Preflop Odds Heads Up 3

[SS] “Fig, I’ve got a bet for you”, Stan the Stat propositioned Figaro the Fish while retrieving a set of three dice from his left pocket.

[FF] “Okay…”, Figaro responded warily.

[SS] “I’ll bet you even-money on who can roll the higher number with one of these dice”, Stan offered. “And because I like you, we’ll roll as many times as you want, and you can pick whichever die you want each time.”

[FF] After examining the dice, the Fish decided, “Okay, I’ll take this one, the only one with nines on it.”

[SS] “Very well, I choose this one with nothing higher than a seven”, the Stat countered.

[LL] Leroy the Lion chuckled, “You’re not really going to con him are you?”

[SS] “No, just for fun; we’re all friends here.”

[LL] “Fig, it doesn’t matter which die you pick, these are nontransitive dice, where each die beats one of the other two and loses to the other one.”

[FF] “How is that possible?”

[SS] “It’s like Roshambo, where rock beats scissors beats paper beats rock. The six faces on each die add up to 30, but they’re set up so you win by a little and lose by a lot, allowing the second player to win five-ninths of the matchups. C > B > A > C:”

Side 1 Side 2 Side 3 Side 4 Side 5 Side 6
Die A 1 1 6 6 8 8
Die B 2 2 4 4 9 9
Die C 3 3 5 5 7 7

[SS] Reaching into his right pocket and pulling out some more dice, Stan continued, “Or consider this set of four, called Efron’s dice:”

Side 1 Side 2 Side 3 Side 4 Side 5 Side 6
Die D 0 0 4 4 4 4
Die E 1 1 1 5 5 5
Die F 2 2 2 2 6 6
Die G 3 3 3 3 3 3

[SS] “The edge is even bigger here as the dominating die wins two-thirds of the time. G > F > E > D > G.”

[FF] “Wow, that’s amazing. Can I borrow those for the next time I go to a bar?”

[LL] “Do you really like to get beat up?”

[SS] “What’s cool is that you can do the same thing with Texas Hold ‘Em hands, although your edge is much smaller.” Having rummaged through a deck of cards and pulled out six of them, Stan continued, “For example, consider these hands: A♣K♦, J♥10♥, and 2♠2♥. Which would you prefer, Figaro?”

[FF] “Definitely the Ace-King at a full table.”

[SS] “Nope, you’re heads up.”

[FF] “I’d still take it.”

[SS] “Well, then I’d select the Twos and have a 53.0% to 47.0% edge on you. Similarly, the Jack-Ten suited has 53.7% equity against the Twos, and the Ace-King offsuit does even better against the Jack-Ten at 58.8%.”

[LL] “The Ace-King is an obvious win over the Jack-Ten, while the other two matchups are classic races. Both connectors mainly need to pair up to beat the Twos, but the Jack-Ten makes enough straights and flushes to tip the scales, while the Ace-King doesn’t.”

[FF] “Pretty cool, indeed.”

Related Links:

  • While it’s not blindingly obvious how to use it, the table at the top right of the Mathematrucker Matchups page lets you see any family of heads-up matchups (by gap and suitedness). Just tap on the proper square in the grid. In the answer cells, a pink background means the hands on the left column are winning and a blue background means the hands on the top row are ahead! Tap on the percentage to display the winning hand type breakdown on the left.

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