Game Trees Unraveled

Hello, everyone!

I’m back for the October 15th issue of the Solver School Newsletter.

Last time, I introduced foundational game theory concepts, such as Nash equilibriums and the differences between simultaneous and sequential games. 

Today, I will dive deeper into sequential games and further explore game trees. Understanding game trees is critical for grasping how solvers calculate their mathematical solutions.

Let’s jump in!

Game Trees 101

A game tree represents all possible sequences of actions players can take in sequential games like poker. The game starts with an initial node where the first player acts. From there, it branches out to other nodes, with each player's possible moves. Finally, it ends in terminal nodes representing the outcomes.

Game trees are best visualized in a graphic like the one below:

The tree above models a simple toy game called the Shove-Fold game. Within the game are two players, Adam and Beth, playing a simplified version of Texas Hold’em.

Here are the rules:

• Both players start with 10 big blinds (BB)

• Adam posts 0.5 BB, and Beth posts 1.0 BB for blinds

• Adam acts first, choosing to shove all-in for his remaining 9.5 BB or fold

• If Adam folds, Beth wins the blinds in the pot

• If Adam shoves, Beth can call or fold

• If Beth folds, Adam wins the blinds in the pot

• If Beth calls, they deal the remaining cards and the best hand wins.

The tree can be represented by three different types of action points:

• Nodes at which Adam has to make a decision (A)

• Nodes at Beth has to make a decision (B)

• Endpoints that indicate the hand is over and no more choices can be made (1, 2, 3)

We can calculate the expected value (EV) at each endpoint and node to help inform the strategies that Adam and Beth can choose in this game.

The Math Behind the Trees

Let’s go deeper into the math.

We can represent the three endpoints by the following equations:

Endpoints #1 and #3 are straightforward, as Adam or Beth win the blinds when the other player folds. Their EV is represented by their net gain, with Adam standing to gain 1.0 BB (Beth’s big blind) and Beth potentially gaining 0.5 BB (Adam’s small blind).

Endpoint #2 is a bit more complicated, but not much more so. EV is the expected outcome minus the investment. Since both players are guaranteed to reach showdown at this endpoint, the expected outcome for each player is their equity multiplied by the pot size (20 BB in this case). The investment is the full 10 BB from their stacks.

We have two additional constraints within this system of equations:

• The combined equity values for Beth and Adam must sum to 1

• The combined EV for Beth and Adam must sum to 0

This is a zero-sum game. With only two players playing, one person’s gain is the other’s loss.

The nodes can be calculated as well using the following equations:

The nodes can be calculated by multiplying the EVs of the node’s endpoints by the corresponding frequencies at which that branch of the tree is taken.

As with the first set of equations, we have two constraints:

• Adam’s combined frequency of shoving and folding must sum to 1

• Beth’s combined frequency of calling and folding must sum to 1

What’s interesting about the equations above is that Node A is controlled by Adam’s decision to shove or fold, and Node B is controlled by Beth’s decision to call or fold.

Additionally, we can also see that Node A depends on Node B. As a result, we have to apply some substitution to get a final representation for Node A’s EV.

The EV for the player controlling the node is represented as a decision. We assume both players are rational actors (I know this is not always true) and will choose the path yielding the highest EV for their particular hand.

Taking it Further

Now that we’ve mathematically represented the Shove-Fold game, we can assume some information and perform the calculations.

Suppose Adam shoves the following range of 634 hands (47.8% of all hands):

22+, A2s+, K2s+, Q4s+, J5s+, T8s+, 98s, A2o+, K3o+, Q5o+, J8o+, T9o

Beth holds JTs and doesn’t know what she should do.

From the equation for Node B, we can see that Beth has a choice to give up her posted big blind and lose 1 BB or risk all 10 BB at showdown.

Rearranging the equation, we can determine that Beth needs at least 45% equity to call against Adam’s range, as a hand with 45% equity should expect to lose 1 BB on average.

From this point, we can use an equity calculator to determine which hands Beth would want to continue with against Adam’s shoving range.

Entering Adam’s range and Beth’s hand yields an equity of 46.24%. Plugging into the equations above, Beth can expect to lose 0.752 BB by calling with JTs against Adam’s shoving range. She should call since that is greater than her alternative of losing 1.0 BB from folding.

And in case you are curious, here is Beth’s overall calling range (378 hand combos, or 28.5% of range):

22+, A2s+, K5s+, Q9s+, JTs+, A2o+, K7o+, QTo+

Calculating More Complex Trees

Poker is a bit more complicated than the Shove Fold game.

There are a lot more possible nodes and endpoints. Think about how the following factors can complicate the visualization of the game tree above:

• Unless you’re playing heads-up, Texas Hold’em is generally played with more players, all of whom have decisions to make

• Each player can act on multiple streets, with action preflop, on the flop, the turn, and the river

• 22,100 unique flops, along with 49 additional turns and 48 additional rivers, can come

• Players can choose many different bet and raise sizings for each action

It’s easy to see how this simple game tree can quickly get complicated.

The game is so complex that modern computers can’t come close to solving them completely. There are more possible decision points in the game tree of one hand of Texas Hold’em than atoms in the universe.

But a solver can help us calculate much more complex versions of game trees that more closely model the actual game.

With this understanding of game trees, you’ll have more context on how solvers generate outputs. When you comprehend the logic behind their algorithms, you can better configure scenarios and interpret solver solutions. And that starts with visualizing the game structure.

Parting Thoughts

Today’s newsletter explored game trees — visual representations of the various decisions made in poker. While solvers utilize complex algorithms, game trees demonstrate the general structure of the game and the parameters through which solvers must operate.

For a deeper dive into game trees and other advanced solver concepts, check out the Solver Masterclass — the most comprehensive solver education material available. In module 1, I have a 30-minute video that goes beyond this introductory example to explore game trees in more detail.

If you’d like to see this Game Tree video for FREE, all you have to do is refer 2 others to subscribe to the newsletter. Click on the link below to do so.

You can also check out the rest of the Solver School website. I have a library of long-form content about solvers and data analysis that you can read for free. I also sell solver and data educational courses to help level up your solver game.

That wraps it up for today! In the next newsletter, out October 30th, I will walk through a fun exercise demonstrating how solvers work to solve full game trees and calculate EV.

Until next time…
Michael Lukich

P.S. I always appreciate your feedback! Feel free to reach out directly at [email protected].