Software Review Thread - For Tools Sticky. (SPAM within?)

[DevinLake]

[ QUOTE ]
For heads up your opponents strategy doesn't matter since any deviation they make from equilibrium will only increase your EV

[/ QUOTE ]

How do you quantify if this is the case?

For instance, if an opponent isn't shoving as wide as NE would dictate, you are gaining EV from the extra blinds you get. But, if you don't stray from NE and call based on NE, he will be gaining EV on your calls.

Which effect is stronger?

[plexiq]

About the nash-calculator:
If you have any suggestions to improve the usability, please let me know.

@Devin:
[ QUOTE ]
For instance, if an opponent isn't shoving as wide as NE would dictate, you are gaining EV from the extra blinds you get. But, if you don't stray from NE and call based on NE, he will be gaining EV on your calls.

Which effect is stronger?

[/ QUOTE ]

By definition, in a Nash Equilibrium no player can increase his own EV by changing his strategy (if all other players stick to the Nash Strategy).

So, to answer your question: The second effect must not be stronger (otherwise we don't have a real NE). [The effects could theoretically cancel each other out and the strategy-change may be EV-neutral. But for NLHE-HU Push/Fold a deviation from the NE is typically -EV if your opponent plays Nash.]

[IFoldPktOnes]

[ QUOTE ]
[ QUOTE ]
For heads up your opponents strategy doesn't matter since any deviation they make from equilibrium will only increase your EV

[/ QUOTE ]

How do you quantify if this is the case?

For instance, if an opponent isn't shoving as wide as NE would dictate, you are gaining EV from the extra blinds you get. But, if you don't stray from NE and call based on NE, he will be gaining EV on your calls.

Which effect is stronger?

[/ QUOTE ]

I thought I might check the maths myself just to be sure. Say you have 10BBs each heads up and you are on the BB. You both play optimaly so SB pushes 59.0% and you call 37.6%. This gives you a range-vs-range equity of 54.9% when you call.

Now calculating your EV in #BBs:
EV = P(opponent push)[P(call)[P(win)*EV(win)+P(lose)*EV(lose)]+P(fold)EV(fold)] + P(opponent fold)*EV(opponent fold)
= (.59)[0.376[0.549*10+.451*(-10)]+0.624*(-1)] + 0.41*0.5
= -0.151 + .205 = +0.054 BBs

This time your opponent changes strategy and pushes 22.7%, you continue calling with a 37.6% range. Your new range-vs-range equity is 45.1%.

EV = P(opponent push)[P(call)[P(win)*EV(win)+P(lose)*EV(lose)]+P(fold)EV(fold)] + P(opponent fold)*EV(opponent fold)
= (.227)[0.376*[0.454*10+0.546*(-10)]+0.624*(-1)] + 0.773*0.5
= -0.220 + 0.386 = +0.166 BBs

(I ignored card removal effects)

As expected the equity you gain from your opponent folding the SB too often more than makes up for the fact you are calling with a range that is way behind. However, if you had adapted to your opponents new range your EV would be even higher. You might wonder if your opponent limping in a certain % of the time might improve his tight-pushing strategy by not giving up the SB as often. But by pushing all his best hands he has weakened his limping range enough so that you can push very wide (often ATC) profitably if he limps. In most cases this strategy would be even worse for your opponent.

Edit: Forgot to carry the one