What Is Convergence?

Have you ever been training a spot or hand with the PLO Trainer (more precisely, preflop and multi-way) and wondered or thought "why do the hands here seem so strange, this doesn't make any sense, why does the trainer/solver want me to take this action in this spot with this hand"?

It's likely due to a convergence issue. When a simulation is first setup, it assigns each player a uniform random strategy and computes the EV loss against each other players current strategy for each hand (and flop, turn, river) throughout the game tree which is quite complex. As the strategy changes for each player, the solver re-calculates the best strategy for a specific part of the game tree and repeats until it finds the optimal solution.

These constant re-calculations are called iterations - and the number of iterations required varies depending on many factors - primarily the number of players in the simulation, and the stack size of each player compared to the size of the pot. Solvers also take rake into account (along with dead money such as straddles, antes, and more), which is how it comes to a conclusion on the optimal strategy with a specific hand (or buckets of hands).

The primary reason that players see an action that leaves them questioning what is going on, is because to a solver this spot simply doesn't happen. Largely due to the ranges in play combined with rake - certain parts of the game tree just wouldn't happen. For example, you are playing a 2/5 8-Max live PLO game with everyone at 100bb. After many factors - an early position opening range only consists of 12.2% of hands - in PLO, there are 270,725 starting hand combinations. This means there are roughly 33,029 starting hands that are profitable (or +EV) to open from an early position in this game.

The next player to act (Early Position +1 or Under The Gun/UTG+1) only has a calling frequency of 6.3% when the simulation is run - meaning that roughly 17,056 hands are a profitable (or +EV) call when facing the previous action. In reality - poker players and especially live poker players are going to be a lot looser in this spot. They call with hands that aren't profitable given multiple factors - optimal ranges in play, stack sizes, rake, position and more.

As you go deeper and deeper into the game tree - with players taking actions than happen less and less frequently, these spots are not converged. Simply put, in a human environment this specific game tree could happen - but to a solver after millions of calculations someone should never be in this spot. Given enough computing power and running a simulation for long enough in theory would produce an accurate result - but this just isn't realistic.

Hands will appear in mixed frequencies (meaning that a player would take a specific action x% of the time) - but given long enough these results would eventually converge to either 100% or 0%.

It's important to utilize critical thinking in these spots and understand which hands make sense to continue in a certain part of the game tree while understanding how a solver comes to a conclusion for each specific hand.