GTO, Solvers, and Game Theory Explained

November 3, 2022 Poker Strategy

Welcome to the first in a series of articles on computers, poker, and game theory. In future articles, I’ll be covering how solvers are changing the poker landscape, the threat they pose to online poker, and how you can use them to improve your own game. But before we get to any of those things, it’s important to understand how these things work and what they can, and can’t, do.

Game theory is a branch of mathematics that studies competitive games where participants must make decisions that materially affect the outcome of the game. It sounds complex, and parts of it can be, but learning some of what the game theory wizards have figured out isn’t too tough. I promise, I won’t make you do any math.

Game Theory Optimal (GTO) is a term used to refer to a strategy that can’t be defeated. These are sometimes referred to as Nash solutions after famed mathematician John Nash, though the two are not the same thing. When we refer to a GTO play, it refers to a play that can’t lose over the long run against any other strategy.

Be aware that the Nash solution, which is the GTO play against other players who are also playing perfectly, isn’t always the most profitable play. If your opponents are making mistakes, you can often alter your strategy to make more money from them than you would using the Nash solution. Let’s look at a simple example to illustrate this.

A Guide to GTO

Our example starts with a very simple game. The card game we referred to as “war” when I was a kid. Two players split a deck of cards in half and then they each flip over a card. The highest card wins and gets to keep both cards. If the cards are the same, each player “antes” another card and then flips over another card, and the winner keeps all six cards. Eventually, one player has all the cards and they win the game.

There is absolutely no strategy in this game. Nothing you can do will increase your win rate. Thus, there would be no way to apply game theory to it. But if we add a simple choice component, we can illustrate a couple of different points about the basics of GTO poker.

Let’s add a betting component to our game. Each player goes through the deck and writes a number from one to five on each card. That’s the number they’re willing to wager on that card. Each time two cards are flipped, the loser pays the winner the lowest of the two numbers written on those cards. So, if you flip over a queen with a five written on it and I flip over a jack with a two written on it, I’ve lost and owe you \$2. Don’t spend it all in one place.

Now think about how you might choose the numbers to write on your cards. Obviously, better cards get bigger numbers, but can we easily find the GTO solution for this game? Give it some thought; it’s an excellent exercise in finding a solution to a simple wagering game.

I’ll wait while you consider your solution.

This is war. (Image: How to Play Stuff)

Let’s start with a simple Nash solution. The easiest solution is to write the number one on every card. There’s no solution that can beat this in the long term; every other solution breaks even against it. None of them lose, but none of them win either. You have effectively removed the skill component of the game.

But there’s a better solution, which is also a Nash solution. Yes, there can be more than one. Can you figure out what the better Nash solution is?

Writing the number five on the aces in your deck and one on everything else is also a Nash solution. There’s no way to exploit this strategy, but it makes money against worse strategies than our first solution.

Picture a deck where someone had written a three on every card. The first solution (one on every card) breaks even against that deck because the wager on every card is one. But the second solution wins three dollars when you have an ace and the opponent has a non-ace card. Over time, you’ll make money consistently against the “all threes” deck if you choose the better Nash solution.

I don’t have a Ph.D. in game theory, but I’m fairly certain that these are the only two Nash solutions to this game. But, that doesn’t mean they’re always the best way to play. If your opponent made an error, as the “all threes” opponent has, then you can exploit this and alter your strategy to make even more money. We’ll refer to this as an “exploitive solution.”

If you were to write a three on your kings and queens, you would certainly make even more money against the “all threes” deck. When you flip up a king or a queen, you’re going to win more often than you’ll lose, so you’ll be making money against the mistakes deck. But, if your opponent then switches to a 5/1 Nash deck, they’ll be making money from you on the occasions when they flip up an ace and you flip a king or a queen.

This illustrates the difference between exploitive and unexploitable solutions. An exploitive solution is, by its very nature, also exploitable when it runs into a Nash solution. I like to think of them as a shield and a sword. The Nash solution is your shield. You can’t be hurt when you’re using it. And the exploitive solution is a weapon. You can hurt your opponent with it much better than you could by bashing them with a shield, but it also exposes you to their weapons.

Reviewing game theory terminology

A Nash solution is any strategy that can’t lose over the long run against another strategy. Sometimes, there are multiple Nash solutions as we saw in our example above.

An exploitive solution can make more money against an opponent who’s made a mistake, but it will lose against a Nash solution.

Now, let’s look at how a computer might solve this game. While there may be a simple and elegant mathematical solution for this game, I’m just not smart enough to know what it is or to tell a computer how to use it. But, I can tell a computer to play millions of hands of this game with every possible solution. At modern processing speeds, a computer can simulate billions of hands in a very short time and would come up with the 5/1 solution as the best way to play because it would be the solution that won the most against all the other solutions, and which never lost over a large sample of hands played.

This brute force approach is how a poker solver works. It’s a lot more complicated than our little game of War, but a solver is basically trying every option to see which one wins the most, or loses the least, in any given situation.

A typical poker solver. (Image: PioSolver)

That extra complication means that a solver takes a lot longer to solve one poker hand than it would take to solve the entire game in War. The fastest current solver (Pio Solver 2.0), running on a very fast computer, will solve one hand from start to finish in about 20 minutes for one possible flop. If you want to solve for every flop, that’s 1,755 times as long, or about three-and-a-half weeks. And that’s just for one hand and one possible bet size.

I’m simplifying a bit here. If you’re a solver nerd, cut me some slack; I’m getting the basics across to people who aren’t familiar with solvers yet.

A solver starts by running all of the possible solutions and finding the Nash solutions. But you can, in the most advanced solvers, also enter in your opponent’s actual ranges, or at least your best guess, so that the solver can run against that particular range and even develop a likely strategy for beating that player going forward.

This saves it some time because it doesn’t have to calculate your opponent’s perfect strategy and hand range. It can, instead, simply find the best strategy against their actual range and strategy.

This means that a solver can, theoretically, give you the exploitive solution as well. Not just the shield, but also the armor. In practice, most people find the Nash solution and learn what they can from that. We’ll talk more about learning from solvers in another article in this series, but for now, I’ll stick with an explanation of how all this works and why it matters.

The solver solution

Speaking of why it matters, let me just leave this simple statement here.

If you’re playing No-Limit Hold’em and aren’t working with a solver or a coach who’s using one, the game is leaving you behind. And, you’re going to fall farther and farther behind every day. The game will pass you by and opponents who are studying will tear through you like a Rottweiler puppy wrecking a cheap pair of flip-flops.

The good news is that there are lots of coaches and training sites working on making this information accessible. Many sites are using these solvers to calculate solutions and to save them in huge databases of “presolves,” which are hands that have already been solved and don’t require that 20 minutes of processing time to find the solution again.

Some examples of these presolve databases include Simple Post Flop, GTOWizard, GTOx, GTO+, Odin, DTO, and a number of others. The pricing varies widely, from free options to \$500 or more.

With these databases of presolved solutions, you can find the answer to most of your questions when it comes to Nash solutions, but if you want to find an exploitive solution against a particular opponent who doesn’t play well, you’ll probably need to run it yourself on Pio, Monker, or another solver, and wait for a computer to grind out a solution. Most players don’t bother with this much at this point, but I’m confident that these exploitive solutions will be important in the near future.

So what do I want you to take away from this article?

I want you to have a basic understanding of how a solver works, how people use them, and to understand the terminology that I’ll be using in future articles. And, just wait until we get into real-time analyzers and screen scraping, and all the other shady things that are happening in online poker games. You may even be playing against a solver and not even know it if you are playing online.

I’ll throw in answers to a few common questions here and wrap this up so I can get to work on the next article in the series, which will cover the easiest ways you can learn from solvers and GTO solutions.

FAQ

Are there solvers for other games?

There are solvers available for PLO that are publicly available, and a few places that have a good number of presolves, though they can be cumbersome and, sometimes, the specific spot you’re interested in won’t be available yet. There are also solvers that exist for Omaha/8, various Stud games, triple draw, No-Limit Single Draw, and a few other games, though these are privately held and tough to access. The people who own mixed-game solvers aren’t interested in sharing them at this point, and they’re only accessible to a select few. I expect this to change soon with batches of presolves for mixed games scheduled to appear in the coming year on a well-known training site.

How can I ever remember all the solutions?

You can’t. They haven’t even all been calculated yet. But, you’ll learn common trends like when to use smaller flop bets and bigger turn bets. And, hopefully, you’ll learn why a solver is doing certain things that help you think about the game more clearly.

Can I use a database of presolves while I play online so I can play perfectly?

First of all, please don’t. It’s cheating and you’ll be banned from the site and your funds confiscated if the site catches you, which has happened a bunch of times recently. Sites are getting very clever with the ways they catch people. I’ll cover more about this in the article about online poker. It is a thing some people are doing, but it’s unethical.

How do I know if someone is using a solver to play against me online?

Unless you’re a real expert, you probably won’t know. Play on websites you trust and if one opponent seems to be crushing you consistently, avoid them. Remember that you play poker for money, not for a fair game. If you’re making money, then the possibility that someone may be cheating shouldn’t scare you away. And if you aren’t making money, it’s time to find a different game, whether you’re being cheated or not. I’ll also cover this topic more completely in part three of this series.

Written by

Chris Wallace

Professional poker player, HORSE world champion, author.

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