I'm testing Piosolver, something gave me a doubt.

Why do we use a range of villains to base our GTO strategy?
When we assume a range of the opponent, are not we playing exploratively?
(we also entered into a guessing game, since in theory we do not know exactly with the villain plays)

A game based on GTO, should not assume a random range of the opponent ?
That is, in the GTO we work with the possibility that he is making the most crazy moves possible with any type of hand.

For example, establish that the range of Opponent EP is 22+, Ts +, A2s +, AJo+, KQo +.
For me it seems that it would be analogous from the assumption that an opponent in stone-paper-scissors, uses only paper.
Nothing against this assumption, but if we start from it, we are no longer basing ourselves on a GTO game.
We are exploiting the opponent, and also opening a breach to be exploited ourselves.

It sounds like you are fearing being exploited by someone playing an unpredictable style, but in my opinion you are overthinking it. For example, let's say a person opens 34s UTG. In a way he is exploiting you if you assume he can never have 34s. And he will occasionally win big on flops like 2-5-6, when you think "hm, 34 and 22 are unlikely, what are the odds he has 55 or 66?" The thing is, the person will lose much more preflop and on other flops where he is in a poor position, to the point where the edge gained by being unpredictable is far outweighed by having too weak of a holding.

The problem with assuming a 100% random range is that you will certainly end up underestimating your opponent's hand strength if you do so. For example, when someone opens UTG and the flop comes AK3, this person clearly has a range advantage over most callers. Realistically most people are playing 15% to 20% of their hands UTG, many combinations of which contain an ace, king, or broadways with backdoor flush combinations. If you assume they have 100% of their hands, you will be way overestimating their frequency of 2-7, 2-6, etc.

I believe PIO has a weighing feature where you could type in a percentage for each hand. So if you're worried 54s will open UTG, you could just put in a small weight of 15% or something to account for the times this would happen. That sounds like the most realistic solution to your fear.

there are no GTO ranges for more than 2-handed games, so you will have to assume ranges preflop.

Just treat solvers as calculating "maximally exploitative" solutions to a "sub-game" (or toy games) of poker. You could get Pio to give you the optimal strategy against someone playing ATC, and it would be quite different to the optimal strat vs someone with a "realistic" range. It would be different again if villain is a nit whose range is AA and nothing else.
The specific equilibrium for range v range, gives you a GTO strategy, but it doesn't give you the (overall) GTO strategy for the entire game of poker (which might require different pre-flop ranges to whatever you've input in Pio).

To put it another way, if you're playing rock/paper/scissors and your opponent only plays paper, the best way to play against him is to always use scissors. You don't need a computer to find that "max EV" equilibrium. It's just obvious. But if you put this game into Pio, it would suggest that playing 100% scissors is the best strategy against a player that won't alter his "100% paper" strat.
In a sense, "GTO" (as far as solvers are concerned) equals "Max Exploit". You just have to provide ranges so that they can calculate what they're trying to exploit. Since no one has a full solution to pre-flop, you have to make your best guess and crunch the numbers from that point.

I’m not sure this question makes much sense, but I’ll ask it anyway.

Which, if any of the following, are equivalent to Max Exploit?

Max EV
Min Regret
Villain Indifference
MaxMin Gain
MinMax Loss

Just as observation-curiosity, these doubts about the opposing range came to me as follows:

Watching players, they say something like "This villain is unknown, so I'm going to play based on GTO ..."
And soon after, they were thinking about the opponent's range: "The villain probably have this cards or the other ..."
Which did not make sense to me, because if he's playing based GTO, why the hell is he thinking about his opponent's range? A game based on GTO, should not you disregard the opponent's hole cards? Why is he wasting time thinking about his opponent's range, if this should not affect his strategy?
At first, I thought this was because a GTO-based game was very complex, and players use plausible ranges of the opponent to facilitate their decision making, although a truly GTO-based strategy should disregard the opponent's cards.

The problem was when I started testing on the solvers, I saw that they were also considering the opponent's cards.
So I thought, why the hell even solvers are using opponent ranges? Should not they work with random ranges of opponent?
And this bugged my mind xD

Sure they exist, unless you mean we don't know what they are, in which case I'd agree.

i think they don't necessarily exist, if i understand it correctly, the existence of an equilibrium is only mathematical proven for 2 player games. let's say we are 3 handed, player A and B play "balanced" ranges, so that no player can increase his EV. now player C can still go ahead and change his ranges in a way that he doesn't gain EV, but the expectation of A and B can still shift, so that they are incentivized to find a new equilibrium strategy.

the way i understand it, there can be multiple equilibria in 6 max.

The Nash Equilibrium existence proof only requires a finite number of players, it's not limited to two players. So yes, NEs still necessarily exist for poker variants with more than 2 players.

Your other points are correct, they don't conflict with the existence of a NE though.

An expert on game theory might have strict definitions for all of those things, but to a lay person like me, they are - to all intents and purposes - the same thing.
i.e. Maximizing gains is basically the same as minimizing losses, and maximizing profit (or minimizing your opponent's share of the pot) is the aim of the game.

Just think it through on a basic level. If someone open jams 50bb but only ever has AA, your best strategy is pretty obvious (you fold everything except the other combo of aces). If that same villain shoves 50bb with an exploitable range of 6 combos of AA and 12 combos of 72o (so he's unbalanced in favour of air), you have >50% equity with almost 45% of hands. (Interestingly, 92s and K7o are calls because they dominate 72, but KQo is a fold. 95s is also a call vs such a weird opponent. Go figure).
Presumably, the best way for villain to balance his AA jams would be with Ax combos. e.g. Villain could jam 6x AA, 4x AKs, 4x AQs, and you can still only call with AA. If he adds AJs to his jamming range, then you can add AKs (but not KK) to your calling range.
By a process of trial and error, you can find the optimal "defence" against a known shoving range. If you make the mistake of presuming villain has range X, when he actually has range Y, you can lose quite badly, and losing isn't optimal. So playing a strat vs "all possible strategies" is not guaranteed to beat each of those strategies. There's a specific strat for each opponent.
What we call the Nash Equilibrium arises when both players are maximizing their own EV. If one of them has a sub-optimal strat (or range), then it can be exploited, and you should therefore alter your own range/strat in order to do so. Solvers are just looking for the best strat given the assumption that villain is also playing his best strategy. If his range is terrible, you can maximize EV by exploiting it. Solvers are basically looking for those exploits. You have to give them the range of the opponent, or they have no data to work with.

That's probably just because GTO is so ill-defined that most people in poker just generally use it to mean whatever they want it to mean in poker contexts.

In studying game theory and nash equillibriums in poker, the strategies and payoffs are known to all players in the game. This includes imperfect information about what is included in a players range at a given point in the hand, but should that range change the strategy will also change at that point forward.

However, if you knew the true nash equillibrium for poker and a player tries to use another strategy that is not nash, the nash strategy will not do any worse, but it also might not do any better. So in that sense the opponent's range does not affect your minimum payout.

But if you are trying to find the nash equillibrium the opponent's range WILL change the other strategy.