Implicit Collusion and Going too Far
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 <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.
