cEV, $EV, cash vs tournament, fallacies, and gigabet: musings

[aaronbeen]

Reading various posts about stack sizes and tournaments and cash games has made me think about the underlying assumptions and models used in poker. Of particular interest is the idea of expected value. In a cash game of course it makes sense to calculate an expected value. With basic assumptions (such as playing adequately bankrolled) the idea of averaging an infinite number of trials to determine whether the outcome is desirable makes perfect sense.

In tournaments, as we have recently seen, this is not so simple. I claim that the difference between cEV and $EV and the problems with ICM and with gigabet’s approach are to some extent due to an underlying problem: it is a fallacy to averaging and comparing averages, especially in a discrete system.

Allow, for a moment, an example which my explain my basic idea. Take an imaginary vitamin which we will assume results in a healthy normal body at its normal dosage. Call this vitamin "stackium." Tragically, a human who does not get any stackium will shrivel up and shrink, often to the height of only a few inches. A human who gets double his daily stackium requirement will double in height, often to excess of twelve feet.

Take a population and give half of them no stackium at all and half of the population a double dose. The expected result of a day of eating for a member of this population will be exactly the daily requirement. Yet no member of our population will be of normal height. Consider the life style of a member of the sample population. Both the miniature people and the giants will experience life very differently from an average human despite the fact that the population has received the average amount of stackium and has the average height. This freakish community will hardly resemble a normal group healthily consuming their stackium.

Now consider a different sample group where members are given doses of stackium according to a standard normal distribution. Here many of the oddities of the previous example are not present.

If the reader has made it past my elaborate fantasy it should (hopefully!) be apparent that many decisions in tournaments have discrete results that cannot simply be averaged.

The concept of EV makes perfect sense for simple situations such as betting on the outcome of a coinflip or whether the turn and river bring a spade, but it cannot always be applied to a tournament.

ICM and other models are really ways to tell us more about the life styles of the miniatures and the giants than the average amount of stackium consumed can say. As tournament models evolve and thinking becomes more advanced I predict standard cEV calculations to be of less and less use in certain situations, and a more advanced understanding of where discrete outcomes really lead to become more prevalent.

[aaronbeen]

good morning bump

[A_PLUS]

I think a lot of what you hoped to get across was lost in your wording. Instead of just your stackium story, why dont you lay out exactly what you are trying to say in poker terms, then maybe I will follow your story.

[SoBeDude]

I'm going to take a shot at paraphrasing. I guess it's also a test to see if I followed his line.

I think he's saying that EV leads you to decisions that are "on average" correct. But you may be very far from an "average" situation.

Thus he is implying that your decision, IN ISOLATION, may be incorrect if you used EV to make it.

But until we start playing poker with cards face up, I think he's barking up the wrong tree.

All we can do is put people on ranges of hands, and make the correct EV decision against that range.

-Scott

[aaronbeen]

Another try:
You can't describe the result of a play with the weighted average of all the possible results when particular outcomes are qualitatively different. The example of people with different heights experiencing life differently was an analogy to different stack sizes playing differently throughout the remainder of a tournament.
The ICM example of this would be that a 50% chance of busting and a 50% chance of doubling is not at all comparable to no change in chips - because a player risks "bubbling out." The "blocks of chips" example would be plays where some of the possibilities of {fold, call and win, call and lose} give the player fundamental advantages or disadvantages in later play.

[A_PLUS]

So all you are saying is that stacks of different sizes play differently. So having double your amount of chips may more than doubles your $EV. Basically chip value is not linearly related to $EV.

If you are saying something beyond that please explain it with a little more poker detail to me. thanks

[aaronbeen]

[ QUOTE ]
Basically chip value is not linearly related to $EV.

[/ QUOTE ]

I'm going a bit further than this. There are common situations where no valid mapping from cEV to $EV can possibly exist. A bit pedantic perhaps but from a mathematical stand point it is an important difference. The variable cEV fails to take into account a lot of information about what is really happening. Knowing the expected chip outcome is like knowing the mean of a distribution.
It follows from this that not only are conceptual approaches valid and valuable, they are often going to be all we have.

[A_PLUS]

[ QUOTE ]
There are common situations where no valid mapping from cEV to $EV can possibly exist.

[/ QUOTE ]

Of course this is true if TCs are your only variable. But whatever factors you think effect the value of a play, why couldnt they be included in a model to map CEV to $EV?

Also, the fact that it is the mean of what could be a very skewed distribution, doesnt mean it isnt valuable. I dont think anyone makes strict CEV decisions late in a tournament. The ICM, doesnt do this either.

[set57hike]

If I read your other comments correctly, I think you are simply saying that the MEAN outcome (the expected value) doesn't tell us enough. For example, we might care about the variance of an outcome too. Or we might care about the probability that a really bad or a really good outcome occurs.

In a tournament, we might call in a large multiway pot with a marginal hand, even if the play is EV- in terms of chips. Why? In spite of the EV being negative, there is a significant positive probability that we will win the hand and have a dominant position in the tournament as well as knock out some competitors. The outcome where we lose isn't so bad, since we already held a significant chip lead over our competitors.

Here is another simple example. Suppose a drawing will be held that will pay $1001. The tickets cost $1 each and there will be 1000 tickets sold. We are allowed to buy 1 ticket. Clearly the drawing is EV+. However, we may rationally decide not to play. Why? Because 99.9% we will lose $1. This is not captured in the expected value calculation.

Really, expected value is just a mathematical construct that is used to summarize a random distribution. Sometimes the summary isn't sufficient.

[LearnedfromTV]

This seems to be a convoluted way of saying that chips have different marginal utilities for different stack sizes. A player with 4000 chips looks at a 50/50 bet for 500 chips differently than a player with 1000 chips. Each receives no change in cEV from taking the bet but that isn't enough to know if it's a bet either should take. The other stack sizes, the blinds, the payout structure are relevant also. In fact, for a 0 cEV situation, those are the only relevant variables as to whether the bet is +$ev. At times, the non-cEV variables can make a -cEV move the right one or a +cEV move the wrong one. This seems straightforward/believable in general, but very complicated to actually work out in detail. It appears this is what Gigabet's thread is about.

One thought I have about why the extra chips that give you a deep stack are capable of generating more chips in the hands of a good player: there are significant constraints put on tournament players vs. say ring game with deep money. In a ring game one can make more plays, see more flops with more implied odds. Similarly, hands that aren't playable with 12xBB are with 30xBB, even when your opponents are shorter stacked than you, partly because you can exert FE pressure, partly because if you hit a set when a small-medium stack hits top pair, you can stack him because this is his best opportunity and he can't get away from it, and probably for a lot of other reasons I'd understand better if I were a better player.