This question has been asked before here and here. But what i am looking for is the specific calculation methodology or pseudocode.

Additionally my understanding itself of Nash equilibrium is this context may be flawed

My attempt so far is as below.

Assume a HU No Limit Hold'em game where each player has 10bb to start with and each player is only allowed to fold or push all in. In this scenario the equations I get are as follows

EC = expected chips

  • Assume player_1 is in SB and player_2 is in BB
  • Then scenario is player_1 SB stack 9.5bb
  • player_2 BB stack 9.0bb
  • pot 1.5bb


EC(player_1 )          = EC(player_1 folds) + EC(player_1 shoves)
EC(player_1 shoves)    = 1.5 * prob(player_2 folds) * (1- prob(player_2 shoves))
                         + 20 * prob(player_2 push) * prob(player_1 win)
                         + 10 * prob(player_2 push) * prob(player_1 tie)
                         + 0 * prob(player_2 push) * prob(player_1 loss)
prob(player_1 win) + prob(player_1 tie) + prob(player_1 loss) = 1

And at Nash Equilibrium I am supposed to equate and solve for EC(player_1 folds) <= EC(player_1 shoves)

This above calculation is suppose to give me the ideal Nash Equilibrium Push rage. but given the equations how do I calculate prob(player_2 push)? since I don't know prob(player_1 win/tie/loss)? am I supposed to assume AA for player 1 and 100% for player 2? the calculate same for KK the AKs AKo so on and so forth?

That does not seem right as Nash Equilibrium is supposed to work with both players playing optimally? So what should I use as starting ranges? Or should i run iterations on player 1 all hands vs 100% range for player 2, Then +EC hands determined for player 1 vs individual hands for player 2 to determine a intermediate optimal call range for player 2? and then keep repeating the process till I reach the absolute end after which no further changes occur in player 1/2 ranges?

  • I think the median 2-card hand is something like Q8 maybe, so maybe run that thru for the opponent hand and see what that yields.
    – Brian
    Feb 12, 2019 at 3:44
  • Have you tried <a href="cardrunnersev.com"> cardrunners ev</a>? This is a paid software ($75) which can calculate Nash Equilibrium. I'm not sure if you can extract the logic from it, but you could at least compare your results.. Mar 6, 2019 at 5:58
  • Already have other softwares to verify.. Mar 11, 2019 at 12:36
  • @SiddharthChabra - Have you ever heard of the AKQ game? I'd check out Chapter 13 of the Mathematics of Poker if not... It may lend a lot of insight to this question. Feb 9, 2021 at 14:09


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