Reference
Frequency Lock
How Exploit calculations work.
Frequency Lock Overview
Frequency lock fixes one player's action frequencies across their whole range and then solves for an equilibrium strategy. Exploit strategies are calculated with this feature.
For example, you can specify that the opponent bets only 20% of the time on this River. The solver then calculates the best strategies for both players, assuming the opponent keeps that frequency. Besides values you enter yourself, you can also use MDA AI predictions as the specified frequencies.
In a standard GTO calculation, both players choose the best strategy from all possible strategies. With frequency lock, only the locked player's choices are restricted to strategies that meet the specified frequencies. The other player can choose any strategy without restriction.
The result is an equilibrium of this restricted game: a state in which neither player can improve their EV by changing their strategy within the strategies allowed to them.
Standard GTO
OOPAll strategies
Best responses to each other
IPAll strategies
Frequency lock
OOPOnly strategies meeting the frequencies
Best responses to each other
IPAll strategies
- Available strategies
- Unavailable strategies
- Strategy chosen at equilibrium
What Is Fixed and What Is Optimized
Only the action frequencies of the whole range at the specified nodes are fixed. Which combos take those actions is not fixed.
The same 30% Bet can be made up of strong hands only, or of strong hands mixed with bluffs. Among the allocations that meet the specified frequency, the solver calculates the one with the highest EV.
- Frequency of the whole range at the specified nodes
- Fixed
- Allocation of the combos that make up that frequency
- Optimized
- Locked player's strategy at other nodes
- Optimized
- Other player's strategy
- Optimized
- Bet portion
- Check portion
How Frequencies Are Counted
Frequencies are counted against the locked player's range that reaches the node. Each combo's weight is its initial range weight multiplied by the probability of the player's own actions leading to that node. The opponent's action probabilities and the probabilities of Turn and River cards are not included in the weight.
If the strategy before the locked node changes, the range that reaches the node also changes. The frequency is kept for the range that reaches the node as a result of the calculation.
For example, if a combo with weight 3 and a combo with weight 1 reach the node and only the first one bets, the Bet frequency is 75%. It is not 50%, the simple average by number of combos.
- Combo A (weight 3): always Bet
- Combo B (weight 1): always Check
What Gets Recalculated
Only one player can be locked per calculation. Multiple nodes of that player can be locked at the same time.
The calculation re-solves the whole game from the start node to the end. A lock affects not only the nodes after the locked node but also earlier nodes and other lines. The locked player adjusts their strategy at earlier nodes in anticipation of the restrictions at later nodes, and the opponent changes their strategy in response.
- Start node
- Locked node
- Recalculated area
- Effect on earlier nodes
Comparison with the Maximally Exploitative Strategy
A maximally exploitative strategy maximizes EV after fixing the opponent's entire strategy combo by combo. The fixed opponent does not adapt to your strategy.
Frequency lock fixes only the frequencies of the whole range. The opponent chooses the best combo allocation while keeping those frequencies, and also responds to your strategy.
The strategy obtained with frequency lock is therefore the best response to the toughest opponent among those who keep the specified frequencies.
| Maximally exploitative strategy | Frequency lock | |
|---|---|---|
| What is fixed for the opponent | Action probabilities for each combo | Frequency of the whole range at the specified nodes |
| Opponent's adjustment | None | Best response while keeping the frequencies |
| Required information | Opponent's strategy for each combo | Opponent's action frequencies |
| EV against the assumed opponent | Maximum | Up to that of the maximally exploitative strategy |
| When the opponent's range composition differs from the assumption | Can lose heavily | Keeps a minimum EV as long as the frequencies are right |
Example: An Opponent with a Low Bet Frequency
Suppose the opponent tends to bet only 20% of the time on the River.
To find the maximally exploitative strategy, you would also have to assume what that 20% is made of. If you assume they bet only strong hands, folding all your bluff catchers is best. But if the opponent also mixes in bluffs, those folds become a large loss.
Frequency lock assumes the opponent builds that 20% in the best possible way. The resulting Call and Fold strategy never falls below a certain EV, however the 20% is made up.
- Frequency lock strategy (guaranteed EV)
- Maximally exploitative strategy (assumes strong hands only)
- Loss when the assumption is wrong
Reading the Results
The locked player's combo allocation shown in the results is not a prediction of the opponent's actual hand composition. It is the toughest allocation under the specified frequencies.
As long as the opponent keeps the specified frequencies, your EV does not fall below the EV in the results, within the calculation's accuracy. If the opponent does not use the best allocation, your EV can be higher than the results. If the opponent's actual frequencies differ from the specified values, this lower bound no longer holds.
With rake, the two players' payoffs do not add up to a constant, so this lower bound is not strictly guaranteed.
How Results Are Verified
During the calculation, the current strategy is checked regularly for two things: frequency and optimality. The strategy checked is the one that would be returned as the result if the calculation stopped at that point.
The calculation ends once both criteria are met, and the result shows "Calculation complete." If the iteration limit is reached before the criteria are met, it shows "Iteration limit reached."
Calculate strategy
- 1Frequency checkWithin 0.001 percentage points of the specified value
- 2Optimality checkUpper bound on exploitability at or below the target
If either check fails, the calculation continues
Frequency Check
The actual frequency of each lock is calculated from the strategy to be returned and compared with the specified value. The allowed difference is 0.001 percentage points.
Frequencies are calculated directly from the strategy that is actually displayed and saved. The result value in "Target ○% · result ○%" on the results screen is this measured value.
If no range reaches a locked node as a result of the calculation, its frequency cannot be defined. This case is not treated as meeting the criterion either.
Optimality Check
This check measures how much each player could improve their EV by changing their strategy. For the locked player, it searches the strategies that keep every lock, including changes to the strategy before the locked nodes. For the other player, it searches all strategies without restriction.
The average of the two players' improvements is the exploitability. The optimality criterion is met when the exploitability is at or below the value set in Accuracy in the calculation settings. This value is a percentage of the Pot at the start of the calculation.
The locked player's improvement is evaluated with an upper bound that is guaranteed never to be below the true value. If the upper bound is at or below the target, the true improvement is always at or below the target too, so the accuracy is never reported as better than it actually is.
True improvement≤Upper bound≤Target
- Meets the criterion
What Exploitability Means Here
Exploitability here shows how close the strategy is to the equilibrium of the frequency-restricted game. It does not include EV the locked player could gain by breaking the frequencies.
So a small exploitability does not mean the strategy is close to an unrestricted standard GTO strategy. The farther the specified frequencies are from GTO, the more exploitable the locked player's strategy becomes in the unrestricted game.
References
The core of the calculation is based on the following research.
- Martin Zinkevich, Michael Johanson, Michael Bowling, Carmelo Piccione. Regret Minimization in Games with Incomplete Information. NIPS 2007.
- Noam Brown, Tuomas Sandholm. Solving Imperfect-Information Games via Discounted Regret Minimization. AAAI 2019.
- Trevor Davis, Kevin Waugh, Michael Bowling. Solving Large Extensive-Form Games with Strategy Constraints. AAAI 2019.
- Daphne Koller, Nimrod Megiddo, Bernhard von Stengel. Efficient Computation of Equilibria for Extensive Two-Person Games. Games and Economic Behavior 14(2), 1996.