Ultimate Texas Hold'em (UTH) is among the most vulnerable novelty games to hole-card play. Between the dealer's two cards and the five community cards, the player has seven opportunities to get some extra information. With the ability to make a Play bet that is 4x the player's original Ante bet, any extra information can be leveraged into a big edge. In this post I am going to examine the case when the player knows one of the two dealer's hole-cards before making his pre-Flop decision. See this post if you need a reminder of the way UTH is played and its basic strategy.
For those of you who like big numbers, the program I wrote to analyze UTH when the player knows one dealer hole-card considered 21,269,384,136,000 different hands (21.2 trillion hands). It took my program about 4 days to complete the cycle running four concurrent processes. The output my program produced was an itemization of each starting hand consisting of the player's two initial cards and the observed dealer's hole-card. It then gave the expected value for the three possible decisions: check, raise 3x and raise 4x. I imported this output into a spreadsheet to determine the best strategy in each case and to compute the overall player edge. Note that "raise 3x" was never the right strategic choice.
To speed things up, my program considered equivalence classes of starting hands so that not every possible starting hand was evaluated. For example, if the player has (2c,Ad) and the dealer has Jd, that is clearly the same as the player having (2d,Ah) and the dealer having Jh. Similarly, player = (3c,3d), dealer = 7c is the same as player = (3c,3d), dealer = 7d. Altogether there were 5083 equivalence classes. In the spreadsheet there is a column called "Perms" that lists the number of starting hands that are equivalent to the given sample hand from its equivalence class.
Here is the spreadsheet that gives the output produced by my program:
UTH_DHC_Computer_Perfect_Strategy
In particular,
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The player edge in UTH knowing one dealer hole-card is 13.5922%.
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The player makes a 4x pre-Flop raise on 34.5762% of his hands.
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The player checks pre-Flop on 65.4238% of his hands.
I note that this edge (13.5922%) is not in James Grosjean's Beyond Counting or Exhibit CAA. As far as I know, this is the first time this number has been published.
Here are some other facts about hole-carding UTH, knowing one dealer hole-card:
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The best starting hand for the player is player = (Ac,Ad) versus dealer = Ah (and its permutations). This hand has EV = 3.9913.
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The worst starting hand for the player is player = (2c,7d) versus dealer 7c (and its permutations). This hand has EV = -1.0840.
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The hand with closest strategy decision is player = (3c,Jd) against dealer 2d (and its permutations). Checking has EV = 0.167868. Raise 4x has EV = 0.167828. The correct decision is to check. The difference between these decisions is 0.0041%.
I am not going to attempt to cull the spreadsheet above to give a descriptive pre-Flop strategy. The dedicated reader is invited to create his own strategy based on the work I provided. However, I do want to compare computer-perfect play with James Grosjean's pre-Flop strategy as given in his book Exhibit CAA.
Out of respect for Grosjean's work, I will not repeat the simple description he gives for his UTH dealer hole-card pre-Flop strategy. Instead, I went through each of the 5083 starting hands in my spreadsheet and recorded Grosjean's pre-Flop strategy decision in the column just to the right of computer-perfect strategy. The following spreadsheet contains the details:
Using this spreadsheet for my analysis, here are some facts about Grosjean's pre-Flop strategy (assuming computer-perfect Flop strategy and Turn/River strategy):
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Grosjean's pre-Flop strategy yields a player edge of 13.4316%.
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Grosjean's strategy is 0.1607% weaker than computer-perfect play.
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Out of the combin(52,2)*50 = 66,300 unique starting hands, Grosjean's strategy is correct for all but 2240 of these hands.
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Grosjean's strategy gets 96.621% of the pre-Flop decisions right.
Here are three sample situations where Grosjean's strategy differs from computer-perfect strategy:
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[Unsuited] Player = (9c,Kd), dealer = Qd. Correct is to raise 4x with EV = 0.1886. Grosjean's strategy advises checking on this hand, with EV = 0.1780.
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[Suited] Player = (8c,Qc), dealer = 9d. Correct is to check with EV = 0.2890. Grosjean's strategy advises raising 4x on this hand with EV = 0.2819.
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[Pair] Player = (6c,6d), dealer = 8h. Correct is to Raise 4x with EV = 0.5028. Grosjean's strategy advises checking on this hand, with EV = 0.4958.
An AP shared with me that the advantage player gets an edge between 12% and 13% against UTH knowing one dealer hole-card by using Grosjean's strategy. Given that Grosjean's Flop and Turn/River strategies are also slightly sub-optimal, and based on the results above, this is a reasonable conclusion. The simplicity and power of Grosjean's strategy is extraordinary, as is just about everything Grosjean does. That said, some concrete numbers are finally available for this enigmatic opportunity.
As far as game protection, I recommend that UTH be hand dealt, using the following procedure:
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After removing the cards from the automatic shuffler, and after each player has made their Ante bet, a card is burned and two cards are dealt face-down to each player.
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The players then look at their cards and make their pre-Flop check/raise decisions.
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Another card is burned, then three Flop cards are dealt and turned face-up.
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The players who have not yet raised then make their Flop check/raise decisions.
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Another card is burned, then two Turn/River cards are dealt and turned face-up.
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The players who have not yet raised then make their Turn/River raise/fold decisions.
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Another card is burned, then two cards are dealt to the dealer and turned face-up.
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All necessary cards have now been dealt, and the wagers can be resolved.
By avoiding dealing cards until those cards are necessary for the next wagering step in the game, there will be no hole-card opportunities. By burning the top card before cards are dealt, there will be no top-card (or marked-card) opportunities. I have observed that this dealing procedure is commonly used in Las Vegas.
