Poker — continuation betting
This article summarises the results of some number crunching done using the GTO+ solver, to investigate the oft heard recommendation to continuation bet 1/3 of pot on the flop with 100% frequency. The output suggests that we should actually be betting a lot bigger, but checking back a lot more too.
- Setting the scene
- The experiment
- The results
- Discussion
To set the scene, we’re playing Texas no limit holdem. We raise pre-flop and get called by one opponent. The flop comes down, we’re in position, and he or she checks to us. How often should we make a continuation bet and if so what size?
A common heuristic is to bet 1/3 of pot with our entire range. This rule of thumb has the virtue of simplicity. It’s certainly easy to remember. And for some players who currently only bet when they’ve hit the flop, it may present an improvement on their current strategy. If they are playing opponents who will never ‘float’ the flop, and will fold if they haven’t hit, it’s likely to be a profitable play.
On the other hand, any rule in poker that suggests to ‘always do x’ is likely to have shortcomings, almost by definition. As Daniel Negreanu says, poker is a game of “no always — no nevers”!
The experiment (in brief)
GTO+ is a poker solver, the cheapest on the market, though I’m confident its game theory optimal solutions are as good as anything else out there. It requires a LOT of inputs though, namely:
- The ranges of the two players: I gave the in position player (IPP) a fairly wide range of 33% of all starting hands, and the out of position player (OPP) a very wide range of 66% of all starting hands. I then repeated the experiment where the OPP raises preflop with TT+, AKs, AQs and KQs, so these hands are no longer in his or her range. These were then the starting ranges for the 2nd experiment.
These starting ranges may not appear that realistic, but my game does play super-wide, and the aim was to investigate the betting where the IPP has just a range advantage (1st case) and both range & nut advantage (2nd case).
2. The size of the pot on the flop was 10 chips, effective stacks of 100 chips, no rake.
3. The betting structure: the OPP always checks (no donk betting). Thereafter I considered various combinations of checking, or betting ⅓, ⅔, 3/3 or 4/3 of pot on the flop. This is where solvers are more like model simulations than real depictions of the game of poker; it’s not possible to allow the solver to just bet anything. You need to specify the betting structure all the way down the decision tree (all possible ‘nodes’, in the parlance).
It would be fine if this heroic guessing as to players’ likely actions on the river had little bearing on the results, but sadly the output is very sensitive to the input. Anyway, apart from the flop decisions, I instructed GTO+ to bet 50% of pot on later streets and to ‘get the money in smoothly’.
4. Flop sample: pre-defined sample of the 19 flops that most accurately reflect the 1755 functionally distinct flops.
Obviously just 19 flops aren’t that reflective, but considering GTO+ takes 2–3 minutes to solve each flop, multiplied by 19 flops, multiplied by 16 betting combinations, not to mention a few mistakes along the way, my speedy Mac Studio computer chugged on for well over a day to complete all the calculations.
The results
The following graph summarises the expected value (EV) of the various betting strategies, averaged over the 19 flops:
The IPP, having the range advantage has equity (EQ) of 55% (54.84% to be exact) — the orange dotted line. He or she would expect to win 5.5 chips out of the 10 pot if the game went to the river with no more betting taking place. This might seem surprisingly low given that the IPP is playing half as many hands as the OPP; one might expect the IPP to have 66.6% EQ vs 33.3% EQ for the OPP i.e twice as much equity. But poker is a game of relatively small edges.
On the other hand, that 55% compounds over time. Play a game of successive coin flips with a biased coin that shows up heads 55% of the time, and the players probably wouldn’t even suspect the coin was weighted over 10 tosses. But over 100 tosses the player with the edge will be ahead more than 80% of the time.
The IPP’s EV (the orange solid line) is always higher than the EQ, starting at 5.72 if they bet 2/3 pot all the time, up to 6.28 if they mix checking and betting pot (labelled 100%, x). The EV outperforms the EQ both by virtue of IPP’s starting range advantage, and because they are in position. Similarly, the OPP’s EV tends to underperform their equity.
Again, this difference of 0.5 chips between best and worst strategies may not seem like a lot, but that translates to about 25 big blinds per 100 hands, which is an edge from just one spot (viz. pre flop continuation betting strategy) that would exceed the overall win rate of good winning player.
The keen eyed may note that increasing the number of betting options tends to increase the EV except that somehow [100%, x] performs best of all. How can restricting the IPP’s options improve their win rate? Actually it can’t. All the runs were done to a certain target level of accuracy (Target dEV = 1%) and this must have penalised the options that required the solver to balance over multiple bet sizes. When I reran “⅓,⅔, 100%,x” at Target dEV = 0.25% (which took over an hour), its EV drew level with “100%,x”). In reality the latter does — and must — sit (marginally) below the options that allow 100% bet size plus other bet sizes. But I didn’t adjust the graph so as to illustrate a further pitfall of interpreting solvers.
Turning to the blue line above, if the OPP has 66% of all starting hands, excluding some of the best hands (TT+, AKs, etc) then the IPP’s EQ increases from 5.5 to 5.6 chips (again, by less than one might think). However, the IPP’s EV increases by a larger margin viz. the blue solid line vs the orange solid line. This is to be expected, as EVs vs EQ have a gearing effect, but it also reflects that the IPP now has a range and nut advantage.
What does nut advantage mean? Here’s the equity graph for one of the 19 flops (J♠9♦2♦), which happens to favour the IPP best (60% EQ):
This graph ranks all possible hands in each players’ range against the opposing range, in order of equity. So, for example, IPP’s median strength hand is Q♦T♣, which has a healthy EQ (57%) and EV (8.4) against OPP’s range. Correspondingly the OPP’s 50th percentile hand, Q♠8♦, shown in blue, is in relatively bad shape. Note that the green line sits above the blue line at the very left of the graph indicating that IPP has the nut advantage. The IPP has pocket JJ for a set, but this is no longer in the OPP’s range.
Going back to the results graph, giving the IPP the nut advantage doesn’t change the order of preference for betting strategy, interestingly, though the IPP really likes to bet big now, pot or more. This is as expected: the IPP can pressure the OPP both with better hands on average, as well as the threat of monster hands.
For the betting strategies that involve multiple bets, the solver has determined the balanced betting frequency. If the IPP has the range advantage it’s as follows:
Surprisingly, the IPP checks a lot! (When we allow it to). And the checking frequency increases as the bet sizes get bigger e.g. the [100%, x] strategy involves checking 60% of time and betting pot 40% of the time. Note that this is an average across flops and actual hand. For this strategy the 100% of pot betting frequency ranges from 13% (on a A♠4♠4♦ flop) to 55% (on A♠9♦8♦), while if you hold AA you’re probably betting it with high frequency regardless of the flop! The solver doesn’t like to slowplay, prefering to balance value bets with bluffs.
If the IPP has both range and nut advantage the betting frequency looks like this:
With the additional equity and the nut advantage the IPP checks less and bets more liberally.
Discussion
Let’s agree that having more than 2 bet sizes (excluding checking) is too hard for a human to implement. The 100% or check strategy performs surprisingly well, but betting 100% would be quite confronting in my home game — I would get too many folds with my strong hands. The solver makes no psychological distinction facing bets of 80% of pot, 100% of pot or 400% of pot, it just adjusts its pot odds accordingly.
I actually recommend the [⅓, ⅔, x] strategy, even though it sacrifices EV against the strategies that bet 100% pot. The solver of course plays a super complicated mix of these bets, even mixing a particular hand, which a human couldn’t hope to learn and emulate, or even fully understand:
Amazingly, in this scenario not a single hand is a pure check, and some hands like 86s are an almost equal mix between checking, betting 3.3 and betting 6.6. The closest the solver solution comes to a pure bet is 33 for bottom set on this Q♠J♦3♣ board; it bets 6.66 chips 94% of the time. The reason (I guess) it does this mixing is for board coverage, so as to be perfectly balanced: it can show up at any node of the game tree with the nuts, medium hands or air.
A replicable strategy is to try to decide whether a board favours your range, and whether globally it prefers a small (1/3) or a large (2/3) bet. If it prefers a large bet, say, then sub-divide your range into the hands that prefer to bet and those that prefer to check. Generally speaking the hands that prefer to bet have high equity or no equity at all. Bet those hands 100% of the time. Check the medium strength hands 100% of the time.
Ah, but how to determine which boards prefer large bets and, within that sub-set, which particular hands prefer to bet? If I knew that I’d be at the casino rather than writing these articles!
(Or, at least, it’s something to explore in another article; this one is long enough already. Thanks for reading if you got this far).