Mixed Strategies

[Jeff W]

What follows is a raw, unfinished essay on Mixed Poker Strategies:

There are two poles of poker strategy:

Simple<--------------------------->Mixed

Simple strategies are strategies where you always play the same way with a given hand. If they have AA UTG, they always raise it(or alternately they always limp-call it, or limp-reraise it, or whatever). Some players are more expoitable than others. For example, you may play against a loose passive opponent who limps and calls raises with 73s, but occasionally he may limp-reraise it.

When you hear people talking about "mixing it up" or "playing deceptively" they are often talking about Mixed strategies. A mixed strategy can describe a set of diverse actions that you take with a given hand range or it can describe a set of diverse hands that you have given a certain action. The second is more important than the first and it's what I'll focus on.

Example hand: you call a raise in the BB with your normal calling range. The flop comes 7h-6h-5c. You can either check or bet. Lets say that when you always check here with any hand. Your check reveals no further information. When you called pre flop in the BB, you announced a certain range of hands(if you only call AA and KK pre flop, your range is AA-KK, if you call any hand, your range is any hand). Your flop check repeats what you said pre flop.

Your opponent bets. This bet reveals information about his hand, but we'll ignore that for now. You have 3 options: call, raise, fold.

If you fold, the information you give isn't particularly useful, but you have announced that you had a hand that didn't meet your standards for continuing on the flop.

Lets say that you raise and your raise represents a set or better 1/5 of the time, a pair 2/5 of the time and a draw 2/5 of the time. This is an example of a mixed strategy.

What you're doing is you're making all of your hands look alike. It's kind of like using the play action pass in american football. The quarterback looks like he is handing the ball off whether the team is running the football or passing. What this does is force the defense to respect the possibility of a pass when the quarterback goes to make a handoff and therefore running the play action pass should increase the team's EV(yards) on running plays, because the defense can't cheat and send the safeties to help defend the run.

The same holds true in poker, by playing your made hands the same way as bluffs and draws, you can use your made hands to increase the EV of your draws and your draws/bluffs to increase the the EV of your made hands. If you're using a good mixed strategy, the relationship between made hands and draws/bluffs will be commensal.

The toughest players in the toughest games approach a strategy that is prudential, or non-exploitable. They play all of their hands in such a way that they force your hand range to make 0 EV decisions against their hand range. This is an important point to emphasize--they cannot see your hand, they can only "see" your range of hands.

Of course, a tough player playing against a predictable opponent(one whose strategy is not mixed enough), they will take actions that exploit that opponents predictable weaknesses. If they are playing against someone who bet-folds for value on the river too much, they will bluff check-raise whenever the pot is laying sufficient odds, even if that strategy is not prudential.

Very good post and something I need to work on

Lozing

Is the idea of the simple---mixed continuum that where you deviate from the simple-"optimum" strategy insofar as it creates the most +ev deceptive strategy (while without departing from the simple "optimum" to the point of self-defeat-> ie -> the donks who check-call their monsters and fast-play their air/marginals)

[Jeff W]

Please clarify your question.

I will note that I expressly used the term "mixed" instead of optimal, because a prudential strategy may be suboptimal EV-wise compared to a simple exploitative strategy.

[redbeard]

you might even be able to discuss the "shenia" theory in there somewhere too. but all in all i think it looks like a very good start.

[Victor]

you are smart.

this is how i try to play too. particularly i almost always cr the flop or turn from the bb or sb with some sort of value. basically, out of position i utilize the cr.

wow. i just realized, i should donk some.

[Jeff W]

[ QUOTE ]
you might even be able to discuss the "shania" theory in there somewhere too.

[/ QUOTE ]

I already did, that's basically what the whole post is about.

[Jeff W]

[ QUOTE ]
this is how i try to play too. particularly i almost always cr the flop or turn from the bb or sb with some sort of value. basically, out of position i utilize the cr.

[/ QUOTE ]

I do the same thing. I pretty much always check and then raise or call HU on the flop if I'm going to continue.

[ QUOTE ]

wow. i just realized, i should donk some.

[/ QUOTE ]

Maybe, but I use flop donks mainly with my strong hands against predictable opponents who misread the donk-bet and give too much action. I think always checking the flop is a very difficult strategy to counter.

[Victor]

[ QUOTE ]
. I think always checking the flop is a very difficult strategy to counter.


[/ QUOTE ]

.19/.11, 150k. /images/graemlins/confused.gif

[Guy McSucker]

Nice post. This is what Abdul Jalib was talking about all those years ago when talking about balanced strategies: taking some slightly suboptimal plays for the purposes of hand-disguise. He went into the preflop possibilities in some detail.



Balanced preflop strategy according to Abdul (http://web.archive.org/web/20030623140048/posev.com/poker/holdem/strategy/balance-abdul.txt)

Guy.

[dave44]

[ QUOTE ]
The toughest players in the toughest games approach a strategy that is prudential, or non-exploitable. They play all of their hands in such a way that they force your hand range to make 0 EV decisions against their hand range. This is an important point to emphasize--they cannot see your hand, they can only "see" your range of hands.

[/ QUOTE ]
Really good post Jeff. I realized the point you make above recently and have began tailoring my game toward it. Once you get there the key point, and the skill I'm still we're all trying to build, is hand reading ability. The better a grasp you can have on your opponents range of hands, the more +EV your decisions will be. Understanding how they will react to your actions with certain hands is also the type of information that will lead to higher +EV decisions. These are the skills I notice in the posts of some of the top players on the forum and hopefully experience will bring them to all of us.

This is all fine and good, but counterproductive unless you're playing against opponents who actually notice. Aren't too many of those -- online anyway -- until at least 30/60.

[dave44]

[ QUOTE ]
This is all fine and good, but counterproductive unless you're playing against opponents who actually notice. Aren't too many of those -- online anyway -- until at least 30/60.

[/ QUOTE ]
I disagree. How could developing an unexploitable strategy be counterproductive? I don't think we're giving up much if anything in order to do this either.

[Surfbullet]

[ QUOTE ]
Nice post. This is what Abdul Jalib was talking about all those years ago when talking about balanced strategies: taking some slightly suboptimal plays for the purposes of hand-disguise. He went into the preflop possibilities in some detail.



Balanced preflop strategy according to Abdul (http://web.archive.org/web/20030623140048/posev.com/poker/holdem/strategy/balance-abdul.txt)

Guy.

[/ QUOTE ]

His analysis is very thorough and an inspiring work. I've been looking through it recently and theres alot to be gleaned from it.

I think the 2+2 mantra of raise-or-fold preflop in terms of opening hands developed as the perfect way to balance - every hand in the hand range looks exactly the same. Overlimping is risky in that it gives away the weakness of our hand but it should be done with multiway hands where we are looking to flop a set or big draw so we don't really care if our hand range is narrow.

Regarding the flop... I employed an always-check strategy at 5/10, it suited me quite well and I think it did well there though it was pretty basic. At 10/20 i've alternated between donking and c/ring (doing both about equal amounts arbitrarily) and really focusing on making the hand ranges for each completely random.

I shouldn't be saying this, but I will anyway. Many multitabling TAGs(myself included, historically!) get caught up in the established "best line" that it becomes painfully easy to read. A good example of this is the WA/WB line of c-c, c-c, bet.

Eg:

Hero raises in CO. Gets 3bet by a TAG on the button. Both know the other's raising standards are lighter than normal because of position, but still reasonable. 20-25% of hands for hero, maybe 10% of hands for villain.

Flop: A92

Now, if hero always uses the WA/WB line with his medium->weak aces, he'll do pretty well with them, villain will continuation bet often on the turn to protect a medium/weak strength hand or get a free showdown. The problem is that any other line does not contain these hands, and becomes much more exploitable. A flop donk is almost never an ace...likewise, neither is a c/r. This removes a huge amount of semibluff power because our hand ranges have been polarized between check-call and check-raise, with check-raise being nothing left but 2nd pair + worse and the rare monster.

It's important to balance the ranges within check-call and check-raise. If we often check-call medium strength pairs to induce continuation bets, we have to occassionally check-call top pair or better so our hand range doesn't become a clear "3rdPTK or worse" when we c-c the flop.

I really like the way you articulated your thoughts, Jeff. I remember discussing this exact idea with you during HULA...the c/r as a default for blind defense does very well in terms of minimizing information given and is not too far from optimal.

Surf

[Surfbullet]

[ QUOTE ]
[ QUOTE ]
This is all fine and good, but counterproductive unless you're playing against opponents who actually notice. Aren't too many of those -- online anyway -- until at least 30/60.

[/ QUOTE ]
I disagree. How could developing an unexploitable strategy be counterproductive? I don't think we're giving up much if anything in order to do this either.

[/ QUOTE ]

Check out my long post. To truly become unexploitable we have to randomize between c-c and c-r with all our hands (monsters, strong + weak draws, pairs). This means missing out on some EV with lines that are clearly optimal against oblivious players - like the WA/WB line, or the flop c/r with top pair.

Surf

[Subfallen]

[ QUOTE ]
[ QUOTE ]
This is all fine and good, but counterproductive unless you're playing against opponents who actually notice. Aren't too many of those -- online anyway -- until at least 30/60.

[/ QUOTE ]
I disagree. How could developing an unexploitable strategy be counterproductive? I don't think we're giving up much if anything in order to do this either.

[/ QUOTE ]

Simple explanation. Let's say you're playing Rock/Paper/Scissors. The optimal (unexploitable) strategy is to choice your action randomly.

But what if you're facing someone who just plays Rock every time? Clearly now the correct choice is to play Paper 100%.

Anyways, just take Hock_ at face value, he plays g00t.

[sweetjazz]

[ QUOTE ]
I disagree. How could developing an unexploitable strategy be counterproductive? I don't think we're giving up much if anything in order to do this either.

[/ QUOTE ]

Weak opponents don't purposefully exploit anything you do. They play a certain strategy regardless of what you do. In this case, devising a strategy that is unexploitable is a serious mistake. Your strategy should, instead, be based entirely on exploiting the mistakes of your opponents.

Convergence toward unexploitable strategy will only occur at tables where most or all players are quite good.

[Monty Cantsin]

[ QUOTE ]
I disagree. How could developing an unexploitable strategy be counterproductive?

[/ QUOTE ]

Choosing an unexploitable strategy can be very -EV. Profit in Poker comes from taking advantage of other people's mistakes. An unexploitable strategy only guarantees that you can break even minus the rake.

Consider some analogies:

Roshambo (rock/paper/scissor) bot tournaments always have some intentionally "broken" bots thrown into the mix, bots whose strategies are always play rock, bots who always choose rock then paper then scissors then rock in an fixed loop, things like that. A "perfect" bot is one who makes totally random moves. You can never beat a perfect bot, it will always play you to a draw. But the perfect bot will never win the tournament because it is incapable of taking advantage of the broken bots that are present.

A good bot will be able to pick up on the fact that it's up against a broken bot and take advantage of that fact. In doing so, it becomes exploitable. The good bot is more exploitable than the perfectly random bot but it is also more "profitable" and will place higher in the tournament.

Another example is an expert Go player playing a handicap game against an amateur. The expert must make overplays in order to win, he can't only make safe plays that have no weakness. In a sense his strategy is exploitable but provides additional profit in this particular environment.

The situation with online poker is particularly interesting because we see so many different players and go up against so many different strategies that we often have to model our opponent's strategy based on an average of all the different strategies we come up against.

There is a decent percentage of players in the online environment who will limp re-raise with 54 and cold call with 73. In a very real sense these players are mixing strategies for you. As long as you are an unknown to your opponent he has to assume that you are capable of doing these things and you can optimize your expectation without worrying about becoming too predictable.

TeddyKGB2005 is paying your Shania tax for you. Straightforward non-mixed-strategy play is extremely robust in this kind of environment.

Obviously, there are also players who do pay attention to each opponent and model their opponents' strategies based on individual histories. The smaller the player population and the more observant the players the more valuable mixed strategies become.

But you will never be in a situation where it is profitable to be %100 unexploitable.

/mc

EDIT: I knew someone would say the same thing better and faster in the time it took me to type this.

[IGMorton]

lol

i am perplexed by that strange collection of words he put together. it doesn't even qualify as a run on sentence /images/graemlins/smile.gif

i'm sure he had a point, or 2, or even 3. it looks like somebody ran it through bable-fish a few dozen times.

[MAxx]

I enjoyed your post. Things Ive thought about, but not in an organized fashion.

[sweetjazz]

There seems to me two ways to mix strategies:

The first is what (I think) Jeff suggested, namely choose your plays so that your hand range is just such that your opponent has the same EV no matter what he does. Theoretically, at least, one can make oneself unexploitable.

However, there is another way of mixing strategies, perhaps labeling it "changing gears" is appropriate. (Mike Sexton would no doubt approve.) Here the intention is to play in a certain style so that your opponent gets a read on you, and then switch it up. Start a session intending to fastplay all your big hands. Keep note of how often you get such hands to play and how often you show them down after fastplaying. Use your judgment to estimate how quickly others are catching on. Then as soon as you think they have recognized an exploitable strategy, switch it up. Fastplay draws and put pressure on people to fold, wait for the turn to raise with big hands, etc.

In the end, of course, you will come up with a long range mixed strategy. But you hopefully gain a bigger advantage because opponents spend more time with a false impression of how you play than they do with an accurate one. (Of course, you could also get to metalevel where you don't change gears at a certain time because your opponents think you will, etc. etc.)

I feel like this latter approach is, to some extent, a more realistic model of how poker is actually played. Our goal is both to be able to go to a mixed strategy but also to develop strategies that exploit our opponents. In some sense, against thinking players, the goal is to balance our two needs. Sometimes we must give up on trying to exploit our opponents in order that we not be exploitable, and other times we open ourselves up to exploitation by trying to exploit another player. In the latter instance, we either expect the player not to be able to take advantage of it (weak players) or we must do something to try to trick him so that he cannot take advantage (strong players).

[dave44]

Yea I got a little mixed up there. The RPS example is perfect. Another example in poker would be a player who never bluffs- there is no need to call down with enough hands to ensure his bluffs would be 0 EV in the long run. By calling down too much, you would be playing suboptimally.

The goal then I would say is to lay the groundworks for an unexploitable strategy, but always be adjusting it according to the players you are up against and their weaknesses.

[Monty Cantsin]

[ QUOTE ]

The goal then I would say is to lay the groundworks for an unexploitable strategy, but always be adjusting it according to the players you are up against and their weaknesses.

[/ QUOTE ]

Similarly, Iocaine Powder (http://dan.egnor.name/iocaine.html), the champion Roshambot, has a pure random choice option that it can fall back on if it senses it's getting outplayed.

But there's a much better method for protecting yourself against observant opponents who are capable of exploiting your strategies: table selection.

/mc

[cartman]

[ QUOTE ]
set of diverse hands that you have given a certain action

[/ QUOTE ]

Against observant opponents I think that this essential. I have experimented--as I'm sure we all have--with various approaches particularly to heads up confrontations. I find that without even realizing it I eventually gravitate to a cohesive strategy in which my play on the flop is very similar if not identical whether I have a made hand or a draw. The example you give about always either checkraising or check-folding the flop is excellent and may very well be optimal. The key I think is that checkraising is correct or nearly correct for practically all the hands that you will choose to continue with. So masking your hands in this fashion requires very little if any sacrifice. All your opponent knows is that you have something in the range of hands that you don't fold.

Another approach might be to checkraise with most of your hands, but to check-call with both the strongest and the weakest hands that you don't fold. This strategy is less mixed than the always checkraise approach because your flop action allows your opponent to put you on one of two hand ranges: 1)weakest hands that you don't fold plus strongest hands and 2)everything else that you don't fold. There is some merit to this strategy, but it is also has some serious drawbacks. When you check-call the looming threat of a turn checkraise may cause him to give you see some free river cards with your gutshots for instance. But it will also allow him to frequently check behind when you really do have a monster. The most dreadful consequence however is that when you do checkraise the turn he can pretty confidently put you on a monster. There are many TAGs who adhere to this line of thought, whether intentionally or not, whose flop checkraise allows me to immediately rule out a very strong hand because they ALWAYS wait until the turn when they are in fact strong.

There are a considerable number of LAGs who use the exact type of mixing you describe by always check-calling the flop and then checking the turn heads up against a preflop raiser. They then begin foaming at the mouth at the prospect of their favorite play--the turn checkraise. Unfortunately for them, we have a lot of options when they check to us on the turn. But when we do in fact bet and get checkraised on the turn, we don't know much about their hand because they take this same line with monsters, pure bluffs, and everything in between.

One last situation to which this concept is directly applicable is head up in position after raising preflop. There are some TAGs who almost always just call when their opponent checkraises the flop and then subsequently raise the turn with a wide variety of semibluffs and made hands as well as some pure bluffs. Again we are confronted with situation in which we know very little about their hand range because they play so many hands identically.

Cartman

[Jeff W]

[ QUOTE ]
But there's a much better method for protecting yourself against observant opponents who are capable of exploiting your strategies: table selection.

[/ QUOTE ]

Past 5/10, you're going to be playing against other solid, observant opponents no matter how good your table selection is. Against these players, you have two choices: be unexploitable or be exploited.

[Subfallen]

[ QUOTE ]
Again we are confronted with situation in which we know very little about their hand range because they play so many hands identically.

[/ QUOTE ]

However unless they weight their raising frequencies to compensate for the disparities in hand strength and likelihood across the range, this action is also exploitable.

[cartman]

[ QUOTE ]
[ QUOTE ]
Again we are confronted with situation in which we know very little about their hand range because they play so many hands identically.

[/ QUOTE ]

However unless they weight their raising frequencies to compensate for the disparities in hand strength and likelihood across the range, this action is also exploitable.

[/ QUOTE ]

You are right and I don't think the strategy of always just calling the flop checkraise and raising a ton of turns is a particularly good one. The problem is that this is a very suboptimal line for many of the hands these guys do it with. So disguising their hands in this fashion comes at a significant sacrifice of EV.

Jeffs example of always checkraising or check-folding the flop is a very good strategy because checkraising is probably correct or nearly correct for the vast majority of the hands that he chooses to continue with.

Cartman

[bobbyi]

[ QUOTE ]
[ QUOTE ]
you might even be able to discuss the "shania" theory in there somewhere too.

[/ QUOTE ]

I already did, that's basically what the whole post is about.

[/ QUOTE ]
As I learned game theory, "mixed strategies" is not the correct term when discussing what we call Shania. Shania refers to playing different hands the same. In game theory, a mixed strategy means playing the same hand differently. These are two different ways of balancing your game. You are lumping them both under the heading of "mixed strategy" which I think is incorrect.

When we talk about a "strategy" in game theory, it means a mapping from all possible states of the world to the action that you take. What do you do with AA UTG in a ten-handed game? If you raise, that's a pure strategy. If you raise 70% of the time and limp the other 30%, that is mixed strategy. If you raise 100% of the time but you also raise 100% of the time with JTs to cover for it, your strategy is still pure because you have a direct mapping from a state to an action rather than having the action decided probabilistically.

The reason that mixed strategies are interesting in game theory is that there are some games where the optimal strategy is mixed. An obvious example is rock-paper-scissors where the optimal strategy is to play each 1/3 of the time.

Regardless, this is semantics. The actual content of your post was very good.

[bobbyi]

[ QUOTE ]
Choosing an unexploitable strategy can be very -EV.

[/ QUOTE ]
Disregarding rake and the threat of collusion, an unexploitable poker strategy is necessarily +EV (or 0 EV if your opponents are playing the same). What you are suggesting is that against specific opponents there may be other strategies with even higher EV, which is true.

[Monty Cantsin]

[ QUOTE ]
[ QUOTE ]
Choosing an unexploitable strategy can be very -EV.

[/ QUOTE ]
Disregarding rake and the threat of collusion, an unexploitable poker strategy is necessarily +EV (or 0 EV if your opponents are playing the same). What you are suggesting is that against specific opponents there may be other strategies with even higher EV, which is true.

[/ QUOTE ]

Yes, I meant comparatively -EV.

/mc

[Jeff W]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
you might even be able to discuss the "shania" theory in there somewhere too.

[/ QUOTE ]

I already did, that's basically what the whole post is about.

[/ QUOTE ]
As I learned game theory, "mixed strategies" is not the correct term when discussing what we call Shania. Shania refers to playing different hands the same. In game theory, a mixed strategy means playing the same hand differently. These are two different ways of balancing your game. You are lumping them both under the heading of "mixed strategy" which I think is incorrect.

[/ QUOTE ]

Bobby,

You're right that I was too fast and loose with the term "mixed strategy"--it has a strict game theory definition. However, I'd like to try to reconcile Shania with true Mixed Strategies as they are defined in game theory.

To play perfect poker, it is probably necessary to play the same hands differently AND play different hands the same way.

Those two ideas are really one and the same. If you play the same hands differently, you are going to end up playing a lot of different hands the same way. The optimal strategy is going to be one where the probabilities for each action for each hand coalesce to give you the highest possible EV for all situations across all hands.

This was the original way I was going to approach the essay, but I changed my mind halfway through because it's easier to explain it by going backwards.

P.S. Good posts everyone.

[Monty Cantsin]

[ QUOTE ]
[ QUOTE ]
But there's a much better method for protecting yourself against observant opponents who are capable of exploiting your strategies: table selection.

[/ QUOTE ]

Past 5/10, you're going to be playing against other solid, observant opponents no matter how good your table selection is.

[/ QUOTE ]

Yes, I agree. And I'm totally in favor of mixing up your play, shifting gears, and so forth.

But my point was that the only time %100 unexploitability is the best strategy is when you are against perfect opponents.

[ QUOTE ]
Against these players, you have two choices: be unexploitable or be exploited.

[/ QUOTE ]

I think you actually have a range of choices which corresponds to a wide spectrum of exploitable to non-exploitable strategies.

Shooting for %100 unexploitability is playing in a way that assumes your opponents are playing optimally. Clearly this is not the way to maximize your EV in real-world poker.

So, again, I'm saying I agree you should reduce your exploitability by using mixed strategies. But I don't think you should shoot for %100 unexploitability.

/mc

[bobbyi]

I agree with everything you say. It's just a question of semantics. Personally, I think redefining the term "mixed strategy" to include Shania confuses the issue and takes away our ability to refer specifically to mixed strategies in the traditional sense.

I use the term "balancing" in the way that you are using "mixed strategy": this is, in my opinion, the proper blanket term for all ways of making yourself unexploitable even if an opponent knows what your strategy is. "Mixed strategy" and "Shania" refer to two specific mechanisms by which you can balance your game.