As you explain it here this is very clear, and there is no confusion in my mind about it at all. And this would make sense when considering Nash ranges if I am correct about how they are derived. It would be simply impossible to beat a Nash range long term mathematically. 

I am not sure why I got wrapped around the idea that a Nash range "needs to be complemented" by a villain Nash range. I think I read it somewhere and it stuck. It sounds about right anyway, but for sure I can see how false it is now. I am not even entirely sure what the sentence even means. Maybe whoever wrote it was making the point that if villain is way out from Nash, then it will be more +ev for hero to move to exploit the deviation than stick to Nash?

As per usual TPE gets me there. I can't tell you how efficient this site is for clearing things like this up for me.

And you are quite right...I am interested in this. I am not quite sure what that says about my personality. I never used to be into anything math related. Maybe I am looking for a Nash equilibrium with my wife whereby I could at least guarantee 0 ev in our arguments instead of always being on a downswing...

I think you should read up a bit on Nash equilibria, as it's clearly a topic you're interested in. Your reading wouldn't have to be poker specific, but if you want a poker book, Mathematics of Poker is the place to go.

The central idea is that your opponent can't improve on his results vs an equilibrium strategy, no matter what he does. So in Rock, Paper, Scissors, choosing your throw at random is an equilibrium strategy. It is NOT dependent on anything your opponent does. He could plays Always Rock, or Always Paper, or any combination, and no matter what, your (and his) chances of winning would be the same.

Now, if you knew that his strategy were going to be to play 90% Rock, then you could exploit that by playing 100% Paper. As long as his strategy didn't change, you would now be doing better then if you played your equilibrium strategy. However, you are of course now vulnerable, because at any time he could change his strategy and turn things around on you. This is an example of an exploitable strategy.

So here goes...the official Riceman definition of "unexploitable".

I Googled the definition and it came back with the ingenious and highly counter intuitive and obscure "that which cannot be exploited". Great. So then I Googled "exploit" and it came back as "make full use or derive benefit from", and gives as synonyms "capitalise from" and "draw benefit from". Perhaps more relevantly it gives another definition: "a software tool designed to take advantage of a flaw in a computer or software system".

I am unsure how Nash ranges are derived, but my guess would be that computers have calculated the average range/ band of hands that will fall from each position. It is possible to find yourself ahead of the range of course, but this will be an exception.

I might be way way out and talking an absolute load of baloney mind you!

So in a Nash poker context the Riceman definition of exploitable is:

Perceiving a flaw in villain strategy (a deviation from Nash ranges) and moving to take advantage of said flaw (deviation).

Hold on...I don't know what the hell I defined there...I actually lost my mind for a second there and spouted out some pure jibberish! It isn't even a definition.

Unexploitable you say?

"A system or strategy which cannot be taken advantage of". Or...

"In non-cooperative gaming a mutually dependent optimal strategy whereby neither party can increase their expectation by changing strategy". Wait a minute... that sounds familiar! Hey! I just invented the Rice Equilibrium! 

But I have a better idea Andrew...

Could you please define unexploitable as we use the term in poker? At least, if you think my definition is lacking somehow... 

The Riceman said
I always thought that for a hero Nash range to be valid, it needs to be complemented by a villain Nash call range. So this was a mistaken understanding?    

Correct (that this was a mistaken understanding) (depending on what exactly you mean by "valid"). I think it would help if you tried to define, as specifically as you can, what "unexploitable" means.

So if villain is shoving way wider or tighter than Nash, if I use a Nash call range, this will still be an unexploitable range?

Similarly, if I am shoving a Nash range, and villain is calling way tighter than Nash or way wider, my shove will still be an unexploitable shove?

I always thought that for a hero Nash range to be valid, it needs to be complemented by a villain Nash call range. So this was a mistaken understanding?  

Still a lot to work with it but getting the idea feels  good ! 

please enlighten me if I got this right.

Question No.1

If Nash shoving range suggest that I shove 35.5% range of hands in certain circumstance. 

Does it mean if I do shove 35.5% in exact spot for 1 million times ,

even if my opponent knows exactly that I am shoving strictly by Nash shoving range which is 35.5% at this specific circumstance,

there is nothing my opponent can really do rather than calling strictly by Nash calling range.

So, that 35.5% shoving range will keep me in close to 0% equity even after 1 million times regardless of what my opponents do? (being inexploitable?)

Question No.2

If I exactly know that my opponent is calling by certain range, say 46.1%. Can I exploit him/her by shoving wider/tighter than actual Nash shoving range? (being exploitative?)

Cheers !