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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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.

Austin D.

Is there anything else like a work book in given scenarios where we can apply this math at our own pace. I am struggling just understanding and applying. Definitely a novice but I'm not completely incompetent. Any suggestions?

A fellow student

Like most things you learn by doing. If you get too caught up in the mathematics you are exploitable.

A fellow student

his explanation of game theory optimization using the rock paper scissors analogy as well as psychologically and mathematically deviating from the standard model is amazing!

Joel M.

The video and the written lesson seem to conflict unless I'm mising something. Daniel references the odds of winning as 36% or roughly 2:1 after the flop, and analyzes that value vs. the pot odds in the video. In the written handout, it's telling you to calculate your hand odds for your NEXT card at 4:1. To me as a novice, it seems that would lead to playing way too tight and folding too often. Shouldn't we be following the video and only dropping to 4:1 after the turn??

A fellow student

Page 19 of Chapt 5 in the discussion book (Column 2) says "For example, if there is $100 in the pot and the bet is $100, then the pot is now $200 and you have to call $100. So, the ratio is 200 (pot) : 100 (to call), and you have 2:1 odds (33percent). Maybe I am missing something, but my 7th grade math says 2:1 is the same as 2/1 and that is 100% not 33%. What is up with this math that I am missing?

Andrés D.

Help please :D? Simple question, in what moment should I use the fold frequency formula? example: we are in a 6 handed table, lets say big blinds are 500. so ok everyone is just calling the big blind, we are in the button. But then, someone raises to 2,000, and the following player also calls the 2000, and now is our turn to choose . My question is, should I use the fold frequency formula based on the person who raised to 2000? or should I base it on the player who called?? because the pot size is different at those 2 moments. When the first person raised, the pot size was 3250 ( small blind 250 + BB 500, +500, +2000) After the other person called, the pot size was 5250 ( small blind 250 + BB 500, +500, +2000, +2000)

A fellow student

Also, when calculating the pot odds, i have read that we should include our own call amount in the calculation. So the pot is $100 and the raise is $100 so the pot is $200. We should include our $100 call which makes $300/$100 so 3:1 as opposed to 2:1. I’m bringing this up because I’ve read some contradictory articles.

A fellow student

Hello, I’ve been reading that we should only multiply our outs by 4 on the flop only if the player is all in and by 2 on the flop on a regular bet. This is because the pot odds will change on the turn since the player can bet again. So if we multiply our outs by 4 on the flop and the player is not all in then we aren’t getting the true pot odds for the situation. Can anyone agree or disagree with this? Thanks.

Andrés D.

hey guys! Is the formula " if Pot odds ≤ % of outs then you continue" correct? ( if pot odds are smaller or equal to the %of outs then you continue; were "% of outs" means the formula Daniel says about outs X 4 for flop and X2 for turn)