Nash Poker Strategy

2021年6月14日
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PusherAKQJT98765432A20+20+20+20+20+20+20+20+20+20+20+20+20+K20+20+20+20+20+20+20+20+20+20+20+19.919.3Q20+20+20+20+20+20+20+20+20+20+16.313.512.7J20+20+20+20+20+20+20+20+18.614.713.510.68.5T20+20+20+20+20+20+20+20+20+11.910.57.76.5920+20+20+20+20+20+20+20+20+14.46.94.93.7820+18.013.013.317.520+20+20+20+18.810.12.72.5720+16.110.38.59.010.814.720+20+20+13.92.52.1620+15.19.66.55.75.27.010.720+20+16.3*2.0520+14.28.96.04.13.53.02.62.420+20+**2.0420+13.17.95.43.82.72.32.12.02.120+***1.8320+12.27.55.03.42.51.91.81.71.81.620+1.7220+11.67.04.62.92.21.81.61.51.51.41.420+CallerAKQJT98765432A20+20+20+20+20+20+20+20+20+20+20+20+20+K20+20+20+20+20+20+17.615.214.313.212.111.410.7Q20+20+20+20+20+16.113.010.59.98.98.47.87.2J20+20+19.520+18.013.410.68.87.06.96.15.85.6T20+20+15.312.720+11.59.37.46.35.25.24.84.5920+17.111.79.58.420+8.27.05.85.04.34.13.9820+13.89.77.66.66.020+6.55.64.84.13.63.5720+12.48.06.45.55.04.720+5.44.84.13.63.3620+11.07.35.44.64.24.14.020+4.94.33.83.3520+10.26.85.14.03.73.63.63.720+4.64.03.6418.39.16.24.73.83.33.23.23.33.520+3.83.4316.68.75.94.53.63.12.92.92.93.13.020+3.3215.88.15.64.23.53.02.82.62.72.82.72.615.0*63s:7.1 - 5.1, 2.3**53s: 12.9 - 3.8, 2.4***43s: 10.0 - 4.9, 2.2
These Push Fold Charts will help you understand when you need to either fold or shove your entire stack when you are short stacked in poker tournaments. Please use these charts as a study tool to help you improve your understanding of push fold scenarios. Free Book: Jonathan Little’s new book –. ’If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.’
Constantinos Daskalakis: On the Complexity of Approximating a Nash Equilibrium. In the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011. ACM Transactions on Algorithms (TALG), 9(3): 23, 2013. Special Issue for SODA 2011. Pdf; Constantinos Daskalakis and Christos Papadimitriou: Approximate Nash Equilibria in Anonymous Games. To find the strategy for SBs T3s, check the green area (suited hands) of the ’Pusher’ chart. The value for T3s is 7.7bb, and since that is larger than the current effective stacks the hand is a push in the Nash Equilibrium strategy. Push Fold Chart: Nash Poker Chart (Poker Shove Chart) The purpose of this comprehensive article is to serve as the only resource you’ll ever need to use in relation to the use of charts in poker. Consisting of contents ranging from bankroll suggestions to poker preflop charts to pot odds charts, this incredible resource has it all!PusherAKQJT98765432A20+20+20+20+20+20+20+20+20+20+20+20+20+K20+20+20+20+20+20+20+20+20+20+20+20+20+Q20+20+20+20+20+20+20+20+20+20+18.615.514.5J20+20+20+20+20+20+20+20+20+18.515.412.410.3T20+20+20+20+20+20+20+20+20+13.612.48.97.4920+20+20+20+20+20+20+20+20+16.58.06.04.7820+20+14.915.420+20+20+20+20+20+11.83.33.0720+18.311.89.710.412.416.820+20+20+15.93.02.5620+17.411.37.56.76.58.513.520+20+18.8*2.4520+16.410.37.14.94.23.83.33.020+20+**2.4420+15.09.56.44.63.22.82.62.52.520+***2.2320+14.38.65.84.23.02.32.22.12.12.020+2.1220+13.38.15.53.72.72.21.91.81.91.71.620+CallerAKQJT98765432A20+20+20+20+20+20+20+20+20+20+20+20+20+K20+20+20+20+20+20+20+17.416.415.113.912.912.3Q20+20+20+20+20+18.514.912.011.410.29.78.88.2J20+20+20+20+20+15.412.210.08.17.97.06.66.4T20+20+17.815.120+13.010.68.57.26.05.95.55.1920+19.813.510.99.720+9.58.06.75.64.84.64.3820+16.111.38.87.76.920+7.46.45.54.74.03.9720+14.39.27.46.55.75.420+6.05.54.74.03.6620+12.88.46.25.24.94.64.520+5.54.74.23.7520+11.87.85.94.54.14.04.04.120+5.14.53.9420+10.57.15.44.33.73.53.63.73.820+4.23.8319.19.96.75.04.23.63.23.23.33.53.320+3.6218.19.46.44.83.93.43.12.93.03.13.02.916.9*63s:8.9 - 4.7, 2.8**53s: 15.2 - 3.8, 2.9***43s: 11.5 - 4.7, 2.6
The tables show the highest effective stack size in big blinds where a hand can be pushed or called. Values bigger than 20 are only displayed as 20+, which means the hand can be played for any stack size of 20bb or less. (Playing push-or-fold is typically only recommended up to around 10bb, depending on the situation. Using push-or-fold for >20bb is almost certainly a bad idea.)
Some hands have gaps in their pushing strategies, they are marked with * and the details are displayed below the charts. For example 63s is included in the pushing strategy for stack sizes between 7.1 and 5.1bb, and stacks below 2.3bb.
SB (9.0bb): T3s
BB (6.0bb): Q2o

The effective stack size is the smaller of the two, before posting any blinds. So the relevant stack size for both players in this hand is 6.0 big blinds.
To find the strategy for SB’s T3s, check the green area (suited hands) of the ’Pusher’ chart. The value for T3s is 7.7bb, and since that is larger than the current effective stacks the hand is a push in the Nash Equilibrium strategy.
To determine if the BB should call with his Q2o, check the orange area (offsuit hands) of the ’Caller’ chart and locate Q2o. The value there is 5.6bb. The effective stacks in the current hand are larger than the value for Q2o, so this hand is a fold in the Nash Equilibrium strategy.Nash Poker Strategy Cheat
This example refers to the no ante solution. If the table values don’t match the example then make sure the ’No Ante’ tab is selected above.
The solution available on this site was independently calculated and published in 2007 as part of a coding experiment, inspired by the discussion of the topic in Mathematics of Poker.
Similar charts have been published by various sources, including SNGPT and Mathematics of Poker, and there are some minor differences between them. The exact Nash solution is considerably more complex and the chart only shows a simplified version. Most hands are actually played with mixed strategies for at least a few stack sizes, and many hands have multiple smaller gaps in their strategy, similar to the 63s, 53s, 43s hands. These details are omitted in the chart to allow a compact representation of the strategy.
The differences in published solutions are most likely due to choices made when simplifying the exact solution into easy-to-use charts. Due to the considerable number of rather arbitrary choices when simplifying into chart form, it is actually quite unlikely that two independently created charts would perfectly match.
The file linked below contains the unsimplified push-or-fold equilibrium strategies for all stacks of 1bb ≤ stacks ≤ 200bb in steps of 0.05bb. This data can be used to derive the charts. For practical purposes the chart form should be completely sufficient though.
The data linked here was re-calculated with a later version than the original chart data, so there may be some very slight differences.
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