Hey guys so I've started doing a bunch of mathematical theory work so I can apply it to my new favourite game 2-7 SD.

I am having trouble getting as grasp of using equity calculations. Should I be working out what the EV is for a particular range for all potential actions or just for a particular decision relative to a different decision? So the EV of a check v.s a bet.

Also to get a full equity equation should I include the EV of my whole range before a particular action or just the EV of the range that preforms a particular action?**
Eg: OOP should I calculate the EV of my bet range (value+bluff) independently and maximize it or should I include the check range and consider that I will have to defend my check optimally and then get the EV of all actions then maximize that to get good ranges/bet sizes?


For the more mathematical:
So moving on I have been working on a toy game to find out what kind of range + bet sizing I should use IP on a river decision. So the way I see it to do so you should create an EV [f(s)] equation with respect to S which is a bet sizing then find that size so that EV is maximized (You do this by finding S where f'(s)=0 ).
*S is in units of pot sized bets.

So I set up a no limit [0,1] toy game where the OOP Villain check's his whole range and Hero IP can bet any amount S then villain can only call.
[0,1] - Means a continuous range from nuts (0) to a hand that loses against everything (1).

So the idea is I will bet with some value range 0-X and will bluff some range from Xb-1 (size of this range is X*S/(1+S)).
I then created an EV equation for Hero's betting range and this is the part I need help with:
- Got an EV for each part of a range assuming villain is calling optimally (Minimum Defense Frequency)
- You get 0 EV when hero has X (0-X) value range v top of villains
- You get +S*(prob of Villain having X) EV when hero's value range is vs villains calling range worse than X
- + 0EV when hero value range is v villains folding range.... is this right???? **
- You get -SEV*(prob of villain having calling range) when hero has his bluffing range vs villains calling range
- You gain +1EV when hero has bluffing range v villains folding range..... is this right?**

To sum this all up then you multiply the the EV of the value range by the probability of having a value hand (how do you get it?**) and multiply the EV of the bluffs by the probability of having the bluffs and sum them. When I did this I just said that the probability was the size of the range, i.e If X is .3 or top 30% of range then the probability of it is 3/10 and the same the the bluff range. But this seems wrong.

For those of you who are interested the final solution is:

The EV is maximized for this game for Hero who is IP is when X = 1/(2S+2) .... this means you bet bigger with a stronger range and smaller with weaker one, quite trivial. When X = top 30% then optimal bet size is 0.24.


So if any of you could help me that would be very appreciated. I don't want to move onto more complex toy games until I fully understand these EV calculations. I particularly would like clarity in spots in this post where I marked a '**'.


TL;DR - What do you include in EV calculations for a spot and what is the EV of a successful bluff? Is dead money considered and why is it not included in the MoP in most cases except for when a bluff works?

Only have time to answer this right now: You can only win what's not yours.
You can only lose what is yours.
Money already in the pot isn't yours, therefore you can win it but you can't lose it.

That makes sense. So then should the EV of value betting and getting called be S+1?

Also I have been trying to edit my post as the solution I gave its not correct and the values are wrong.
It is the answer to the Partial differential of the EV equation (d/ds)f(s,x) = 0. Basically with this equation you can use it to find what range you should bet to maximizes your EV if you use a constant bet size. So if you only have 50% pot size bet left you should bet the top 33% of your range and the bottom 11% as bluffs.

This is of course only if I set up the EV equation correctly which most likely I didn't.