Monday, August 31, 2015

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!



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