Saturday, April 28, 2012

Nash Equilibrium in Heads Up Poker, Applicability

In Mathematics of Poker, Bill Chen has provided approximated Nash Equilibrium Shove/Fold, and Call ranges with respect to Effective Stack Size, for Heads Up (1 on 1) No-Limit Texas Holdem poker. The next logical thought would question whether if this is practical for actual playing. It isn't likely for the significantly large sample size required, and a highly probable negative EV (Expected Value).

Likely Negative EV

1) The above solutions are approximated, not analytical, making the player's EV uncertain at best.
    Analytical vs. Numerical Solutions Reference

2) These approximations were done without considerations of the rake.

3) While an analytical Nash Equilibrium solution offers the minimum payoffs, it doesn't have to be positive; i.e. the guaranteed minimum payoff could very well be negative. This would be something like the Blackjack Basic Strategy, where you're expected to lose money over time at best.

Sample size, volatility of returns

There are combinations 5 card hands available.

With 2 players, there would be
= 1.3378456e+8 ≈ 133,784,560 combinations of 7-cards. 

This means a HUGE volume of hands are required to ever reach within EV range. Even if the player has EV = 0, this implies short term volatility of game results (risk of ruin) would likely get so large that it becomes highly undesirable for the professional player.