No Limit Texas Hold’em Tournament Strategy:

A Manually Computed Monte-Carlo Simulation

Jason W. Barnes and David S. Choi

Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721
jbarnes@c3po.barnesos.net, dchoi@lpl.arizona.edu

ABSTRACT
We will be holding at least one No Limit Texas Hold’em poker tournament around the campfire in the evening
on the Sentinel Peak field trip. All are invited to play. There will be an entry fee of $5. In this handout we
describe poker for the interested and scientifically literate beginner, starting with hand ranks and probabilities.
We then describe the game of Texas Hold’em, the variation we will be playing, and No Limit Hold’em tournaments
in particular. We conclude with a short primer on Hold’em strategy and tactics.
Subject headings: Monte Carlo Simulation – Poker – Texas Hold’em – No Limit Hold’em Tournaments

1. INTRODUCTION Some of the low-entropy hands are multiple instances of

a single number card, like, say, four 8s, or 3 queens. The
The game of poker is a study in probability and psychol- more complex groupings are a full house, which is three
ogy. Understanding probability allows proper computation of any one number card and two of any other; a straight,
of betting, payoffs, and when to cut your losses; psychology which is 5 continuous cards of any suit such as 3, 4, 5, 6, 7;
gives insight into what’s your opponent’s hand, whether a flush, which is any 5 cards all of the same suit (e.g. 3h,
he’s bluffing, and whether you can get her to fold without 7h, 10h, Qh, Ah); and a straight-flush, both a straight and
having to even show your hand. Master both, and you’ve a flush simnultaneously (7d, 8d, 9d, 10d, Jd). For straights
got a chance. only, an Ace can be either high (10, J, Q, K, A) or low (A,
2, 3, 4, 5), but NOT both at the same time (NO K, Q, A,
However, poker is also a game dominated by stochastic- 2, 3 wraparounds!). In all other cases, Aces are high. The
ity. Even the best players won’t win every game – not even order of poker hands and their probabilities are listed in
close. Professional poker players can make money, but the Table 1.
standard deviation of their winnings is high. In a single
game, or single tournament, anything can happen. Frequently, two of the same type of hand will show down
against one another. In this case, the hand that involves
To our knowledge, poker cash games were played on higher cards in its important elements will win: Qs Qd 8c
only one previous LPL fieldtrip, the Southern New Mexico 9c 10s beats 8h 8s Qs Ks As. Similarly Jc Jd Jh 2c 5c
trip in Spring 2001. However, it was tragically interrupted beats 10d 10h 10s As 2h in three of a kind; 9d 10c Jc Qc
by the great Jim Richardson Meteoriteathon for the Third Kd beats 3s 4h 5s 6h 7s in straights; Ac 9c 6c 3c 2c beats
Millenium, and so was not as successful as it might other- Kc Qc 9c 3c 2c in flushes (watch this one!); and 8c 8d 8h
wise have been. 7d 7h beats 5d 5h 5s Ac As in full houses.
In case of a tie between the important elements in a
2. POKER: HANDS AND PROBABILITIES
hand, the hand with the highest card not involved in mak-
ing the low-entropy hand wins. This deciding card is then
All possible groupings of 5 playing cards are equally called the ’kicker’. 8c 8d 8h 8s 10s beats 8c 8h 8s 4s, Ac
probable to occur in any given random 5-card sample. A Ad Kc 10c 4s beats As Ac 10c 4s 2d, and Ac 10d 4d 3d
hand with the Ace, King, Queen, Jack, and 10 of spades 2c beats Ac 9s 8s 7s 6s. If the two highest non-important-
(As, Ks, Qs, Js, 10s) is exactly as probable as 3 of clubs, element cards are the same, then neither is the ’kicker’,
5 of diamonds, 8 of spades, 9 of clubs, and King of hearts and the process proceeds down to the next highest card to
(3c, 5d, 8s, 9c, Kh). In the game of poker, the goal is to determine the winner.
obtain into your hand one of several low-entropy groupings
of cards. The lower the entropy of these groupings (and If two hands are identical, they split the pot. Note that
therefore the lower your probability of drawing them ran- this CAN happen, because there are no suit rankings
domly), the more valuble they are. Hence the first hand in poker. 3c 3d As Ks Js and 3s 3h As Ks Js are the
listed above, As, Ks, Qs, Js, 10s, will very likely make you same; they tie and split the pot. The 3s does NOT beat
a large amount of money, while the second hand, 3c, 5d, the 3d.
8s, 9c, Kh, will get you a whole lot of nothing.
A poker hand is always composed of exactly 5 cards.
3. NO LIMIT TEXAS HOLD’EM TOURNA-
MENT PLAY

3.1. Texas Hold’em

The game of Texas Hold’em starts with every player

receiving two cards. These are referred to as your hole
cards. Then there is a round of betting. Here, the player
must decide whether they be in or out of the pot. The
two (sometimes one) players to the left of the dealer have
forced bets before the cards are dealt; these bets are called
blinds. During the first round of betting, the player must
decide whether they want to fold their hand, call and
match the blind, or raise the bet. Action then ensues.
Once all of the bets are collected, the dealer takes the
next card and “burns” it by placing it in the discard pile.
This is done to prevent the possibility of cheating. Three
cards are dealt face up; this is known as the flop. A round
of betting then ensues starting with the player to the im-
mediate left of the dealer. During the betting round, a
player can check (not bet) if that option is available, bet,
raise, or fold. When that is over, another card is dealt
face up; this is the turn, after which is another round of
betting. Finally, a fifth and final card is dealt face up, the
river. A final round of betting ensues. When all the bet-
ting is over, the player with the best five-card poker hand
wins the pot.

3.2. Tournament Rules and Style

There are two ways of playing poker. One is a cash

game where the chips have a cash value, and what you
win or lose in a pot corresponds 1:1 to your profit/loss for
the session. The other, and arguably more exciting way
to play, is a tournament. Here, the chips do not have an
explicit cash value. Every player initially pays a buy-in
and receives the same amount of chips. Our tournament
buy-in is $5. In contrast, the buy-in for the main event
of the World Series of Poker (the one you see featured on
ESPN) is $10,000.
Once all of the players have assembled, the poker tour-
nament begins. The blinds for a hand start at a low num-
ber, and increase after a fixed period of time. This struc-
ture is illustrated in Table 2, and for our tournament, the
blinds increase every 20 minutes. In our tournament, if a
player loses all of his chips during the first hour of play,
the player is allowed to continue playing in the tournament
as he has the option to rebuy for the same price as the
buy-in and receive the same initial amount of chips. The
objective of the poker tournament is to win all of the chips;
then, you will be declared the winner and receive a per-
centage of the total prize pool (buy-ins plus rebuys). The
payouts are 50% for 1st, 30% for 2nd, and 20% for 3rd.

4. BASIC STRATEGY AND TACTICS

You should fold way more than half the time. It costs a
lot to see a hand to a showdown at the end – if you start
poorly and end up pushing a bad position to its logical
conclusion, you will almost certainly get an early exit from
the tournament. The biggest mistake most people make in at least, make them pay for their pathetic low-probability
Texas Hold’em is to not fold often enough (Proctor 2005, dreams.
personal communication). Your goal should be to WIN the
BIG hands. Don’t sweat the small stuff, and don’t mind If you’re the adventurous type, you might even consider
losing small hands. going all-in right off the bat on something like this. How-
ever, there’s a good chance everyone else will just fold and
let you just win the blinds, which is an awful waste of KK.
4.1. Pre-flop This trick is not for the faint-hearted, though, as even 22
has a 20% chance of taking you down. 80% is a bet I’d
After you get your first two cards, evaluate them and make any day, but its still one that I’d lose one workday a
decide whether to even bother playing this hand. Although week.
you might get lucky, if your hold cards are 3d 8c, you would
be foolish to bet on such a low-probability event as winning If you have a middling pair (77-TT), you should proba-
that hand. Good hole cards comprise: a pair, face cards, bly raise to double the big blind, just to scare out people
suited cards, and/or connecting cards. One thing to keep with nothing at all. Be careful on this one, I (JB) seem to
in mind though is that playable hands increase in general keep losing my shirt on them. A simple QJo (o means ‘off-
as the number of total players at the table decreases, and suit’, while QJs means ‘suited’, the same suit and not just
as your position relative to the button (dealer) changes. spades in this case) has about a 45% chance of beating you.
Unfortunately, this topic is beyond the scope of this hand- If two opponents stay in, your chances of winning plummet
out. dramatically. You might try to bet high and scare every-
one but one opponent away, but if that opponent then has
4.1.1. Pairs a high pair you could exit the whole tournament early.
Better to play your middle pair moderately. Stay in for
It is easy to see that starting with a pair is a good low-lowmiddle bets, but don’t go throwing your money af-
thing. It already beats half of the hands out there, and ter massive raises or reraises – its probably just not going
has a chance to go on to become three or four of a kind, a to make it.
full house, or two pair. It means that while other players Small pairs like 22-66 are a crap-shoot waiting for an-
might be trying to cobble together a complex, low proba- other one to pop up to give you 3 of a kind. There’s about
bility hand like a flush, you’re sitting pretty knowing that a 12% chance that it will happen. Not that good if you’re
you have a hand that can win when your opponent’s flush only playing against one or two others. I’d call the BB, but
doesn’t pan out. not much else. If someone else raises to double the BB they
The probability of getting a pair in the hole is 5.88%, could very well have a higher pair than you, in which case
about 1 in 20. In a 10-person game, there’s about a 50% your chances of winning are those 20% (yuck), or they’ve
chance that someone got dealt a pair in each hand. Multi- got two face cards, in which case you’ve got a formal 55%
ple people with pairs is not that improbable in this context. - 45% advantage if there’s only one, but much worse with
Don’t get too excited – its easy to overbet a pair. multiple opponents. If you don’t come up with that three
of a kind, these are almost guaranteed second-best, and
The best hole-pair is AA, known as “pocket rockets” or therefore losing, hands.
“bullets”. Its a good hand. However, it loses a lot more
often than you might think. Bet it hard, but don’t push 4.1.2. Face Cards
it farther than it will go. If the flop breaks against you
(say, 7c 6c 8c), be VERY cognizant of how your opponents Two face cards (Jack through Ace) can be a pretty po-
bet, and if you think they’ve got something that will beat tent hand. In particular, AK (“big slick”) can get you
you (3 of a kind or up, in this case a shot at a flush or a places. If another face card that matches yours pops up,
straight) give up and go home. Remember, the very worst you’ve got a formidable pair, and a high kicker to boot in
possible hand you can have in poker isn’t 2c 3d 4d 5h 7s. case someone ELSE has your same face card pair. Plus
The worst hand is the second-best one. That’s the one you’ve got a shot at a high straight.
that you’ll bet a lot of money on and lose. Some people
will bet everything they have on that AA even when they The other reason that face cards, and AK in particu-
can see plenty of ways to get beat, and someone betting as lar, are valuble is in case the winner has the most prob-
if she has made one of those. If you do that, you deserve able poker hand: nothing. With fewer players, or with
to lose. only a few remaining in the hand to see the flop, there’s
still a reasonably high probability that nobody will end up
Good pairs are AA, KK, QQ, probably JJ, too. First with anything interesting, and that you’re AK will take
thing you want to do with these: raise the big blind. If the pot on high-card alone. Additionally the dealer could
the BB is 20, raise to 40 (maybe higher, like 80 or 100). flop something phenominal like QQQ, in which case unless
You are confident that you can win against what any one someone else is holding on to a queen or a pair, your AK
particular opponent has at the moment. However, a lucky has a good chance of breaking the resulting QQQ tie.
flop for someone else can beat you. Solution: minimize the
number of people that can get lucky on the flop. If you Formally, AK has only a 45% of beating a pair, and
don’t want to get beat by some loser that stays in 3c 5c this probability rings true in heads-up play (one on one).
(or can stay in for free as the big blind) and gets a flop 3d However, in a game with 10 people where 8 fold and you
3h 5s, get them to fold before they even see the flop. Or, have AK up against your opponent’s 99, the AK has better
chances: those 8 people might not have folded had they had probability, bet the minimum necessary to stay in until you
an A or a K in their hands! Therefore the folders probably see the flop. If the betting gets too heavy, fold. Especially
siphoned off some low cards, enriching the deck in AK and on low suited connectors, don’t push them too far. Try to
increasing your chances of getting a high pair to take down get in cheap to see the flop, but if it doesn’t go well cut
the 99. your losses and get out.
I would bet AK pretty hard, about as hard as QQ. Raise Connectors themselves, like 89 offsuit, seem like they
to double the BB at least, and scare out those people trying might be valuble. I (Barnes) never play them. Without
to see a lucky flop. However, be much more careful with your chance at the extra flush draw, these will bleed you
KQ, KJ, or QJ. Someone else at the table probably has at dry seeing cards with too few out there that can help you.
least one Ace in their hand, and that one will make the Clearly with connecting face cards (Section 4.1.2) its a
difference quite a few cases. different story, as the pair and kicker possibilities come
into play.
AX, where X is anything else, has a surprising amount
of promise. Ace with 2-5 has a shot at the low straight as
well as the high pair or strong kicker. However, its kicker 4.1.4. Trash Hands
quality is lacking when compared to A with 9 or so through
Q. Trash hands are hands such as 72o, T4o, A5o, 84s, 63s
that are better left unplayed. These hands are classified as
I almost always at least stay in to see the flop with an “dominated hands” (such as KX, where X is a low rank),
Ace and anything else of the same suit. If there’s a flush “gap” hands where it’s possible (but unrealistic) to make
in that suit in Hold’em, nobody else’s flush can beat your a straight (such as 59o), and seemingly random hands like
Ace-high flush. Its a great defense against the second-best 72o. Yes, any two cards have a chance at winning, but
hand. these hands are likely to get you in trouble more often
than not. The key to becoming a successful hold’em poker
4.1.3. Suited Cards and Connectors player is recognizing these trash hands, and folding them.

Having two cards of the same suit gives you a chance 4.2. Post-flop
at the flush draw while simultaneously setting you up for
a lucky flop that will allow you a good hand that nobody The flop is the most critical phase of a hold’em hand.
sees coming. However, low suiteds lose an awful lot of the It is here where you have to make the decision whether or
time. 3c 5c might seem good, suited and close enough for a not to continue in the hand. Did you hit your face card
straight maybe, but even if you get the flush, how sure are giving you top pair? Bet hard, and make it expensive for
you that Mandy doesn’t have Ac Qc? She very well might, speculative hands (those drawing for a flush or straight)
and then as second-best hand you bet and lose a fortune. to stay in the hand. Conversely, if you’re in a drawing
With suited cards both under 9 that aren’t connected (next situation, see if you can get away with seeing the next
to each other, see below), I would strongly consider cutting card for as cheaply as possible. (NB: There are certain
my losses. You might stay in to see the flop to see if a bunch situations where it is advantageous to bet or raise with a
more of your suit come up, and then see how everyone else drawing hand. For example, if you have 4 to a flush and
bets to see if THEY are going for your flush, TOO. But hold an A or K. This also depends on the size of the pot
don’t put in much on one of these hands, it ain’t worth it. and bet, and whether your opponents are likely to call or
fold.)
High suited cards are a different story. If one of your
suited cards is a face card, you’re in business as you have a If the flop gave you a whole lotta nothing, you’re prob-
good shot at winning a flush-flush showdown. Bet enough ably better off just checking/folding instead of trying to
to stay in, but no more until your 5th suited card falls. bluff and attempting to buy the pot. If you choose to
Then bet like there’s no tomorrow, assuming it looks like bluff, I recommend that you insure that you have a fairly
there are no pairs on the board (a pair on the board could high number of “outs” (cards that will give you a winning
give someone a full house or 4 of a kind). Of course, that’s and/or stronger hand) that you can at least hope for in
not a very subtle bet, as once that third heart falls and case your bluff does not work.
you bet to the hilt, if I don’t have at least a flush that I
think can beat you, I will sooo fold. Of course, this leads 5. Future Work
to bluffing possibilities (to be discussed in the forthcoming
Intermediate Strategy article for the fall field trip).-
Coming Autumn 2004: sections on Position Relative to
Suited connectors are a hand that I personally like a Dealer, Chip Position, Betting tactics such as slow-playing
lot. This is when your two suited cards are next to one and the check-raise, Bluffing, Advanced Flop Strategies,
another, like 8d 9d, or 10s Js, or even 5s 6s or so. The Turn and River Play, Pot Odds and Implied Pot Odds,
advantage to these is that not only do you have the flush and Ideal Playing Styles.
draw, and the chance at a lucky and unusual flop, but you
also have a good shot at a straight. While none of those are
Vegas Baby! – DC
likely things to happen, if they DO happen, you are likely
Develop your own strategies, and be unpredictable some-
to make BANK on them, so under the right circumstances
times. Also, since Dave and I wrote this handout, you
suited connectors can rock. However, since they are so low
know how we play. Use that against us. I would. – JB