Theoretical ramblings

I know this forum isn't really for talking about poker theory, but I'm hoping that this can be counted as a blog and left where it is. I don't feel it would be any more appropriate in any other forum.

I like to think about poker. To be honest, I probably spend a lot more time thinking about poker than playing. When you play poker you spend the vast majority of the time making fairly straightforward plays and only occasionally come across really interesting decisions. But thinking about poker you can spend as long as you like pondering the most interesting of situations and decisions and in doing so can gain a some great insights into why poker is played the way it is. Having spent (wasted) so much of my time thinking about these things, I thought I might as well share my thoughts, in the hope that they might provide an alternative perspective on poker to some people, or at least generate some interesting discussion.

I will try to post an entry to this blog whenever I think about something which I think is interesting and I haven't seen covered anywhere else.

1. A useful thought experiment and the value of a polarised range

Balance is something that gets talked about a lot in poker. I'm not going to go into a lot of detail about balance just now, save to define a balanced play as one where across the range of hands with which you make this play you will make the same amount of net profit irrespective of your opponents actions. ie. on average you don't care what your opponent does. You are therefore unexploitable (but will not fully exploit your opponent's mistakes).

For those of you who are not totally sure what people mean by the idea of balance the following example should help:

Imagine a river situation where your range is split into 50% nut hands which are ahead of your opponents entire range and 50% air which are behind your opponents entire range. Both of you are exactly aware of each other's ranges. You are first to act and have two choices; bet pot or check. When you check your opponent will always check back since you will never call him with worse. If you always checked here then you would each win the pot 50% of the time. I'll use this as an EV base of 0. First thing to do to change this is to value bet our nut range of course. However when we do this our opponent should always fold to our value bets, meaning we don't actually make any more money. So we decide to throw in some bluffs to take advantage of this. If we bluff too much then our opponent should call, if we bluff too little then he should fold. However there is a mid-way point where it doesn't actually matter whether our opponent calls or not. This is the point of balance.

So what is this point? How often should we bluff? Well when our opponent folds we win a pot-sized bet from every bluff, and win nothing from every value bet. When our opponent calls we win a pot-sized bet from every value bet and lose a pot-sized bet from every bluff. Since we're balanced, we don't care whether or not our opponent calls, meaning we must make the same amount whether he calls or folds. Therefore

(profit from folds) number of Bluffs x 1psb = (profit from calls) number of valuebets x 1psb - number of bluffs x 1psb
Therefore number of Bluffs = (1/2) x number of valuebets.

So to be balanced you need to bluff half as often as you value bet.

So how much EV have we gained by doing this? Well, since you're balanced, you don't care whether or not your opponent calls. So for simplicity, we can just assume he folds every time. This means by bluffing half as often as value betting, we have increased our expected value by 50%. This is equivalent to us winning 75% of the pots And this result is independent of what our opponent does. Even if our opponent is the best player in the world, he only has 25% equity here despite having the best hand half the time! Pretty neat eh?

So why do we have so much equity here? It's not really to do with the whole balance idea, that's just a useful way of explaining things. The reason is that our range is polarised, while our opponent's is well-defined. The value difference is real and it's huge. Think a bit about the implications of this when it comes to check-calling multiple streets. That's what my next entry is going to discuss.

For now though there is another interesting application of the idea of a balanced river betting range. The set-up for this experiment is going to be exactly the same river situation as before, except this time you can choose your bet-size. Given that we are going to be consistent about our bet-size (otherwise we are not balanced), what is the best bet-size to use, and how can we work this out? Well earlier on we calculated the balanced number of bluffs for a potsized bet. Given that we are bluffing a balanced amount, we don't care whether or not we get called. So for the purposes of expected value calculations, we can assume that our opponent always folds, and our gain in equity is simply equal to the pot multiplied by the balanced number of bluffs. Therefore the more we can bluff (in a balanced way) the more money we make.

We therefore need to choose the bet-size which allows us to bluff with the greatest frequency (and be balanced). Let's try working out the balanced number of bluffs for a half psb. If we do this then profit from folds is still number of bluffs x 1psb but profit from calls is (1/2)psb x no. value bets - (1/2)psb x no. bluffs.

Equating these as before we find that number of bluffs = (1/3) number of value bets. So betting half-pot, even in a totally balanced way, is simply less profitable than betting full pot. Everybody already knows that it's better to bet something comparable to the pot rather than to make very small bets, but hopefully this will give you a better understanding of why.

So what happens when we bet more than the pot? Well actually, in this particular situation you'll find that the larger your bets the more bluffs you can get away with, and so in this situation you should in fact overbet-shove all your nut range and nearly as many bluffs. (Note that there are diminishing returns with increased bet-size. Betting 1/2 pot allows you to bluff with 1/3 as much as your value range. Betting full pot takes that up to 1/2. You need to bet an infinite amount for that figure to approach as many hands as your value betting range.)

Here we seem to have parted from the realms of realistic poker; how can overbetshoving be the optimal play here? Well the answer is because your range is totally polarised. This won't usually happen in poker. Usually your opponent's range overlaps with your value range, and overbetting will just mean that he'll only call when he's beating you. But occasionally your range will be very polarised and you should start to bet more; overbetting is the optimal play if your range is entirely air or stone cold nuts.

So what can we learn from all this when it comes to actually playing poker?

1. Polarised range good, condensed range bad. Just because you are ahead half the time doesn't mean you have 50% equity in the pot.

2. The amount of bluffs you can get away with making is dependent on your bet-size. If you bet larger you should bluff more.

3. Against an opponent who is well-adjusted to your play you can never bluff more hands than you value bet on the river.

4. You should bet larger with a highly polarised range (relative to your opponent's range) and smaller with a less polarised one.

There may be more things to be drawn from this aswell, I'd be very interested to hear any ideas.

Halleluja!

Great read! I like polarized ranges, but I love dynamic ranges!!!

2. A polarised range with multiple streets of action

Today I'm going to extend the model of balanced play with a polarised range over multiple streets. Let's imagine that you again have a range that contains a certain number of nut hands along with a bunch of absolute air (far more air than nut hands), while our opponent's hands are all of medium value. This time however, action starts on the flop. For the purposes of this model we're assuming that hands will not change in value across later streets, the only real purpose of having multiple streets here is to allow multiple betting rounds. For simplicity we will also assume that you always bet the size of the pot.

Since your opponent is never going to bet his bluff-catchers, you obviously should bet all your nut range at every opportunity. But how much should you bluff to be balanced? Remember first that to be balanced you are bluffing with a frequency such that it makes no difference whether your opponent calls (on average). The way to work out how often you should be bluffing on the flop is to work backwards from the river.

River: As discussed last time, you bet pot with 100% of your value range and 1/2 as many bluffs. Against this range it makes no difference whether your opponent calls or folds. They therefore have zero equity against a river range this size. However, for your turn bet to have been balanced, you need to not care whether your opponent calls it. so given that your opponent is calling a pot-sized-bet on the turn, he needs to have exactly 1/3 equity at this point. This means that your value range and half as many bluffs only accounts for 2/3 of your range on the river. The other 1/3 comes from more air which you check and give up with. This air you give up with comes to a range 3/4 the size of your value range.

Therefore on turn you are betting 5/4 times as many bluffs as value bets. Since it doesn't matter whether opponent calls flop bet either, he must have 1/3 equity on calling flop bet. Since it doesn't matter whether he calls the turn bet, we can assume he folds every time, therefore we must give up 1/3 of the time, so that it doesn't matter whether he calls the flop. Since you're betting 9/4 of your value range on the turn (value range + 5/4 bluffs) you must give up turn with a further 9/8. Therefore on the flop we have to bet that 9/8 on top of the 9/4 (18/8) of our value range. Therefore we bet a total of 27/8 of our value range.

Once again, it doesn't matter whether he calls this bet or not, so we can assume that he never does. That means we win the pot with 27/8 times our value range. We've more than tripled our showdown value in equity!

Now let's look at the flipside. We're now the guy calling down. What we've just said is that your opponent can bet 27/8 times his value range on the flop and be perfectly balanced. ie. It makes no difference whether we call. So even though he's bluffing more than twice as often as value betting, a fold is just as good as a call. Now imagine that your opponent is not balanced and he's only bluffing exactly twice as much as value betting. The fewer bluffs in his range mean that even though we're ahead 2/3 of the time it becomes an easy fold. Strange but true...

So here we've shown that a polarised range is far more valuable in relation to the pot size with multiple streets of action remaining. This has a result for bet-sizing which is more intuitive than for the river decision discussed last time. Because overbetshoving a polarised range on one street can not be done with more bluffs than value-bets, doing this can not do any more than double your expected value (see last entry). As we've just shown, with multiple betting rounds we can more than triple our expected value. Therefore we should size our bets such that we can get a decent sized bet in on each street. And it just so happens that the common practice of betting around about the pot on each streets is usually about the right amount to do just that when 100bb deep.

Another application of this discussion is in thinking about the number of hands you can get away with bluffing on the flop. Assuming pot-sized bets, we've shown that you can bet 1/2 as many bluffs as value bets on the river, 5/4 as many on the turn, and almost 2.5 times as many on the flop. But this is assuming that all your value betting hands are strong enough to bet 3 streets. In real poker your range is not this polarised and some hands are only worth one or two streets of value. I'm going to hypothesise that the number of streets of value you can try to get from each hand in your range impacts directly on the number of bluffs you can afford to make (in a balanced way). So I will suggest that for every 1 value street hand you can add 1/2 a bluff. For every 2 street value hand you can add 5/4 bluffs. For every 3 street value hand you can add 19/8 bluffs.

As an example let's say we have a flop where we hit 5% nut hands, 10% 2 value-street hands, 15% 1 value-street hands and 70% air. Assuming for argument's sake that we bet all of our value range on the flop then that allows us to bet with 5x(19/8) + 10x(5/4) + 15x(1/2) = 32% of hands as bluffs. So we bet this flop a total of 62% of the time.

Of course the exact numbers I've got in here are derived using a simplified model that does not factor in draws, opponent's aggression etc, so this isn't going to tell you the exact number of bluffs you should make. But the central point holds and is that:

1. For every value betting hand in your range you should add bluffs.
2. You should add more bluffs for every stronger value betting hand than you should for each weaker value betting hand.

Stack size also effects the equity to be gained from our polarised range in a similar way. If there are only 1 or 2 psbs left in effective stacks then you can't afford to bluff as often, since you won't have as many betting rounds to exploit your advantage.

In summary:
Polarised range even more important with multiple betting rounds remaining
Selective double and triple barrelling is very profitable and allows you to raise bluff frequency several fold
Just because you're usually ahead doesn't mean you should call
More and stronger hands in your value range mean you can bluff more
You should bluff more on early streets with greater stack/pot ratio.

3. Thoughts on the balanced range model, +Now & Next theorem

In my first post in this column, I defined a balanced range as one where on average you didn't mind how your opponent played, since they could in no way exploit you, and left it at that. Today I'm going to elaborate on this and discuss in more detail the relevance of the balance range model to real poker. I'm also going to try to clarify some misinterpretations of the concept which lead to faulty or incomplete reasoning in decision making.

Another way to look at the balanced range model is as a theoretical construct in which you attempt to maximise your expected value vs an opponent who is perfectly and instantaneously adjusted to the optimal play vs your true range. Such an opponent forces you to play in a "balanced" way, because he will immediately be exploiting any imbalance. But to what extent is such an opponent realistic?

The key here is "instantaneously" adjusted. What I mean by this is that he knows when you decide to raise a hand that it's in your raising range. He knows if you fold that it's not (or sometimes isn't). If you normally raise only AA utg but this time decide to also raise 72o if you get dealt it then he's calculating on a range of AA +72o. This is obviously impossible. Your opponent can only judge your range against previous actions in previous hands, he can't know if you've suddenly changed your range (unless you give away some kind of tell).

Thus the balanced range model, apart from assuming your opponent to know everything there is to know about your game, takes no account for the dynamic nature of poker (more specifically of dynamic ranges).

What other flaws does the balanced range model have? A pretty major one is that it focuses on minimising the amount your opponent is able to exploit you, rather than exploiting your opponent's mistakes. This can be fine if you believe that your opponent is going to be one step ahead of you, but if this is the case you should perhaps find a new opponent. Of course that doesn't mean the balanced range model isn't useful to consider in general, but it is important to remember that the most balanced play is unlikely to be the optimal one, and that maximising EV, not
being balanced, is the ultimate goal in poker.

As a simple example of this I'm going to return to the simple scenario outlined in my first post, of the river decision with one player having a polarised range (50%nuts 50%air), and one player a condensed one(100%bluff catchers). (You should probably read this now if you haven't already.) As we established, the polarised player's most balanced range is to bluff half as often as his value range. It can similarly be determined that a balanced calling range is to call half of the time in this situation. (You can check this, but I figured doing the maths again would be boring.)

Now imagine you're playing this exact situation a million times against the same opponent, half of the time with the polarised range, half with the condensed one. If you do the balanced thing every single time (ie making decisions by appropriately weighted chance) then you will break-even, irrespective of your opponent's actions. We know this because by definition, we don't care what he does. If he is being exactly balanced then we're doing just the same thing as him, therefore we must break-even, and if he's doing something different then it makes no difference to us, therefore we must still break-even. Obviously the smart thing to do is to perhaps start out balanced if we don't know anything about the player, then unbalance ourselves appropriately to take advantage of his patterns of play.

Of course this trivial example sounds obvious, but the point is that the point of balance is something to be aware of, but not to strive towards. Knowing what the balanced percentages are allows you to know which side of this your opponent is on, which side you are on, and in what way you are each vulnerable to exploitation. Being on the point of balance wins you nothing.

A lot of the value of the balanced range model is in optimising things like betsizing. I've spent some time on this in the two previous posts without really explaining what I was doing. What I was really doing there was finding the optimal bet size for the worst case scenario of playing against a perfectly balanced opponent. This bet size will not necessarily be best for playing against a real, exploitable opponent, but it's going to be a pretty good starting point against anyone reasonably competent and capable of thinking beyond first level. ie. Someone who is going to notice and start thinking why if you bet different amounts depending on your hand strength.

I saw an amazing thread on 2+2 related to this idea (linked from an article in the high stakes forum on here) discussing the relative merits of different betsizes on a river bluff vs a very weak opponent. The crux of the debate was one poster proposing that a small bet was the best play, since it folded out all the same hands that a shove would, with another insisting that the bluff had to be a shove, since this was the optimised balanced play and since this is what the player would have done were he value betting.

Now although this second poster was in some ways thinking on a much higher level, his thinking is not joined up. He has failed to relate the more advanced concept of balance back to what it's all about: Expected value. Of course in this kind of situation, against a decent opponent you would shove all-in. Ask a good player why and they'll usually say something like "balance" or "I'd shove all-in if I was value betting". But that isn't actually a reason. There is only one possible reason to make a larger bluff, and that is because you think they'd call a smaller bet more often. But then you have to ask why they'll call more often. The reason for this is twofold, one reason is pot odds, the other is that your line doesn't make sense for a value betting hand. Why doesn't it make sense? "balance" or "Because you'd shove all-in if you were value betting". So what our good player says is true, but it misses out a couple of simple but important stages in reasoning.

What our 2+2er has forgotten then is to think through these stages. He hasn't thought about whether the opponent will work out that your line makes no sense. He's so used to playing against people who will spot an obviously unbalanced play a mile off that he's forgotten that this stage of reasoning even exists and has somehow come to think of playing in a balanced range optimising way as the ultimate goal, rather than just a way of increasing EV vs
well balanced opponents.

I feel like what I've written in the last couple of paragraphs is not totally clear, so I'm going to go over it again in a slightly different way, since i think it's important:

Poster 2 is advocating the balanced range optimising play. Using the kind of techniques I used in my first two posts to determine optimal play vs a perfectly-adjusted opponent you would find that you should bluff here a certain % of the time and the optimal betsizing would be all-in (Consistently betting smaller could also be profitable, but less so as discussed in 1st entry). Against this theoretical opponent if you bluff half pot and shove for value then you will have exactly defined your hand and you opponent will be able to call all your bluffs and fold to all your value-bets thus totally destroying you. A real, but thinking, opponent will also be able to work out what you're thinking a lot of the time and will also win a lot of money against you. However the opponent you're playing against here is far from perfectly adjusted and may not be thinking at all about whether your bet makes any sense. If he is equally likely to call bets of different sizes then of course it is better to bet more for value and less as a bluff.

I hope at least one of those explanations makes sense to you. Of course the point I'm trying to make here is not what is the best bet-size. It's quite possible that the opponent in question is smart enough to work out that the smaller bet makes no sense. It's also possible that he'll call a lot more just because it's cheaper. The point I'm trying to make is that considerations of balanced ranges and in this case balanced range optimisation can make even very good players lose sight of what they're really trying to achieve.
Judging by the amount of arse-kissing that went on further down the thread this guy is probably successful at very high stakes and yet appears to have a fundamental misunderstanding about basic poker theory.

This kind of misunderstanding is really common and it arises due to to vaguaries in the way poker players express themselves (see above comments by "our good player") and also due to the complications driven by metagame considerations.

What I mean by this is the fact that sometimes our decision in a particular hand has an effect on not only our expected value in this hand, but also the way opponents will react to us and hence expected value in future hands.

In general I think the importance of this is over-estimated, but of course it cannot be completely neglected. Resolving the kind of confusion and misconceptions that are caused by all this lead me to conceive:

Now and Next Theorem

There are two times you can make money:
1. Now, in this hand, with the actual hand you are holding.
2. Next, in all future hands, with all possible holdings.

And making more money is better than making less money

Ok so it's a bit of a joke calling this a theorem since it's more of a truism, but I wanted to have a theorem. The point of this "theorem" is that any time you make a decision you have to relate all of your reasoning back to either NowEV or NextEV.

If you plan to take a NowEV hit in order to increase NextEV, remember to consider:
1. Your whole range of future hands
2. Whether your opponent is really paying any attention
3. Whether you're going to play enough hands with the opponent (or other observing opponents) to make the short-term loss worthwhile.

Also remember that if you think a certain action increases your NextEV relative to another, what you're saying is that either one action makes your opponent a better player who is harder to exploit, or the other action tilts the player into adjusting badly and being easier to exploit. ie. a player adjusting against your probably isn't going to be any easier to exploit, unless they make a horrible job of it.

So people please don't make statements like "I bluff in order to make money from my good hands" without backing them up with an EV statement. "I bluff because you believe your opponent will overadjust to this and start to call down ridiculously light, thust giving me a ton of value on my better hands" is much better (NextEV). But remember in this case you also lose some NextEV where bluffing becomes less profitable. More likely to be true: "I bluff because my opponent expects me not to bluff often and therefore I will take advantage of this and take the pot down a lot" (NowEV). "If he works this out and seems likely to over-adjust then I will exploit him by value-betting more thinly and bluffing less."

Apologies for the length of this post. I promise the next one will be shorter! If I had to shorten this all down to one sentence I'd go with "Whatever theoretical tools you use to make your decisions in poker they must relate back to expected value in this hand, with your actual holdings or to a lesser extent in future hands . And you'd better have a damn good reason for sacrificing any value on this hand."

Damn, that was two sentences. I'm really not good at this conciseness thing...

4. The Science of slow-playing

Slow-playing is a much-maligned but important part of poker strategy. Because most beginners make a habit of slow-playing too much, and because at microstakes most players are very passive, the general message you're likely to get on these boards is just "don't do it". Although it's not a strategy that should be used a lot, there are times when it's a very useful weapon to have. I'm going to take a look at when slow-playing is a good strategy, first from the perspective of an individual hand and then with the idea of strengthening ranges.

So, you've flopped the nuts. Whoopdedoo. Now it's value time. So how do we best get value out of our hand? Quite simply, we work out whether our opponent is likely to put more money in the pot calling bets, or betting himself. The first step in answering this question is of course our opponent. If he's very passive, bets small, is incapable of bluffing or value-betting then of course we shouldn't slow play. But if he's a fairly competent player then the answer is going to be bit closer.

Next we have to look at ranges. Namely his likely range and our perceived range, the one he'll be putting us on. If our perceived range is more polarised than his, then he is unlikely to bet. (If you don't understand why this is the case then read ABCD theorem, or one of dozens of other articles/posts/books, or just think about it for a bit.) Therefore in this situation we should bet ourselves.

If our perceived range is less polarised than his, then he is likely to bet. If we bet in this situation, then he will become wary of why we are betting our condensed range, and our perceived range may change to one far more heavily weighted towards our few nut hands, which is going to cost us money. Therefore in general we should be more inclined to slowplay.

Now let's look at a balanced range analysis of the situation. When we have a polarised range, betting becomes very profitable, as discussed in the first 2 articles on this thread. But when we have a condensed range, what happens if we bet the few nut hands in it. Well it allows us to bet a few of our weakest hands as bluffs, but then leaves a large and even more condensed range behind. That means that, relatively speaking, our opponent now has a much more polarised range. This means that over 3 streets he will be able to destroy us with balanced barelling and triple his equity as discussed in my second article.

If we check our nut hands here however, his value hands lose a lot of their value. A lot of the 3 street value hands become 2 street value hands etc, which forces him to bluff less and gives a lot more of your weaker holdings a chance at showing down. Basically, our nut hands add a lot more value to our checking range than they would to our betting range

So we've come to the same conclusion twice: The more polarised your opponent's range is relative to your own, The more you should consider slow-playing.

Although I chose to approach this topic from the point of view of just one part of your range (nut hands), a more general result can be found: When our range is more polarised than our opponent's we should be more inclined to play aggressively. When our range is less polarised than our opponent's we should be more inclined to play passively.

Have been doing a bit more thinking about post 3. and realised that the whole concept I'm trying to get across can be expressed far more simply:

"Any action that is 'good for your range' against a well-adjusted opponent must also be profitable in isolation"

Why? Because if your action isn't profitable then your opponent doesn't need to adjust to it, as he can now make more money against this hand, and the same against the rest of your range, by not adjusting.

Quote:

---

Originally Posted by knaplek

Have been doing a bit more thinking about post 3. and realised that the whole concept I'm trying to get across can be expressed far more simply:

"Any action that is 'good for your range' against a well-adjusted opponent must also be profitable in isolation"

Why? Because if your action isn't profitable then your opponent doesn't need to adjust to it, as he can now make more money against this hand, and the same against the rest of your range, by not adjusting.

---

ok, but say we make a bluff that is in isolation break-even
we haven't shown down hands in the same situation before
if our bluff gets called, it will make our opponent suspect that we've been bluffing a lot more in the previous hands when we could have just been running well

because poker is a game of incomplete information we can manipulate our opponents into thinking they're being exploited when in fact we're bluffing a small percentage of the time and betting for value a large percentage of the time

or the opposite can be true
we can make a river bet that is break even at best to make villain believe we are going for really thin value every time, when in fact the hands we haven't shown down were bluffs

What you're saying is true, however the idea behind what you're suggesting is that your opponent will over-adjust to a bluff or a thin value-bet, it's not a fundamental concept so much as an attempt to put your opponent on a kind of tilt against you.

I could equally well say that you should check down that break-even bluff. Your opponent could then think that you're only ever betting for value and you can steal a lot of pots by bluffing.

Or we could check down the thin value-betting hand, to make our opponents think our range is really polarised when we value-bet, which will cause him to either call down really light - so we can get loads of thin value, or fold everything - so we can make a ton of bluffs.

So basically it comes down to individual opponent tendencies. Yes there are likely to be times against certain opponents when it's worth giving up a little bit of value now in order to make them play badly against you later. But the way in which you choose to do this is entirely based on your opponent's psyche, as supposed to some theoretical range-optimising play.

My point is that we want our opponent to adjust to us incorrectly by misinterpretting our previous actions. So if we run well, we might want to keep pressuring our opponent until he adjusts. When he does, he will feel like we've been bullying him all along. From his perspective there is no difference from us bluffing him 5 times in a row and vbetting 4 times and bluffing once.

But from a theoretical standpoint we are more likely to be just crazy aggressive than to get all of these hands in a row if we show a bluff once.

5. The funny thing about tournament play

Is that not all chips are worth the same. I know that, you know that, boring, yawn. But the implications of this are surprisingly interesting. And I'm not talking about ICM, because that's pretty dull too. What I'm going to talk about is the changes in the fundamental theory of the game that are brought about by this fact.

The first thing you can say about this is that each poker hand is no longer a zero sum game. Oh sure, the overall tournament is (minus the rake) but on a hand by hand basis, what one player gains in a heads up pot is not equal to what the other player loses. In fact it's possible for both players to take an expected value hit when the money goes in.

But how does this affect the idea of "balanced" play, which seems to have been the central theme of this thread. I came up with two earlier definitions of balanced play, which for cash games were effectively equivalent.

1. A play where across the range of hands with which you make this play you will make the same amount of net profit irrespective of your opponents actions. ie. on average you don't care what your opponent does.

2. A theoretical construct in which you attempt to maximise your expected value vs an opponent who is perfectly and instantaneously adjusted to the optimal play vs your true range.

The first version is not totally robust and doesn't actually exist in a tournament scenario. So the two definitions are no longer equivalent. Therefore for this thread we are forced to use the second definition.

The kinds of effects I'm talking about are most prevalent the flatter the prize structure. In fact a winner takes all tournament should play just like a cash game. So for the other end of the spectrum I'm going to use double-or-nothing sit'n'gos since they most effectively illustrate the point I'm trying to make.

First example: 6 man tourney, top 3 paid equal, bottom 3 leave with nowt. 4 guys left, blinds 200/400, ante 50. You have 2000 chips and have posted the BB. Guy on your right has 2500. Other two player have 1000 and 500 chips. They both fold and the big stack in the small blind pushes all in. What is the correct "balanced play".

Answer: Call with any two.

I'm not kidding.

That's because your theoretical "balanced" opponent knows what you're going to do before he makes his decision to push. If he knows you'll fold everything, like a typical ICM machine, then he can push with anything. But if you call with any two cards then he has to fold everything, even AA, because of the equity he loses in a race with you.

Here's the maths: If he folds he has 93% of the prize money in equity (ICM). If he wins all in vs you this goes up to 100%. If he loses it goes down to 44%. So he'd need to win 88% of the time to make for a good push. AA vs ATC is 85%.

Therefore if you call with any two here then he won't have pushed in the first place!

Of course I'm not advocating that anyone should go actually call with AA in this spot, let alone ATC, but this example illustrates some interesting points.

1. The static balanced range model is a useful concept but not the be-all and end-all of poker. I've made this point a few times and I think this effectively illustrates that.

2. Meta-game is potentially far more important in tournament play than cash games.

You don't often see the effect of this second point because in many tournament situations you're playing people you very likely won't play again, so you can't think too much of future hands. But if you played against the same people again and again then it would become seriously important. Let's say you played a series of high stakes hyper-turbo DON tournaments with the same group of sng pros online. And you decide to play a really loose style, calling super-wide, nothing like what an ICM calculator would tell you to do. At first you'd lose horribly. But then your opponents would start to adjust and wouldn't push into your blinds since they would know they're getting called. Ultimately, assuming each opponent stuck to playing correct ICM poker, applying accurate calling ranges to you, you would run out a big winner.

Obviously in a cash game this could never happen. In a cash game bad play is exploitable. You make money when your opponent makes mistakes. It generally doesn't matter what your opponent thinks your style of play is in cash game, as long as he's wrong (or you can adjust to make him wrong). In tournaments not all mistakes are exploitable.

Another product of the difference between cash and tournament play is the ability to team-up. If me and a bunch of my friends invited a top professional to one of our cash games, there is nothing we could do to avoid him destroying us over the long run, without cheating in some way. But if we invited him to come and play turbo sngs or DONs then we could beat him easily. We'd just have to all agree to make calls and pushes vs him which would be -ev for ourselves but also -ev for him. Everyone else at the table would benefit and in the long run we'd all win apart from him.

So, apart from emphasising the general weirdness of tournaments, particularly sngs/dons/satellites, what from all this is actually relevant to your poker?
1. Get past the idea of static balanced ranges, particularly for sngs. Sometimes it's just stupid.

2. If you're a high stakes SNG player, think more about your meta-game. If you can convince the regs you play with that you call crazily wide then that could be more valuable than making correct ICM decisions that make you look like a push-over. Certainly consider calling when the decision is borderline and your opponent is likely to be paying attention