Understand Game Theory in Poker Solvers

Hello, solver enthusiasts!

It's Michael Lukich again with Issue 3 of the Solver School Newsletter!

Last time, I covered the basics of what a poker solver is. Today, I’ll continue to peel back the onion and dive into some foundational game theory concepts critical for getting the most out of your solver.

Game theory provides the backbone for how solvers generate their mathematical solutions. While solvers employ complex algorithms, game theory gives context for interpreting their outputs. Grasping key ideas like Nash Equilibrium allows you to configure scenarios better and comprehend the meaning behind the outputs.

So, in this newsletter, we'll explore game theory fundamentals and how they relate to solvers. By the end, you'll have the baseline to connect the dots between game theory and solver outputs. Let's jump in!

Game Theory Defined

Game theory is a framework for modeling strategic decision-making between independent actors. It tries to determine the optimal moves players should make to maximize their payoffs, assuming rational opponents also try to maximize theirs.

Game theory has many applications beyond poker, like business, politics, and biology. But it's particularly useful in poker because the game revolves around decision-making with imperfect information against opponents trying to outplay you. Understanding game theory provides a mathematical perspective on finding the most profitable strategies.

There are two main types of games: simultaneous and sequential. Let's look at examples of each.

Simultaneous Games

In simultaneous games, players make decisions at the same time without knowledge of the other's choice. One common example you might be familiar with is the game of Rock Paper Scissors. Both players announce their decision simultaneously with rock beating scissors, paper beating rock, and scissors beating paper.

Another common example is the Prisoner’s Dilemma. In this game, detailed in the graphic below, two prisoners are arrested for a crime and separated into two rooms without communication. The police interrogate both prisoners to attempt to obtain a confession.

• If both stay silent (or cooperate with the other prisoner), they will receive a short 1-year sentence.

• If either prisoner betrays the other by confessing their role in the crime, the police will go easy on them with no jail time, charging the other prisoner a full sentence of 5 years.

• If both prisoners betray the other, neither gets off easier, and they each get a lengthy sentence of 3 years.

While both prisoners could benefit from keeping their mouths shut and accepting a shorter prison sentence, betraying the other will always yield a better option for them and is the dominant strategy.

Because of this, mutual betrayal is the Nash equilibrium in this game. In other words, it’s the solution where neither player can unilaterally increase their payout by changing their decision.

Poker doesn't involve much simultaneous action. But these simple games help illustrate foundational concepts like dominant strategies and Nash equilibriums.

While basic, this shows how game theory finds "solved" strategies. Poker scenarios are more complex, but the fundamentals remain the same.

Sequential Games

Unlike simultaneous games, sequential games involve players acting in a set order. One player acts at a time, with later players having the additional information of the earlier player’s decision(s). This structure aligns with most poker scenarios.

Sequential games are often represented as "game trees" that map out all possible sequences of actions. I’ll cover the mechanics of game trees in more detail in the next newsletter issue, but I will introduce the concept today.

One of the simplest game trees we can envision is a simple poker game called the Shove-Fold game, as visualized in the graphic below.

The game works as follows:

• There are two players — the Small Blind and Big Blind.

• The Small Blind acts first and can shove or fold.

• If the Small Blind folds, the game is over, and the Big Blind wins.

• If the Small Blind shoves, the Big Blind now acts and can call or fold.

• If the Big Blind folds, the game is over, and the Small Blind wins.

• If the Big Blind calls, the player with the better hand wins.

Of course, an actual poker game is more complicated than this, but the idea remains the same. Each player makes their decision sequentially, with the players who act later benefitting from the additional knowledge of the decisions of the earlier players.

Considering all players' possible moves and their implications lets us deduce mathematically sound strategies. While we can solve a game tree, such as the one above, by hand, an actual poker hand can get much more complex. This is where solvers come in; they can churn through calculations more effectively than we can.

Nash Equilibrium

As mentioned above, the "solved" strategy for a game is called the Nash equilibrium. Both players reach a state where they maximize their payoffs, assuming their opponent is also playing optimally.

In poker, a Nash equilibrium is a bet sizing and action frequency strategy for all hands within their range where neither player can improve by changing their plan. If my strategy is unexploitable, I can't do better by altering it even if I know yours. And vice versa.

Of course, no human players play Nash equilibrium strategies. However, these strategies provide a mathematical blueprint that is theoretically optimal. And solvers calculate approximations of these equilibriums for the scenarios we define.

Parting Thoughts

Game theory gives us the foundational logic behind solver outputs. While the algorithms are complex, game theory provides mathematical insights about optimal decision-making against rational opponents.

Understanding key concepts like dominant strategies, game trees, and Nash equilibrium can help you better structure solver inputs and interpret the meaning of the complex solutions solvers provide.

If you want to geek out on game theory and solver strategy, check out my long-form articles and training courses for sale on the Solver School website. My flagship product, The Solver Masterclass, dives deep into the underlying game theory concepts that will make you a solver expert.

You can also follow me on Twitter and YouTube, where I share more solver-focused and poker strategy insights.

That's all for today! In the next issue, dropping on October 15th, I’ll continue to explore using game trees to model poker situations. This will prime you for unlocking even more value from your solver.

Until next time.
Michael Lukich

P.S. As always, I'm eager to hear your thoughts. If you’d like to reach out, you can reach me at [email protected].