Mistakes and the Fundamental Theorem

[Stephen Keller]

I am reading through Theory of Poker for the first time, encouraged by lurking on this board to really understand the fundamentals and math of the game. One thing I am having a bit of trouble with is what Sklansky means by other players making mistakes according to the Fundamental Theorem and how you always gain by this.

I understand why it usually benefits you, but there is one instance in which I am a little confused. In TOP, he gives an example in the chapter about TFOP (example six, p. 23) where you want your opponent to call your two pair with a four-flush when he is getting only 3-to-1 odds on his draw with one card to come. However, you'd want your opponent to fold when he is getting 5-to-1.

I understand why drawing to the flush is a bad idea with only 3-to-1 odds (ignoring implied odds), and so I see how the first case is a mistake for my opponent. But how does it benefit me when he calls in the first case but harm me when he calls in the second?

After all, my odds of winning the hand if he calls are the same regardless of the size of the pot. Shouldn't I always want him to fold, thus not giving him a chance to beat me?

I know that I am not quite getting a simple concept. I can sort of see how this works, but I'm a little unclear. The only thing I can think of is that the situation is analgous to the following example: since I am ahead in the hand, right now the pot is "mine" since I would win if we didn't draw more cards.

Thus his calling my bet is similar to my offering him a bet outside of poker. If I were to offer him 3:1 odds on a roulette wheel with 5 spaces (a 4:1 shot), it would be the same as his drawing a flush with only 3:1 odds. Likewise, I would not want to give him 5:1 odds on the same roulette game. Does this example correspond to the poker scenario?

The only reason I am hesitant to really accept the fact that the two situations are the same is because it seems like I am coming out ahead in the long run even if he calls with 5:1 pot odds. If there is $50 in the pot and a $10 bet to him and we play the hand over and over, he'll win $60 one out of five times, spending $50 to win $60, for a profit of $10. I, however, will win the $60 pot four out of five times, for a total of $240, though I spent $10 each time (giving me a profit of $190).

It seems like his call there is profitable for the both of us, while I know that giving him 5:1 odds on my roulette example has a negative expectation for me. This tells me that I'm thinking of this situation in the wrong way still. Any help would be appreciated.

[Cyrus]

"In TOP, [Sklansky] gives an example in the chapter about TFOP (example six, p. 23) [actually p.22] where you want your opponent to call your two pair with a four-flush when he is getting only 3-to-1 odds on his draw with one card to come. However, you'd want your opponent to fold when he is getting 5-to-1.

I understand why drawing to the flush is a bad idea with only 3-to-1 odds (ignoring implied odds), and so I see how the first case is a mistake for my opponent. But how does it benefit me when he calls in the first case but harm me when he calls in the second?"

When your opponent calls and his drawing odds (say 3:1 against) are more than compensated by the pot odds he's getting (say 5:1), he is having the best of it. As you acknowledge somewhere in your post, if you two play this hand many times over, your opponent gains and you lose.

If your opponent knows what you have and, thus, can calculate precisely what are his odds, he will make the call -- and will have the best of it. (We are assuming he's a rational person.) This is why it is imperative for you to fool him about what your hand actually is and make him fold. This is what the FTOP is about.

It's true that your opponent's call gives him a chance to beat you. Even though that chance is small, it is always higher than zero (since you don't have the nuts). But, even so, you WANT your opponent to be calling when he has the worst of it : when he calls by putting in an additional $10 bet, and you have an edge of say 35%, he is virtually handing over to you 35% of that $10 bet. Your EV in the scenario whereby he's calling is thus higher (by +$3.50) compared to the scenario whereby he just folds.

--Cyrus

[Stephen Keller]

Could I say, then, that while my EV is still positive whenever my opponent is on a draw against me in this situation, that my EV is higher when the pot doesn't favor my opponent's call? If that's so, I think I'm understanding it, then.

It's not as if it's ever unprofitable for me when I hold two pair against my opponent's four-flush, but in a smaller pot my EV is higher when he calls compared to a large pot because the amount I stand to lose if he makes his draw is proportionately less comapred to the amount he has to call.

Does this make sense?

[Cyrus]

The pot is $20. If you bet another $10 and your opponent folds his hand, that bet wins you the $20 pot that you were contesting. (And your opponent would be correct to fold if he knew that he is a 5:1 dog to win the pot.)

If you had fooled him instead into calling your $10 bet, then that bet would win you the $30 which you are now contesting 5/6ths of the time and a loss of that $10 bet 1/6th of the time. That's an EV on the round of abt $23.34.

[Mike]

If I understand your first question correctly and the sceond scenario's intent(s) as you wrote it:

Your opponent hurts you because he is getting proper odds to draw to a flush, eg, 5:1. Your odds to draw to the full house you need to beat him the times he makes his flush are not there. He is getting 5:1 and you are not getting ~ 12:1 you need to continue drawing to the full house.

Hope this is what you are asking....

[Nottom]

[ QUOTE ]
I, however, will win the $60 pot four out of five times, for a total of $240, though I spent $10 each time (giving me a profit of $190).


[/ QUOTE ]

Correct, but if he folds everytime you profit $250. So him folding is preferable for you.

[CrisBrown]

Hi Stephen,

Because his odds of hitting a flush, coupled with your odds of not filling, are something on the order of 1:4. If the pot offers only 3:1, it's a mistake for him to call (he's getting insufficient pot odds) so you want him to call. If the pot offers 5:1, he is correct to call (he's getting sufficient pot odds), so you want him to fold.

Cris

[andyfox]

Let's say the odds of your opponent winning a hand are 4:1 against.

There's $30 in the pot and he can call for $10. Four times he calls and he loses $10 for -$40. One time he calls and wins $30. He's net -$10, which means you are plus $10.

If there's $50 in the pot and he can call for $10, four times he calls and he loses $10 for -$40. But now the one time out of 5 that he hits he's $50, for a net of +$10, which means you are minus $10.

Your odds of winning the hand indeed remain the same, as do his. But the amount of money in the pot determines whether or not you or he are getting the best of it. While you will indeed win the hand 100% of the time if he always folds, if he's putting money into the pot not getting proper odds, you end up ahead. That is, when he folds, you win zero additional dollars. But when he calls and, in the long run, loses more money than he would by folding, you win.

Imagine that, instead of 4:1 odds, he was a 1,000,000:1 dog to win the hand. You'd want him to call even though it would reduce your chance of winning the hand from 100% TO 99.999999%. Even though his calling always reduces your chance of winning below 100%, sometimes the reduction is worth it if your opponent will not win enough to cover his losses.

[Nottom]

Correction: You would profit $200 not $250, either way its better for you if he folded everytime when he was getting the odds to draw.

[LetsRock]

OK, I tried work some numbers out but it just started hurting my head, I understand how the theory works but I can't prove it. Let's try this.

This may be a bit obtuse, but lets compare it to buying something. Let's say the normal value of a hamburger is $4. If you buy a hamburger for $4 you get proper "value" for your money and the burger man breaks even. If you can convince the burger man to let you have it for $3 he loses a dollar every time you make this deal. If you offer him $5 for it, you lose a dollar every time you have a burger. Obviously it's to your advantage to buy as many hamburgers as you can at $3 ea., and avoid the days when they're $5. Every time you buy a $4 burger, you're both getting a "fair deal.....

So...everytime you give your flush draw opponent 5:1 odds to call, he's getting a good "bargain" to try. Everytime he calls with only 3:1 odds, you're making money. Everytime he does makes the call at 4:1 you're both getting a "fair" deal.


Geez, now I'm hungry..... /images/graemlins/wink.gif

[Copernicus]

Without reading through all the replies, at least one of which probably has a good explanation, maybe this brings a different approach to it. Your original reasoning as to why it shouldnt matter is actually quite close. Why risk him taking the pot at all...hope for a fold 100% of the time. However, it misses one point, which is that for him to call he has to risk additional money that you dont get if he folds. It is the relative value of that bet that swings your preference from a call when he has improper odds (because you are better off with a smaller chance( vs 100% if he folds) of winning a bigger pot) to a fold when he has proper odds, because you prefer the certainty of taking down the bigger pot, to taking the risk of winning an even larger pot.

Putting numbers to it, when he is getting 3/1 odds, his call increases the value of the pot by 33%, and you have 79.5% (assuming all 9 flush outs are winners) chance of winning 133% of the pre-call pot. When he is getting 5/1 odds, his call increases the pot by 20% and you have the same 79.5% chance of winning 120% of the pre-call pot.

.795*1.333=1.06 (greater than the 1 you win from a fold)
.795*1.200=.954 (less than the 1 you win from a fold)

The breakeven point is where you are expected to win 1 even if he calls, which is 1/.795 or 1.2578...ie his call of 1 increases the pot by .2578, which means there was 3.879 bets in the pot to begin with (ie his odds are 3.879/1), and 1/4.879 is his break even probability = .205, which is, of course the inverse of your break even probability of .795.

[Louie Landale]

You are a 4:1 favorite and bet on the second to last betting round. The opponent "should" call in the big pot but not in the small pot.

[1] Your EV is clearly bigger in a big pot that the opponent correctly draws for, than in a small pot he correctly folds for: getting 4/5ths of a big pot is better than getting 100% of a small pot.

[2] The DIFFERENCE between the opponent calling and his folding is BIGGER in a big pot than for a small pot: In a small pot the most you make is his 1bet if he calls incorrectly. In a big pot you can gain multiple bets when he folds incorrectly.

- Louie

BTW, [2] suggests that a tight image that gets opponents to incorrectly fold in big pots is potentially more profitable than a wild image that gets opponents to incorrectly call in small pots; at least in games like Holdem that feature lots of medium and big pots.

[MicroBob]

[ QUOTE ]
Could I say, then, that while my EV is still positive whenever my opponent is on a draw against me in this situation, that my EV is higher when the pot doesn't favor my opponent's call? If that's so, I think I'm understanding it, then.


[/ QUOTE ]

yes, this is the correct way of viewing it i believe.
this will be similar if not identical to an example poointed out earlier but i'm going to do it anyway. i still consider myself a novice for the most-part so bear with me. i am doing this example as an exercise for myself here as much as anything else.....input welcome and appreciated.

imagine that it's not a 4-flush draw but rather a really bad chase.
in a 10/20 gane with a pot of $40 you have AA and your opponent has 22 and the board is A86
the only way your opponent can win here is to hit runner-runner deuces.
if your opponent folds immediately you have a 100% chance of winning the $40.
if your opponent bets into you then you have a 99% chance of winning $50 or more. obviously he is not getting the correct odds to call here and you welcome his call because it means more money for you in the long-run (and probably in the short run). in other words, a 99% chance at $50 or $60 is better than a 100% chance at $40 because if you keep making this bet over and over again you'll be richer in the 99% scenario.

however, if Bill Gates walked up and dropped $1-million into the pot for the winner of that hand then your opponent would obviously be correct to stay in for a $10 bet as he now is getting pot odds of 100,000-to-1 and he has a 100-to-1 shot (or whatever it is) of hitting his runner-runner quads. in this scenario, you are obviously hoping he folds, even though his bets are likely to get you extra money on this hand (the extra $10 or $20 he puts into the pot) you know that he is getting sufficient odds to call and you want him to make the incorrect play and fold. a 100% chance of $1-mil is better than a 99% chance at $1-mil+20 because if you played it out over and over again you would eventually lose to his runner-runner quads on one of the hands and that would mean that in the long-run you make only $980k per trial (or something like that) if you did it over and over.

thanks for posting the question though....it got me thinking and helped force me to analyze some things about the FTOP and pot-odds in general.
obviously, the 4-flush vs. 2-pair is just a less-exaggerated version of this. you can be at an advantage either way but the pot determines whether you want him to call or not.

[MrGrob]

Yes -- you are +EV either way, but you are going to win more if he folds in the 5 to 1 scenario, and gain more when he calls in the 3 to 1 scenario.

The theory states that if someone makes a play other than the one they would make if they knew your cards, you gain. So, if someone has AKs and is on a flush draw (and he thinks he has over card outs as well) you are hoping that he calls you in the 3 to 1 scenario, as you GAIN here by his calling...he thinks he is sitting better then 3 to 1, or even, but is not -- same as in a situation where he has, only a flush draw with 54s or something, and he thinks that maybe his flush is no good if he hits, or maybe he is low on chips and does not want to chase, though the pot is laying him 5 to 1, you GAIN HERE by his folding and not chasing you down when he had the odds to call.

In both the cases, if he KNEW what you had, and was able to make the right mathematical play, you would not gain as much as when he goes against the best paly for whatever reason.