Central limit theorem logic is behind the seemingly tautological statement of i choose the best EV choice because the sum of many different trials or a large enough ensemble of the same trial for many observers will have the larger expectation. In the very large N limit basically the resulting normal distribution looks like a pin function. Its very sharp around the average. _|_ rather than _/\_ . So the random result is so close to the average that the typical results of a better EV strategy are all better than almost all results of the other inferior local EV strategy. In the large N limit having selected the locally best EV option tends to give almost always a better result and this is clearly something we can all agree on now has value. There has to be special value at having a sum that is typically better than the alternative. And the term typically can be made astronomically certain basically rendering the possibility of a locally worse EV strategy leading now trivially unimportant. In the end you want to be basically ahead of the alternative and this is almost certainly ensured by the best EV local choice.
Additionally the probability for any spectacular result happening will also be larger for the higher EV ensemble eventually so the best EV strategy will end up leading in all kinds of human metrics of leadership even eliminating the essence of utility theory. In the end in the large N limit utility theory becomes irrelevant even if locally it wasnt! Thats why playing the lottery is wrong for the public for example and casinos or banks can be made to operate in such way that have unimportant risk of ruin . In the large N limit it is no longer a gamble it is a regular paycheck!

...Sweet Caroline!

There is no EV, or probability, that can be stated for any one-time event. Those concepts have no meaning whatsoever in that context.

If there are 100 people and 50 play a high EV situation every time while the other play a low EV decision everytime. Depending on the amount of decisions, there could easily be some people in the low EV group that make more money than the people in the high EV group but on average it is more likely that the high EV group will make more money. Even if the edge is very small. Could the guy that makes the worst decision every time make the most money? Yes, but he is the least likely to do so. I def so what the original poster is saying about how exact situations dont repeat all that many times in poker, but it doesnt matter at all. Maxaon, if you dont believe us, then lets flip coins where i get 2 to1 odds like Alberto mentioned.

EV is as real as real money. So if you are calling allin with naked flush draw you are buying 13$ (EV) for 17$ (cash). If you are buying lottery ticket you are paying 6$ (cash) for 3$ (EV) etc.
But this two kind of money are equally real!
Also if there are two ways to play a poker hand:
option 1) you can buy 50$ pot (EV) for 20$ (cash) (low variance)
option 2) you can buy 120$ pot (EV) for 40$ (cash) (high variance)
You should play like in option 2, not because you will win more in long term, but because EV is real money.

Hope this makes any sense

The question is not whether we like EV or not, the question is why we should like it for one-time events.

I thought this has been answered. The parallel universe explains it imo. Unless someone can refute it

Im not too sure what EV means to everyone else but to give u what i think of EV and why i think it is soo important to my game and a big component in my decision making at the tables is that its just poker but broken down differently then b4 with mostly numbers. For like mad years poker has been played by feeling people out making adjustments and knowing some basic odds and having a feel or instinct for the game after playing thousands of hands by a lot of great players imo, but now EV has sorta renamed the game and I think its actually literal. Its like just a title attached to the whys of poker mathmatically. Some ppl consider EV strictly math some ppl consider it a mix of things like psychology and philosophy and some ppl consider their EV based on feel. So really what EV is imo is just making the best decision in the moment with what we have for the long run or big picture no matter if u use a equation or feel to make it. Now although not any 2 situations are identical in poker, countless hands or situations are very similar which would show profit very fast cuz its not just working in one specific absolute spot but many that fall into a similar category in which EV (mathmatical) doesnt change all that much

WHAT? lol

EV means "Expected Value". It originates from maths and is a term that comes from statistics. It is used in poker because poker is a game, and statistics can be applied to game theory

rustybrooks is right by reading your comments it appears that you believe optimum strategy is not to make decisions based on the highest expected winnings. Thus, it seems you would prefer the lower ev option. That is not rational in any way. one is not able to predict the outcome of a future event when lacking data as in this case (as we don't know the next card). However, by playing the best long term option you will usally, but not always, make the best short term decision. To post an obvious example, if you call a preflop allin against kings and lose your play is still best, because you the 1:1 odds give you great money against your actual % to win

Making a play that is not the most +EV may be optimal if he is playing out of bankroll.

David S has a chapter about this in DUCY? I can't remember the exact chapter, but it has to do with betting your entire net worth on a single 50/50 trial with a 2:1 payout. He makes an argument that your current financial situation is more important than the EV and you shouldn't take the bet under certain situations. (He, for example, wouldn't take this bet himself if I remember the chapter correctly..)

There are numerous examples when a decision based only on EV is probably not wise but they almost always are based on utility. The gain is winning $X + $Y (the pot + call of your bet, for example) does not match the pain of losing $Y (the bet).

A typical poker example would be a satellite tournament where there are three players remaining and the top two get into the WSOP. You are the chip leader by a wide margin and hold AA. Your two opponents are all-in. You should fold even though it would be a +EV decision to play.

If the situation is not repeated, it depends on how much of your roll the bet is for and what you're actual odds of winning are. EV means nothing when the situation is not repeated.

For example, say you have $100,000 in life savings. You are offered a bet where you are asked to pick a number between 1 and 1 million. If you win, you will be paid
$200,000,000,000. If you lose, you owe $100,000. Expected Value says you should take the bet but actually taking it would be foolish.

The parallel universe is just another way of turning a one-time event into an event that happens an indefinite amount of times, which is basically evading the question. Secondly, these other parallel universes don't really exist, or at least do not interfere with my life in the current universe, so how do these parallel universes exactly influence my decision here?

They are imaginary (or real, it actually doesn't matter, because they can be seen as dependent universes but different ones).

The point I make is that 1 decision really is the long run.

In these parallel universes, we see that lots of different outcomes to the coin flip do actually occur, and so you can say that of the set of all parallel universes,( because as the number of universes tend to infinity, the probability of the coin landing heads is 50%), the probability that heads will be landed on in any randomly chosen universe is 50%.

Similarly, for a poker hand, if your EV for each decision is a%, b%, c% etc, then you can easily work out the total EV of a given when line given the bet sizes, and since EV is really just a linear combinator of %'s over some field, then we can deduce that if the %'s average out as the number of universes tend to infinity, then so does the EV (algebra of limits). Then, we can think of the EV of 1 decision as a linear combinator of %'s for one decision, and thus "%'s for one decision are the probability of winning the hand" implies "EV is the expect amount of winning in the hand".

The second blue writing is the same as the first

I tried in my long post to cover many cases of this problem even including utility theory concerns and risk management issues. Then i argued that if you see your life as a sequence of trials even if different and you took the approach to select the best EV each time rather than the worse or even a second best or whatever inferior but systematically did so or at least with some fixed probability to not select the best each time, you will end up due to central limit theorem with a normal distribution in the end that has as average the sum of averages and which has standard deviation that can be made negligible with respect to the difference of the 2 average sums (the one that is the result of always selecting the best EV and the one that is random or whatever systematic way negliecting the best EV). What this means is that in the the large n limit (of many different even trials) the resulting 2 sums differ by each other typically by margins that are much larger than the standard deviation of the sum of the best EV always. So practically it is true that the "best EV always" sum is tending to be 100% of the time better than all other strategies sums. This is why the best EV each trial even if it happens only once is the best approach.



However i will try now in doing something similar to give you another reason that is simpler using the famous Ancient Greek method of proof by contradiction ie reductio ad absurdum ( εις άτοπον απαγωγη ).


Take 2 distributions f1 and f2 . They can be discrete of not. We know that <f1> > <f2>
where <f>= EV of f= expected value of f.

As an example from poker it may be that we have to decide between going all in when facing a bet or just calling. One method may have EV of 1.5bb the other EV of 0.8bb for example . We want to know why choose 1.5 and not 0.8 . These 2 by the way may have different standard deviations . One may be very large compared to the other for some cases. I will not get specific here.


In any case evf1>evf2 without regard for all other than simply to state the standard deviations are both finite.

Lets say that it is to our advantage to select f2 with inferior EV rather than f1 for that one trial.


Ok if that is true it is going to be always true. So if we did it once and then faced with identical trial later we would have no reason to change and claim its better now to do something else. The same reasons that lead us to choose f2 in that initial trial should remain valid and we must choose f2 again.

If it is true that it is to my best interest that one time or the next one to select f2 it will remain true for all future trials as well. So in principle i can construct a sequence of trials all of which select f2 . I can also select an identical sequence where someone else selects always f1 just for comparison purposes.

So obviously i expect after many trials n>>1 to have a sequence that behaves better than the alternative that selects always f1. Well by a well known result from central limit theorem i know that in the large n limit the f2 strategy has expected value evf2*n and standard deviation n^(1/2)*s2. Similarly the f1 strategy has expected value evf1*n and standard deviation n^(1/2)*s1. The difference of the 2 is also a distribution that has average value n*(evf1-evf2) and standard deviation n^(1/2)(s1^2+s2^2)^(1/2) . In the n->infinity limit the fraction (or %) of these differences that is larger than 0 tends to 100%. So in the large n limit we find that strategy 1 is "always" better than strategy 2 . But by the original assumption that it was to our best interest to select f2 for that one trial we had deduced that selecting always 2 would lead to better sum overall too.

That is opposite to the above conclusion and therefore the original assumption that it is to our best interest to not choose the best EV f1 is wrong.

Thats why we must always choose the best EV (risk issues and utility issues aside) when we have a single trial.

Of course to most people this is trivially obvious but true it still requires a proof like this is.

this is basically what I said. Your "trials" are equivalent to my parallel universes [bijection]

Yes but you realize that the original poster wasnt asking this. Also you know what proof is , right? Until one is given nothing can be accepted as truth however intuitive. He was asking for a proof of why in trials that happen only once it is still to our advantage to select best EV moves. I never had to use mutiple trials for the observer in question. I constructed trials others do based on his belief that its ok to choose a sub optimum EV choice. This is very different in that i am totally free to do whatever i want now. I am not forcing anything on him , i am just using his convictions to arrive at false statement.

Additionally you still do not prove anything by saying that if you do it many times elsewhere you converge to the EV. This is tautological. He knows the average is the maximum when you select the best EV by definition. He doesnt connect other than intuitively why average maximization equals best choice for the one time. It is a legitimate question although almost nobody would question such selections ,they are still not based on a proof until you deliver one.

My answer used in both occasions central limit theorem which is a totally different result than basically saying the average converges to EV after many trials. That alone is not enough. How it converges matters too. It is the normal properly of the theorem that is of value here in making claims of practical value.

The practical aspect of this is that if you had different trials forever you still create a superior sum by selecting always best EV and it has the property that it deviates from other sums in ways that grow stronger than the standard deviation of the differences making the dominating nature of such choice convincing fast . Still that was not a rigorous result yet until the introduction of the last post only because anyone can come and tell me that this guy plays only one game in his life and doesnt have a sequence of choices so my claim for convergence per central limit thorem never materializes. I evade all this by forcing the convergence on others that simply use as basis his claim for that one time. And there i definitely have the freedom to go to infinity if i wanted.

How about when you play with a huge fish that you are sure you wont ever see again, whom is such a spewtard. Do you still push the small edges or do you prefer to milk the bigger ones?

You introduce a utility theory issue here. If you are not going to see him again but you will see others like him even only 1 times each and you play poker long enough and the bankroll you have is huge and you do not have such issues in mind as tilt etc you go for it even with small margins.

But if your bankroll is sensitive to fluctuations and you are a very good player in that table you probably give it up because you will likely allow this idiot to give you a better one time chance than if you took the risk and he won and left. You keep them happy at the table for defeating you a bit and maybe you get them later. If you think others may get him first that is fine too. Clearly if others are willing to open up to risk they become targets of yours too if you land a great spot. So let the fish open up the table for you and if he leaves so be it. The expected profit by not engaging him marginally if he has to leave after that is probably superior to the edge you give up because of what you allow to possibly happen in the future . This is no proof though only an intuitive guess.

Jewbison you cannot pull infinite Universes out of your ass and use them as proof. We have no evidence of their existence nor do they affect us.

Masque I think you may have answered the question but I have trouble understanding some of your posts. What we're trying to do here is show that even if we play each game only once we will be playing many different games and choosing max EV for all of them will be a good strategy. Let Xn be the random variable representing the game we play. Our total utility over j games is U(j) = X_0 + X_1...+ X_j. We need to show that lim U(j) j-> inf = EV(X_0 .... X_j). I believe the central limit theorem shows that U(j) will have a normal distribution with an average of the total EV. So what you're saying that as the number of trials increases the sigma of the distribution decreases and therefore the sum of the random variables converges to the EV?

Ok, believe what you want, but it's almost infinite universes, not infinite universes, fyi

I read up to here in your long post, which is long btw. I suggest you use the phrase "without loss of generality" more, instead of explaining absolutely everything...


Basically, I think that you are assuming too much when trying to get from:

to

.

However, my "almost infinite" universe explanation tries to cover this step, which is the main step, imo. But apparently I'm literally talking out my arse

And add the fact that he will leave the table immediately if he takes a stack off you.