Information Reduction through Strategy Restrictions and Balancing
version 1.1
Copyright 1998, 1999, Abdul Jalib
Permission to copy for personal use.
[This is a technical report on play balancing. If you are looking
for general concepts on how to play, you've come to the wrong place.]
I've been using Texas Turbo hold'em to examine preflop strategies
that disguise my holdings without sacrificing much expected value (EV)
to opponents who aren't paying attention anyway. Turbo isn't
ideal for this task, but in any case, I mostly wanted to
share the general concepts of this approach. The two main means I have
identified of avoiding giving away information preflop are:
1) to impose general restrictions (e.g., never more than 3-bet)
2) to make suboptimal plays within those restictions in order to
balance the probabilities of the properties of your hands in
different situations.
The easiest way to restrict your preflop opening play is to always make
the same initial play, either always open-limping or always open-calling
no matter what the hand.
OPEN-LIMPING
I tried to reduce preflop under the gun (UTG) strategy to just 3 options
for opening the pot and responding to subsequent action:
1. Open-limp, then raise if it's raised behind, but if it's
raised and reraised behind then just call.
2. Open-limp, and if it's raised 1 or 2 (or more) times, then call.
3. Open-limp, and if it's raised once then call, but fold for two more bets.
Relative to also allowing similar open-raise options, Turbo simulations
estimate that this restricted strategy reduces EV UTG in a *tight*
10-handed $10-$20 game from $1.14 to $1.04 per hand. The corresponding
always-open-raise strategy won $1.01.
Here are the 3 options again but with hands filled in, and these hands
have been juggled around a bit, making some suboptimal plays in order
to balance the information. Asterisks (*) denote hands to which I have
assigned suboptimal plays.
1. Open-limp, then raise if it's raised behind, but if it's
raised and reraised behind then just call:
QQ, JJ, TT, 99*, AKs, AQs, AJs*, KQs, QJs* AK
2. Open-limp, and if it's raised 1 or 2 (or maybe more) times, then call:
AA*, KK*, 88, 77, ATs, JTs, T9s*, AQ
3. Open-limp, and if it's raised once then call, but fold for two more bets:
66*, A9s, A8s, KJs, KTs, QTs*, J9s*, AJ, KQ*
With this restricted strategy, there are only 4 possible ways in which
I can see the flop:
A. I open limp, an opponent raises, I reraise.
=> I have a hand in group #1.
B. I open limp, an opponent raises, another opponent reraises, I call.
=> I have a hand in group #1 or group #2.
C. I open limp, an opponent raises, I call.
=> I have a hand in group #2 or group #3.
D. I open limp and nobody raises.
=> I have any playable hand: group #1, #2, or #3.
I'll try to quantify how much information I reveal by computing
various conditional probabilities:
PREFLOP ACTION SCENARIO
A B C D
total number of distinct hands 10 18 17 27
% chance pocket pair 43 46 29 34
% chance suited 36 31 35 35
% chance offsuit nonpair 21 23 35 30
% has pair >= 2 with 2 on board 43 46 29 34
% has pair >= 3 with 3 on board 43 46 29 34
% has pair >= 4 with 4 on board 43 46 29 34
% has pair >= 5 with 5 on board 43 46 29 34
% has pair >= 6 with 6 on board 43 46 27 33
% has pair >= 7 with 7 on board 43 45 21 29
% has pair >= 8 with 8 on board 43 39 18 27
% has pair >= 9 with 9 on board 40 36 21 28
% has pair >= T with T on board 28 37 28 28
% has pair >= J with J on board 29 31 31 31
% has pair >= Q with Q on board 24 35 35 31
% has pair >= K with K on board 29 25 26 27
% has pair >= A with A on board 36 36 33 34
Scenario D, where I limp in and no one raises, has the
same probabilities as the priors, since this strategy employs
open-limping with everything playable. The differences in the
figures for the other scenarios show how much information I'm
giving away. Note that the probability that I have a pair of
kings or better with a king on board remains nearly constant
across all the scenarios, and similarly constant for aces for
when an ace flops. Thus I give away almost no information for
those particular cases. In general, I'm trying to play each
hand fairly appropriately and yet give away minimal information
for each scenario. The probabilities for having a pair
with a low board in scenarios A and B, where I put in 3 bets,
are a bit different from the probabilities for scenario C,
where I'm raised but elect not to reraise, but of course
giving away a certain amount of information is unavoidable.
All the changes I made would cost about 15 cents in the Turbo
simulation, bringing the EV down from $1.09 to $0.94. The
limp-calling with aces and kings instead of limp-reraising is
responsible for almost half of this 15 cents in reduction.
Turbo players almost make little use of information from previous
rounds for future rounds. Whether these sacrifices
to preserve information would be worthwhile against human
opponents is unknown.
OPEN-RAISING
If you want to consider open-raising all the time rather than open-limping,
then one problem is that there are very few hands that you can 4-bet with
after you've open-raised and it's been reraised - pretty much just
AA, KK, and QQ. AKs, AQs, and AK can 4 bet with small losses (possibly
a profit in some games) relative to their optimal plays. Avoiding the
4 bet play altogether, however, costs 4 cents.
1. Open-raise, then reraise if it's raised back to you:
AA, KK, QQ, AKs*, AQs*, AK*
2. Open-raise, then call if it's 1 or 2 more bets bet:
JJ, TT, 99, AJs, KQs, JTs
3. Open-raise, then call if it's 1 more bet back, fold for 2 more bets:
88, 77, ATs, A9s, KJs, KTs, QJs, AQ, AJ, KQ*
There are four possible scenarios where I can see the flop:
A. I open raise, an opponent reraises, I cap it.
=> I have a hand in group #1.
B. I open raise, an opponent reraises, and another caps it, I call.
=> I have a hand in group #1 or #2.
C. I open raise, an opponent raises, I call.
=> I have a hand in group #2 or #3.
D. I open raise, and no one raises.
=> I have any playable hand (group #1, #2, or #3).
PREFLOP ACTION SCENARIO
A B C D
total number of distinct hands 6 12 16 22
% chance pocket pair 47 53 31 35
% chance suited 21 29 33 29
% chance offsuit nonpair 32 18 37 35
% has pair >= 2 with 2 on board 47 53 31 35
% has pair >= 3 with 3 on board 47 53 31 35
% has pair >= 4 with 4 on board 47 53 31 35
% has pair >= 5 with 5 on board 47 53 31 35
% has pair >= 6 with 6 on board 47 53 31 35
% has pair >= 7 with 7 on board 47 53 28 34
% has pair >= 8 with 8 on board 47 53 22 29
% has pair >= 9 with 9 on board 47 51 19 27
% has pair >= T with T on board 47 47 20 28
% has pair >= J with J on board 47 43 27 33
% has pair >= Q with Q on board 53 33 27 34
% has pair >= K with K on board 68 40 20 32
% has pair >= A with A on board 60 36 30 39
The modifications this time add up to about $.19, bringing
it down to $.90 per hand. This is lower than the $0.94 of my open-limping
strategy, but it would be a mistake to place too much faith in the
simulation numbers, as it's hard to say whether the Turbo simulations
are realistic enough. Mostly, I wanted to discuss the general concepts.
All the numbers in the tables are not simulation numbers, they're
exact computations, except for the hopefully small perturbations caused by
the fact that your opponents actions also convey information. You
can use the same approach to evaluate, and balance, the preflop strategy
of your choosing.
The probability of having a pair of kings or better with
a king on board in scenario C is pretty low. I'd suggest backing off
with AK versus a 3 bet about 1/3 of the time in order to boost scenario
C's king (and ace) pair probability. This will actually save you some
money in raw theoretical terms too, since 4 betting with AK was a
suboptimal deviation in the first place.
As you can see by how different the columns in the table are from
each other and in particular from the last column (the priors),
this strategy gives away a ton of information. Perhaps a better
job could be done in balancing the information.
BOTH OPEN-LIMPING AND OPEN-RAISING
Allowing both limps and raises for your opening causes problems.
First, you just inherently give away more information. Second,
if you are going to limp-reraise with your best hands, then it's
hard for you to be able to call with your weaker hands when you
open-raise and it's raised and reraised behind you.
Here is an attempt to balance the hands for this case:
1. Open-raise, then call if it's 1 or 2 bets back:
JJ, 99, 88, KQs, QJs*, KTs, QTs, JTs*, J9s*, T9s*, 98s*, AK
2. Open-raise, then call if it's 1 more bet back, fold for 2 more bets:
77, AQ, AJ, KQ
3. Open-limp, then raise if it's raised back, but if it's raised twice
make it four bets.
QQ, TT, AKs, AQs*, AJs*
4. Open-limp, then if it's raised once call, but fold for two more bets.
ATs*, A9s, A8s, A7s, KQs*, KJs,
5. Open-limp, then if it's raised once call, but if it's raised twice
make it four bets.
AA*, KK*
This strategy results in 6 preflop scenarios:
A. I open-raise, an opponent reraises, I call.
=> I have a hand in category #1 or #2.
B. I open-raise, two opponents reraise, I call.
=> I have a hand in category #1.
C. I open-raise, and nobody raises.
=> I have a hand in category #1 or #2.
D. I limp, an opponent raises, I call.
=> I have a hand in category #4 or #5.
E. I limp, two opponents raise, I reraise.
=> I have a hand in category #3 or #5.
F. I limp, and nobody raises.
=> I have a hand in category #3, #4, or #5
PREFLOP ACTION SCENARIO
Prior A B C D E F
total number of distinct hands 29 16 12 same 8 7 13
% chance pocket pair 29 23 29 as 33 67 40
% chance suited 41 31 52 A 67 33 60
% chance offsuit nonpair 29 46 19 0 0 0
% has pair >= 2 with 2 on board 29 23 29 33 67 40
% has pair >= 3 with 3 on board 29 23 29 33 67 40
% has pair >= 4 with 4 on board 29 23 29 33 67 40
% has pair >= 5 with 5 on board 29 23 29 33 67 40
% has pair >= 6 with 6 on board 29 23 29 33 67 40
% has pair >= 7 with 7 on board 30 21 29 43 67 46
% has pair >= 8 with 8 on board 28 18 31 43 67 46
% has pair >= 9 with 9 on board 29 18 32 43 67 46
% has pair >= T with T on board 27 18 31 43 64 43
% has pair >= J with J on board 29 22 21 43 60 41
% has pair >= Q with Q on board 32 28 15 43 56 38
% has pair >= K with K on board 28 25 26 48 38 33
% has pair >= A with A on board 35 28 15 52 40 48
Again, you shouldn't place too much faith in this strategy from Turbo,
as it was based on a small number of trials and Turbo is not totally
realistic. I don't think the hands are well balanced to avoid
information leakage. In general, it's going to be hard to balance
six different scenarios. On the other hand, this approach gets a lot
of bets into the pot at times.
This strategy results in an EV of $1.00, just 9 cents down from
optimal, and actually this is without 4-betting, so it actually
does better than $1.00. But it does give away tons of information.
COMPARISON TO S&M PREFLOP STRATEGY
S&M recommend a mix of raising, limp-reraising, and limping,
with some mixing of those plays for individual hands. For
example, they recommend open-raising with T9s 1/3 of the time,
limping the other 2/3's, in early position. Perhaps you could
open-raise and open-limp without giving away much information
if you mixed up which you did on many particular hands. You'd
have to be careful, however, and the systematic procedure I've
presented should help you develop a mixed preflop strategy that
hides information.
I think S&M are generally wrong in their overall preflop
game plan in tight games. S&M suggest raising with your really
strong hands (AA, KK, QQ) and often limping with your
marginal hands (JTs, T9s, 88.) It shouldn't be hard to see
why this is a mistake in a tight game. In a tight game, you've
got a good shot at taking down the blinds if you raise. If you
succeed, you win $15. This is a disaster for AA, KK, and QQ,
which have EV's in excess of $20, pushing $60 for AA, according to
Turbo. (And in fact Mason Malmuth seems to realize this, having advocated
in a recent 2+2 forum article open-limping with AA UTG in a very tight
game, if I recall correctly.) Stealing the blinds is a huge coup for
maginal hands like JTs, T9s, and 88. Therefore, if you don't have to worry
about giving away information, then you should actually open-limp (and
then reraise) with your huge hands, and open-raise with your weak hands,
in tight games. Aside from the hand-waving argument, that's what the
Turbo simulation results suggest. On the other hand, in many loose games
you should open-raise with your AA, as the idiots will call 2 bets cold
almost as readily as 1 bet.
CONCLUSIONS
Perhaps this whole exercise seems exceedingly paranoid to you.
After all, how many opponents watch your play that carefully? Well,
I play against a lot of the same people a lot of the time, and some
of them are pretty sharp. It's quite likely that some of them could
pick up some partial heuristics, such as knowing when I'm not too
likely to have a king based on my actions, that would allow them to
use some of the information I'm giving away. And even if my opponents
are mostly unconscious in the $20-$40 games I find myself playing most
of the time now, I always try to gear my game for my ultimate goal of
playing much higher games against much stiffer competition.
Perhaps this whole exercise seems exceedingly anal to you.
After all, shouldn't our play be constantly changing depending upon
the exact circumstances? I can't really argue with that, but you
have to first make yourself undefeatable before you worry about
defeating the enemy, and so you have to worry about information
leakage.
About Pokerator
