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.

{ The Hold ‘Em at Home blog is brought to you by THETA Poker Pro, the strongest, fastest, and most configurable Texas Hold ‘Em game for iPhone, iPad, iPod touch, and Apple TV. }

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