Monday, July 16, 2007

Poker is a Game of Chance

Bean-counters from the tax department state this. It has to be correct, since the Law says so (the Dutch Supreme Court ruled it in 1998.) Many other people also agree with this, like for instance the group of people that buy state lottery tickets and have an estimated IQ of over 120. Those with a lower IQ would probably call it a game of luck instead.

“Poker is a game of incomplete information.”

Interesting statement, or at least, it must be, since it’s been written in every article on poker which is over one page long.

Let’s see if any of both of these statements are true, by taking this single hand as example.

Remember though, this is not about the actual hand, it's about the methods used to analyze other factors in a poker game than basic odds and hand strength. Feel free to let me hear your thoughts on this.

I also posted this on Internet Texas Holdem, click link to see the original post:

Available information:
I only played a few hands at this table, about 1 orbit, so I don’t have any reads yet on the other players.
I have slightly over a full stack, and the big stack is sitting two seats to my left, the small stack two to my right. Lousy table selection by me.

Full Tilt Poker - No Limit Hold'em Cash Game - $0.50/$1 Blinds - 6 Players - (LegoPoker HH Converter)

Soultwister (SB): $103.90
BB: $180.40
UTG: $81.70
MP: $100.00
CO: $58.50
BTN: $100.00

Preflop: Soultwister is dealt
Ace of Clubs Queen of Clubs (6 Players)
UTG calls $1.00, MP raises to $5.00, 2 folds,

What happened so far? UTG limped, which is a weak play in 6-max in my eyes. A player behind him raised, which doesn’t say much about his hand. He could be raising here for value with a strong hand, or just raising a decent hand to isolate against the limper.

The size of the current pot tells me I should fold or reraise, getting 1.66:1 pot odds out of position. But if I call, the big blind is getting good pot odds so may call with any two cards, and the player UTG will most likely call for the same reason. So my estimated pot odds of calling here would be between 2.66:1 and 3.44:1. Good enough for a call. Folding here isn’t bad either. Reraising is just too risky without more information, or at least would be a high variance play OOP without info. Not my style.

Soultwister calls $4.50, BB calls $4.00, UTG calls $4.00

Flop: ($20)
King of Spades Jack of Clubs 4 of Clubs (4 Players)
Soultwister bets $12.00, BB raises to $34.00, 2 folds, Soultwister raises all-in to $98.90,

Very nice flop for me, even while I just have ace high now. How many outs do I have to hit the stone-cold nuts if that is what I need to win? 9 clubs and 3 10’s, so 12 outs to the nuts. That gives me about 45% chance to hit the nut flush or nut straight by the river.

I can check here and decide what my best options are when the action gets back to me, most likely checkraising. I would have chosen that option if UTG had been the initial raiser, since that increases the chance to get more players/money into the pot. It’s unlikely that this flop will get checked through, so my relative position is great right here.

The other option is betting out small into the PF raiser. This gives UTG good odds to tag along if he has any broadway cards and add some more money into the pot, while MP may interpret this bet as an informational bet, semi-bluff, value bet or pot sweetener. And he’ll be correct.

After my donkbet the BB reraised me, which generally means he must have a very strong hand, since he doesn’t seem to be afraid of preflop raiser.

I made a mistake here, I figured my 12 outs and current pot size would probably give me good odds to be in a profitable all-in situation, so put my chips in without realizing that I was already getting 3:1 pot odds. With 12 outs, I only need to get 2.92:1 to make my hand on the turn. So there was no need to make a move here at all.

When I go all-in and get called, the total pot will be $217.8 while I invest $86,9 here. Which is 40% of the total pot, while my outs tell me I’m winning 45% of the time. So with that information readily available, I’m making money here?

If I just look at my outs I win (0.45*217.8 )-86.9=$11.11. Folding earns me $0. So with that info I’d be losing $11.11 by folding compared to pushing all-in if villain never folds.

Pot odds can be deceptive though since there are more hidden factors. Is this situation really as profitable as it seems?

First, the rake was not substracted from the total pot. The rake is $3 maximum on full tilt at this limit, I get about a third back thanks to rakeback, so $2 total rake which lowers my winnings from earlier calculation to (0.45 * 215.8 )-86,9=$10.21. Small difference in large pots, but can have quite some impact on small pots where the odds seem marginally +EV.

Second, I won’t be winning 45% of the time at all. Against villains hand range, his sets/two pairs could improve to a full house or quads. But how likely is that to happen?
cards win %win lose %lose tie %tie EV
Ac Qc 325 32.83 665 67.17 0 0.00 0.328
Kc Kd 665 67.17 325 32.83 0 0.00 0.672

EV: -$16.11

cards win %win lose %lose tie %tie EV
Ac Qc 423 42.73 558 56.36 9 0.91 0.432
Kc Ad 558 56.36 423 42.73 9 0.91 0.568

EV: $6.33

cards win %win lose %lose tie %tie EV
Ac Qc 409 41.31 581 58.69 0 0.00 0.413
Js Kc 581 58.69 409 41.31 0 0.00 0.587

EV: $2.23

There’s always a chance here that villain is making a move here with a draw or bluff, actually making me a favourite, but to calculate my odds I’m going to keep it simple and not take other hands into consideration, so let’s say he has at least top pair, two pair, or a set, and has one of these hands: JJ, AK, 44 or KJ. AA/KK look quite unlikely here because of his preflop play. And if I’m already making a worst case scenario, let’s scrap KQ/K10 from his range as well.

With the pot size I needed to win 40% of the time here to make this move profitable. There are 6 possible combinations for him to have a set if I take KK out of his hand range, 6 for the other hands mentioned. If each combination is just as likely that leads to (32.8+32.8+43.2+41.3)/4= 37,5% chance to win here.

So instead of earning $11.11 here like the first rough estimation I’ll be losing $5.98 here if I’m called all the time.

Does that make pushing all-in a –EV move? That depends on how often he needs to fold to make this profitable. If I move all-in here, I bet $86.9 (1.32 times pot) to win $64 (current pot – rake) if he folds. He will be getting (152.9/64.9) = 1 in 2.36 to call. What hands could he possibly fold here? Perhaps AK or KJ.

Time for some real math using the Fimbulwinter-method.

A=Current pot size – rake
X=Villain’s folding percentage / 100.
B= Winning chance
C= Total pot if called – rake
D= Your bet/raise

EV = Ax + (-x ) ((B*C )-D )

EV = 64x +(-x)((.37,5 )(215.8 )-86.9 )
EV= 64x +(-x )(-5.98 )
-x(-5.98 )=5.98x
EV = 64x +5.98x -5.98x = 0
5.98 = 64x +5.98x
X = 5.98/69.98 = 0.085

He only has to fold 8.5% of the time here to make this profitable. A random villain would lay down his hand more often than that for his remaining stack. So without trying to guess how often villain is actually reraising me here with a hand like KQ, K10, Q10, aces or a pair of deuces, this move is +EV right away.

BB calls $64.90

Turn: ($217.80)
Ace of Diamonds (2 Players - 1 All-In)

River: ($217.80)
2 of Spades (2 Players - 1 All-In)

Pot Size: $217.80 ($3 Rake)

BB had
King of Clubs Jack of Diamonds (two pair, Kings and Jacks) and WON (+$110.90 )
Soultwister had
Ace of Clubs Queen of Clubs (a pair of Aces) and LOST (-$103.90)

So is there any chance or luck involved? Short term indeed, but who cares what cards show up after you put your money in, poker is all about long term results. And in the long term, luck is a neglectable factor. And the chances are known.

Incomplete information? With the information available in this hand I could make no mistake that would make me lose more money than folding, even when having the worst hand
. I should have just called on the flop against an unknown, and reraise against some regulars when I know more about their post-flop tendencies. More information would just allow me to make slightly better estimates of expected value.

So what IS poker? I think it's just a game in which you have to minimize your mistakes and try to induce your opponents to make their mistakes larger. If you just do that, and also keep taking the dead money on the tables often enough, it should be a profitable game. And dead money on poker tables is quite common if you realize we live in a world where people who are considered to have an over-average intelligence buy state lottery tickets.

Yours truly,

"Keep your friends close, and your enemies to your right."


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