(Theory) Tournament Hand Rankings v1

[IFoldPktOnes]

I'm tired of Slansky hand rankings for tournament poker, so I thought I would try and come up with my own...

(skip down to my results section if this is too boring)

Theory:
The key assumption behind the popular Slansky hand rankings is that your opponent knows your exact hand. Obviously this is unrealistic so I set about coming up with a ranking system where your opponent only knows what %range of hands you are on.

My system is based on a simple push/call game. There are no blinds, however the caller needs >=50% equity to call (I'll call this the threshold%). So if Player1 pushes with an X% range, Player2 (the caller) knows the value of X and therefore comes up with an optimal call range, Y%. Player2 uses a range that attempts to counter Player1's calling range and both players must follow a strict hand ranking system.

How accurately does this ranking system apply to tournament poker?
This is largely effected by the choice of the threshold%. I made the arbitrary choice of 50% as a starting point, and in some situations this is a good approximation to real sit n' go play. More specifically, these hand rankings are most accurate when the effect of the bubble is exactly counteracted by the size of the blinds. More extreme bubble situations, or situations with small blinds are not well represented by my hand rankings. However, hand rankings for these situations could be determined simply by repeating my method for a new threshold%.

Here is an example using PokerStars structure where my hand rankings are most accurate:

Big Blind is t200 with t25 antes

CO (t3844)
Button (t3844)
SB (t1968)
BB (t3844)
Co folds, Button folds, SB pushes...

In this case BB needs 49.995% equity to call the push by SB, and my ranking system would represent near-optimal ranges. If someone could calculate the exact nash equilibrium in this case it would be interesting to compare the results to the ranges predicted by my hand rankings.

The Results:
Note: I did all this by hand using pokerstove and it is quite likely I made some small errors.

1. Calling Strength Hand Rankings

If Player1 is pushing a range less than or equal to this X% then you have to fold the corresponding hand. This percentage gives you a rough idea of relative hand strength for calling.

X%_____Hand
0.00%__AA
1.81%__KK
3.47%__QQ
5.13%__JJ
6.03%__TT
6.49%__AKs
6.49%__AKo
8.90%__99
9.80%__AQs
12.52%_88
13.42%_AQo*
14.33%_AJs
15.54%_77
17.04%_ATs
17.35%_AJo
17.65%_KQs
21.27%_ATo
22.17%_66
22.47%_A9s
26.40%_KJs
26.40%_A8s
26.40%_KQo
29.11%_A9o
30.62%_55
30.62%_A7s
31.52%_KTs
34.24%_A8o
34.24%_A6s
36.35%_A5s
38.16%_QJs
39.37%_KJo
39.37%_A7o
40.57%_44
40.87%_A4s
42.99%_A3s
43.29%_K9s
43.29%_KTo
43.29%_QTs
46.00%_A2s
46.30%_A6o
48.11%_A5o
49.92%_QJo
50.23%_K8s
50.23%_JTs
51.73%_K9o
52.64%_A4o
54.15%_Q9s
55.05%_K7s
56.26%_QTo
57.16%_A3o
59.58%_K6s
60.78%_K8o
61.69%_A2o
63.50%_J9s
64.40%_33
64.40%_K5s
66.21%_Q8s
66.52%_K7o
67.42%_JTo
68.63%_Q9o
70.44%_K4s
72.55%_K6o
72.55%_T9s
74.66%_Q7s
74.66%_K3s
77.38%_J8s
78.28%_Q8o
78.28%_Q6s
78.28%_K5o
79.19%_K2s
81.90%_J9o
83.71%_Q5s
84.62%_K4o
86.43%_T8s
87.33%_J7s
89.14%_Q4s
89.14%_Q7o
90.95%_K3o
91.86%_J8o
91.86%_T9o
93.67%_Q6o
93.67%_Q3s
95.48%_22
95.48%_98s
97.29%_J6s
97.29%_T7s
97.29%_K2o
99.10%_Q2s
99.10%_Q5o
100.0%_J5s
100.0%_T8o
100.0%_J7o
100.0%_Q4o
100.0%_97s
100.0%_J4s
100.0%_T6s
100.0%_J3s
100.0%_Q3o
100.0%_98o
100.0%_87s
100.0%_T7o
100.0%_J6o
100.0%_96s
100.0%_J2s
100.0%_Q2o
100.0%_T5s
100.0%_J5o
100.0%_T4s
100.0%_97o
100.0%_86s
100.0%_J4o
100.0%_T6o
100.0%_95s
100.0%_T3s
100.0%_76s
100.0%_J3o
100.0%_87o
100.0%_T2s
100.0%_85s
100.0%_96o
100.0%_J2o
100.0%_T5o
100.0%_94s
100.0%_75s
100.0%_T4o
100.0%_93s
100.0%_86o
100.0%_65s
100.0%_84s
100.0%_95o
100.0%_T3o
100.0%_92s
100.0%_76o
100.0%_74s
100.0%_T2o
100.0%_54s
100.0%_85o
100.0%_64s
100.0%_83s
100.0%_94o
100.0%_75o
100.0%_82s
100.0%_73s
100.0%_93o
100.0%_65o
100.0%_53s
100.0%_63s
100.0%_84o
100.0%_92o
100.0%_43s
100.0%_74o
100.0%_72s
100.0%_54o
100.0%_64o
100.0%_52s
100.0%_62s
100.0%_83o
100.0%_42s
100.0%_82o
100.0%_73o
100.0%_53o
100.0%_63o
100.0%_32s
100.0%_43o
100.0%_72o
100.0%_52o
100.0%_62o
100.0%_42o
100.0%_32o

*AQo is slightly undervalued by this method, and is actually stronger than 88 for 12.22% < X < 12.97%


2. Pushing Strength Groupings
This part of the results is best interpreted in groupings.

3.17%
{QQ+,AJs+,AKo}

9.80%
{44+,ATs+,KTs+,QTs+,JTs,AQo+}

14.93%
{22+,A9s+,K9s+,Q9s+,J9s+,T9s,AJo+,KJo+}

29.71%
{22+,A2s+,K4s+,Q6s+,J7s+,T7s+,96s+,86s+,75s+,65s,5 4s,A8o+,KTo+,QTo+,JTo}

49.92%
{22+,A2s+,K2s+,Q2s+,J4s+,T6s+,95s+,85s+,74s+,64s+, 53s+,43s,A2o+,K7o+,Q9o+,J9o+,T8o+,97o+,87o,76o}

70.14%
{22+,A2s+,K2s+,Q2s+,J2s+,T2s+,92s+,83s+,73s+,62s+, 52s+,42s+,A2o+,K2o+,Q5o+,J7o+,T7o+,96o+,86o+,75o+, 65o,54o}

90.05%
{22+,A2s+,K2s+,Q2s+,J2s+,T2s+,92s+,82s+,72s+,62s+, 52s+,42s+,32s,A2o+,K2o+,Q2o+,J2o+,T3o+,95o+,84o+,7 4o+,63o+,52o+,43o}

(copy/paste into pokerstove to view these better)

[IFoldPktOnes]

Here is the full list of Player1's "pushing strength" hand ranking system.

I include this for interest. These rankings should not be directly applied because this set of hand rankings is derived from a very specific and tailored counter-range which makes it less suitable for comparing relative hand strength.

The X% on the left is the tightest range that includes the corresponding hand.

X%_____Hand
0.45%__AA
0.90%__KK
1.21%__AJs
1.51%__AQs
1.81%__AKs
2.71%__AKo
3.17%__QQ
3.47%__KQs
3.92%__66
4.37%__55
4.83%__JJ
5.13%__QJs
5.58%__77
6.03%__TT
6.49%__99
7.39%__AQo
7.69%__ATs
7.99%__QTs
8.30%__KJs
8.60%__JTs
8.90%__KTs
9.35%__88
9.80%__44
10.71%_KQo
11.01%_J9s
11.92%_KJo
12.22%_T9s
12.52%_K9s
12.97%_33
13.42%_22
14.33%_AJo
14.63%_Q9s
14.93%_A9s
15.23%_T8s
15.54%_98s
15.84%_A5s
16.14%_K8s
16.44%_A8s
16.74%_K6s
17.04%_Q8s
17.35%_K5s
17.65%_A4s
17.95%_K7s
18.25%_J8s
18.55%_87s
19.46%_ATo
19.76%_A7s
20.06%_97s
20.97%_QJo
21.27%_76s
22.17%_JTo
22.47%_A3s
22.78%_86s
23.08%_T7s
23.38%_J7s
23.68%_65s
23.98%_A2s
24.28%_54s
24.59%_A6s
25.49%_A9o
26.40%_A8o
27.30%_QTo
27.60%_75s
27.90%_K4s
28.81%_KTo
29.11%_Q7s
29.41%_Q6s
29.71%_96s
30.62%_A7o
31.52%_A5o
32.43%_T9o
32.73%_T6s
33.03%_K3s
33.33%_Q5s
34.24%_J9o
35.14%_A4o
35.44%_64s
36.35%_A6o
37.25%_A3o
38.16%_A2o
39.06%_Q9o
39.37%_85s
40.27%_98o
40.57%_J6s
40.87%_K2s
41.18%_95s
42.08%_K9o
42.38%_Q4s
42.68%_J5s
42.99%_53s
43.29%_Q3s
44.19%_T8o
45.10%_87o
45.40%_Q2s
45.70%_74s
46.00%_J4s
46.30%_43s
47.21%_97o
48.11%_76o
49.02%_K8o
49.92%_K7o
50.23%_63s
51.13%_J8o
51.43%_J3s
51.73%_T5s
52.64%_Q8o
53.54%_K6o
53.85%_84s
54.15%_J2s
55.05%_86o
55.35%_52s
56.26%_K5o
57.16%_65o
57.47%_T4s
58.37%_K4o
59.28%_T7o
59.58%_T3s
59.88%_T2s
60.18%_73s
60.48%_94s
60.78%_93s
61.69%_75o
62.59%_54o
63.50%_K3o
64.40%_96o
65.31%_Q7o
66.21%_K2o
66.52%_42s
67.42%_Q6o
68.33%_J7o
68.63%_62s
69.53%_Q5o
69.83%_92s
70.14%_83s
70.44%_82s
70.74%_32s
71.64%_85o
72.55%_Q4o
73.45%_T6o
74.36%_64o
74.66%_72s
75.57%_J5o
76.47%_J6o
77.38%_Q3o
78.28%_74o
79.19%_53o
80.09%_95o
81.00%_Q2o
81.90%_J4o
82.81%_43o
83.71%_T5o
84.62%_63o
85.52%_J3o
86.43%_T4o
87.33%_84o
88.24%_J2o
89.14%_T3o
90.05%_52o
90.95%_94o
91.86%_T2o
92.76%_73o
93.67%_42o
94.57%_93o
95.48%_92o
96.38%_62o
97.29%_83o
98.19%_82o
99.10%_32o
100.0%_72o

Notice the %'s on the left here also correspond to the % ranges in the first hand rankings chart so you can work out the (near) optimal push/call combinations for Player1 and Player2.

[IFoldPktOnes]

I cross-posted this in the Poker Theory Forum, with just a little more info on the method I used:
http://forumserver.twoplustwo.com/showfl...c=#Post11320771

[DevinLake]

How is this applicable?

Ie, when are we playing without blinds? Also, when does cEV trump $EV?

[IFoldPktOnes]

When I say there is no blinds I mean its like a cash game with no blinds, the caller needs at least 50% equity.

It's not based on cEV, It's based on EV according to ICM for specific bubble/blind size conditions: "when the effect of the bubble is exactly counteracted by the size of the blinds" making the push/call tournament game analogous to a push/call cash game with no blinds.

Stack sizes, blind sizes and opponents ranges all change relative hand strength. Repeating this ranking method for different threshold% values can account for different blind/stack sizes. Opponents push range is also incorporated in the Calling Strength hand rankings, It's the % on the left.

My Pushing Strength hand rankings however are not applicable and are simply theoretical, they assume your opponent calls optimally and that they know your range exactly. As groupings I think they can be used as rough guidelines.

Edit: I also gave an example in my OP where BB should exactly follow my calling strength hand ranking system.