Heads Up No Limit

Jan 26, 2018 Because of its large size and strategic complexity, heads-up no-limit Texas hold’em (HUNL) has been the primary benchmark and challenge problem for imperfect-information game solving for several. When it is played with just two players (heads-up) and with fixed bet sizes and a fixed number of raises (limit), it is called heads-up limit hold’em or HULHE (19). HULHE was popularized by a.

Introduction

Expert Heads Up No Limit

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.

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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

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 player should make the 3X raise with any pair except deuces. Otherwise, use 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, as follows:

Medium Raise: Make the 2X raise with any of the following:

• Two pair or better.
• Hidden pair*, except pocket deuces.
• Four to a flush including a hidden 10 or better to that flush

* Hidden pair = Any pair with at least one card in your hole cards (thus the pair is hidden to the dealer).

Small Raise: Make the 1X raise with any of the following, otherwise fold:

• Hidden pair or better.
• Less than 21 dealer outs beat you.

For a more powerful small and medium raise strategy, I recommend the James Grossjean strategy card for Ultimate Texas Hold 'Em.

Analysis

Heads Up No Limit Strategy

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

Heads Up No Limit Solved

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.

Heads-up No-limit Texas Hold’em

Internal Links

Heads Up No Limit Hold'em

Detailed calculations on some starting hands.

Heads Up No Limit Hold'em Book

Extneral Links

Play Heads Up Hold 'Em at the Galaxy Gaming web site. One click and you're playing.

Heads Up No Limit Hold'em

Written by:Michael Shackleford