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#1
10-30-2006, 09:20 AM
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Basic Theory: -Expected Value-
edited to fix glaring typo in 2nd example and to clear up AA v KK example - edits are in italics (matrix128)
Expected Value - is commonly referrred to as EV. from here on in positive Expected Value is +EV and negative Expected Value is -EV. Poker is a game in which skill will beat luck every time assuming that you play for long enough. While it's true that any two cards preflop can win any given individual hand and that luck is a large part of this game if you hold any aspirations whatsoever to beat Poker overany significant amount of time/hands you must learn to make +EV plays and not make -EV plays. EV is simply what you expect to make on average with any particular play. here is a simple example Hero(100BB) has A[img]/images/graemlins/spade.gif[/img] A[img]/images/graemlins/heart.gif[/img] and raises preflop to 4xBB from the CO. Villain(100BB) calls from the BB and both see a HU flop of 9[img]/images/graemlins/club.gif[/img] 3[img]/images/graemlins/diamond.gif[/img] 6[img]/images/graemlins/heart.gif[/img] Villain tells us he has black Kings (he's not lying) and then raises all-in and Hero calls. Villain tables K[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/club.gif[/img] (disregarding how good the play is in this hand) what is the EV of calling knowing we are against specifically K[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/club.gif[/img] ?) If we punch those numbers into Pokerstove we get this output.. Board: 9c 3d 6h Dead: equity (%) win (%) tie (%) Hand 1: 08.3838 % 08.38% 00.00% { KcKs } Hand 2: 91.6162 % 91.62% 00.00% { AcAs } we can see here that if this hand goes to showdown (as it is going to) that Hero will win on average ~92% of the time. so if we run this hand 100 times Hero ought to expect win 92 times and lose 8 times. there are ~200BB at stake so Hero wins 18400BB the 92 times his AA holds up - and loses 1600BB the 8 times he loses the hand. Total net win of 168BB/hand. This play is +EV and has an EV of 168BB *every* time you make it. It's important to note that EV and actual results can vary massivley over any short term period. e.g. if we actually ran the hand above 100 times you might win all 100 times - does this mean the EV has changed? or you might be unlucky and lose 25 times in 100 - does this mean the play is now less EV? - no EV remains 168BB per hand. Everytime you make this play you "earn" 168BB and the more times you repeat this the closer your actual real results will get to the "perfect average" of winning 92% of the time. Once you have played enough hands (an infinite amount) your total actual results will equal the sum of all of the total EV of the plays you have made. The closer your total number of hands gets to infinity the closer your actual results will get to this theoretical figure. So in theory every time you make a -EV play and get chips in when you are an underdog you a "losing money" regardless of the actual results of the hand - and conversely everytime you get chips in when you are a favourite in a hand you are winning money. If you added up all the "Sklansky Bucks" (theoretical EV money) you made in the long run and compared this amount to your actual winrate - after playing an infinite amount of hands these two numers will be identical - and the more hands you play the closer these two numbers will get to each other. Lets look at a more complicated example, in our simple example above we knew villains exact hand before calling so we don't have to put him on a range (which affects the EV of our play) in practice we never know what particular hand we are against when we make our decisions. This is a real hand from my database. Poker Stars No Limit Holdem Ring game Blinds: $0.10/$0.25 6 players Stack sizes: UTG: $27.85 UTG+1: $24.65 CO: $28.95 Button: $23.95 Hero: $25.15 BB: $27.80 Pre-flop: (6 players) Hero is SB with 2[img]/images/graemlins/spade.gif[/img] 2[img]/images/graemlins/diamond.gif[/img] UTG calls, 2 folds, Button calls, Hero calls, BB checks. Flop: J[img]/images/graemlins/club.gif[/img] 2[img]/images/graemlins/heart.gif[/img] 5[img]/images/graemlins/heart.gif[/img] ($1, 4 players) <font color="#cc0000">Hero bets $1</font>, <font color="#cc0000">BB raises to $3</font>, UTG folds, Button calls, <font color="#cc0000">Hero raises to $8</font>, <font color="#cc0000">BB raises all-in $24.9</font>,Button folds, Hero calls. Turn: 9[img]/images/graemlins/heart.gif[/img] ($53.8, 1 player + 1 all-in - Main pot: $53.8) River: 9[img]/images/graemlins/spade.gif[/img] ($53.8, 1 player + 1 all-in - Main pot: $53.8) Results: Final pot: $53.8 - it's the flop action I am interested in here. In real life we don't know what sepcific hand we are facing at the point in time where we make a decision. What hand does BB have here? is my hand strong enough to call his all-in? and how do we work out the EV of this play?? The answer is to put BB on a range of hands - if we re-run this hand 1000 times say sometimes he has AA and we are a huge favourite, sometimes he has 55 and we are a huge underdog, he might also have JJ-KK, AJ,KJ,J2,52,J5, Ax[img]/images/graemlins/heart.gif[/img], or he might be bluffing. In this particular case his range is wide because there was no preflop raise. Also we are not saying htat he will always play every hand in this range exactly this way - but that he isn't playing any other hand apart from the ones in this range in this fashion. Against most of these hands I am a favourite, and against some of them I am an underdog. I have no way of knowing what hand he has and certainly don't have time at the table to put the numbers into Pokerstove so we just make an educated guess. I play using the general rule that I should never fold a flopped set for ~100BB. The reason being that no matter the flop if we can get all the money in on the flop we are almost always a favourite to win the hand at the showdown vs our opponents range of hands. So I happily call his all-in. But have I made a +EV play and will this earn me money in the long run??? Lets put his range and my hand into pokerstove and see... Board: Jc 2h 5h Dead: equity (%) win (%) tie (%) Hand 1: 78.7155 % 78.72% 00.00% { 2d2s } Hand 2: 21.2845 % 21.28% 00.00% { JJ+, 55, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o } (T7ss is included in this range to represent a bluff) and the numbers say that on this wide range of hands my play is +EV and that calling his all-in here means that vs that range I expect to win ~79% of the time. The actual results don't matter, as long as my range is accurate, and what cards come on the Turn or on the River don't matter either (as the decision is already made by then) if I make this play everytime it is +EV and in the long run I expect to win ~170BB everytime I make this play. As this play costs me 100BB to make I make a profit everytime here of 70BB, whether BB shows me JJ for top set or A [img]/images/graemlins/heart.gif[/img] 8 [img]/images/graemlins/heart.gif[/img] for a busted flush draw I still "gain" ~70BB everytime I make the play. Whenever you determine at the table that a play is +EV you should make it EVERY time. If you don't you are losing money in the long run. Do You See Why? Ultimately it is EV that will decide what your true winrate is, you can't beat it, or get around it in the long run eventually your total real results will match your expected results. Closely tied in with EV is variance - a lot of people misunderstand what variance is and try to avoid it. But you shouldn't. The very very best players at poker don't care about variance and try to make every single +EV play that they can (this is the main reason why they are such big winners) Variance is simply how much your actual results can vary from the statistical EV results in the short term. It's the reason that a 20x buyin roll is recommended. So that you don't go broke in the short term making +EV plays that you lose in the short term because the real results vary from the Expected results. Variance is neither good or bad - and the bigger bankroll you have to absorb variance the more you ought to be willing to risk on a marginal +EV play. Lets say you determine that a play is +EV and you'll win 51% of the time, the more money you stake on this play the more you stand to win in the long run. 51% of 200BB is more than 51% of 20BB - though in the short term real results will vary lots and you stand a great chance of losing this particular bet if you can afford it (have a large enough bankroll) you should bet as much as you can on this 51% shot. As a final thought here is an exercise you can try when you next get a big losing session. Review all the hands in the session and for each hand you play work out a range of hands for each villain, run the numbers into pokerstove and see how much you made in EV. I do this sometimes and often find out that I had a +EV session that in real results lost me lots of real money. If most of the losiung sessions you have are +EV you are paying well and eventually real results will catch up with your EV results and you will be a long term winner, so despite losing now in the short term you can be happy that in the long run you're still winning [img]/images/graemlins/smile.gif[/img] 
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#5
10-30-2006, 11:55 AM
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Re: Basic Theory: -Expected Value-
[ QUOTE ]
Total net win of 168BB/hand. This play is +EV and has an EV of 168BB *every* time you make it. [/ QUOTE ] Two out of three of these statements are wrong. Do You See Why? Hint #1: this play is +EV. Hint #2: [ QUOTE ] Villain(100BB) calls from the BB and both see a HU flop... [/ QUOTE ]
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#7
10-30-2006, 12:13 PM
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Re: Basic Theory: -Expected Value-
[ QUOTE ]
Board: 9c 3d 6h Dead: equity (%) win (%) tie (%) Hand 1: 51.9765 % 51.73% 00.25% { 2d2s } Hand 2: 48.0235 % 47.78% 00.25% { JJ+, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o } (T7ss is included in this range to represent a bluff) [/ QUOTE ]I guess this should be:[ QUOTE ] Board: Jc 2h 5h Dead: equity (%) win (%) tie (%) Hand 1: 51.9765 % 51.73% 00.25% { 2d2s } Hand 2: 48.0235 % 47.78% 00.25% { 55, JJ+, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o } (T7ss is included in this range to represent a bluff) [/ QUOTE ] Great post btw [img]/images/graemlins/smile.gif[/img]
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#8
10-30-2006, 12:16 PM
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Re: Basic Theory: -Expected Value-
[ QUOTE ]
[ QUOTE ] Total net win of 168BB/hand. This play is +EV and has an EV of 168BB *every* time you make it. [/ QUOTE ] Two out of three of these statements are wrong. Do You See Why? Hint #1: this play is +EV. Hint #2: [ QUOTE ] Villain(100BB) calls from the BB and both see a HU flop... [/ QUOTE ] [/ QUOTE ] it'd be so much better if you just point out what statements are wrong and then I can correct them/you as appropriate [img]/images/graemlins/smile.gif[/img] in my simple example numbers aren't exact, and re-reading it now I could/should have written it better to stop people nit picking at semantics...
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#9
10-30-2006, 12:18 PM
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Re: Basic Theory: -Expected Value-
[ QUOTE ]
Flop: J[img]/images/graemlins/club.gif[/img] 2[img]/images/graemlins/heart.gif[/img] 5[img]/images/graemlins/heart.gif[/img] ($1, 4 players) <font color="#cc0000">Hero bets $1</font>, <font color="#cc0000">BB raises to $3</font>, UTG folds, Button calls, <font color="#cc0000">Hero raises to $8</font>, <font color="#cc0000">BB raises all-in $24.9</font>,Button folds, Hero calls. [...] Lets put his range and my hand into pokerstove and see... Board: 9c 3d 6h Dead: equity (%) win (%) tie (%) Hand 1: 51.9765 % 51.73% 00.25% { 2d2s } Hand 2: 48.0235 % 47.78% 00.25% { JJ+, AhKh, AJs, J5s, J2s, Ts7s, 52s, AJo, J5o, J2o, 52o } (T7ss is included in this range to represent a bluff) [/ QUOTE ] Shouldn't this board supposed to be Jc 2h 5h? Nice post anyway, thanks. Best wishes.
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