|Equity/Tie % Tables|
|Win % By Hand|
|52%||< EQUITY >||48%|
|Unique Texas Hold'em Matchups:|
Preflop High Hand Equity and Tie Percentages
|by Mark Bowron
After Vanessa Selbst made her "first light shove" of the 2012 WSOP Main Event holding Q♥7♠ in middle position she got called by Greg Merson in the big blind holding A♣8♣. Both players caught a piece of the 7♣2♣T♠ flop.
Evidently aware that many if not most people would probably think otherwise, Selbst instantly declared "I'm behind! I'm behind!" This was correct. The flop had only reduced Merson's 66‑to‑34 preflop advantage to 55‑to‑45.
Aside: this website is more theoretical than practical. It just displays the win and tie percentages for every unique preflop matchup. To reach the level of poker‑odds knowledge Vanessa Selbst displayed in that memorable hand, I recommend beginning with Phil Gordon's Little Green Book: Lessons and Teachings in No Limit Texas Hold'em.
Getting back to the hand, the turn came 6♥. This gave Vanessa her first lead in the hand: she was now a 59‑to‑41 favorite to win. But the turn card also gave Merson four additional outs. One of them (the 9♦) hit on the river. Merson had Selbst covered, so she was out in 73rd place ($88,070). Greg went on to win the tournament ($8,531,853).
Now let's talk preflop matchups.
It turns out that there are 47,008 unique two-player ("heads up") matchups preflop in Texas hold'em, where two matchups are considered identical if one can be obtained from the other by permuting suits. Tables containing high hand equity (win percentage plus half of tie percentage) and tie percentages for these matchups are available for viewing on this website by clicking the links on the right. These tables were created using a combination of tools including the excellent poker probability software at propokertools.com.
The number 47,008 was first announced in print by Card Player Magazine columnist W. Lawrence Hill (1944-2005) in his article Hold'em Matches and Mismatches (Sep 6, 1991, p. 30). Poker's "mad genius" Mike Caro launched a series of $100 prize problems on RGP (rec.gambling.poker) on Oct 15, 1997. His first prize problem asked for an enumeration of the unique matchups with proof. Erik Reuter quickly supplied a concise demonstration and took down the $100 prize. A more recent proof by Stewart N. Ethier appears on pages 716‑718 in his 2010 book, The Doctrine of Chances: Probabilistic Aspects of Gambling.
I have also written up an enumerative proof. The main difference between my proof and Reuter's is he looks at all possible rank distributions first, then all possible suit distributions within. My proof takes the reverse approach by looking at all possible suit distributions first, then all possible rank distributions within. Reuter's method is simpler, but both work.
Answers to a few natural questions about the 47,008 matchups are given below. Most if not all of these results have appeared before in other sources. Remember, two matchups are considered identical if one can be obtained from the other by some permutation of the suits (for example by changing hearts to spades, spades to diamonds, diamonds to hearts, and leaving clubs unchanged).
Which is the most lopsided matchup?
Answer: 2♠K♥ vs K♠K♦ (94.92% high hand equity)
Which is least likely to end in a tie?
Answer: 8♠7♥ vs A♦A♣ (0.23% tie probability)
Which is most likely to end in a tie?
Answer: 3♠2♥ vs 2♠3♥ (98.57% tie probability)
Among matchups where both hands have equal winning percentages (such as the one above), which is/are least likely to end in a tie?
Q♠6♠ vs Q♥6♥, J♠6♠ vs J♥6♥, J♠5♠ vs J♥5♥,
T♠6♠ vs T♥6♥, T♠5♠ vs T♥5♥, T♠4♠ vs T♥4♥
Each of the above matchups has exactly the same (85.68%) chance of ending in a tie. Their rank combinations happen to be the only ones that prohibit ties when the board forms a flush or straight flush in a suit held by one of the players. (To satisfy this property, a rank combination has to be six or fewer ranks apart, with one rank above 9 and the other below 7.)
When the list is sorted by high hand equity, where does the largest jump occur?
8♠J♥ vs J♠J♦ (90.63% high hand equity)
K♠J♥ vs K♦K♣ (90.29% high hand equity)
This jump (0.34%) may not look very large, but remember, the list contains 47,008 entries. It turns out that the second-largest jump (0.26%) occurs between the two most lopsided matchups:
2♠K♥ vs K♠K♦ (94.92% high hand equity)
2♠Q♥ vs Q♠Q♦ (94.66% high hand equity)
How many different high hand win-tie percentage combinations exist among the 47,008 matchups?
How many different high hand equity percentages exist?
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