two cards and profit. If we are on the river this calculation makes sense... but what if we are before the river, well now, even when we call, our opponent will still win sometimes with his bluffs when the river helps him. Say our opponent will win 10% of the time when we call his bluffs. Now, when we fold he wins 100, when we call he only loses 70 (10% of the original hundred + extra hundred we give him - 90% of his 100) so he makes profit if we fold 70/170 = 41% of the time or more. Clearly in this situation, facing a bet of 100 into 100 it is /absolutely incorrect/ to say that we should fold 50% of the time as Daniel claims. The concept he is discussing suggests we absolutely cannot fold 50% of the time or more. Wiser students should also take this as an indication of the overall quality of the course, there are some basic game theory concepts which Daniel does not understand, but can be learned from any introductory textbook on the subject along with many poker books. Unfortunately it seems that this misinformation has been allowed to enter the material here, and therefore probably also elsewhere in the course. Advanced point: There are situations where earlier street action and the board dictates that our opponents range is so far ahead of us that we can fold more than MDF suggests, that occur in the equilibrium (or GTO) strategies. Understanding this requires quite some work however.

A fellow student

He did scissor hand for paper and rock hand for scissors. I paused the lesson to find a deeper meaning of that action. Then I understood that "you should never let your opponent see your real hand" was the deeper meaning that he wants to convey. Am I right in assuming that?

A fellow student

Does fold frequency make any sense? If I have the right odds - Why should I fold pots where im getting the right odds? Also, by the formulae that fold frequency will never be more than 50%. Which means even with the lousiest of hands, I should call 50% of the times! Makes no sense. Can someone explain the nuances of this?

Mark G.

I have seen this same formula presented as the breakeven percentage, and I'm not sure I can wrap my mind around the difference because Daniel only gave two examples.

Ome R.

I don't really get the purpose of the fold frequency yet. To me, it does not yet incorporate into your range of hands vs their range of hands and the player tendencies. Per example, if I have AQo and my opponent has 10s. The flop comes: and it is 4,7,J and my opponent bets with a pot of 1000 with 500 (making it 1500). This would give me a fold frequency of 25%. But: If the flop would have been 4,7, Q with the same numbers, my fold frequency would be the same, but my change to win would be way higher. So I don't really get it...

Del J.

Stupid question, but are you supposed to do these calculations with every bet? Do you guys use the calcs every time you are facing a bet? Or do you only do them when you are facing a difficult decision? Thanks!


This sort of explains GTO but i would like to go much more in depth on the maths. Can you recommend any books on GTO? Or would the Pokerstars School be a good place to learn GTO?


Quick shortcut for calculating fold frequency for fractional bets (1/2 pot, 2/3 pot, etc.): Leave the numerator, then add the numerator to the denominator, that's your answer. 2/3 pot bet gives you 2/(2+3) = 2/5 or 40%. 1/2 pot bet gives you 1/(2+1) 1/3 or 33%. 1/4 pot bet gives you 20%, etc. This also works for multiples of the pot, e.g. 3x pot is really 3/1; 3/(3+1) = 3/4 = 75%. Of important note, this is also the required amount of fold equity you need to make a profit when you are bluffing! So if you bluff 3x the pot, your opponent must fold 75% of the time for you to be profitable.