Thin call?

[Aisthesis]

Ok, I think we're on the same page.

First, a question: Is the general definition of Nash equilibrium the following?

If player A has strategy S1 and player B has strategy S2 in a given game situation, then S1 and S2 (i.e., only the PAIR of strategies) are in a Nash equilibrium iff there is no strategy S1' such that A can increase his EV against S2 by adopting S1' rather than S1 AND there is no strategy S2' such that B can increase his EV against S1 by adopting S2' rather than S2.

After your post, I was thinking the definition had to be something like that--in which case they indeed are in a Nash equilibrium but, as you say, an unstable one. The weird part here is that against S2, any arbitrary S1' is just as good as S1. But if S1' is chosen differently from S1, then B can choose S2' so that S1' is much worse than S1.

The only quibble I have with what you said above is that EV(LP) is negatively correlated with call. Again, if EP calls too much, LP can again make a strategy change of never bluffing and thus justify his call, right? If EP calls more than half the time on one of the scare-cards, I feel sure LP can make a profit by value-betting only.

On your comments having to do with practice, I agree completely.

[grizy]

NOt in this case. Call=1 gives you an EV like 178.. less than the 210 needed.

[Aisthesis]

Sorry, but I'm still not following you on this point.

Call=1 and LP knows this. Then, LP value bets only. So:

Value-bet: 9(+1050)
Everything else: 31(-210)

EV: (9,450 - 6,510)/40 = $23.13

[grizy]

I was talking about when EP was blocking LP's straights.

[Aisthesis]

Ok, now I'm with you. I think we're in agreement the whole way in that case.

[grizy]

[ QUOTE ]
Ok, now I'm with you. I think we're in agreement the whole way in that case.

[/ QUOTE ]

No, we're not, I am telling you it's still possible for the call to be profitable even when EP has blockers.

And I am also telling you it's not necessarily true if EP has blockers he always should call. Yes, if he has blockers and opts to always call on river, then LP cannot call turn profitably. This says nothing about about whether calling the river is the more profitable move.

You need to work through your payoffs, chart them if you have to. You are seriously getting your causal relationships confused.

[jimmyfingers]

results?

[Aisthesis]

Well, in actuality, if you figure in all of the various blocker scenarios (the outs and pseudo-outs for LP are only 9 and 12 about 1/4 of the time), if EP just sticks with 50% calling for both straights (unless his side-cards happen to make him the nuts, in which case he obviously calls 100%), then LP still shows a loss.

And EP can improve over that strategy by calling and folding selectively depending on which straight he has blockers for--e.g., he has KKQ3, then he calls the big straight 100% and the little straight never, but if he has KK83, he calls the little straight always and the big straight never (since he doesn't know which draw LP has, he has to do the same on both). While on that strategy, EP loses money on the big straight when he has KKQ3 (even though LP has fewer outs, he gets great implied odds) but gains it back with interest when he has KK83.

I have gone through and charted the payoffs--not sure why you think I haven't. Have you? And which causal relationships am I getting confused?

I do agree that this says nothing about whether LP can call the turn profitably or not. It does say that he can't call the turn profitably IF EP will call either straight 50% of the time and has some set 100% of the time.

[Aisthesis]

My opinion (I'm not sure whether there's agreement here or not): Not unless EP plays badly, defined as calling or folding too often when one of the straight hits.

[Aisthesis]

Working through all the cases of various blockers and correcting both the flop pot-size (RT was right that $210 isn't the pot if there are 3 limpers who all call--it should be $275) and giving a little more stack-depth to make a full-pot river bet possible, I get the following when LP value bets always and bluffs with a frequency of 3/8:

1) My suggested optimal strategy for EP (selective calling depending on which blockers EP has): LP loses on average $32.46.

2) EP still just calls either straight 50% of the time (except when his side-cards makes him the nuts, in which case he always calls): LP loses $13.16.

3) grizy's suggested optimal strategy (EP's behavior doesn't matter when he has no blockers, but with any blockers for either straight he always calls): LP loses $12.41.