Strategy & Theory intermediate
Optimal With Respect to What Model?
I want to step back from poker for a moment, because the GTO confusion is one specific instance of a much larger pattern, and seeing the larger pattern is what lets you stop being captured by future instances of it. The GTO thing is not a one-off. It is a particular case of something structural, and once you can see the structure, you start catching the same move everywhere.
Here is the structure. Optimality is always relative to a model. There is no such thing as optimal in the abstract. There is only optimal given a specification of the problem.
There is no "optimal" floating in the air
When someone tells you a strategy is optimal, your brain hears something like best possible, full stop, true everywhere. But that is not what optimal can ever mean. Optimal is always optimal given a specification — and the specification includes the players, the payoffs, the information structure, the rules, the time frame, the constraints. Change any of those and you change what is optimal.
Which means every claim of optimality is secretly a claim about which specification we are using. The specification is doing all the work. And the specification is usually unstated. That is the important part. The unstated specification is where the smuggle happens. Someone hands you a clean answer — this is optimal — and the model that the answer is optimal with respect to never gets said out loud. You accept the answer and never audit the model, because you were never shown the model. It was folded into the word.
GTO's hidden model
So apply that to GTO. What is the unstated specification underneath the claim that GTO is optimal? It is the abstract two-player, zero-sum game, without rake, between perfectly rational opponents who both commit to the equilibrium. That is the model. That model produces a specific, precise notion of optimality, and that notion is GTO.
Now hold the model up against the real environment of poker. The real environment is a raked, often multi-way game, full of imperfect opponents who do not commit to the equilibrium, where you have partial information that grows hand by hand. The model and the environment do not match. They are not even close. The model is two players; the real table is often more. The model is zero-sum; the real game is negative-sum because of the rake. The model assumes both opponents play the equilibrium; the real opponent has never played the equilibrium in his life.
The real environment has a different notion of optimality, which is some form of exploitative, adaptive play. The industry has been quietly substituting one model for the other, and the substitution is exactly what the word "optimal" has been hiding. You were sold the optimization of the model as if it were the optimization of the real thing.
The optimization of the model versus the optimization of the phenomenon
This pattern is not unique to poker. It exists everywhere mathematics meets a market. In every such case, a precise mathematical object is computed given a model, the model is then conflated with the underlying real-world phenomenon, and the customer is sold the optimization of the model as if it were the optimization of the phenomenon.
The optimization of the model is what we have. The optimization of the phenomenon is what we want. These are not the same thing, and conflating them is one of the most common errors in all of applied mathematics.
Optimal nutrition is optimal with respect to a model of human physiology — a model that is partial and revised every decade. Optimal investment strategy is optimal with respect to a model of markets — a model that famously fails in exactly the moments that matter most. Optimal control in engineering, optimal portfolio theory in finance, the loss you are minimizing in a machine learning system — same trick, every time. A clean answer to a question about the model, sold as a clean answer to a question about the world. The model is always partial. Often it is wrong in ways the optimization itself cannot see, because the optimization is trapped inside the model and cannot look out at the gap between the model and the world.
The cruelty of the poker version
The poker version of this error is in some ways particularly cruel, because the error is being committed by smart people who pride themselves on rigor. The smart, rigorous poker player has been led to believe that GTO is the rigorous answer, and that exploitative play is the unrigorous alternative — the loose, feel-based stuff old-school players did before the real math showed up.
This framing has it exactly backwards. GTO is rigorous within its model. Exploitative play is rigorous in the real world. The exploitative pro is doing more applied math, in a real sense, than the GTO pro, because the exploitative pro is constantly updating his model in real time against incoming evidence under conditions of partial information. That is what applied math actually looks like. The clean closed-form answer is the textbook version. The messy, real-time updating against live data is the practitioner version. Most mature domains have figured out that the practitioner version is the one that wins in the real world. Poker, partly because of its content industry, has been slower to admit it.
Practitioner versus customer
So here is the move that frees you, and I want you to hold it longer than anything else: optimality is a model-dependent claim, not a fact about the world. Every time you hear the word "optimal" — in poker or anywhere else — ask, with respect to what model?
That question is the whole practice. The practitioner who has internalized it stops being captured by the word. He asks: what is the model? What are its assumptions? Where does it match my situation, and where does it not? The questioning is where the actual skill develops. The accepting of the answer is where the laziness lives.
Most of modern intellectual life is a sequence of accepted answers to questions whose assumptions were never audited. The practitioner audits the assumptions. The customer accepts the answer. You have been a customer of the GTO framing for years, and the customer relationship has produced predictable outputs — predictable leaks. The practitioner relationship produces different outputs. And the shift from customer to practitioner is mostly a shift in the questions you ask. The questions are free. The shift is available to you starting tonight.
The next time the word "optimal" comes up in your thinking, pause. Ask, with respect to what model. If the answer is the Nash equilibrium in a frictionless two-player game, then notice that you are talking about a model, not the real game. The pause is the practice. Over months, the pause dissolves the bonding between GTO and optimal in your mind — and once that bonding is dissolved, you can finally use the solver as a tool without being captured by its own description of itself.
This article is drawn from the audio lesson The GTO Illusion.