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FToP in multi-way pots

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TylerK

07-05-2005, 01:15 AM Post subject: FToP in multi-way pots

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This probably goes better in a different forum, but I thought it would be cool if someone actually, you know, read it.

In ToP, Sklansky devotes a grand total of about a page and a half to talking about the FToP in multi-way pots. Specifically, he says that in some cases (and gives an example), you prefer one opponent to play correctly (according, again, to the FToP) because it causes a more profitable mistake from an additional opponent in the pot. Or, in other words, in some cases you prefer one opponent to play correctly because your expectation from the resulting actions of additional players in the pot is greater than your expectation would be from the first opponent playing "incorrectly."

I need more discussion about this, because I'm sort of half-forming thoughts and not really getting anywhere. Anybody want to give it a go?

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Demiparadigm

07-05-2005, 03:11 AM

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Morton's Theorem:

I usually enjoy reading Mike Caro's Card Player column. One from last June made a big impression on me. In it he says:

_The real low-limit secret for today_. The most important thing i can teach you about playing the lower limits is that you usually should *not* raise from early positions, no matter what you have... because all of those theories of thinning the field and driving out opponents who might draw out on you don't hold true in these smaller games [where] you're usually surrounded by players who often call with nearly hopeless hands.... Which is better, playing against a few strong and semistrong players with possibly a small advantage for double stakes, or playing against a whole herd of players, mostly weak, for single stakes? Clearly, when you're not likely to win the pot outright by chasing everyone out, you want to play against weak opponents, and the more the merrier. So, why raise? There, I've just described one of the costliest mistakes in low-limit poker. The mistake is raising when many potential callers remain behind you, thus chasing away your profit. Don't do that.

Until recently, this made a lot of sense to me. After all, the Fundamental Theorem of Poker states (roughly) that when your opponents make mistakes, you gain, and when they play correctly, you lose. In holdem, if all of those calling stations in the low-limit games want to chase me with their 5 out draws to make trips or 2 pair when I flop top pair best kicker, and they don't have the pot odds to correctly do so, that sounds like a good situation for me.

Yet, it seems like these players are drawing out so often that something must be wrong. Hang around the mid-limits, holdem or stud, for any length of time and you're sure to hear players complain that the lower limit games can't be beat. You can't fight the huge number of callers, they say. You can't protect your hand once the pot has grown so big, they say.

At first, I thought these players were wrong. They just don't understand the increased variance of playing in such situations, I told myself. In one sense, these players are right, of course. The large number of calling stations combined with a raise or two early in a hand make the pots in these games very large relative to the bet size. This has the effect of reducing the magnitude of the errors made by each individual caller at each individual decision. Heck, the pot might get so big from all that calling that the callers _ought_ to chase. For lack of a better term, I call this behavior on the fishes' part _schooling_. Still, tight-aggressive players are on average wading into these pots with better than average hands, and in holdem when they flop top pair best kicker, for example, they should be taking the best of it against each of these long-shot draws (like second pair random kicker). In holdem, the schooling phenomenon increases the variance of the player who flops top pair holding AK, but probably also _increases_ his expectation in the long run, I thought, relative to a game where these players are correctly folding their weak draws.

Thinking this way, I was delighted to follow Caro's advice, and not try to run players with weak draws out of the pots where I thought I held the best hand on the flop or turn. This is contrary to a lot of advice from other poker strategists, as Caro points out, and I found myself (successfully, I think) trying to convince some of my poker playing buddies of Caro's point of view in a discussion last week.

Well, some more thinking, rereading some old r.g.p. posts (thank you, dejanews), a long discussion with Abdul Jalib, and a little algebra have changed my mind: I think Caro's advice is dead wrong (at least in many situations) and I think I can convince you of this, if you'll follow me for a bit longer.

What I'm going to tell you is that if you bet the best hand with more cards to come against two or more opponents, you will often make more money if some of them fold, *even if they are folding correctly, and would be making a mistake to call your bet.* Put another way, *you want your opponents to fold correctly, because their mistaken chasing you will cost you money in the long run.* I found this result very surprising to say the least. I've never seen it described correctly in any book or article, although at least a few posts to this newsgroup have concerned closely related topics.

I'm no poker authority but I think this concept has got to lead to changes in strategy in situations where players are chasing too much (and yes, Virginia, this happens not only in the 3-6 games, but also in the higher limits from time to time. Curiously, I have several friends who play very well who often complain that they can't beat 20-40 games when they get loose like this, or at least don't do as well in these games as they do in tighter games. hmmm....).

Let's look at a specific example.

Suppose in holdem you hold AdKc and the flop is Ks9h3h, giving you top pair best kicker. When the betting on the flop is complete you have two opponents remaining, one of whom you know has the nut flush draw (say AhTh, giving him 9 outs) and one of whom you believe holds second pair random kicker (say Qc9c, 4 outs), leaving you with all the remaining cards in the deck as your outs. The turn card is an apparent blank (say the 6d) and we1ll say the pot size at that point is P, expressed in big bets.

When you bet the turn player A, holding the flush draw, is sure to call and is almost certainly getting the correct pot odds to call your bet. Once player A calls, player B must decide whether to call or fold. To figure out which action player B should choose, calculate his expectation in each case. This depends on the number of cards among the remaining 46 that will give him the best hand, and the size of the pot when he is deciding:

E(player B|folding) = 0

E(player B|calling) = 4/46 * (P+2) - 42/46 * (1)

Player B doesn't win or lose anything by folding. When calling, he wins the pot 4/46 of the time, and loses one big bet the remainder of the time. Setting these two expectations equal to each other and solving for P lets us determine the potsize at which he is indifferent to calling or folding:

E(player B|folding) = E(player B|calling) => P'_B = 8.5 Big bets

When the pot is larger than this, player B should chase you; otherwise, it's in B's best interest to fold. This calculation is familiar to many rec.gamblers, of course.

To figure out which action on player B's part _you_ would prefer, calculate your expectation the same way:

E(you|B folds) = 37/46 * (P+2)

E(you|B calls) = 33/46 * (P+3)

Your expectation depends in each case on the size of the pot (ie, the pot odds B is getting when considering his call Setting these two equal lets us calculate the potsize P where you are indifferent whether B calls or folds:

E(you|B calls) = E(you|B folds) => P'_you = 6.25 Big bets.

When the pot is smaller than this, you profit when player B is chasing, but when the pot is larger than this, your expectation is higher when B folds instead of chasing.

This is very surprising. There's a range of pot sizes (in this case between 8.5 and 6.25 big bets when the turn card falls) where it's correct for B to fold, and you make more money when he does so than when he incorrectly chases. You can see this graphically below

Code:

                                      | 
                        B SHOULD FOLD | B SHOULD CALL 
                                      | 
                                      v 
                             | 
                  YOU WANT B | YOU WANT B
                     TO CALL | TO FOLD 
                             | 
                             v 
+---+---+---+---+---+---+---+---+---+---> POT SIZE, P, in big bets 
0   1    2    3    4    5   6    7   8    9 
                             XXXXXXXXX 
                                   ^ 
                       PARADOXICAL REGION

The range of pot sizes marked with the X's is where you want your opponent to fold correctly, because you lose expectation when he calls incorrectly.

This is an apparent violation of the Fundamental Theorem of Poker, which results from the fact that the pot is not heads up but multiway. (While Sklansky states in Theory of Poker that the FToP does not apply in certain multiway situations, it would probably be better to say that it in general does not apply to multiway situations.) In essence what is happening is that by calling when P is in this middle region, player B is paying too high a price for his weak draw (he will win the pot too infrequently to pay for all his calls trying to suck out), but you are no longer the sole benefactor of that high price -- player A is now taking B's money those times that A makes his flush draw. Compared to the case where you are heads up with player B, you still stand the risk of losing the whole pot, but are no longer getting 100% of the compensation from B's loose calls.

These sorts of situations come up all the time in Hold'em, both on the flop and on the turn. It1s the existence of this middle region of pot sizes, where you want at least some of your opponents to fold correctly, that explains the standard poker strategy of thinning the field as much as possible when you think you hold the best hand. Even players with incorrect draws cost you money when they call your bets, because part of their calls end up in the stacks of other players drawing against you. This is why Caro's advice now seems wrong to me, in general. Those weak calling stations are costing you money when they make the mistake of calling too much. In practice, when you flop a best but vulnerable hand, the pot size is rarely smaller than this middle region, where you actually want your opponents to call. Normally, the pot size is such that you want them to fold even if they would be wise to do so. In loose games, the pot size will often be at the high side of the scale, where you would love for them to fold, but they have odds to call and their fishy calls become correct.

This brings up another interesting point. In our three-handed example, both you and player B are losing money when B chases you incorrectly (both your and his expectations would be higher if he folded). This implies that player A is benefitting from his call, since poker is a zero-sum game (neglecting rake, etc). In fact, player A is benefitting _more_ from B's call than the magnitude of B's mistake in calling (since you are also losing expectation due to B's call).

Because you are losing expectation from B's call, it follows that the _aggregate_ of all other players (ie, A and B) must be gaining from B's call. In other words, if A and B were to meet in the parking lot after the game and split their profits, they would have been colluding against you.

I don't really know Roy Hashimoto or Lee Jones, but I suspect that this situation might be what Roy had in mind when he first described what he calls "implicit collusion" in games where there are many calling stations: one fish makes a play which reduces his overall expectation and all fish benefit by more than the magnitude of the first fish's mistake. That's collusion, just as if a player reraises with the worst hand to trap a third player for more bets when the first player's buddy has the nuts. Of course no one realizes there's collusion going on in these situations, so the collusion is implicit. (I'd sure like to hear from Roy or Lee on this point, because I think there's a significant difference between what I've called 'schooling' and what I've called 'implicit collusion', and that the two concepts are often confused with each other, but I'd hate to further confuse the issue by misappropriating someone else's label for this phenomenon.)

There was an interesting thread on this group last year started by Mason Malmuth called 'Going Too Far,' about the appropriate strategy changes in a game where many players are calling too loosely not only before the flop but also on the later streets. I suspect that the phenomenon described here (where both the leader and the chasers are giving up expectation to the player who is drawing to a very strong hand) lies behind the correct response to his discussion in that thread. One strategy change he mentions is that you'd like your starting hand to be suited in games like these. In light of what I've presented here I can not only understand this strategy change, but can see others as well. If this has made sense to anyone who can think of other strategy changes resulting from these ideas, let's hear them.

Finally, having criticized something by one of the famous poker authors, Abdul is encouraging me to go for broke <g>: It seems pretty clear that Sklansky also missed this idea, at least when he was writing Winning Poker, the precursor to Theory of Poker. First, he mentions that the Fundamental Theorem applies to all two-way and nearly all multiway pots. While I haven't proven it, it seems likely that nearly all multiway pots will contain some sort of region of implied collusion where the leader would prefer that players fold correctly, ie where the Fundamental Theorem breaks down. Later, in the chapter "Win the Big Pots Right Away," Sklansky makes his ignorance of this concept explicit. Discussing a multiway seven stud hand in which your hand is almost certainly best on fourth street he writes:

You must ask yourself whether an opponent would be correct to take [the odds you are giving him] knowing what you had. If so, you would rather have that opponent fold. If not -- that is if the odds against your opponent1s making a winning hand are greater than the pot odds he1s getting -- then you would rather have him call. In this case, instead of winning the pot right away, you1re willing to take the tiny risk that your opponent will outdraw you and try to win at least one more bet. ...you would not want to put in a raise to drive people out. (p. 62)

Slowplaying is certainly correct in some cases, but your 'druthers' in a multiway pot can never be decided so simply as by asking whether each of your individual opponents has the right pot odds to chase you.

To win in poker you only need to be one step ahead of your opponents. Two steps may be detrimental.

Demiparadigm

07-05-2005, 03:41 AM

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Fighting the School of Fish

Morton shows that there is a certain range of pot sizes for which it is correct for the chaser to fold (odds wise), but you make more money when he folds than when he incorrectly chases.

How can this be? Morton's great insight is that in a multiway pot, the player with the best hand is no longer making 100% of the profit on the incorrect chasing by the fish. In a heads up situation, the player with the better hand takes 100% of the incorrect chaser's bets, since there are only two players. But in a multi-way pot, the player with the current best hand is dividing up the incorrect calls with the player holding the best DRAW.

Let's look at a simplified example:
You hold 77 in early position and the button holds AK hearts. You raise and 4 players (including the button) cold call. The flop is Qh Jh 7c. Assume that our set of sevens represents the current best hand. Obviously, the button has the best draw. Morton's idea is that for every incorrect call the 3 players between us and the button make, the potential profit is divided between us and the player on the button. We can picture half of these calls going into our stack, and half of them going into the button's stack.

Now let's pretend that two of the players are colluding. Player C1 holds 98o and player C2 holds QJc. So Player C1 has a gutshot (4 outs) and Player C2 is drawing to the full house (4 outs, since another 7 gives us quads). With 5 BB in the pot on the flop, C1 (about 11:1) and C2 (about 7:1) individually are not getting the proper pot odds to draw (considering only the turn, for simplicity). However, together C1 and C2 have 8 outs, and are getting 4.7:1. So if these two fish play together, implicitly (implicit collusion) or explicitly (collusion), they are making the correct call.

This is the notion of "implicit collusion" and "the schooling of the fish." Although individually, C1 and C2 are making the incorrect play, from your perspective, the 2 headed fish have enough outs that they are getting the proper odds to draw. To make things worse, their calls are divided between you and the button, who has the best draw. Morton's Theorem says that we want the fish to fold in a multiway pot, even if they are getting the incorrect odds to draw out on us.

This "schooling" makes it very tough for the player with the best starting hand to win, and leads to the bad beats we see all the time at the lower limits. If 5 players at a table never fold, then together they have many, many outs.

To win in poker you only need to be one step ahead of your opponents. Two steps may be detrimental.

Demiparadigm

07-05-2005, 03:48 AM

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Poker Schooling

Schooling is a belittling term used to describe what loose poker players do as a defense mechanism. (Sheep flock; fish school.) If a pot is fairly small on the turn in Holdem, and the player with the best hand bets, any single player with a only gutshot draw will be making a significant mistake by calling. But now suppose several other people call too, with different gutshot draws. Because these other players are playing bad also, now the pot has grown to the point where the gutshot draws are getting better pot odds on their calls. These bad calls "school" together and miraculously become not-so-bad calls!

Schooling is part of the reason many reasonable players complain that they are unable to beat loose games. Everybody going to the river, sucking out every possible draw, how can a sensible player make a hand "hold up" and beat such a game? Well, it’s not hard really. A winning player merely wins money differently (and with higher variance) in these games. Schooling is actually profitable to good, winning players, but it does take a little analysis to see why. One column can't do justice to this topic, but maybe an example will help some people start having the right idea on how to view schooling.

Suppose you are playing $10/20 Holdem. In the big blind you have A9 (suits don’t matter here). Six people limp in, you check. The flop is AT5. Not so great, but you bet to see what happens. All six of your opponents call. Uh-oh, you start thinking about checking and mucking on the turn. But the turn card miraculously comes another Ace! You bet $20 into the $140 pot.

Via the magic of being able to make this stuff up, it turns out our six opponents have KQ, KJ, QJ, 43, 42, and 32. Of the 34 possible remaining cards in the deck, only 2 make a winner for each individual opponent. That’s 16-1 against them. When it comes to the first player, let’s say the KQ, he has to put in $20 at $160. He’s only getting 8-to-1 on a 16-to-1 draw. Bad call. But now as each subsequent player also calls, when it gets around to the 32, he has to put in $20 at a $260 pot. He’s getting 13-to-1 on his 16-to-1 draw. His call is not nearly so bad as the KQ’s call! That’s schooling, but the schooling of the other players has now also turned the KQ’s call into not nearly so bad a call -- likewise for all the other players.

But we don’t care about them, we care about our A9. If everybody had folded when we bet the turn, we get the $140. After 100 times, we’d be $14,000 ahead. But now what about when they all call? It turns out that A9 will end up winning about 65% of the time. So, after a hundred times, 65 times we get another $120 (six turn calls of $20 each), assuming nobody ever tries to bluff or calls a bet by us on the river. The 35% of the time we lose, we lose our $20 turn bet, plus any action on the river. Just to pick some numbers, I suggest we lose one bet on the river 50% of the time (when the river card is a king, queen or jack) and two bets the other 50% of the time (when the river card comes a four, three or deuce). So we lose an average of $30 on the river -- $50 total that 35% of the time the school draws out on us. What this works out to be is a decent extra profit per hand for the A9. The schooling helped our opponents, but it is still more profitable for us for them all to call -- to the tune of about $11.50 a hand. (65 wins of $260 = $16,900. 35 losses of $50 = $1750. Total profit = $15,150, or $1150 more than the $14,000. Also note that the 35 times we lose, we lose the $20 we invested in the pot to that point, or $700. However, that is not what we are analyzing here. We are looking at our situation on the turn. That $20 is already in the pot. It isn’t ours anymore. The before the flop action and flop calls by the other players have their own schooling ramifications.)

Now some people might prefer getting the $14,000 profit after 100 incidents of hands like this with everybody folding when our A9 bets the turn -- zero variance, win 100% of the time. It is about $1150 more profitable though for the A9 to live with the variance of having everybody calling. Most important, the fact that all these folks are calling/schooling is not a dramatically bad thing. A good player playing properly will do just fine against schooling opponents.

But it’s not that simple. If we change the 43 and 42 to 77 and 66, now we are going to win only 59% of the time, with that other 6% (the difference between our 65% and 59%) of the wins going to the 32. The 32 now snares a bunch of the profit in the hand, to the point that we would prefer that everybody would fold, and we just take the $140 each time. However, the A9 is still making money from people playing poorly by calling the turn bet, it just so happens that sometimes the main beneficiary of schooling is the best draw out there (the 32), not the best hand. Sometimes the second best hand benefits the most (in this case the 32 goes from a losing hand to a profitable one when everybody else calls), it all depends on the actual hands and how good their draws are, and how strong the best/most-likely-to-win hand is.

Schooling games give good players two main ways to win -- by either playing the best made hand or the best draw. There is more money to be made overall, but you have to make sure your game adapts to get the profit from both these ways.

You beat a schooling game the same way you beat any other game -- play smart, appropriate poker.

To win in poker you only need to be one step ahead of your opponents. Two steps may be detrimental.

Demiparadigm

07-05-2005, 03:55 AM

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To win in poker you only need to be one step ahead of your opponents. Two steps may be detrimental.

TylerK

07-05-2005, 01:52 PM

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Thanks for the info, good stuff!

TylerK: its just gambling if i want to worry about money i'll go to work lol

TylerK

07-05-2005, 03:30 PM

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So we can start by restating the "converse" of the fundamental theorem more correctly:
"Conversely, every time opponents play their hands differently from the way they would have if they could see all your and your opponents' cards, you and/or the other players in the pot gain; and every time they play their hands the same way they would have played if they could see all your and your opponents' cards, you and/or the other players in the pot lose."

So far so good?

TylerK: its just gambling if i want to worry about money i'll go to work lol

elipsesjeff

07-05-2005, 05:42 PM

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I read teh first paragraph above about not raising with good hands and I decided that was bunk!! Screw him.

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TylerK

07-05-2005, 05:58 PM

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Quote:

Originally Posted by elipsesjeff

I read teh first paragraph above about not raising with good hands and I decided that was bunk!! Screw him.

It's leading up to I think Caro's advice is dead wrong,, keep reading.

TylerK: its just gambling if i want to worry about money i'll go to work lol

Fnord

07-05-2005, 07:00 PM

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Quote:

Originally Posted by elipsesjeff

I read teh first paragraph above about not raising with good hands and I decided that was bunk!! Screw him.

In a schooling game you should be more selective about raising from EP.

I think in these games you're better off limping big offsuit unpaired hands other than AK/AQ. Also, I wouldn't open 99 and below for a raise as I'd rather invite others into the pot and play set.

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