Firstly, you did make a mistake in your post. You used x as both the frequency of betting and the frequency of calling a set, without saying you were doing so.

I'll be using your well written out post to do my own math, but generalizing all the assumptions instead of using the ridiculous 10% call/5% bet assumption. Before we start:

- b is the frequency an opponent bets on the river with a hand we scoop.
- x is the frequency an opponent bets on the river with a set
- y is the frequency an opponent calls on the river with a set
- c is the frequency an opponent calls on the river with a hand we scoop.

To avoid confusion (and to get you up to speed), I'll go through the same scenarios and hand frequencies as you did. If required, I'll move onto the general case of variable hand ranges by player type later on.


1) check-fold:


25% of the time, he will bet A2 and you will fold. Net amount collected from the river on: 0.
25% of the time, he will have a set.
-of that 25%, the (x) times he will bet, you will fold, and you will collect 0.
-of that 25%, the (1-x) times he does not bet, you will collect 800.
50% of the time, he will have a hand that you scoop, as we've agreed.
-of that 50%, b% of the time he will bet, you will fold, and you will collect 0.
-of that 50%, (1-b)% of the time he will check, and you will collect 1600.


Expressed mathematically, we have:

.25*0 + x*.25*0 + (1-x)*.25*800 + b*.5*0 + (1-b)*.5*1600 = 200 - 200x + 800 - 800b

= 1000 - 200*x - 800b


2) check-call:

25% of the time, he will bet A2 and you will call. Net amount collected from the river on: -600.
25% of the time, he will have a set. Whether he bets or checks, you get the same net amount collected from the river on, namely 800.
50% of the time, he will have a hand that you scoop.
-of that 50%, b% of the time he will bet, you will call, and you will collect 2200.
-of that 50%, (1-b)% of the time he will check, and you will collect 1600.


Expressed mathematically, we have:

.25*-600 + .25*800 + b*.5*2200 + (1-b)*.5*1600 = -150 + 200 + 1100b + 800 - 800b

= 850 + 300b


3) lead:

25% of the time, he will have A2 and you will lose 600.
25% of the time, he will have a set.
-of that 25%, the (y) times he will call, you will collect 800.
-of that 25%, the (1-y) times he folds, you will collect 1600.
50% of the time, he will have a hand that you scoop.
-of that 50%, you've allowed that c% of the time he will call, you will collect 2200.
-of that 50%, then, (1-c)% of the time he will fold, and you will collect 1600.

Expressed mathematically, we have:

.25*-600 + (1-y)*.25*1600 + y*.25*800 + c*0.5*2200 + (1-c)*0.5*1600 = -150 + 400 - 400y + 200y + 1100c + 800 - 800c

= 1050 + 300c - 200y

To summarize:

Check-fold: 1000 - 200x - 800b
Check-call: 850 + 300b
Lead: 1050 + 300c - 200y

No we have an linear optimization problem. Which option is greater for various values of x,c,and b? I slept through linear programming classes, so I'll just plug in some values and see what comes out.

We'll start from the bottom end of the scale (him being scared shitless), and work our way up. Player characteristics are in bold.

-----------------------------------------------------

1. Rock

He's scared shitless (x=0,b=0,c=0,y=0)

check fold = 1000
check-call = 850
lead = 1050

Lead > check-fold which is >> check call.

Of course, this ignore the fact that being a rock, and having raised the turn, his hand ranges are not the same as our assumptions. This will be dealt with later in the general case.

---

2. Loose, Passive, loves hand

He's too scared to bet, but sometimes looks you up on the river. If he's calling 2 pair, he's calling a set more often. (x=0,b=0,c=20%,y=50%)

check-fold = 1000
check-call = 850
lead = 1050 + 300*0.2 - 200*0.5 = 1010

Very close, slightly in favor of lead. However, a passive raising the turn will have a very tight hand range (far more sets and A2s), which pushes this a long way into check/fold.

--

3. Semi tight, Passive, loves hand

He will never call a worse hand on the river, and won't bet, but will make a crying call with a set. (x=0,b=0,c=0,y=50%)

check-fold = 1000
check-call = 850
Lead = 1050 + 0 - 100 = 950

The opposite of the rock situation, check-fold > lead. This is further magnified by the points in (2) above.

4. Scared, slighty aggressive

He rarely takes a stab if checked to, and never calls without a set and rarely with one. He takes a stab twice as often with a set as without (x=10%,b=5%,c=0,y=10%)

check-fold = 1000 - 200*.1 - 800*0.05 = 940
check-call = 850 + 300*0.05 = 865
lead = 1050 + 300*0 - 200*0.1 = 1030

Well in favor of leading.

--

5. Semi aggressive / bluffer

He's scared by that river, and will definitely fold 2 pair to your bet (what is he beating?), but he sometimes takes a stab when checked to (why would you check the wheel?). (x=30%, b=20%,c=5%,y=20%)

check-fold = 1000 - 200*0.3 - 800*0.2 = 780
check-call = 850 + 300*0.2 = 910
lead = 1050 + 300*0.05 - 200*0.2 = 1025

Still well in favor of leading.

6. Aggressive

He will take a stab on the river often, knowing you're probably weak, but won't call less than a set, because he knows you wouldn't call the turn and bet the river without a big hand. He will call a set fairly often based on the pot odds (x=50%,b=50%,c=0%,y=50%)

check-fold = 1000 - 200*0.5 - 800*0.5 = 500
check-call = 850 + 300*0.5 = 1000
lead = 1050 + 300*0 - 200*0.5 = 950

In this case, check/call > lead.

-------------------------------------

Conclusion

You should lead against rocks and scared semi aggressives.
You should check-fold against both tight and semi-loose passive players. Once you account for the fact that the turn raise significantly alters their range toward a set or A2, it's a clear check/fold.
You should check call against moderately to very aggressive players.

What does this mean about Hero? Given his read on Villain, leading is marginally best.

However, the following statements made by Wintermute are incorrect:

1. You can never fold.

Clearly check/folding is a superior option against a broad range of the most common type of players.

2. If you decide to call, you must lead.

Clearly, this is incorrect against moderately to very aggressive players. Which Villain may or may not be.



So I'm happy with my victory in this thread. Leading is only marginally best given our read on Villain, but it is better, so Wintermute wins the hand and gains a few bucks in EV.

Meanwhile, I correctly pointed out gaping holes in Wintermute's conceptual model. One of which resulted in him owning himself quite badly.

The final score reads like this:

Wintermute: 1
Philip: 1,000,000,000,000,000,000 x 10^12000

The end

Phil,

Terrific analysis. I agree with everything in your mathematical description of the river decision, and you math looks right to me.

I find it interesting that you would declare victory after your math shows that leading is most often correct (by a significant margin), and that when it is incorrect, it is by only a slight margin, and never comes close to being the 3rd best option. Those conclusions sound to me like you would have to agree that in general, leading is best--this was the general thesis of my arguments.

(By the way, if you want to pick out specific things I wrote, show that they are not explicitly true for all cases, and then claim victory, how would you defend what you said in your 2nd post on this argument, where you said "The river is not a push but a clear check/fold." By your own analysis, a check-fold is almost never correct. I am not going to pretend that you are dumb and hold you to that statement, because I realize different assumptions were then in play, and that you were being bold for emphasis. Shame you are too obtuse to do the same in return.)

Anyhow, good discussion. You were valiant in defeat, and I accept your surrender.