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Last Updated: December 9, 2013

Heads Up Hold 'Em

Introduction

2013 Global Gaming Expo.

Introduction

Heads Up Hold 'Em is an Ultimate Texas Hold 'Em variant by Galaxy Gaming, based on Texas Hold 'Em. The player may raise his bet one time, and has three opportunities to do so. The earlier he raises, the more he can bet. The main differences between Heads Up Hold 'Em and Ultimate Texas Hold 'Em are in the former the player may raise only 3x his Ante bet before the flop, but the game includes bad beat bonuses for losing with a straight or higher.

Rules

Following are the rules for Heads Up Hold 'Em. For those used to the terminology in Ultimate Texas Hold 'Em, what is called the Blind there called the Odds bet here.

  1. The game is played with a single ordinary 52-card deck.
  2. The player must make an equal bet on both the Ante and Odds.
  3. Two cards are dealt face down to the player and dealer. The player may look at his own cards.
  4. The player can check or make a Play bet equal to three times the Ante.
  5. The dealer turns over three community cards.
  6. If the player previously checked, then he may make a Play bet equal to two times his Ante or check again. If the player already made a Play bet, then he may not bet further.
  7. Two final community cards are turned over.
  8. If the player previously checked twice, then he must either make a Play bet equal to exactly his Ante, or fold, losing both his Ante and Odds bets. If the player already raised he may not bet further.
  9. The player and dealer will both make the best possible hand using any combination of their own two cards and the five community cards.
  10. The dealer will need at least a pair to open.
  11. The following table shows how the Ante, Odds, and Play bets are scored, according to who wins, and whether the dealer opens.

    Scoring Rules

    Winner Dealer Opens Ante Play Odds
    Player Yes Win Win See rule 12
    Player No Push Win See rule 12
    Dealer Yes Lose Lose See rule 12
    Dealer No Push Lose Lose
    Tie Yes or No Push Push Push

    This is each player's betting area.

  12. Winning Ante and Play bets pay 1 to 1. Winning Odds bets pay according to value of the hand and whether it wins or loses. The following pay table shows what winning Odds bets pay.

    Winning Odds Bet Pay Table

    Hand Pays
    Royal Flush 500
    Straight Flush 50
    Quads 10
    Full House 3
    Flush 1.5
    Straight 1
    All Other Push

    Galaxy Gaming, the game owner, has four different pay tables for losing blind bets, as follows.

    Losing Odds Bet Pay Table

    Hand Pay Table
    1 2 3 4
    Straight Flush 500 500 500 500
    Quads 50 50 50 25
    Full House 10 10 10 6
    Flush 8 6 5 5
    Straight 5 5 4 4
    All Other Loss Loss Loss Loss

  13. In addition, there are two side bets, that pay based on the player's cards only, the Trips Plus and Pocket Bonus. These side bets are explained after the analysis of the base game.

Strategy

The following table shows when to make the large 3X raise.

The strategy for the medium and small raise are the same as in Ultimate Texas Hold 'Em, except the player should never fold with four of a kind. For that strategy, I recommend the James Grossjean strategy card.

Analysis

The following table shows all possible outcomes of each hand, what it pays, the probability, and contribution to the return under the losing Odds bet pay table number 1 (the one that goes 500-50-10-8-5). The lower right cell shows a house edge of 2.36%.

Return TableExpand

Player Raise Dealer
Qualifies
Winner Pays Combinations Probability Return
Fold -2 5,498,078,560,920 0.197674 -0.395349
Less than pair 1 No Dealer -2 60,518,663,424 0.002176 -0.004352
Straight flush 1 Yes Dealer 498 20,279,100 0.000001 0.000363
Four of a kind 1 Yes Dealer 48 1,726,735,980 0.000062 0.002980
Full house 1 Yes Dealer 8 10,082,720,220 0.000363 0.002900
Flush 1 Yes Dealer 6 49,072,032,216 0.001764 0.010586
Straight 1 Yes Dealer 3 47,952,010,720 0.001724 0.005172
Less than straight 1 Yes Dealer -3 3,006,630,550,164 0.108098 -0.324295
Anything 1 Y/N Push 0 455,081,939,824 0.016362 0.000000
Royal flush 1 No Player 501 6,914,880 0.000000 0.000125
Straight flush 1 No Player 51 279,004,320 0.000010 0.000512
Four of a kind 1 No Player 11 - 0.000000 0.000000
Full house 1 No Player 4 - 0.000000 0.000000
Flush 1 No Player 2.5 43,096,215,600 0.001549 0.003874
Straight 1 No Player 2 145,034,240,580 0.005214 0.010429
Less than straight 1 No Player 1 816,981,676,824 0.029373 0.029373
Royal flush 1 Yes Player 502 46,580,760 0.000002 0.000841
Straight flush 1 Yes Player 52 2,023,968,588 0.000073 0.003784
Four of a kind 1 Yes Player 12 964,337,328 0.000035 0.000416
Full house 1 Yes Player 5 46,108,374,192 0.001658 0.008289
Flush 1 Yes Player 3.5 202,016,746,236 0.007263 0.025421
Straight 1 Yes Player 3 421,268,280,080 0.015146 0.045438
Less than straight 1 Yes Player 2 1,600,861,520,204 0.057556 0.115113
Less than pair 2 No Dealer -3 11,023,268,784 0.000396 -0.001189
Straight flush 2 Yes Dealer 497 16,724,460 0.000001 0.000299
Four of a kind 2 Yes Dealer 47 311,575,460 0.000011 0.000527
Full house 2 Yes Dealer 7 18,705,932,580 0.000673 0.004708
Flush 2 Yes Dealer 5 27,615,003,664 0.000993 0.004964
Straight 2 Yes Dealer 2 21,031,977,440 0.000756 0.001512
Less than straight 2 Yes Dealer -4 1,979,644,169,384 0.071175 -0.284699
Anything 2 Y/N Push 0 214,144,135,720 0.007699 0.000000
Royal flush 2 No Player 502 11,938,680 0.000000 0.000215
Straight flush 2 No Player 52 397,598,400 0.000014 0.000743
Four of a kind 2 No Player 12 - 0.000000 0.000000
Full house 2 No Player 5 - 0.000000 0.000000
Flush 2 No Player 3.5 45,718,738,920 0.001644 0.005753
Straight 2 No Player 3 53,068,201,380 0.001908 0.005724
Less than straight 2 No Player 2 1,207,385,216,712 0.043410 0.086819
Royal flush 2 Yes Player 503 147,692,880 0.000005 0.002671
Straight flush 2 Yes Player 53 3,016,851,612 0.000108 0.005749
Four of a kind 2 Yes Player 13 20,440,911,312 0.000735 0.009554
Full house 2 Yes Player 6 320,575,227,408 0.011526 0.069155
Flush 2 Yes Player 4.5 183,447,763,404 0.006596 0.029680
Straight 2 Yes Player 4 158,035,798,360 0.005682 0.022728
Less than straight 2 Yes Dealer 3 2,415,318,761,280 0.086839 0.260516
Less than pair 3 No Dealer -4 66,873,993,600 0.002404 -0.009617
Straight flush 3 Yes Dealer 496 14,499,400 0.000001 0.000259
Four of a kind 3 Yes Dealer 46 316,891,120 0.000011 0.000524
Full house 3 Yes Dealer 6 13,387,474,080 0.000481 0.002888
Flush 3 Yes Dealer 4 20,484,007,080 0.000736 0.002946
Straight 3 Yes Dealer 1 22,371,396,720 0.000804 0.000804
Less than straight 3 Yes Dealer -5 3,136,124,565,400 0.112754 -0.563771
Anything 3 Y/N Push 0 223,641,379,520 0.008041 0.000000
Royal flush 3 No Player 503 86,472,360 0.000003 0.001564
Straight flush 3 No Player 53 180,911,880 0.000007 0.000345
Four of a kind 3 No Player 13 - 0.000000 0.000000
Full house 3 No Player 6 - 0.000000 0.000000
Flush 3 No Player 4.5 41,791,833,360 0.001503 0.006762
Straight 3 No Player 4 64,386,219,840 0.002315 0.009260
Less than straight 3 No Player 3 1,317,173,128,560 0.047357 0.142070
Royal flush 3 Yes Player 504 556,552,440 0.000020 0.010085
Straight flush 3 Yes Player 54 1,444,036,640 0.000052 0.002804
Four of a kind 3 Yes Player 14 21,003,399,360 0.000755 0.010572
Full house 3 Yes Player 7 261,421,403,040 0.009399 0.065793
Flush 3 Yes Player 5.5 199,160,655,360 0.007160 0.039383
Straight 3 Yes Player 5 195,058,020,480 0.007013 0.035065
Less than straight 3 Yes Player 4 3,140,424,343,760 0.112909 0.451635
Total 27,813,810,024,000 1.000000 -0.023584

As shown in the lower right cell, the house edge is 2.36%. This is the expected loss to the Ante wager only. For example, if the player bets $5 on both the Ante and Odds, then his expected loss would be $5 × 0.023584 = 11.79¢.

The average final wager per hand is 3.67 units. That makes the element of risk 2.36%/3.67 = 0.64%. This means for every dollar you wager in the game, on anything, other than the side bets, you can expect to lose 0.64¢.

By comparison, the element of risk in Ultimate Texas Hold 'Em is 0.53%.

The standard deviation, relative to to the Ante bet, is 4.56.

The next table shows the house edge and element of risk under all four losing Odds bet pay tables according to the pay table for a losing Odds bet.

House Edge Summary

Hand Losing Odds Bet Pay Table
1 2 3 4
Straight Flush 500 500 500 500
Quads 50 50 50 25
Full House 10 10 10 6
Flush 8 6 5 5
Straight 5 5 4 4
All Other Loss Loss Loss Loss
House edge 2.36% 3.06% 3.73% 4.55%
Element of Risk 0.64% 0.83% 1.02% 1.24%


This is full table cloth.

Trips Plus

The Trips Plus bet will pay according to the poker value of the player's hand regardless of the value of the dealer's hand. Following is an analysis of the most common pay table.

Trips Plus Return Table

Hand Pays Combinations Probability Return
Royal flush 100 4,324 0.000032 0.003232
Straight flush 40 37,260 0.000279 0.011140
Four of a kind 30 224,848 0.001681 0.050420
Full house 8 3,473,184 0.025961 0.207688
Flush 7 4,047,644 0.030255 0.211785
Straight 4 6,180,020 0.046194 0.184775
Three of a kind 3 6,461,620 0.048299 0.144896
All other -1 113,355,660 0.847300 -0.847300
Total 133,784,560 1.000000 -0.033363

The next table shows four known pay tables for the Trips Plus.

Trips Plus Pay Tables

Hand Pay Table
1 2 3 4
Royal flush 100 100 100 100
Straight flush 40 40 40 40
Four of a kind 30 30 30 30
Full house 9 8 8 7
Flush 7 6 7 6
Straight 4 5 4 5
Three of a kind 3 3 3 3
All other -1 -1 -1 -1
Total -0.74% -1.74% -3.34% -4.34%

Pocket Bonus

The Pocket Bonus bet will pay according to the value of the player's two hole cards. The follow tables show what each two cards pays, the probability, and contribution to the total return for each known pay table for the Pocket Bonus.

Pocket Bonus — Pay Table 1

Hand Pays Combinations Probability Return
Pair of aces 30 6 0.004525 0.135747
Ace & face suited 20 12 0.009050 0.180995
Ace & face unsuited 10 36 0.027149 0.271493
Pair 2s - Ks 5 72 0.054299 0.271493
Loser -1 1,200 0.904977 -0.904977
Total 1,326 1.000000 -0.045249

Pocket Bonus — Pay Table 2

Hand Pays Combinations Probability Return
Pair of aces 25 6 0.004525 0.113122
Ace & face suited 20 12 0.009050 0.180995
Ace & face unsuited 10 36 0.027149 0.271493
Pair 2s - Ks 5 72 0.054299 0.271493
Loser -1 1,200 0.904977 -0.904977
Total 1,326 1.000000 -0.067873

Pocket Bonus — Pay Table 3

Hand Pays Combinations Probability Return
Pair of aces 30 6 0.004525 0.135747
Ace & face suited 20 12 0.009050 0.180995
Ace & face unsuited 10 36 0.027149 0.271493
Pair 2s - Ks 4 72 0.054299 0.217195
Loser -1 1,200 0.904977 -0.904977
Total 1,326 1.000000 -0.099548

Acknowledgments

I would like to thank Charles Mousseau and Stephen How for their assistance, which confirmed my analysis. Thanks to Charles also for his advice on the medium and small raise strategy.

Detailed calculations on some starting hands.


Written by: Michael Shackleford

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