Implicit Collusion and Going too Far Andy Morton 4/3/97 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                                   |                     B SHOULD FOLD | B SHOULD CALL                                   |                                   v                          |        YOU WANT B TO CALL| YOU WANT B TO FOLD                          |                          v +---+---+---+---+---+---+---+---+---+---> POT SIZE, P, in big bets 0   1   2   3   4   5   6   7   8   9                          XXXXXXXXXX                            ^                    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 :  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.