Heads up problem vs blind man....

[lorinda]

The statistics regarding the values of hands heads up vs two random cards got me thinking.

Many here will know the old one about a NL heads up game with no blinds and one player goes all-in blind every hand, what are his opponents chances of winning, and what is his best strategy.

Answer, wait for aces, 84.9% (since a tied pot puts you back where you were)

But what about a more real scenario, Ive kept the numbers limited to make it easier to discuss.

Blinds 1/1, Stack 50/50

One player goes all in blind every hand, what is best strategy for opponent.

Some hands that you might want to play with chances of winning:

AA 84.9
KK 82.1
QQ 79.6
JJ 77.2
TT 74.7
99 71.7
AKs 66.2

The full table can be foundhere (http://gocee.com/poker/he_ev_wins.html)

Remember, each time you fold, you effectively have 1% less chance of winning, so the best result will come from the %age of hands you are willing to play crossed with the %chance of winning the pot, and I can't quite figure it out.

What I am pretty sure of is that the result will be a little scary for those who insist that morons should never win.

Lori

[irchans]

Lorinda,

I think I have the answer. I have put my guess at the optimal strategy into
the table below. Column 1 is the number of chips in front of the opponent.
Column 2 is the weakest hand the opponent should play with that number of
chips. Column 3 is the number of hands that are equal to or better than
the weakest hand. Column 4 is the probability that the opponent will win
with that number of chips using this strategy.

There were two surprises in this table for me. First, I did not expect the
opponent to play lots of hands when he held most of the chips. I guess the
opponent just wants to put the small stack allin, even with inferior cards,
just hoping to end the game. Second, I was surprised that the expert player
beats the wild player only 61% of the time when they both start with 50.


Irchan's Guess at the Optimal Strategy

<pre><font class="small">code:</font><hr>
Chips WHnd #hnd ProbWin
1 32o 169 0.015
2 32o 169 0.031
3 52o 166 0.047
4 52s 155 0.062
5 83s 139 0.078
6 T3o 129 0.093
7 96o 119 0.108
8 T3s 113 0.123
9 T5s 106 0.139
10 87s 101 0.154
11 Q4o 95 0.168
12 T8o 91 0.182
13 T7s 85 0.196
14 Q6o 82 0.21
15 22 80 0.223
16 Q4s 77 0.237
17 K4o 75 0.251
18 Q5s 72 0.265
19 K2s 71 0.278
20 K3s 66 0.291
21 T9s 63 0.303
22 A2o 60 0.316
23 A2o 60 0.328
24 K5s 56 0.339
25 Q9o 55 0.350
26 A3o 54 0.362
27 K8o 52 0.373
28 K8o 52 0.385
29 K8o 52 0.396
30 A4o 50 0.408
31 A4o 50 0.42
32 A4o 50 0.432
33 A2s 48 0.444
34 A2s 48 0.455
35 A5o 47 0.466
36 QTo 45 0.477
37 44 40 0.488
38 K9o 39 0.498
39 QJo 37 0.508
40 A4s 36 0.518
41 A4s 36 0.528
42 A4s 36 0.538
43 A7o 35 0.548
44 A5s 34 0.557
45 A5s 34 0.567
46 QTs 33 0.576
47 A6s 32 0.585
48 A6s 32 0.593
49 88 28 0.602
50 88 28 0.61
51 A6s 32 0.619
52 A5s 34 0.628
53 A5s 34 0.638
54 A4s 36 0.647
55 A4s 36 0.657
56 K9o 39 0.667
57 K9o 39 0.677
58 JTs 43 0.687
59 QTo 45 0.697
60 A5o 47 0.707
61 A5o 47 0.716
62 A5o 47 0.726
63 A5o 47 0.735
64 A2s 48 0.744
65 A2s 48 0.753
66 A4o 50 0.762
67 A4o 50 0.771
68 A4o 50 0.78
69 A4o 50 0.788
70 A4o 50 0.796
71 K8o 52 0.804
72 K8o 52 0.812
73 A3o 54 0.820
74 Q9o 55 0.827
75 Q9o 55 0.834
76 K5s 56 0.841
77 K7o 58 0.848
78 A2o 60 0.856
79 K6o 65 0.863
80 Q6s 68 0.871
81 K2s 71 0.878
82 Q5s 72 0.886
83 K4o 75 0.893
84 Q4s 77 0.9
85 22 80 0.908
86 Q3s 83 0.914
87 T7s 85 0.921
88 K2o 87 0.927
89 T8o 91 0.933
90 Q4o 95 0.939
91 T5s 106 0.946
92 T3s 113 0.952
93 J2o 121 0.959
94 75s 125 0.964
95 85o 136 0.97
96 52s 155 0.976
97 62o 167 0.982
98 32o 169 0.988
99 32o 169 0.994
</pre><hr>

(For example, if the opponent had 71 chips in front of him before the game
and the wild player raises allin, then the opponent should call with the best
52 hands = AA KK QQ JJ TT 99 AKs 77 AQs AKo AJs ATs AQo 66 AJo KQs ATo A9s
KJs KTs A8s KQo 55 A7s A9o KJo QJs 88 K9s KTo A8o A6s QTs A5s A7o A4s QJo
K8s K9o 44 A3s Q9s JTs A6o QTo K7s A5o A2s K6s A4o Q8s and K8o. If the
opponent uses the strategy above, he will win 80.4% of the time with 71
chips.)

Comments ?

Cheers, irchans

( Cross posted on 2+2, rec.gambling.poker, and pokermonster )

[BB King's]

Many here will know the old one about a NL heads up game with no blinds and one player goes all-in blind every hand, what are his opponents chances of winning, and what is his best strategy.

Call w/ every hand that has a pot equity of 50 % (or better) - that is J5s (or better).

If there are blinds - you should call even more.

[irchans]

If there are no blinds, then calling with J5s+ or better has the highest expectation per hand.

If there are no blinds and each player has a finite stack size, then the strategy of calling with AA only maximizes the probability that you take all of the other guy's money. This is true in spite of the fact that you have less expectation per hand.

[lorinda]

Thanks Irchans, great reply and great work, I will now be able to look for some kind of "counter" to this, and hopefully won't find one /forums/images/icons/laugh.gif

61% is around what I had guessed (65%) and shows that the all-in strategy is a pretty strong one.

Just out of interest, was it a computer sim, or did you use some formula to devise how to do it?

If there are no blinds and each player has a finite stack size, then the strategy of calling with AA only maximizes the probability that you take all of the other guy's money. This is true in spite of the fact that you have less expectation per hand.

Hence the reason that tourney players and ring players are very different animals.

Lori

[lorinda]

Hence the phrase "Never give a sucker an even break"

Lori

[jack1234]

so how can this be practically applied to nl toyrneys with very high blinds at the end to determine the correct mathematical strategy for moving all in with hand x against one (or more) opponents with y times the amount of total blinds in their stack? Is this a good way to approach the question?

See my post in general theory 'the math of moving all in' as theres more specific questions there.


fly me to vegas,
jack

[irchans]

infinitgames@yahoo.com (Irchans) wrote in message
&gt; Lorinda: (edited)
&gt; &gt;Blinds 1/1, Stack 50/50
&gt; &gt;One player goes all in blind every hand, what is best strategy for opponent.
&gt; &gt;
&gt; &gt;The full table can be found here
&gt; http://gocee.com/poker/he_ev_wins.html
&gt;
&gt; I think I have the answer. I have put my guess at the optimal strategy into
&gt; the table below. Column 1 is the number of chips in front of the opponent.
&gt; Column 2 is the weakest hand the opponent should play with that number of
&gt; chips. Column 3 is the number of hands that are equal to or better than
&gt; the weakest hand. Column 4 is the probability that the opponent will win
&gt; with that number of chips using this strategy.
&gt;
&gt; Irchan's Guess at the Optimal Strategy
&gt; [wrong table deleted]

Oops. Barbara Yoon found a mistake in the original post. Here is the
corrected table:
<pre><font class="small">code:</font><hr>
Chips WHnd #hnd ProbWin
1 32o 169 0.016
2 32o 169 0.031
3 62o 167 0.047
4 52s 155 0.062
5 75o 140 0.078
6 65s 128 0.094
7 T2s 118 0.109
8 T6o 112 0.124
9 J5o 107 0.139
10 87s 101 0.154
11 T6s 96 0.169
12 J5s 90 0.183
13 J6s 85 0.196
14 K3o 80 0.210
15 K3o 80 0.224
16 Q7o 77 0.237
17 K4o 75 0.251
18 Q5s 72 0.265
19 K2s 71 0.278
20 33 66 0.291
21 K3s 63 0.303
22 A2o 60 0.316
23 A2o 60 0.328
24 Q9o 56 0.339
25 J9s 55 0.350
26 K5s 54 0.362
27 K8o 52 0.373
28 K8o 52 0.385
29 K8o 52 0.396
30 K6s 50 0.408
31 K6s 50 0.420
32 K6s 50 0.432
33 44 48 0.444
34 44 48 0.455
35 QTo 47 0.467
36 K7s 45 0.478
37 K9o 40 0.488
38 QJo 39 0.498
39 K8s 37 0.508
40 A7o 36 0.519
41 A7o 36 0.528
42 A7o 36 0.538
43 A4s 35 0.548
44 QTs 34 0.558
45 QTs 34 0.567
46 KTo 33 0.576
47 A5s 32 0.585
48 A5s 32 0.593
49 QJs 28 0.602
50 QJs 28 0.611
51 A5s 32 0.619
52 QTs 34 0.628
53 QTs 34 0.638
54 A7o 36 0.648
55 A7o 36 0.657
56 QJo 39 0.667
57 QJo 39 0.677
58 A5o 43 0.687
59 K7s 45 0.697
60 QTo 47 0.707
61 QTo 47 0.717
62 QTo 47 0.726
63 QTo 47 0.735
64 44 48 0.744
65 44 48 0.753
66 K6s 50 0.762
67 K6s 50 0.771
68 K6s 50 0.780
69 K6s 50 0.788
70 K6s 50 0.796
71 K8o 52 0.804
72 K8o 52 0.812
73 K5s 54 0.820
74 J9s 55 0.827
75 J9s 55 0.834
76 J9s 55 0.841
77 K7o 58 0.848
78 A2o 60 0.856
79 J8s 65 0.863
80 Q6s 68 0.871
81 K2s 71 0.878
82 Q5s 72 0.886
83 K4o 75 0.893
84 Q7o 77 0.900
85 K3o 80 0.908
86 98s 83 0.914
87 K2o 86 0.921
88 Q2s 88 0.927
89 J7o 92 0.933
90 T6s 96 0.940
91 J5o 107 0.946
92 95s 114 0.952
93 85s 120 0.959
94 93s 126 0.965
95 85o 137 0.970
96 62s 156 0.976
97 42o 168 0.982
98 32o 169 0.988
99 32o 169 0.994

</pre><hr>

(For example, if the opponent had 71 chips in front of him before
the game and the wild player raises allin, then the opponent should
call with the best 52 hands. The best 52 hands are all the hands
better than or equal to K8o. If the opponent uses the strategy
above, he will win 80.4% of the time with 71 chips.)

The best hands in order are : AA KK QQ JJ TT 99 88 AKs 77 AQs AJs
AKo ATs AQo AJo KQs 66 A9s ATo KJs A8s KTs KQo A7s A9o KJo 55 QJs
K9s A8o A6s A5s KTo QTs A4s A7o K8s A3s QJo K9o Q9s A6o A5o JTs K7s
A2s QTo 44 A4o K6s Q8s K8o A3o K5s J9s Q9o JTo K7o K4s A2o Q7s K6o
K3s T9s J8s 33 Q8o Q6s J9o K5o K2s Q5s J7s T8s K4o Q4s Q7o T9o J8o
K3o Q6o Q3s 98s T7s J6s K2o 22 Q2s Q5o J5s T8o J7o 97s J4s Q4o T6s
J3s Q3o 98o T7o 87s J6o J2s 96s Q2o T5s J5o T4s 97o J4o 86s T6o T3s
95s 76s J3o 87o T2s 96o 85s J2o T5o 94s 75s T4o 93s 86o 65s 84s 95o
T3o 92s 76o 74s T2o 54s 85o 64s 94o 75o 82s 83s 93o 73s 65o 53s 63s
84o 92o 43s 74o 72s 54o 64o 52s 62s 83o 82o 42s 73o 53o 63o 32s 43o
72o 52o 62o 42o 32o

Thanks again to Barbara.

Cheers,
Irchans