Luck adjusted winnings in SNGs

[Parket]

In that particular instance when it went to 3 flips, yes my $EV was $1.50. But in many tournaments my $EV would be just $0.50.

Your $EV IF it went to 3 flips is $2.50, not $1.50. By assuming it goes to 3 flips, it implies you only look at the subpopulation that has HHx, and obviously you then have 50% to win $2 and 50% to win $3.

I just realized I counted these incorrectly. The correct calculation is:
In HHH my $EV is $1.5, HHT my $EV is $1.5, HT my $EV is $1 and T my $EV is $0.5.
This is because my EV is increased by $0.5 according to how many flips I do. This means my EV is 125*$1.5+125*$1.5+250*$1+500*$0.5=$875. So my net won per game is -$0.125.

Interesting way of computing the $EV. Normally, $EV of a game is simply the sum of probability*outcome or 0.5*0 + 0.25*1 + 0.125*2 + 0.125*3 = $0.875. I'm still wondering why both calculations lead to the same result. I believe you're trying to prove here that by entering the HEM $EV values in your formula, you come to the correct conclusion.

Now do you understand why I am correct?

No.
What I still don't see is how apparently the actual outcome of the game (i.e. one particular sample) can change the $EV of the game as a whole. E.g. in this same game, suppose you immediately tossed Tails and won $0. Then according to you (or HEM) the $EV of this game is $0.5. But this doesn't make sense to me, because it completely ignores the possible winnings of future winnings if I had tossed Heads first.
You stated earlier that my mistake is that I overweigh earlier decisions/branches and that order should not be important. My belief is actually that you/HEM make the mistake of modifying the weight of later possible outcomes by a) overvaluating them when they actually occurred, or b) ignoring them completely when an earlier outcome caused it never to happen.
The $EV of the game as a whole should be a constant, regardless of what your actual outcome is when playing 1 sample of the game.

What I'm really interested in by comparing my result with the $EV is whether I ran below or above expectation (like you say, this is essential in hyper HU SNGs) of the game as a whole. Now, suppose you run the above game 100 times and for some reason it comes up 100 times HT, so you win 100x $1. Now, according to HEM the $EV will also be $100 so I'm running right at $EV. THe reality however is that I was running hot, because the $EV of the game is $0.875 and on average I would win $87.5, not $100 !!!

[Parket]

Discussion can be put to rest at last.

I wished I had posted this on 2+2 upfront, because they were immediately able to show the issue. Unlike what you're trying to tell me, the algorithm is not incorrect per se (let alone an abomination). However, the problem with it, is that it is purely a measure for your all-in luck. The HEM approach is not perfect either (already illustrated by the problem with $EV getting less than -1BI or more than 1st prize) but it is a better measure of your EDGE against your opponent(s), i.e. what your expectation is in any tournament.

So, if you just wanted to see how (un)lucky you were, my algorithm accurately depicts this. If you are more interested in what your general expectation is against your opponent(s), then the HEM approach is better. I'll be the first to admit that the latter is the most interesting one for a poker player.

[Zangeeph]

In that particular instance when it went to 3 flips, yes my $EV was $1.50. But in many tournaments my $EV would be just $0.50.

Your $EV IF it went to 3 flips is $2.50, not $1.50. By assuming it goes to 3 flips, it implies you only look at the subpopulation that has HHx, and obviously you then have 50% to win $2 and 50% to win $3.

Well we already know the outcome. It's really not $2.50, it's $1.50. I suggest you post on the probability section of 2+2, there are some smart people over there. It's true that if someone said to you "I played the game and got 3 flips, guess how much I won" you'd estimate $2.50 ($1.50 would be impossible).

Here's a simple probability question for you: I flipped a coin twice, at least one of the flips landed on heads. What is the probability that both flips were heads? (hint: it is not 1/2)

Now do you understand why I am correct?

No.
What I still don't see is how apparently the actual outcome of the game (i.e. one particular sample) can change the $EV of the game as a whole. E.g. in this same game, suppose you immediately tossed Tails and won $0. Then according to you (or HEM) the $EV of this game is $0.5. But this doesn't make sense to me, because it completely ignores the possible winnings of future winnings if I had tossed Heads first.[/quote]
Just think about it in terms of the number of flips - forget about the game. Per flip you make $0.50, so we multiply the number of flips by $0.50. It's that simple.

The $EV of the game as a whole should be a constant, regardless of what your actual outcome is when playing 1 sample of the game.

Yes, but we don't have the complete strategies of both players. So it's impossible to know the $EV of a tournament.

What I'm really interested in by comparing my result with the $EV is whether I ran below or above expectation (like you say, this is essential in hyper HU SNGs) of the game as a whole. Now, suppose you run the above game 100 times and for some reason it comes up 100 times HT, so you win 100x $1. Now, according to HEM the $EV will also be $100 so I'm running right at $EV. THe reality however is that I was running hot, because the $EV of the game is $0.875 and on average I would win $87.5, not $100 !!!

That's right, it isn't perfect. But it's the best we've got. However, the fact is that HEM will (on average) converge to your true ROI faster than any other method we know of, excluding solving the entire game (which is what you've done there - in poker you can't know the exact $EV of the game).

I hope the developers get it right the next time around. They need to do the calculation for each player in every single hand and store it in the database (so that it doesn't need to calculate it all every time you start up PT4). This includes hands where there were no all ins.

[Parket]

Here's a simple probability question for you: I flipped a coin twice, at least one of the flips landed on heads. What is the probability that both flips were heads? (hint: it is not 1/2)

I'm familiar with that one (as well as the Monty Hall problem and a whole slew of other interesting probability puzzles) and I did have a course on probability theory in grad school. Don't see how it's related to the above, unless you somehow wanted to challenge me... :p

[_dave_]

So I take it the "proper" method did not make it in to the presumably-final-before-public-beta release?

just updated and at a quick glance today's results do not match HEM1

[kraada]

Make sure to re-run auto detect on your tournaments - that will reset the values for you (and all tournaments going forward will use the new method).

[_dave_]

Make sure to re-run auto detect on your tournaments - that will reset the values for you (and all tournaments going forward will use the new method).

I did indeed run the auto-update (took quite some time on only ~300 husng!), and even so the discrepancies noticed were from games played + imported only yesterday, after new version install and all old histories updated.

So I take it the new method is in and supposed to be working? I will compare again after today's games and try and find the problems in that case, otherwise I wouldn't bother if the results are not yet expected to match HEM1.

BTW the equity calcs are still some way off e.g. A7o vs KJo, inaccurate small-sample monte carlo still looks to be in use at a guess.

[WhiteRider]

I will make sure the development team see your posts, but I'm not sure the results would be expected to match HEM exactly - that is not what we're aiming for.

[Yogi Rob]

_dave_,

I've tested results with thousands of HU tourneys and the results were very similar to HEM. The only difference should be Monte Carlo imprecision due to pre-flop allins which you've noted. If you're seeing otherwise, please submit some hand histories so I can analyze this.

For multi player SNGs there are a lot more variables, and a lot of hand situations where our algorithm or theirs may disqualify it from EV calcs which would cause many more discrepancies.

[Zangeeph]

_dave_,

I've tested results with thousands of HU tourneys and the results were very similar to HEM. The only difference should be Monte Carlo imprecision due to pre-flop allins which you've noted. If you're seeing otherwise, please submit some hand histories so I can analyze this.

Why use Monte Carlo simulations? In the worse case scenario there would be two opponents all in pre flop (including hero) so that there are 48 unknown cards. With 5 cards to come, that's 48 choose 5 = 1,712,304 possible outcomes. That's a small number especially when there are open source hand evaluators than can do 142,779,680 hands per second. So perhaps you should be enumerating all possibilities. Plus, there's no random number generation so the hands per second would be greater than when using monte carlo. How many simulations are you using for the monte carlo calculations? If it's more than 1,712,304 I will be disappointed.

Monte carlo should only really be used preflop when it's range vs range - having a large range of hands dramatically increases the number of possible outcomes. In PT4, there are absolutely no range vs range calculations when it comes to calculating EV or luck adjusted winnings.

For multi player SNGs there are a lot more variables, and a lot of hand situations where our algorithm or theirs may disqualify it from EV calcs which would cause many more discrepancies

There should be no difference in the result of the calculation. The algorithms should be calculating the same thing. Just as I'd expect my bb/100 to be the same in HEM as in PT4, my luck adjusted winnings should be the same.

Coming up with a novel way to do luck adjusted calculations is certainly not trivial. You would need to be an incredibly good mathematician to do so. If you really do believe that you have a superior algorithm, please do publish your findings. It's really quite necessary that you do this - it would be a worthless statistic without knowing its exact calculations. You can see that HEMs luck adjusted line comes under a lot of scrutiny (as shown by the 1000+ post topic on 2+2), don't expect any less scrutiny for your method.