# Solving Poker - What is Solving?

What does it mean to solve a poker game? How can we “solve” a game that seemingly relies so much on psychology, reading opponents, and deception?

## Definitions

When we talk about a solution to a poker game, we mean playing the game theory optimal (GTO) Nash equilibrium strategy. As discussed in the Game Theory Foundations section, Nash equilibrium is defined as when there is a strategy profile such that no player can unilaterally alter his current strategy to increase his expected utility. The Nash strategies in two-player zero-sum games limit a player’s exploitability at the expense of not exploiting weaker opponents, as each player is minimizing his worst-case expected payoff. This means that in expectation, no strategy can do better against a GTO strategy.

This means balancing one’s playing and and thinking about the range of hands that you play in each situation and how you act with each of those hands. The result is play that in theory is minimizing the worst case outcome and in practice (i.e. in real life as a human who can’t play a true GTO strategy) is minimizing chances of opponents exploiting or taking advantage of leaks and weaknesses. Humans can think about this as how to play if they had to announce their strategy in advance and give it to the opponent. This seems crazy, but emphasizes the balance of such a strategy where actions attempt to make the opponent indifferent, so even if they know your strategy (they of course don’t know your actual cards), they can’t take exploit you.

In small toy games, these strategies are relatively easy to find. In 1v1 Limit Texas Hold’em a very close approximation of this strategy was computed in 2015 at the University of Alberta. In commonly played games like 1v1 No Limit Texas Hold’em and multiplayer No Limit Texas Hold’em, no complete game theory optimal strategies exist…yet. In multiplayer games, the concept is less clear because of the interactions involved with other players and so the solution approach has been to develop agents that perform well, but aren’t necessarily playing some theoretical optimal strategy.

### Measuring Agent Quality

We are often working with only approximately optimal strategies when solving games. While small toy games can be small enough to find completely game theory optimal solutions, it’s important to be able to evaluate approximately GTO agents. We go into this further in Section 4.3: Agent Evaluation, but will mention the three main ways of measuring agent quality here as well:

1. We can look at a given strategy against the “best response” strategy, which is the strategy that maximally exploits the given strategy (i.e. how well can someone do against you if they know your exact strategy). This shows the maximum that your strategy would lose in expectation against such an opponent, which is defined as the exploitability of an agent. Note that a game is in Nash equilibrium if and only if all players are playing best responses to what the other players are doing.

2. We can look at a strategy against a different computer agent, including other approximate equilibrium agents that were somehow computed differently.

3. We can play the agent against humans, though this will require a lot of human time since many hands are needed to properly assess agent quality given the variance inherent in poker.

## Why the GTO Strategy?

GTO makes sense as a formalized way to solve a game because it can’t be beaten! But what if you are playing against a very bad player and a non-GTO strategy would be much stronger? In some sense the GTO strategy is not optimal in this situation, but we distinguish between a GTO strategy and an exploitative optimal strategy. By definition, when solving a game it makes sense to use the game theory optimal strategy because the exploitative optimal strategy is far less robust and while it may be more profitable against certain players, could be far less profitable against other players. So while a best response is the actual optimal strategy for a certain situation, it can’t be generalized in the way that GTO can be.

## Solving Methods

Generally algorithms and commercial programs are built around Counterfactual Regret Minimization (CFR), which is described in Section 4.1. In short, it works by playing the game against itself repeatedly and eventually coming to an equilibrium strategy, similarly to how two humans might play each other repeatedly and start off with some default strategy in mind, but then may reach some sort of equilibrium as they re-optimize based on the opponent strategy.

## The Evolution of Poker Studying

Poker studying has evolved over the last decade or so from:

1. Using simple odds evaluation software that would literally just show the odds of two or more specific hands given a board (or no board for the preflop odds). I remember being especially surprised the first time I saw how close some odds were for hands that seemed so much better than other hands, like AK vs. AJ or AK vs. 98!
2. Software that could run expected value simulations with decision trees. You could provide rule-based strategies at each node in a decision tree and the software would compute which hands were profitable, like calculating which hands could be shoved all-in from the small blind against the big blind if the big blind were calling with a fixed set of hands. This is sort of like the first step in finding an equilibrium, which would entail each player’s strategy getting modified after each subsequent round until an equilibrium was found. But if for example it was known that a specific player or population of players tended to play a certain strategy, there could be a lot of value in finding optimal exploitative counter strategies to them.
3. Solving toy poker games and applying lessons from these games into full games of poker. A lot of important lessons about balance and how to split up a hand range into different actions like betting for value and betting as a bluff were derived from looking at solutions to simple toy games and extrapolating them into full poker environments.
4. Using “solver” software that can actually solve for the GTO play given a specfic situation. Commercial solver software is arguably the most important development in poker over the last decade. The software lets you specify a game tree, generally with highly abstracted bet options (e.g. ~5 total bet options instead of being able to bet any integer or even any amount down to the cent) and then provides a full solution for that situation, which shows exactly how to play each hand in your range of hands.

For this entire time, statistical software that players can use to analyze their own hands and the population of players that they have played against has also been essentially mandatory for serious players (although some sites now don’t allow saving hand histories and some sites use anonymous names so individual opponents cannot be analyzed). This software is super valuable for “leak finding” by running reports and evaluating one’s own play, both on specific important hands and on trends like “folding too much to bets on the turn card”. The software also has a “heads up display” or HUD that shows up on screen while playing with important statistics about opponents like how often they voluntarily play hands, how often they raise preflop, how often they 3-bet preflop, and many more.

As AI continues to improve and computing power increases and as humans study more, AI and humans will both get gradually closer to optimal, but for now, we rely on abstractions and approximations for larger games.

# Useful Poker Math

Standard poker games are played with a 52 card deck with four cards of each value and 13 cards of each suit. The math here is important when playing poker and also makes sense to be familiar with when studying AI poker because it’s important to be able to interpret why solvers do what they do and to make generalizations, not just to try to memorize solver outputs.

## Computing Equities

Equity is the odds of you winning the pot. If there were no additional cards to come, then the player with the better hand would have 100% equity and the player with the worse hand would have 0% equity since those odds would be fixed. Assuming that we have two players and one has a stronger hand and there are still cards to come, the player with the weaker hand needs a certain number of “outs”, or cards to improve their hand.

For example, if the board is A562 and one player has AK and one player has 87, the player with 87 has an open ended straight draw and needs either a 9 (for the 56789 straight) or 4 (for the 45678 straight) to win. Since there are four each of card 9 and card 4, the player has eight total outs to win.

We’ve seen 4 + 2 + 2 = 8 total cards so far (four community cards on the board and two hole cards for each player), which means there are 52-8 = 44 cards left in the deck. The player with the straight draw has 8 outs of the 44 cards so 8/44 = 18% chance of winning. This math assumes that the 9 and 4 cards are still in the deck, even though the opponent could have them – we compute this way to simplify the situation and since we have no way of knowing the opponent’s cards.

The equity is computed as chance of winning * size of pot, so if the pot was \$X, the straight draw player would have 0.18 * x equity and the other player would have (1-0.18) * x = 0.82 * x equity. So with a \$100 pot, the players would have \$18 and \$82 equity.

In this case the player could be certain that completing the straight would result in a win, but sometimes this isn’t the case so you have to be cautious! For example, a player with a 3 on the A562 board could hit a 4 for a straight, but would then still be losing to a player with 87! Equities can be clearly computed when all cards are known, but in a real game situation, each player has to approximate which cards (outs) are helpful. In the straight example above this is pretty clear, but for example if the board was 456Q and you held A7, you could be somewhat confident that a 3, 8, or A would give you the best hand (9 total outs), but in each case you could still be losing! Someone could have 87 and beat you if the 3 comes, someone could have 79 and beat you if the 8 comes, and someone could have a huge number of things that would beat a single pair of Aces.

Briefly, there is also a concept called equity realization, which is the amount of equity a hand is expected to actually realize. An example of this is that in a one on one situation if I have 44 and my opponent has JT offsuit, we are right at about 50% equity each. However, 44 is very difficult to play postflop because unless I hit a 4, there aren’t many good flops for my hand, while JT will frequently have a straight draw (or straight) or at least a pair or might make me fold when high cards come as a bluff.

## Computing the Expected Value of an Allin

A valuable exercise in understanding poker math is to compute the expected value of going allin in a one versus one setting. Let’s once again take a look at the situation of small blind vs. big blind. We’ll use \$1 and \$2 blinds for simplicity. So the small blind posts \$1 and the big blind posts \$2 and now the small blind can either go allin or fold. Let’s assume that both players start the hand with \$40, again for simplicity. Note that if one player started with more than the other, only the smaller stack matters (e.g. if the big blind had more money, they would only have to call \$40 to match the bet, so any additional money over the minimum player’s stack size is disregarded).

There are 3 scenarios that can occur here:

1. The small blind folds
2. The small blind goes allin and the big blind folds
3. The small blind goes allin and the big blind calls

Let’s think about the expected value from the perspective of the small blind allin player.

Case 1: Small blind folds and the EV is 0. This is because we compute the EV from the point of the allin, so even though the small blind posted a \$1 blind, this is a “sunk” cost and is not used in computations. Case 2: Small blind allin and gains the entire \$3 pot when the big blind folds, so the EV is +\$3. Case 3: When the small blind is allin and the big blind calls and the small blind wins the pot, the win is all of the \$80 chips. When the small blind is allin and loses the pot, the loss is the additional $39 bet for the allin. Combining these, the EV = (win %) * (80) + (lose %) * (39). Note that the win is the amount you get back from the pot when you go allin and win and loss is the amount that goes in with the allin action. Putting this all together, we have EV = (opponent fold %) * 3 + (1- opponent fold %) * ((win %) * (80) + (lose %) * 39) Written more generally, we have EV = (opponent fold %) * (pot size before allin) * (1 - opponent fold %)((win %)(total pot size after allin) + (lose %) * (allin bet size put at risk)) Note that fold equity is the name of the equity that you get from the pot when your opponent folds. ## Pot Odds Pot odds means the ratio between the size of the pot and the size of a bet. Continuing with the above example, if the small blind goes allin then the big blind sees a pot of \$42 (the small blind’s \$40 + his \$2 big blind) and has to call \$38 more, so their odds are 42:38. This can be converted to a percentage by taking 38/(38+42) = 38/80 = 47.5%. Note that this is the size of the call divided by the size of the pot if we were to make the call. We can interpret this as the odds needed to win – the big blind needs to have at least 47.5% equity to make a profitable call. (Note that if this was a regular (non all-in) bet then there would be additional action on the hand, which we will discuss in the below Implied Odds section.) Pot odds are super important when deciding whether or not to make a call in poker and it’s important to be able to calculate an approximation on the fly in your head when playing. We use pot odds in conjunction with equity as a factor in deciding whether or not to make a call. In general, if you compute that your equity is higher than your pot odds, then you are making a profitable call. There are important standard pot odds that are useful to memorize – if an opponent bets half the pot, then pot odds are 25%. If an opponent bets 3/4 of the pot, then pot odds are 43%. A bet of full pot gives pot odds of 50% and a bet of 2x the pot gives pot odds of 67%. This intuitively makes sense – when facing lower bets as a percentage of the pot, we need lower odds to make the call because we have less at risk relative to what we can gain. This is something to think about when betting as well – to manipulate the pot odds that your opponent is getting – for example betting larger can protect a made hand against a posssible drawing hand and betting smaller with a very strong hand can “suck in” the opponent since they are getting good odds (but could also backfire if the opponent hits a draw cheaply, which is why bets are often made according to conditions on the board and how many draws are possible). Pot odds are especially useful in the context of catching a bluff with a medium-strength hand since that is when our decision is simply to call or fold. When we have a very strong or very weak hand, we would possibly be considering more options than just calling and folding, like raising for value or as a bluff. ## Implied Odds Pot odds simplify things by considering only the immediate odds of the poker situation, which is valid on a final betting round or when all of the money is in the pot, but not when there is still money and cards to come. Implied odds takes into account the chips that you could win in addition to the current pot. The typical example for this is that if someone bets the pot on the flop and you have a flush draw, you’re getting 50% pot odds and just a flush draw will not have 50% odds – more like 35%. However, when you do make that flush, you can expect on average to win additional money. But beware of reverse implied odds, which refers to losing more than the immediate pot, like if you are drawing to a very small flush, hitting that could be big trouble if an opponent has a higher flush. When you have good implied odds, it can make sense to call without the correct immediate pot odds. If you don’t think you’ll make more money in later betting rounds, then you have low implied odds and it probably doesn’t make sense to call a bet unless you have the correct immediate pot odds. We can compute the additional money needed to win on later streets assuming that we can estimate our equity in the hand. This is the same computation as before for pot odds, but with the denominator adding a variable that represents what we would need to win on later streets. So if the pot is \$100 and the opponent bet \$50, the standard pot odds would be 50/(150+50) = 0.25 (which again, is the bet divided by the current pot and the amount that would be added when calling the bet). Assuming that it’s possible to win \$x more on later streets, we could compute 50/(150+50+x) = equity. Assuming that we have 0.2 equity, that means that we’d have 50/(150+50+x) = 0.2, so 50 = 40 + 0.2x and 50 = x. So if our equity was 0.2, we’d have to win an extra \$50 in later streets to make calling now breakeven. If our equity was 0.25, then we’d have to win \$0 more in later streets (making a call breakeven immediately, with any additional money gained later as positive value). If our equity was only 0.1, then we’d need to win an additional \$300 later, which seems like a lot, but in no limit games with deep stacks, it could make sense to go for unexpected draws to extract a lot of money later! ## Minimum Defense Frequency Minimum defense frequency is related to pot odds and shows the percentage of hands you should minimally continue with – if you continue with less than this then your opponent can always bluff and be profitable! To calculate minimum defense frequency (MDF), we have (pot size)/(pot size + bet size). Recall that pot odds is (bet size)/(pot size + bet size + call size). Suppose that we are in a pot of \$100 on the river in Texas Hold’em and the opponent bets \\$50. This means that our MDF is 100/(100+50) = 0.67. The pot odds are 50/(100+50+50) = 0.25. In words, the MDF means that we should be continuing with at least 67% of hands and the pot odds means that we should have at least 25% equity to call.

So if you fold more often than the MDF then your opponent can exploit you by over-bluffing. However, going by MDF means attempting to be unexploitable, but if your opponent is a new player and never bluffing, then that kind of specific knowledge (read) may be more useful. In general, MDF is useful to keep in mind and to try to optimize hand ranges to get a feel for which hands in your range you should be calling with and is useful as a theoretical concept, but may not be good to use in specific scenarios like exploiting an opponent who bluffs too much or too little or when it’s very unlikely that an opponent can actually be bluffing and your hand isn’t good enough to beat anything but bluffs.

Pot odds, on the other hand, especially on the river (or final bet like an allin) are strictly applicable to the situation and if you determine that your equity is higher than the pot odds then you should call!

I co-authored a 2020 paper with Sam Ganzfried called Most Important Fundamental Rule of Poker Strategy, that extrapolates a minimum defense frequency rule from game theoretic strategies in a way that can be interpreted by humans. The rule does simulations combines minimum defense frequency with a concept called rane advantage, which is how much stronger the range of one player’s hand is compared to the others’. While this can’t be known with certainty, using this approximation in conjunction with MDF can provide a better result than MDF alone.

We generated 100,000 one on one simplified poker games and solved them for Nash equilibrium, while computing the optimal defense frequency for each. We then used machine learning regression to find an equation to optimize getting as close as possible to the optimal defense frequency in terms of the minimum defense frequency and range advantage. We found that in a simplified game with only one pot sized bet possible (meaning MDF was fixed, at 0.5 in this case), using range advantage in addition to MDF instead of just MDF leads to a significant reduction in mean squared error, about a 56% reduction.

In a 100,000 game dataset with three bet sizes, 0.5 pot, 0.75 pot, and pot, we uncover a rule that would extend to different bet sizes. Again, we find that using MDF and RA both as features reduces the mean squared error loss by about 56% compared to using MDF only. We found that the best result was using the strategy to call at least 0.904 * MDF - 0.495 * RA + 0.261. Since we wanted to make this formula human interpretable and since we see that the formula is close to MDF - 0.5 * RA + 0.25, we simplify our suggested result with these easier to remember coefficients. Finally, we noticed that the optimal defense frequency only rarely exceeds the MDF, so we found that using min(MDF, MDF - 0.5 * RA + 0.25) improves about 33% from the models without the truncation. The mean squared error of this approach is 0.0032, about 50% better than linear MDF.

This led to the Fundamental Rule of Poker Strategy: Given minimum defense frequency value MDF when facing a certain bet size, and assuming a range advantage of RA, then you should call the bet with a fraction of the hands in your range equal to min(MDF, MDF - 0.5*RA + 0.25), and fold otherwise. When neither player has a range advantage, then RA = 0.5, and the equation simplifies to min(MDF,MDF) = MDF, which makes sense!

The rule can make using MDF easier because it can provide more theoretical justification to fold given more information about your opponent and your hand ranges. We give an example in the paper:

Suppose we are in a setting where the opponent bets the pot (so MDF = pot/(pot+pot) = 0.5) and we think that the opponent has a range advantage of 0.8, then we would want to call at least min(MDF, MDF - 0.5 * RA + 0.25) = min(MDF, MDF - 0.15) = MDF - 0.15 = 0.5 - 0.15 = 0.35. Suggesting that we should call a minimum of 35% rather than 50%. The only downside here is that this removes the theoretical property of the MDF and relies on an estimate of the range advantage.

## Hand Combinations

Hand combinations are the math behind how likely hands are in poker. As we know, there are 52 cards in a deck with 13 cards of each suit and 4 cards of each type (rank). We can compute 52c2 as 52!/(50! * 2!) = 52 * 51/2 = 1326 combinations of 2-card starting hands.

Each non-paired hand has 4*4 = 16 combinations of which 12 are offsuit (each card a different suit) and four are suited (both cards the same suit). The four suited combinations comes from the four suits in the deck, so one suited combination of each suit. When computing combinations possible given a certain board, we reduce the 4 * 4 multiplication values to account for cards that appear on the board. For example, if the board is J93, we can look at a few combinations:

1. AK is not on the board so still has 4 * 4 = 16 combinations
2. AJ has one card on the board so now has 4 * 3 = 12 combinations
3. J9 has two cars on the board so now has 3 * 3 = 9 combinations

Finally, there are 4c2 = 4!/(2! * 2!) = 4 * 3/2 = 6 combinations of paired cards. Every card of a certain type that appears on the board reduces the combinations. For example if the board is T52, then there are only 3c2 = 3 combinations of each of those cards instead of the normal 6 combinations. If two of a card are on the board then only one pair combination remains (2c2 = 1).

We can use hand combinations to assist in approximating our equity in a hand, which can be used in conjunction with pot odds to make informed decisions, especially on the final betting round.

As opponent hands are more narrowly defined towards the end of a hand, it becomes more possible to count the combinations of possible hands that they might have and in turn compute your own equity.

Updated: