Hier ist ein (englisches) Beispiel, das ich in einem anderen Forum zum gleichen Thema verfasst habe, vielleicht erklärt das einige der Fragen:
Or maybe let´s stop talking about theoretical concepts and take a simple example. We´re at the river and have one psb left. We have 10 nut-combos and 10 air-combos. Villain has a bluffcatching-range that beats our air-hands. Given our ratio of value:bluffs, if the hands got checked down, everybody would take 50% of the pot longterm.
Now, let´s assume, we have seen Villain folding any single time on the river so far. We start exploiting him by betting the river 100%. Villain folds 100%. Our EV in this case is 1*p (which is twice our equity share!), EV of Villain is 0*p.
Now, Villain starts recognizing that we have bet any single river so far - and he starts to feel like Jack Ass. So he starts calling every single time. What happens? Our EV goes down to 0.5p and Villain´s EV goes up to 0.5p, so everybody takes his fair share on the pot.
Is that a Nash Equilibrium - as we both get the same EV now - namely exactly the equity share our ranges have in the pot?
NO!!
It can´t be a Nash, as we have a simple strategy to exploit Villain´s "new leak" (his stationary tendency). We simply stop bluffing. Our EV when we never bluff but valuebet 100% (and get called 100%) is (0.5*2p) + (0.5*0) = 1p. Villain´s EV is (0.5 * -1) + (0.5 * 1) = 0. So we´re at the starting point again.
So, by now Villain knows that he can´t fold 100% (because then we simply bluff 100% - and taking his entire equity share) and he knows he can´t call 100% (because then we simply bluff 0% - and again take his entire equity share). That means, both players draw circles around each other by adjusting to each other´s strategy. Where will they land? How often should Hero bluff - and how often should Villain call?
Let´s start with Villain. He has to call exactly often enough that we don´t make money with our bluffs (we´re indifferent on bluffing or not, so we can´t win money from either bluffing more or bluffing less). As we´re getting 1:1 on our bluffs, we make money (and have incentive to bluff 100%) if Villain folds > 50%. We have incentive to never bluff and only valuebet, if Villain folds < 50%. If Villain calls exactly 50% - we can either bluff 100% or 0% or anything in between. It doesn´t matter.
Now, if Villain calls exactly 50% (to keep us honest) - how often should we bluff now? Similar answer: we should bluff often enough that Villain is indifferent on calling, meaning he has no incentive to bluffcatch more or less than 50%. As we offer Villain pot-odds of 2:1 after us shoving one psb, Villain has to be good 1 out of 3 times. That means, if we would bluff >33% of the times we bet, Villain would have incentive to call 100%, because he makes money every time he calls, if we bluff <33%, Villain would have incentive to fold 100%, because he loses money everytime he calls. That means, we have to bluff exactly 33% of all the times we bet and valuebet 67%. So, our value:bluff-ratio is 2:1 (10 valuecombos and 5 bluffcombos) which is exactly the odds we offer Villain on the call.
OK, summarized:
Hero bets 75% (10 nut-combos and 5 air-combos) and checks 25% (5 remaining air-combos).
Villain calls 50% and folds 50%.
The EV for Hero is:
EV (Hero) = (0.5 * 0.5 * 2p) + (0.5 * 0.5 * p) + (0.25 * 0.5 * -p) + (0.25 * 0.5 * p) + (0.25 * 0) = 0.75p
The EV for Villain is:
EV (Villain) = (0.75 * ((0.5 * 0.67 * -p) + (0.5 * 0.33 * 2p)) + 0.25 * 1 = 0.25p
Et voilà: Nash Equilibrium! Even though, Villain gets less than his "fair share" of the pot, there´s nothing he can do without giving us the chance to increase our profit, up to 1p. Same goes for Hero, if Hero bluffs more or less, Villain can always maximize his profit - and lower Hero´s profit, down to 0.5p.
