The GTO Blackpill
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Tombos21
4
8mo agoI've been working on an experiment recently that I think will fundamentally change the way you look at GTO.
A while back, I posed this question to the community: "Does Playing a Less Exploitable Strategy Give You an Edge?" The just of it was, if two players are both making unbiased mixing mistakes, will the less exploitable one have the edge, on average?
Tbh I wasn't certain what the correct answer was, so I set out to find it. In doing so, I discovered something that reframed the way I see poker.
The "Bad Reg" Strategy
Let's define a "Bad Reg" as a player who perfectly finds every pure move but just randomizes every mixed decision. They are the ultimate button-clicker.
- Pure action - GTO always takes one action (e.g. always calls this hand)
- Mixed actions - GTO mixes between actions (e.g. call 30%, fold 70%)
The Baseline: GTO vs. GTO
First, we need a baseline. We'll use a standard CO vs. BB single-raised pot, 100bb cash. Here are the baseline GTO EVs:
Experiment 1: Rounded OOP vs GTO IP (Known)
Now the fun begins. Let's round OOP's GTO strategy to the nearest 1/2 frequency. So for every decision point in their strategy, from flop to river:
- Low frequency actions become 0%
- Mid-frequency actions become 50%
- High frequency actions become 100%
This is the definition of the "Bad Reg" strategy. They are playing randomly when mixed, but the pure actions stay the same.
Here's an example of what that looks like:
Important note: Do not re-solve after rounding. The point is to measure the original fixed equilibrium strategy vs this newly rounded strategy. After rounding, I simply click "calculate results" to see the EVs of this new matchup:
Results: The EVs are identical (technically the EVs shifted by 0.004 bb, but that's just solver noise; goes away if you solve baseline to higher accuracy).
This was not a surprise to me, in fact it's an expected result from the laws of indifference, something I've written about before. In short, Nash implies that every mixed action should have the same EV, otherwise it's not at equilibrium. Since rounding only changed the frequencies of indifferent decisions, OOP doesn't lose any EV. This is the classic "mixing mistake" (exploitable, but 0 EV loss vs. GTO) vs. a "pure mistake" (playing an action GTO never takes, which does lose EV vs GTO).
In other words, the "bad reg" loses nothing vs GTO.
Up to this point, nothing is new. Pros and game theorists have known this for ages.
Experiment 2: Rounded OOP vs Rounded IP (Novel)
Okay, so what happens if we round both player's strategies? This is where it gets weird.
Now we have two rounded strategies facing each other. Crucially, the mixed actions are no longer indifferent. Facing GTO, some hand had the same EV regardless if it bet or checked. Facing bad reg, one of those actions has higher EV. The indifference argument no longer applies.
Both players are making a ton of mistakes, so one player should "accidentally" exploit the other just by chance, right?
Nope.
Result: The EV remains exactly the same as the GTO baseline.
I've run this experiment on a dozen different spots, in different formations and SPRs, varying levels of complexity, and the EVs stay locked to the GTO baseline.
This is very suspicious to me. Like, I would have expected mixing errors to somewhat cancel out just due to luck, but not this precisely. The mixing errors cancel out to like ~1/1000th of a big blind. Smaller than I can reliably measure. And the further down I solve my baseline, the smaller the error becomes. What in the F***? Why? How? How are two semi-random strategies just by chance arriving at the exact GTO vs GTO EVs? It is completely counterintuitive to me that this should be the case.
Analysis
We already know from Experiment 1 that a GTO player doesn't beat the Bad Reg:
- [GTO vs Bad Reg] EV = [GTO vs GTO] EV
The real blackpill is what Experiment 2 showed:
- [Bad Reg vs Bad Reg] EV = [GTO vs GTO] EV
That's the truly novel part. Unbiased mixing mistakes precisely cancel out in the aggregate.
So... Are Mixtures Useless? (Caveats)
No, but it shows that GTO mixing is 100% defensive.
The only reason to mix is to reduce exploitability. Our "Bad Reg" strategy is wildly exploitable (~15% of the pot in this case). An exploitative opponent will destroy this strategy. But even then, there are ways to reduce exploitability that don't require mixing.
Caveat 1: In reality, most opponents don't make unbiased mixing errors. They lean some direction (e.g. over-folding, over-calling, over-aggressing, etc). When mixing mistakes are biased, they presumably won't cancel out in the aggregate.
Caveat 2 (The Counter-Example): This "error canceling" effect isn't a universal law. I built a simple polarized river toy game (1 value, 3 bluffs, 1/2 pot shove behind), and rounding both strategies does change the EV. In a simple, asymmetrical toy game, the errors don't cancel out. But in the full game with millions of decision points, the they seem to wash out perfectly.
The Blackpill GTO Study Guide (How to Actually Use This)
I've been calling this player a "Bad Reg," but this is actually a very high-level strategy lol. This is someone who loses no EV against a GTO trainer. This is a player who:
- 1. Finds every single pure action correctly.
- 2. Knows exactly what actions to mix between.
So, your first goal should be to become the Bad Reg. If you can get your baseline to that level, you're already playing at a very high level. Then you can focus on exploitation.
Level 1: Master the Pures. "This hand always bets here", "This hand always folds here". These pure actions are the load-bearing walls of your strategy. When you study solvers, focus on finding the pure actions first.
Level 2: Master the Classification. Forget the frequencies for now. Just learn to identify: "Okay, this hand is a mix between calling and folding" or "This one is a mix between checking and betting small". Everything else is noise.
Level 3: Exploit. Once your baseline is prepped, you can start deviating to exploit your opponents. Lean your mixtures in certain directions to exploit the pool and player-specific tendencies....
