The Problem: Do you go all-in or not?

Player T (your opponent) and Player U (you) are heads up at the final table of a no limit Texas Hold’em tournament. All-in heads up confrontations before the flop are common and can determine the outcome of a tournament. Two possible scenarios arise. A) Player T acts first going all-in. Should Player U call Player T’s bet? B) Player U acts first. Should Player U go all-in?

We endeavor to develop a mathematical framework for Player U to judge the profitability of calling or making the all-in play before the flop based on the expected value of his options. The only information available to Player U will be the initial pot size p, the size of the all-in raise r (or his pot odds, (p+r)/r or p/r), and Player U’s assumptions on what hands his opponent will play.

Let S be the set of all playable starting hands (hole cards) that Player U assumes Player T will either go all-in with or call an all-in bet given the current game situation. Player U further assumes that Player T will choose only the ‘strongest’ starting hands. Without one of these hands, Player T will not raise all-in or make the all-in call. Essentially, Player U puts Player T on a hand in S. The set S is Player U’s profile or read of Player T.

Player U assumes that Player T will select the size and members of S, based on the current game conditions. For example, Player T can consider the relative stack sizes, the blinds and antes, whether he (Player T) wants to make a move, his read on Player U, etc. Player U condenses all this information into a specific set of hands Player T will play.

Again, the set S consists of all playable starting hands Player U assumes Player T will play.

As an example, Player U may read the current game situation as follows. Player T credits Player U to be a solid player. Player T has been playing ‘somewhat tight’ lately. Player T’s stack size is large enough with respect to the blinds and ante such that there is no need for him to make a risky play at this time. As such Player T will only get involved if he holds a premium hand. In this instance Player U puts Player T on the following hands for an all-in play:

S = {AA, KK, QQ, JJ, TT, 99, 88, AKs, 77, AQs, AJs, AK}.

(AKs denotes ace-king suited while AK denotes ace-king offsuit.) Ignoring Player U’s hole cards, this represents 72 out of all 1326= 52C2 two card combinations or 5.43% of the1326 starting hands.

Now suppose Player U’s hole cards are AQ. The set S contains only 59 hands since Player U’s cards prevent Player T from holding some of those hands counted above. With only 50 cards remaining in the deck, there are only 1225= 50C2 possible starting hands for Player T. Thus the set S listed above actually contains 4.82% of the 1225 possible starting hands for Player T.

For our purposes later, the 59-member set S will be referred to as the top 5.43% of the 1326 starting hands. Further references to a percentage of the actual 1225 hands will not be used. For simplicity, the goal will be to put Player T on a ‘percentage’ of the 1326 possible starting hands.

Two players remain before the flop. The pot size is p. Player T has hand t and Player U has hand u. The community cards are denoted by c (not really needed but used for notational clarity later). Let W be the set of all 1225 possible starting hands for Player T, given Player U’s hole cards u. S Í W.

A) Player T acts first going all-in.

Before the flop, Player T goes all-in and raises the pot by r. The pot odds for Player U is now (p+r)/r. Should Player U call Player T’s raise assuming that Player T would only have gone all-in with hand t where t Î S? Otherwise Player T would have folded or made some other play.

The expected values for Player U are:

EV( Player U folds) = 0

EV( Player U calls) = (p+r) x Prob( Player U wins ÷ t Î S )

+ (p/2) x Prob( Player U ties ÷ t Î S )

- r x Prob( Player U loses ÷ t Î S ) Eq. A1

Player U should call the all-in raise when

EV( Player U calls) > EV( Player U folds)

( (p+r)/r ) x Prob( Player U wins ÷ t Î S )

+ ( ( (p+r)/r ) – 1 ) x (1/2) x Prob( Player U ties ÷ t Î S )

- Prob( Player U loses ÷ t Î S ) > 0 Eq. A2

Solving for the pot odds (p+r)/r, Player U should call whenever he gets favorable pot odds of

(p+r)/r > [ 2 Prob( Player U loses ÷ t Î S ) + Prob( Player U ties ÷ t Î S ) ] /

[ 2 Prob( Player U wins ÷ t Î S ) + Prob( Player U ties ÷ t Î S ) ] Eq. A3

Prob( Player U wins ÷ t Î S ) = [ S N( u+c > t+c) ] /

for all t Î S

[ S N( u+c > t+c) + N( u+c = t+c) + N( u+c < t+c) ]

for all t Î S

= [ S N( u+c > t+c) ] / [ S 48C5 ]

for all t Î S for all t Î S

= [ S N( u+c > t+c) ] / [ N(S) 48C5 ] Eq. A4

for all t Î S

= S ( N( u+c > t+c) / 48C5 ) / N(S)

for all t Î S

= S Prob( u+c > t+c) / N(S)

for all t Î S

= S N(t) x Prob( u+c > t+c) / N(S) Eq. A5

for all t Î S

Prob( Player U ties ÷ t Î S ) = S N(t) x Prob( u+c = t+c) / N(S)

for all t Î S

Prob( Player U loses ÷ t Î S ) = S N(t) x Prob( u+c < t+c) / N(S)

for all t Î S

N( u+c > t+c) = The number of hands where the two hole cards u will beat the two hole

cards t at showdown for all 48C5 = 1,712,304 possible five community

card combinations.

Prob( u+c > t+c) = Probability before the flop that the two hole cards u will beat the two

hole cards t at showdown for all c’s. (Pre-flop showdown probability)

N(S) = Number of starting hands in S

N(t) = Number of instances of t. N(t) is introduced to allow us to regroup the summation.

t which referred to a specific hand such as Jª8§ (1 instance), can now refer to a

set of hands such as J8 (12 instances). The choice depends upon how we want to

perform the summation and how we can compute the corresponding showdown

probabilities.

Equation A5 is conceptually easier to understand and more useable with existing poker calculators. However, Equation A4 would be a cleaner implementation.
B) Player U acts first going all-in.

Should Player U go all-in, pre-flop, raising the pot by r? Player U is getting pot odds of p/r. Player U assumes that Player T will only call him if Player T holds a hand t where t Î S. Otherwise Player T will fold.

The expected values for Player U are:

EV( Player U checks) = 0

EV( Player U raises all-in) = p x Prob( Player T folds )

+ (p+r) x Prob( Player T calls and loses )

+ (p/2) x Prob( Player T calls and ties )

- r x Prob( Player T calls and wins ) Eq. B1

Player U should raise all-in when

EV( Player U raises all-in) > EV( Player U checks)

(p/r) x Prob( Player T folds ) + ((p/r) + 1) x Prob( Player T calls and loses )

+ (p/r) x (1/2) x Prob( Player T calls and ties )

- Prob( Player T calls and wins ) > 0 Eq. B2

Solving for the pot odds p/r, Player U should raise all-in when

p/r > [ Prob( Player T calls and wins ) - Prob( Player T calls and loses ) ] /

[ Prob( Player T folds ) + Prob( Player T calls and loses ) +

½ Prob( Player T calls and ties ) ] Eq. B3

Prob( Player T folds ) = N(~S) / N(W)

= ( N(W) – N(S) ) / N(W)

Prob( Player T calls and loses ) = Prob( Player T calls )

x Prob( Player T loses ÷ Player T calls )

= Prob( Player T calls ) x Prob( Player T loses ÷ t Î S )

= ( N(S) / N(W) ) x S N(t) x Prob( t+c < u+c) / N(S)

for all t Î S

= S ( N(t)/N(W) ) x Prob( t+c < u+c )

for all t Î S

Prob( Player T calls and ties ) = S ( N(t)/N(W) ) x Prob( t+c = u+c )

for all t Î S

Prob( Player T calls and wins ) = S ( N(t)/N(W) ) x Prob( t+c > u+c )

for all t Î S

N(W) = 1225

We note that the options considered for Player U were either an all-in raise or to check. Player U could have chosen to make a smaller raise instead, and its expected value could have exceeded the all-in raise. However, computing that value is too difficult. Similarly, the expected value of Player U checking cannot really be determined. We assume Player U checks, and if raised he folds. Our criterion is, will the expected value of the all-in raise be profitable (non-zero)?
Simplifying S

Equation A2 gives the mathematical criteria for Player U to call Player T’s pre-flop all-in bet. The equation is in terms of Player U’s pot odds (p+r)/r, Player T’s assumed set S of playable all-in starting hands, and Player U’s particular hole cards u.

Similarly, Equation B2 gives the mathematical criteria for Player U to push all-in pre-flop. The equation is in terms of Player U’s pot odds p/r, Player T’s assumed set S of callable starting hands, and Player U’s particular hole cards u.

Digressing for a moment, for each of the 169 (ignoring suits) possible Texas Hold’em starting hands, we can compute the probability of wins, ties, and losses against a single random opponent. The hands are played to showdown, with no player folding. We can then compute a pot equity value of PE=Prob(wins)+ ½P(ties ) or a return on investment of RI= Prob(wins)-Prob(losses). The 169 starting hands can then be ranked by the pot equity or return on investment. (See Table 1.) Note that a linear relationship exists where RI=2 PE –1 since Prob(wins)+Prob(ties)+Prob(losses)=1. Therefore both rankings lead to the same ordering. This hand ranking forms a rational basis for starting hand selection for two-player all-in confrontations.

Returning to the set S of playable hands for Player T, we shall now further assume that S contains only the strongest starting hands ranked by the pot equity as described above. Consequently, S can be uniquely determined by selecting only its size as a percentage of the 1326 starting hands. S(1.36%) indicates the set {AA, KK, QQ}, see Table 2. Note the probability of being initially dealt one of these hands is 1.36%.

Referring to the earlier example, with hole cards u = AQ, the 59-member set S is denoted as

S(5.43%) = {AA, KK, QQ, JJ, TT, 99, 88, AKs, 77, AQs, AJs, AK}.

We agree that our notation and naming convention refers not to the full 72-member set but the 59-member set. This is done for ease of identification since we can readily refer to Table 1 or 2 to quickly see which hands are involved when a ‘percentage’ is called out. We must however mentally account for conflicts with Player U’s hole cards.

Player U’s profile or read of Player T is now reduced to one variable, the ‘percentage’ of the 1326 starting hands. Equations A2 and B2 are now in terms of only Player U’s pot odds, Player T’s ‘percentage’ of the 1326 starting hands, and Player U’s particular hole cards.

Thus Player U can now

1) Knowing his hole cards u, knowing his pot odds; can deduce how loose a percentage of starting hands Player T must play for him (Player U) to be profitable.

2) Knowing his hole cards u, knowing Player T’s percentage of playable hands, can deduce the pot odds needed to make a profit.

3) Knowing his pot odds, knowing Player T’s percentage of playable hands, can determine if his hole cards are strong enough to play.

The critical assumption made above is that:

Player U assumes Player T is playing uniformly the ‘strongest’ starting hands based on the pot equity rankings for two player games played to showdown.

This assumption is valid even for the ‘loosest’ of players, simply by selecting a larger ‘percentage’ of playable hands. For a player naively unaware of hand selection at all or for one that is bluffing, one would select S(100%). The assumption is invalid if Player T is biased towards a different rank ordering of cards. For example, Player T may feel that the 4ª is his lucky card and will go all-in with it and any other card.

At the final table of a tournament, Player T would most likely be a competent player making rational judgments on hand selection using a rank ordering not too dissimilar to the one in Table 1. So, our assumption of using the pot equity rankings seems reasonable. (The reader is free to substitute his own ranking system as he sees fit.)

Table 1 - Texas Hold’em Heads Up (Two Player) Hand Rankings Base on Pot Equity

This table presents the probability of each starting hand winning, tying, or losing in a two-player game against a single random hand at showdown. The hands are ranked by pot equity, Prob(wins) + ½ Prob(ties). Num refers to the number of distinct hands taking suits into consideration. Cum refers to the cumulative number of hands in the table. The cumulative ‘Percentage’ given is the probability of being dealt that hand or better.

Data for this table has been adapted from http://gocee.com/poker/he_ev_comp.html.

Hand	Prob(wins) %	Prob(ties) %	Prob(losses) %	Pot Equity %	Num	Cum	‘Percentage’
AA	84.9319154847766	0.543595634648892	14.5244888805745	85.203713302101	6	6	0.45
KK	82.1173351632582	0.55668924705531	17.3259755896864	82.3956797867859	6	12	0.90
QQ	79.6319900090219	0.586348104122651	19.7816618868555	79.9251640610832	6	18	1.36
JJ	77.1529795109814	0.632986780337117	22.2140337086815	77.4694729011499	6	24	1.81
TT	74.660264885255	0.703030131403331	24.6367049833417	75.0117799509566	6	30	2.26
99	71.6656519698676	0.783199950571432	27.5511480795609	72.0572519451534	6	36	2.71
88	68.7173824369543	0.891306064095809	30.3913114989499	69.1630354690022	6	42	3.17
AKs	66.2196077236714	1.65004917112754	32.130343105201	67.0446323092352	4	46	3.47
77	65.7253536516785	1.02133828610636	33.2533080622152	66.2360227947317	6	52	3.92
AQs	65.3136986832969	1.79032742803061	32.8959738886725	66.2088623973122	4	56	4.22
AJs	1.99011676545706	1.99011676545706	33.6122625850722	65.3926790321993	4	60	4.52
AK	64.4693779342253	1.70138770895346	33.8292343568212	65.320071788702	12	72	5.43
ATs	63.488915185955	2.22694320348609	34.2841416105589	64.602386787698	4	76	5.73
AQ	63.5087873486512	1.8461040486612	34.6451086026876	64.4318393729818	12	88	6.64
AJ	62.5351750909766	2.0561656417676	35.4086592672558	63.5632579118604	12	100	7.54
KQs	62.4084198476296