There’s a big gap between the mistakes you describe and cepheus’s highly mixed play. No human in game time could randomize like that, so no, it isn’t that meaningful to use removal like Cepheus does. You just have to be sure that you’re somewhat randomizing based on the pot size given your range. Round to 25%s. Each suit you add to a hand gives you 25%. ie if you have AK, 25% will have a heart. If you have a pair use the first two cards on the flop.

This is true, but it's also dependent on playing a non adjusting opponent that won't exploit you. Maybe it's not a big piece of the puzzle? Ok.

From my limited understanding this is correct. Here is how I think about it, which is more or less repeating what has already been said.

I think it easy to apply too much human thinking to a CPU. Value betting and bluffing is just a heuristic we use to simplify and understand why hands get grouped together.

Computers have zero concept of a value bet or a bluff. What they do understand is the resulting EVs from each decision. If a CPU can change its strategy and have a higher expectation it will do so. Once it reaches a state where any change in strategy results in a loss of EV, it has reached equilibrium and of course will refuse to alter its play.

To some extent the end result looks like our notions of value bets and bluffs, but this is not because the computer is actively trying to say xyz hands are value bets and therefore I must match a corresponding amount of bluffs. It is simply because playing in this fashion has the highest EV against a perfect playing opponent. This is to be expected given our knowledge of much smaller toy games as described in The Mathematics of Poker.

I believe this is correct as well, but there's a catch:

one of these things is happening:

a) we play a mixed strategy and get the frequencies wrong with the wrong hands.
b) we play a mixed strategy and get the frequencies wrong with the right hands.
c) we play a mixed strategy and get the frequencies right with the right hands.
d) we play a pure strategy and never get exploited.
e) we play a pure strategy and get exploited.

I may have missed a scenario here or there, but this is the effect in those respective situations:

a) we end up with a very poor strategy.
b) we end up with a pretty good strategy.
c) we end up with a very strong strategy.
d) vs cepheus? we will not be exploited here for our pure strategy. vs good players? this is liability town imo.
e) vs cepheus? wouldn't happen. vs good players? this is when it's time to study frequency based play and at least attempt to implement it.

Thanks for this.

Your example is simple enough for me to utilize once I start playing live poker again recreationally when work/market slows down.

You are the absolute best UpHillBothWays!

Thank you Bob148!

Love the concept of putting this into 5 major buckets of strategy and 5 most likely results.

With that said, I attempted C) for obvious reasons but probably ended up being B) 5-9 handed because I'm a luckbox and an A) 2-4 handed. Although most everyone is terrible HUHU online or live since I game select harshly when I play now.

In my opinion, I figured pros like DonJuan, UpHillBothWays, OntheRail, DeathDonkey, Unguarded, Schneids, BK, NinaWilliams (if still plays LHE) plays C) style always and crushes as a result!

Probably missing a bunch of other crushers/pros that play C) style.

With that said, in a real HUHU game, my personal experience Online and Live is that even with C) strategy, we eventually need to switch gears which may in turn end up being D) OR E) strategy for awhile when you are able to find a game flow to exploit Villain.

I'm guessing my last paragraph is just wrong from a mathematical stand point though lol

All gambits have the potential for payoff, but the term gambit implies that the payoff is unclear. In a cashgame? We never really know when our opponent will quit, making the potential payoff a bit more unlikely.

Gambit cycle looks something like this:

pigeon + hero play a hand of poker.

hero makes a gambit which causes his strategy to lose a certain amount of ev that is equal to or greater than zero. For example if I just called with AA from the big blind vs a button raise.

................. possible outcomes:
............../ ...........................\
pigeon takes gambit.............pigeon declines gambit

if the gambit is declined, there is only ev loss with no potential for recouping the lost ev.

if the gambit is accepted, there is an unclear amount of ev to be recouperated that may or may not completely regain the initial loss of ev. This ev is always greater than zero because cards have equity.

In the example of just calling AA from the big blind, these are the possible ways the pigeon might adjust:

a) no adjustment = immediate loss of ev for AA.
b) play more cautiously postflop in single raised pots = potential gain for weak draws and bluffcatchers in the big blind.
c) 4 bet more preflop = potential gain for big blinds 3 betting strategy.
d) play more aggressively postflop vs big blinds 3 betting stategy = potential gain for big blind's 3 betting strategy.

Since these adjustments are not mutually exclusive, there is actually some decent potential ev here. Vs players that will go into the exploitive deep end, this play can win lots of money. Vs players that don't adjust, this play can lose some money.

These all seem very reasonable. It was priced in for turn, and now priced in for river.

Keep in mind what it thinks your 3b range should look like. Ideally, you should have a decent % of low-low combos. I"m guessing yours may be too high-card / pair heavy.

I don't three bet much out of position. I usually do 77+, A7s+, 67s+, suited one gapers Q10+

That's pretty tight. See
http://poker.srv.ualberta.ca/preflop

Click on 'bet' to see what it does pre flop after first raise

with rake the % should be a bit different but it pretty close with my own solution 3b %. ex. 22 is only 3b at 10%

One big thing is that Cepheus really goes to town preflop w/ suited Aces, suited Kings, and pocket pairs. It's more likely to raise K4s than KTo, for example.

If you do all the combo math out, you'd see that out of 1326 combos, Cepheus is raising 430.38. That's a 32.46% 3 bet, which is probably way higher than most everyone in here has. Which in turn means his range is much weaker when taking a flop OOP passively. So hands we'd consider "far down in our range" and a fold, cepheus considers high enough for a call.

And FWIW this is a great chart to get an idea of how cepheus is playing.

Suited hands / Suited hand 3 bet: 312 combos, 190.37 3 bets (61% 3 bet rate (!) )
Pocket pairs / Pocket pair 3 bet: 78 combos, 73.745 3 bets (94.5% 3 bet rate)
Unsuited hands / unsuited hand 3 bet: 936 combos, 166.27 3 bets (17.76% 3 bet rate).

Given Cepheus propensity to trend towards three betting connectors over big little combos as well, it informs that on a lot of textures (AAXYZ, 3 flush, multiple straights), it seems likely to call down light simply because it won't have a lot of the combos that make big hands on these textures.