Hold'em or Fold'em: Validating a Winning Strategy

Introduction

In previous posts (here and here) I analyzed simulated hold'em poker hands to determine a winning strategy.  Bluffing aside, when should you play and when should you fold in order to have the best chances of winning?

The observations I made do not readily yield a single strategy for play.  They may sometimes conflict or be overly conservative when combined.  For example, my advice for hole cards may offer a more than 50% chance of winning on its own, but when combined with the advice on hand rankings (which also offers a better than 50% chance of winning) it may require the folding of many winning hands.

Still, we need a place to start.  Perhaps the principles I generated can be condensed, simplified, reorganized, and summed up in this way:

A Strategy

• Hole cards
• If your hole cards form a pair, contain an Ace, or have a rounded average rank greater than or equal to 10...
• Continue playing
• Else
• Fold
• Flop, Turn, River
• If you have two pair 5s or better, one pair Queens or better, or three of a kind or better
• Continue playing
• Else
• Fold

Will this strategy work?  Let's validate it by simulating a game against one other player.

The Program

The program remains largely the same as before, except that I have now implemented the above logic into the program.  If one folds on the hole cards, then this is tracked as "Fold - hole cards".  If one passes this test but fails on the flop, turn, or river, this is tracked as "Fold - Flop, Turn, River".  If one passes both tests, then this is tracked as "Play".

I reran 100,000 simulations of hold'em deals to Player1 and Player2.  A successful strategy will have a high percentage of Win/Play combos along with a high percentage of Lose/Fold combos.  That is, we want to play when we will win, and fold when we will lose.  If our strategy is not successful, then there will be a high percentage of Win/Fold combos (we folded when we would have won the hand) and Lose/Play combos (we played when we would have lost).

In other words, the ideal simplified scenario can be represented by the below confusion matrix:

	Play	Fold
Win	50	0
Lose	0	50

In the above scenario, a win is as equally likely as a loss, so each occurs 50% of the time, adding to 100%.  However, we have predicted perfectly, so we won all the hands we have played, and lost only the hands we have folded.  Assuming we have been betting well (betting minimally with hands we will fold, betting maximally with hands we will win), the eventual result will be that we win the game and all of the money.

What does the actual confusion matrix look like?

Validation

The results of implementing this strategy are below:

Player1 Result/Action	Fold - Flop, Turn, River	Fold - hole cards	Play	Grand Total
Lose	10773	32593	4548	47914
Tie	326	3044	816	4186
Win	7158	24708	16034	47900
Grand Total	18257	60345	21398	100000

The table above gives the final result for Player1's hand (left side) and Player1's action based on the simplified principles we gave him (top).  The combinations are displayed in the middle, with row/column totals on the ends.

A simplified version of the table:

Player1 Result/Action	Fold	Play	Grand Total
Lose	43366	4548	47914
Tie	3370	816	4186
Win	31866	16034	47900
Grand Total	78602	21398	100000

Some observations:

• As we would expect, Player1's hands lose and win about equally
• Player1 wins about 75% of the hands he plays
• Player1 only plays 21% of the hands
• Of all the hands Player1 would have lost, he only plays 9% of them
• Of all the hands Player1 would have won, he only plays 33% of them
• 40% of Player1's folds were winning hands
• Player1 only plays 19% of ties.

Strengths of this approach:

• Player1 plays very few of the games he will lose
• The strategy is simple
• The strategy relies on information known at the time of decision

Weakness of this Approach

• Player1 folds quite a few games in which, if he had played, he would have won
• Player1 folds most of the games in which there is a tie (and would split the pot)
• Player1 plays very few games

This is definitely a conservative strategy.  The result of combining the various insights we have gathered in previous posts is too cautious.  Player1 only plays when there is a great chance of winning.  Perhaps with good betting and bluffing, this can be a very effective strategy.  However, it folds a lot of winning hands, in fact, most winning hands. 

The strategy misclassifies (ignoring ties) 38% of all hands.  Counting ties (assuming you should play ties), it misclassifies 40% of all hands, that is, 40% of the time it recommends a fold when you should play and recommends that you play when you should fold.

Is there a better way?  Most certainly.  Let's use machine learning to analyze our data and come up with a more effective model.

Machine Learning: Decision Tree

I use a decision tree to develop a model for hold'em strategy because it is easy to understand the decision paths.  Here is a plot of the model along with the associated splits and ratios of classification:

The details may be difficult to see, so here is the decision tree strategy with words:

• Flop, Turn, River
• If Player1 has two pair or better
• Play
• Else (Player1 has one pair or high card)
• If Player1 has a high card hand
• Fold
• Else (if Player1 has one pair)
• If both of the cards that make up the one pair are in the community cards
• Fold
• Else (if only 0 or 1 of the cards that make up the one pair are in the community cards)
• Play

This model has only 27% misclassification, that is, it predicts a loss when you really win or predicts a win when you really lose, about 27% of the time.  The confusion matrix is below for play/fold and win/lose/tie combinations:

Player1 Result/Action	Fold	Play	Grand Total
Lose	29591	18323	47914
Tie	1452	2734	4186
Win	7429	40471	47900
Grand Total	38472	61528	100000

Some observations:

• Player1 wins about 66% of the hands he plays
• Player1 plays 62% of the hands
• Of all the hands Player1 would have lost, he plays 39% of them
• Of all the hands Player1 would have won, he plays 84% of them
• Only 19% of Player1's folds were winning hands
• Player1 plays 65% of ties.

Strengths of this approach:

• Most folds are losses; most plays are wins
• The strategy is simple

Weakness of this Approach

• The strategy relies on information not known at time of decision (at least before flop)
• Player1 plays too many hands (in particular, too many losing hands)

This strategy is much more effective.  However, it offers no guidance for the hole cards, recommending that one should always play at least through the flop.

Do the hole cards really matter in an overall strategy?  Looking at another decision tree using only the hole card information (max rank, min rank, average rank, pair (yes/no)) reveals that, at best, the model will still misclassify about 43% of the time.  A simple split based on the average rank of the hole cards being >= 7.5 yields only a slight advantage in winning (54%) and correctly classifies only 45% of the time. 

If we force the previous decision to use the hole card information to its full extent, the model becomes incredibly complex (over 1000 nodes) and only improves to about 26% misclassification.  The decision tree looks like this:

Clearly, this isn't helpful.  But we know hole cards are important to some degree, so shouldn't this count for something?  In addition, this strategy plays too many hands, and as a result, plays too many losing hands.

A Compromise

What should we do now?  If we compare the first strategy, we see that the major problem is that one folds too much, particularly, on hands that would have been won (the lower left corner green box).  It plays too little.  In contrast, the decision tree model plays too much, in particular, it plays too many hands that it will lose (the upper right yellow box).

Is there a way to combine both strategies in a way that is more effective than either, is still simple, plays about 50% of the time, and provides more guidance at each stage of the game?  I combined both approaches and played with the logic until coming up with a more balanced strategy:

• Hole Cards
• If the rounded average rank of the hole cards is >= 6 or the hole cards form a pair
• Continue Playing
• Else
• Fold
• Flop, Turn, River
• If you have two pair or better
• Continue Playing
• Else (one pair or high card)
• If one pair, only 0 or 1 of the cards that make up the one pair are in the community cards, and the pair is 5s or better
• Continue playing
• Else (high card, one pair of 4s or worse, or one pair with both pair cards in the community)
• Fold

The confusion matrix is as follows (I validated the approach against a test set and the results were virtually the same):

Player1 Result/Action	Fold	Play	Grand Total
Lose	35010	12904	47914
Tie	2005	2181	4186
Win	13500	34400	47900
Grand Total	50515	49485	100000

Some observations:

• Player1 wins about 70% of the hands he plays
• Player1 plays 49% of the hands
• Of all the hands Player1 would have lost, he only plays 27% of them
• Of all the hands Player1 would have won, he plays 72% of them
• 26% of Player1's folds were winning hands
• Player1 plays 52% of ties.
• The strategy misclassifies 28% of the time.

Strengths of this approach:

• Most plays are wins; most folds are losses
• The strategy is still fairly simple
• The strategy relies more on information known at the time of decision
• Player1 plays about half the time
• The misclassification is low.

This approach does not suffer the weakness of the other two approaches, but represents a balanced compromise between them. While the initial strategy is too conservative, the decision tree strategy is too liberal.  This compromise still has good predictive power, is simple, and useful, and represents a middle ground strategy.

Conclusion

As the foregoing shows, there is no single master strategy for winning at poker.  Any strategy will have strengths and weaknesses.  And different strategies may, for this reason, appeal to different people.  What I have shown above are some data driven strategies designed to maximize strengths and minimize weaknesses, but how they do so differs.  Feel free to choose the strategy (or modified suitably) that appeals to you.

This concludes my analysis of poker hands.  For now at least....

Good luck!

Hold'em or Fold'em: Breaking the Tie and Other Insights

Introduction

In a previous post, I looked at the likelihood of winning with the various kinds of hold'em poker hands one could be dealt.  For simplicity, I ignored ties.  Having had some time to develop my code, I now wish to add tie breaking into the analysis, along with other new insights gathered from capturing more detailed data.  The result will be a more refined set of observations useful for winning at hold'em.

Where does tie breaking matter?  Let's take a look at the chart of Wins, Ties, and Losses against one other player without tie breaking implemented.  This consists of 100,000 simulations.

Player1	Lose	Tie	Win	Lose/All	Tie/All	Win/All
1.Royal Flush	0	0	23	0.00	0.00	1.00
2. Straight Flush	0	5	140	0.00	0.03	0.97
3. Four of a Kind	0	24	150	0.00	0.14	0.86
4. Full House	40	409	2193	0.02	0.15	0.83
5. Flush	80	620	2121	0.03	0.22	0.75
6. Straight	150	768	3050	0.04	0.19	0.77
7. Three of a Kind	733	1001	3184	0.15	0.20	0.65
8. Two Pair	3274	8857	11148	0.14	0.38	0.48
9. One Pair	14830	20968	8441	0.34	0.47	0.19
10. High Card	11655	6136	0	0.66	0.34	0.00

Against one player, tie breaking does not seem to matter as much for a three of a kind or better.  These hands win most of the time (greater than 50%), so if one always played when one had one of these hands, one would win on average.  Similarly, tie breaking does not matter much for a single high card hand since this hand loses most of the time.

Tie breaking does seem to matter for one pair and two pair, however.  Tying is very common for both, and no result (win, lose, tie) occurs more than 50% of the time.  So getting a more refined look at one pair and two pair hands should provide better guidance for what to do with these sorts of hands.

With Tie Breaking

Now consider the updated table against one other player which includes tie-breaking for all types of hands.

Player1 - Tie Breaking	Lose	Tie	Win	Lose/All	Tie/All	Win/All
1.Royal Flush	0	0	23	0.00	0.00	1.00
2. Straight Flush	0	4	141	0.00	0.03	0.97
3. Four of a Kind	11	5	158	0.06	0.03	0.91
4. Full House	160	182	2300	0.06	0.07	0.87
5. Flush	352	89	2380	0.12	0.03	0.84
6. Straight	150	768	3050	0.04	0.19	0.77
7. Three of a Kind	1165	157	3596	0.24	0.03	0.73
8. Two Pair	7079	1212	14988	0.30	0.05	0.64
9. One Pair	24689	1387	18163	0.56	0.03	0.41
10. High Card	14552	418	2821	0.82	0.02	0.16

What has changed?  Obviously, ties are much more rare.  These now only occur when player1 and player2 have the exact same hand (as opposed to the same type of hand, as was previously the case). All hands tie below 7% except the straight, which ties at 19%.

The better hands got better and the worse hands got worse.  Everything above and including three of a kind improved in odds of winning.  On the other hand, the high card  greatly increased in its chances of losing.  Of particular importance given the previous discussion, the two pair now wins the majority of the time (64%) while the one pair loses the majority of the time (56%). 

Thus, based on this updated information, it would appear that against one other player one should (in general):

• play when one has a two pair or better
• fold when one has a one pair or worse

The Tie-Breaker

In the event of a tie in type of hand, what will break the tie?  First, note that 62% of the time, there will not be a tie in the type of hand.  However, in 19% of hands, the high card is what breaks the tie.  After that, the single one pair breaks the tie at 9%, the first pair in a two pair at 4%, and the tie is not broken at 3%.

Tie Breaker	Count
No tie	61980
High Card	19394
One Pair	9438
First Pair	3575
Tie	3454
Second Pair	1915
Pair	161
Three of a Kind	83

Clearly the high card is very important in the event of a tie in type of hand, breaking about half of all ties.  How high of a high card do you need to break the tie?

Tie Breaker - Key Cards: High Card	Lose	Win	Win%
14	0	2629	100.00
13	1204	2110	63.67
12	1614	1543	48.88
11	1601	1168	42.18
10	1430	846	37.17
9	1192	581	32.77
8	973	370	27.55
7	700	235	25.13
6	470	104	18.12
5	326	42	11.41
4	131	9	6.43
2	61	0	0.00
3	55	0	0.00

It looks like a King (13) or better is needed as a high card tie breaker to win more than 50% of the time when depending on this to break the tie.  Let's look at the two types of hands where a tie-breaker matters the most: the one pair and the two pair.

One Pair

Consider the table below of Tie Breakers for Player1's one pair hands:

Tie Breaker - One Pair	Count	Percentage
Not a tie	23271	52.6
1. One Pair	9438	21.3
3. High Card	5307	12.0
4. High Card	3196	7.2
5. High Card	1640	3.7
6. Tie	1387	3.1

In most cases, a one pair does not tie (53%).  After that, it is the pair itself that breaks the tie at 21%, and the highest single card at 12%.  Having a good high card can often prevent a tie or loss when one has a one-pair hand. 

What is a good pair to break a tie with?  Not surprisingly, the higher the pair, the better the chances of breaking the tie in your favor are.  Looking at the table below, we can see that a pair of Aces always wins (722-0) whereas as a pair of 2s always loses (0-785). 

Tie Breaker - Key Cards: One Pair	Lose	Win
14	0	722
13	60	660
12	97	645
11	184	547
10	228	482
9	280	451
8	363	352
7	385	271
6	485	232
5	523	201
4	662	88
3	684	51
2	785	0

Key takeaway:

• One needs to have a pair of 9s or better to win the tie break most of the time.

What about the high card?  Again, not surprisingly, the higher the kicker, the better the chances are of winning the tie break.  The table below shows that, with an Ace breaking the tie, one always wins.  What is odd is the Lose column.  What is going on here?

Tie Breaker - First High Card	Lose	Win
14	0	1388
13	607	692
12	708	335
11	585	145
10	393	54
9	225	21
8	109	1
7	32	3
6	9	0

The lose column consists of all those times when Player2 has a higher kicker than Player1.  So for example, the 708 losses with a Queen (12) consist of all instances where Player2 had a King or Ace.  What is amazing is that one needs to have a King or Ace in order to win with the kicker more often than not. 

Key takeaway

• Don't rely on anything less than a King kicker to win a one-pair tie break depending on the kicker.

Combining these results, we have this takeaway:

• To win a one-pair tie break more than 50% of the time, have at least a pair of 9s and a high card of a King or better.

Two Pair

Consider the table below of Tie Breakers for Player1's two pair hands:

Tie Breaker - Two Pair	Count	Percentage
Not a Tie	14422	61.95
1. First Pair	3575	15.36
3. High Card	2155	9.26
2. Second Pair	1915	8.23
3. Tie	1212	5.21

Most of the time, a two-pair does not tie.  However, 15% of the time it does tie and it is the first pair that breaks the tie.  Interestingly enough, it is the kicker that breaks the tie next most often at 9%, followed by the second pair at 8%. The kicker is slightly more important than the second pair in breaking ties.

Not surprisingly, the better the first pair, the more likely one will win the tie break. 

Tie Breaker - Key Cards: First Pair	Lose	Win
14	0	412
13	69	355
12	125	294
11	171	212
10	205	163
9	225	154
8	246	85
7	218	48
6	191	41
5	161	18
4	116	4
3	62	0

It seems one needs a first pair of Jacks to have a better than 50% chance of winning the tie break.  The lose column is a little difficult to understand at first, but it is combining the likelihood of having a first pair of a certain rank along with the chance of losing.  So for example, it is much more unlikely to have a first pair of 3s than a first pair of Aces because Aces can be a first pair to a second pair of Kings, Queens, etc.  On the other hand, a first pair of 3s can only have a second pair of 2s.  This combination is much less likely to occur.  Also, since a first pair of 3s is the lowest possible first pair possible, it is never going to win a tie break with another two pair hand, whereas a first pair of Aces will never lose a tie break with another two pair hand.

Key takeaway:

• A pair of Jacks is needed as a first pair to win a tie break most of the time against another two pair hand.

What about the kicker?  Again, the better the kicker, the more likely one is going to win when using that kicker.  A Queen or better is needed to win with the kicker more than 50% of the time.  And as above, it is much more unlikely to have a lower kicker than a higher kicker.

Tie Breaker - Key Cards: High Card	Lose	Win
14	0	335
13	76	293
12	129	171
11	143	131
10	134	81
9	117	60
8	104	30
7	73	10
6	71	7
5	52	6
4	37	1
2	53	0
3	41	0

Key takeaway:

• A Queen kicker is needed to win a tie break most of the time against another two pair hand when relying on the kicker to break the tie.

Now to the second pair.  Why is the second pair less important than the kicker?  Consider the table below, sorted by second pair rank descending.

Tie Breaker - Key Cards: Second Pair	Lose	Win	Win%
13	0	37	100.0
12	5	76	93.8
11	15	95	86.4
10	19	106	84.8
9	40	118	74.7
8	46	107	69.9
7	91	103	53.1
6	95	110	53.7
5	111	77	41.0
4	151	57	27.4
3	176	43	19.6
2	237	0	0.0

Because this is the second pair, it is very unlikely that one will have, for example, a pair of Kings as a second pair, because there is only the possibility of having a pair of Aces for a first pair in that case.  In contrast, a second pair of 2s can have 3s, 4s, on up as a first pair, so this is much more likely to be the case.  Thus, low second pairs are much more common than high second pairs.

In addition, in most cases, a two-pair hand will require that one of the pairs is in the community cards.  Doing an analysis on two pair hands reveals that 48% of the time, both first pair cards are in the community cards, 47% of the time one of the first pair cards is in the community cards, and only 5% of the time will no first pair cards be in the community cards. In two pair ties, this changes to 63% for both, 31% for one, and 5% for no first pair cards in the community cards.

For second pair cards, 53% of the time both cards are in the community, 43% of the time one card is in the community, and 4% of the time no cards are in the community.  When looking only at ties, this changes to 67% for both cards, 29% for 1 card, and 4% for no cards.

In other words, when there is a tie, most of the time the first pair will be in the community cards.  For all two pair hands, most of the time the second pair is in the community cards, and with ties, 2/3 of the time the second pair will be in the community cards. 

Combining, these results in the table below, we can observe that 83% of time for all two pair hands for Player1, one of the pairs is in the community.

Two Pair: First Pair Cards in Community	Two Pair: Second Pair Cards in Community	Count	Percentage
1	2	7166	30.78
2	1	6030	25.90
2	2	4183	17.97
1	1	3859	16.58
0	2	1096	4.71
2	0	945	4.06

For two-pair ties, 96% of the time one of the pairs is in the community cards.

Two Pair: First Pair Cards in Community (Ties)	Two Pair: Second Pair Cards in Community (Ties)	Count	Percentage
2	2	3003	33.91
1	2	2433	27.47
2	1	2190	24.73
0	2	466	5.26
2	0	409	4.62
1	1	356	4.02

We now have an answer to our original question: why is the second pair less important than the kicker in breaking a tie?  The answer is that when there is a two-pair kind tie, 96% of the time one of the pairs will be shared.  And chances are that it will be the second pair and second pairs tend to be low in rank.  And since a low ranked, likely shared second pair is not likely to break the tie, we must move to the kicker in order to do so.

Key Takeaway:

• Don't depend on the second pair to break a tie.
• However, a second pair of 6s or better is needed to win a tie break when depending on the second pair to do so.

Combing these takeaways, we can sum up:

• To win a two-pair tie break more than 50% of the time, have at least a first pair of Jacks, at least a second pair of 6s, and a high card of a Queen or better.

To conclude this section, if you suspect you are in a tie break situation...

• have at least a King high card to break the tie
• One pair: a pair of 9s or better with a King kicker is needed to break the tie in your favor more often than not
• Two Pair: a pair of Jacks, a pair of 6s, and a high card Queen, or better, is needed to break the tie in your favor more often than not

Dealt Hole Card Odds

But how does one know which kind of hand one will have until all the cards have been dealt?  One can't know most of the time.  So here are some other insights to help in judging whether or not to play or fold.

Consider the cards that are dealt to player1 (the hole cards).  Below is a heat map of winning percentage against one other player based on the rank of the cards dealt (Note: 14="Ace", 13="King", 12="Queen", and 11="Jack").  I am ignoring suit for simplicity.

Win%	2	3	4	5	6	7	8	9	10	11	12	13	14
2	55	31	29	32	27	34	36	41	44	42	43	47	53
3	32	55	29	31	34	35	34	41	40	44	49	50	56
4	31	29	51	33	35	35	39	36	40	45	47	49	53
5	31	32	35	59	38	38	41	42	44	45	50	54	55
6	33	32	36	38	59	42	40	43	46	47	49	49	60
7	31	35	38	38	40	65	43	48	45	50	49	53	55
8	34	34	38	38	41	46	66	44	50	48	51	56	61
9	39	39	40	40	41	44	44	75	50	52	53	58	61
10	38	39	43	44	44	42	50	51	77	53	57	62	62
11	48	45	44	47	45	47	51	54	51	78	61	61	58
12	44	45	51	47	49	51	53	53	56	57	81	62	62
13	48	51	50	51	54	50	54	58	60	58	56	84	63
14	54	53	56	57	58	58	58	57	61	58	62	63	85

The heat map changes color from red (low values) to yellow (middle values) to green (high values). The heat map is roughly symmetric along the diagonal, since the order of hold cards doesn't matter (e.g., 14, 13 = 13, 14).  The greener, the better.

As we would expect, the higher the cards are, the better the chances of winning are.  The exception to this is the diagonal, representing pairs that are dealt as hole cards (e.g., 2,2).  These are much better than similar hole card ranks that are not pairs (e.g., 2,3).

For simplicity, let's just look at which hole cards win 50% or more of the time.  Here is that map:

Win%	2	3	4	5	6	7	8	9	10	11	12	13	14
2	55	31	29	32	27	34	36	41	44	42	43	47	53
3	32	55	29	31	34	35	34	41	40	44	49	50	56
4	31	29	51	33	35	35	39	36	40	45	47	49	53
5	31	32	35	59	38	38	41	42	44	45	50	54	55
6	33	32	36	38	59	42	40	43	46	47	49	49	60
7	31	35	38	38	40	65	43	48	45	50	49	53	55
8	34	34	38	38	41	46	66	44	50	48	51	56	61
9	39	39	40	40	41	44	44	75	50	52	53	58	61
10	38	39	43	44	44	42	50	51	77	53	57	62	62
11	48	45	44	47	45	47	51	54	51	78	61	61	58
12	44	45	51	47	49	51	53	53	56	57	81	62	62
13	48	51	50	51	54	50	54	58	60	58	56	84	63
14	54	53	56	57	58	58	58	57	61	58	62	63	85

This more clearly shows which hole card combos win more often (green) and which do not (red).  The winning hole cards form an arrow, including the diagonal (all pairs) and the lower right corner.  Based on the above, we can offer another set of general principles.

• if your hole cards form a pair (e.g., 2,2), you should play
• if you have an Ace (14), you should play
• for hole cards of rank1 and rank2, if rounding the result of sqrt((rank1)^2 + (rank2)^2) is >= 14, then you should play.
• Note: since this is hard to do mentally, taking the rounded average of rank1 and rank2 >= 10 works almost as well.

The result of applying these principles looks like this, and is a close mapping to the above result (Note: the value is the rounded result of sqrt((rank1)^2 + (rank2)^2)):

Win%	2	3	4	5	6	7	8	9	10	11	12	13	14
2	3	4	4	5	6	7	8	9	10	11	12	13	14
3	4	4	5	6	7	8	9	9	10	11	12	13	14
4	4	5	6	6	7	8	9	10	11	12	13	14	15
5	5	6	6	7	8	9	9	10	11	12	13	14	15
6	6	7	7	8	8	9	10	11	12	13	13	14	15
7	7	8	8	9	9	10	11	11	12	13	14	15	16
8	8	9	9	9	10	11	11	12	13	14	14	15	16
9	9	9	10	10	11	11	12	13	13	14	15	16	17
10	10	10	11	11	12	12	13	13	14	15	16	16	17
11	11	11	12	12	13	13	14	14	15	16	16	17	18
12	12	12	13	13	13	14	14	15	16	16	17	18	18
13	13	13	14	14	14	15	15	16	16	17	18	18	19
14	14	14	15	15	15	16	16	17	17	18	18	19	20

As might be expected, the heatmap of losing percentage looks like the opposite of the heatmap for winning percentage:

Lose%	2	3	4	5	6	7	8	9	10	11	12	13	14
2	44	63	66	62	66	61	58	55	51	53	52	50	43
3	62	44	67	63	62	60	59	54	56	52	47	45	42
4	64	65	48	59	59	58	55	59	56	50	49	46	43
5	64	61	59	40	56	55	53	54	51	51	45	43	41
6	64	62	57	55	39	53	54	52	50	48	46	48	38
7	63	61	56	56	53	34	53	48	50	46	47	42	41
8	60	59	56	56	53	47	33	51	46	48	47	40	36
9	56	56	56	56	54	51	52	25	47	44	43	39	37
10	56	56	53	52	52	53	45	45	23	43	41	36	35
11	49	49	52	49	51	48	45	42	46	21	38	36	40
12	51	50	45	49	46	45	43	44	41	39	19	35	37
13	48	45	47	43	42	46	41	40	38	39	40	16	35
14	43	43	39	40	39	39	38	41	36	39	37	34	15

Now consider the tying heatmap.  One is extremely unlikely to tie with a pair (the diagonal).  One is more likely to tie (6-8%) in the lower ranks (2-8) and less likely to tie (1-5%) in the upper ranks (9-Ace).

Tie%	2	3	4	5	6	7	8	9	10	11	12	13	14
2	1	6	5	6	7	5	6	4	5	5	5	3	4
3	6	0	4	6	4	5	7	5	4	4	4	5	3
4	5	6	2	8	6	7	6	5	4	5	4	6	4
5	5	7	6	1	6	7	6	5	5	4	4	4	4
6	3	6	7	7	1	4	7	5	5	5	5	3	3
7	6	4	7	6	7	1	4	5	5	4	4	5	4
8	6	7	6	6	6	7	1	5	4	4	2	4	2
9	6	5	4	4	6	5	5	0	4	3	4	3	2
10	6	5	5	4	5	5	5	5	0	4	2	2	3
11	3	6	4	5	4	5	3	4	3	0	2	3	2
12	5	5	3	4	5	3	4	3	3	3	1	3	2
13	4	4	3	6	4	5	5	2	3	3	4	1	1
14	4	4	5	4	3	3	4	2	3	3	2	3	0

Key Cards

By key cards, I mean the rank of the cards that are used to determine the hand or determine a tie break.  For example, a full house of 3-3-3-J-J would have a key card value of 3.  If there is a tie with an opponent of 3-3-3-9-9, then the key card value would be 11 (for Jacks).  Can looking at hands this way tell us anything useful?  Consider the table below:

Player1 KeyCards	Lose	Tie	Win	Win%	Lose%	Tie%	Win/(Win+ Lose)	Lose/(Win+Lose)
14	8283	493	8056	47.86	49.21	2.93	49.31	50.69
13	6122	387	6766	50.97	46.12	2.92	52.50	47.50
12	4785	412	5829	52.87	43.40	3.74	54.92	45.08
11	4169	437	4977	51.94	43.50	4.56	54.42	45.58
10	3568	460	4248	51.33	43.11	5.56	54.35	45.65
9	3262	480	3698	49.70	43.84	6.45	53.13	46.87
8	3086	519	3118	46.38	45.90	7.72	50.26	49.74
7	2888	435	2765	45.42	47.44	7.15	48.91	51.09
6	2656	282	2374	44.69	50.00	5.31	47.20	52.80
5	2528	137	1754	39.69	57.21	3.10	40.96	59.04
4	2326	70	1495	38.42	59.78	1.80	39.13	60.87
3	2222	44	1309	36.62	62.15	1.23	37.07	62.93
2	2263	66	1231	34.58	63.57	1.85	35.23	64.77

The table is sorted by key card rank descending.  Here are some observations:

• The higher the rank, the more likely the key card.  In other words, key cards are heavily distributed towards the higher ranks, and then decrease towards the lower ranks.
• Excepting the Ace, the Win% decreases by generally following rank, that is, the higher the rank, the higher the Win%. 
• A rank of 10 or better is needed to win more than 50% of the time (Aces are the exception).
• The middle ranks tie the most (8 at 8% and 7 at 7%).

This is mostly intuitive.  We would expect the higher ranks to win most often.  We would expect the distribution of key cards to favor the higher ranks (because the key cards are usually the highest ranked cards in the hand that determine the kind of hand).  We would expect most ties to occur in the middle, since higher cards are more likely to win and lower cards are likely to lose.

What remains to be explained is the Ace.  Why is it more likely to lose than win as far as key cards go?  The reason is that the Ace is very likely to be the key card for a high card hand coupled with the fact that high card hands lose most of the time.  Consider the below table:

Player1 KeyCards: High Card Hand	Count	% of Total
14	7251	40.76
13	4186	23.53
12	2297	12.91
11	1369	7.69
10	801	4.50
9	562	3.16
8	461	2.59
7	391	2.20
6	256	1.44
5	158	0.89
4	59	0.33

Ace key cards account for 40% of all high card  hands.  High card hands lose about 82% of the time (from the table near the beginning).  So it should be no surprise that a very high percentage of Ace key cards lose.  In fact, by observing the table below, we see that this is the only instance in which an Ace key card loses most of the time.  Of all the hands in which an Ace is the key card, 78% of the losses occur when the hand is a High Card hand. 

Player1 - Ace Key Card	Lose	Tie	Win	Win%
1.Royal Flush	0	0	23	100.000
5. Flush	36	0	880	96.070
3. Four of a Kind	0	1	13	92.857
4. Full House	5	15	193	90.610
7. Three of a Kind	43	0	342	88.831
6. Straight	18	106	480	79.470
8. Two Pair	561	371	2567	73.364
9. One Pair	1097	0	2830	72.065
10. High Card	6523	0	728	10.040

Key takeaway:

• Key card rank alone is not very useful in determining win% for a hand.
• A key card of rank 10 or better is slightly more likely to win than lose.

Redeeming the One Pair and High Card Hands?

Only the one pair and high card hands lose most of the time when playing against one other player.  But are there types of one pair hands and high card hands that win more than 50% of the time?  By observing the key cards for these two kinds of hands, we discover that in no circumstance is a High Card hand likely to win more than 50% of the time:

Player1KeyCards: High Card Hand	Lose	Tie	Win	Win%
14	6523		728	10.04
13	3584		602	14.38
12	1884		413	17.98
11	1024		345	25.20
10	533		268	33.46
9	331	44	187	33.27
8	232	98	131	28.42
7	186	114	91	23.27
6	127	89	40	15.63
5	93	49	16	10.13
4	35	24		0.00

The best is 33%.  One is still better off folding in any circumstance if one only has a high card hand.

What about the one-pair hand?  Turns out, a one pair hand of Kings or better wins more than 50% of the time:

Player1KeyCards: One Pair Hand	Lose	Tie	Win	Win%
14	1097		2830	72.07
13	1773		2407	57.58
12	2072	89	2160	49.99
11	2254	176	1789	42.40
10	2154	254	1546	39.10
9	2087	268	1359	36.59
8	2019	253	1170	33.99
7	1958	198	1047	32.69
6	1883	94	875	30.68
5	1886	43	807	29.50
4	1805	12	745	29.08
3	1812		702	27.92
2	1889		726	27.76

So one shouldn't always fold if one has a one pair hand.  Instead, one should play with an Ace or King pair hand, and probably also with a Queen pair hand (wins more than loses).

We can update the win, tie, loss by hand table to further refine our guidance.  It is reflected below.

Conclusion

Here is an updated table with win ratios by type of hand:

Player1	Lose	Tie	Win	Lose/All	Tie/All	Win/All
1.Royal Flush	0	0	23	0.00	0.00	1.00
2. Straight Flush	0	4	141	0.00	0.03	0.97
3. Four of a Kind	11	5	158	0.06	0.03	0.91
4. Full House	160	182	2300	0.06	0.07	0.87
5. Flush	352	89	2380	0.12	0.03	0.84
6. Straight	150	768	3050	0.04	0.19	0.77
7. Three of a Kind	1165	157	3596	0.24	0.03	0.73
8a. Two Pair - 5s or better	6101	1187	14495	0.28	0.05	0.67
9a. One Pair - Queens (12) or better	4942	89	7397	0.40	0.01	0.60
9b. One Pair - Jacks (11) or worse	19747	1298	10766	0.62	0.04	0.34
8b. Two Pair - 4s or worse	978	25	493	0.65	0.02	0.33
10. High Card	14552	418	2821	0.82	0.02	0.16

Here is summary of  the suggestions for playing against one other player:

• If one has a two pair 5s or better, one pair Queens or better, or three of a kind or better, then play
• If one has a two pair 4s or worse, one pair jacks or worse, or a high card hand, then fold
• Ties:
• Depend only on a King or better high card to break the tie in your favor
• One pair: 9s or better to win the tie break most of the time
• Two pair: Jacks or better to win the tie break most of the time
• Hole cards
• If your hole cards form a pair (e.g., 2,2), you should play
• If you have an Ace (14), you should play
• For hole cards of rank1 and rank2, if rounding the result of sqrt((rank1)^2 + (rank2)^2) is >= 14, then you should play.
• Note: since this is hard to do mentally, taking the rounded average of rank1 and rank2 >= 10 works almost as well.

Good luck!