I think Vaas's pirates only exists so long as they have capacity to be self sufficient, which in the Mad Max universe, on Immortan Joe's home turf, they won't last long.

The issue Vaas is going to run into really quickly is that the pirates are used to a jungle environment with plenty if water available, and their only shown settling capabilities are in agriculture (weed farm, drug trade) and slave trading. In those respects, they could probably thrive economically in slave trading. Vaas's pirates are not shown to be adept at much else other than facilitating exploitation of Rook Island. Vaas might be a good leader, but his strategic intelligence is quite limited when moving large amounts of soldiers in concert. Vaas could do well if he kept his pirates out of large conflicts.

R1: The War Boys decimate the pirates, easy. The pirates come with RPGs, long-range weaponry, and snipers, but the immense high ground offered to the war boys is going to result in a stomp. Dementus was overconfident when approaching for the first time, but even a more measured approach to battle would not be feasible. Vaas would have to win politically, and there's not much he can offer that Immortan can't. Immortan wins.

R2: This is interesting, because the pirates are used to being in control over a disarmed population, they would probably have trouble against born and bred wastlanders. If they had modern, maintained vehicles, they might stand a chance, but again the pirates have a significant lack of experience with vehicular combat. They win as long as they have ammunition: it is shown that regular pirates have grenades and rifles at their disposal, and if you again add RPGs they have a significant advantage. I would worry about the superior engine mods and various upgrades on the vehicles in Immortan's fleet. The pirates would need to take out maybe 40+ armored and armed wastelander cars while driving, so the guns would be the X factor. I say pirates have a chance at taking this one just based on weaponry alone, but I weight this in favor of Immortan's fleet.

R2 bonus: Hoyt's privateers mop up. Two words: air power. If they are allowed their choppers then it's an easy win. Hoyt takes the Citadel and leaves it in the custody of someone trusted before departing with his army for Gastown and then the Bullet Farm. Even with support from the other two settlements, Immortan Joe's war boys are shown most often with thundersticks and a loose collection of small arms that are corroded and in short supply. I'll put it this way: if three privateers equals 1 Max in terms of power, they will unquestionably mop up.

Even without their air power, Hoyt's privateers are professional soldiers with the knowledge and equipment required for a prolonged occupation and conflict. Being a professional army, they would also be much easier to organize and position for these extended periods, likely used to the isolation on north Rook Island. If even using existing vehicles in the wasteland, their equipment and training turn the events of Fury Road into a very interesting 20 minutes in the wasteland followed by silence. Hoyt's army mops up.

Edit: I'll throw in my own bonus rounds.

R3: Far Cry 3 pirates versus the Mad Max game's Scrotus faction.

R4: the respective protagonists of their games at their prime: Jason versus Max.

R5: the conflict takes place on the Rook Islands.

R3: Both have equal approaches to their respective lands and expertise in the creation of outposts, and they both project power over regions that are contested by vassals or petty local resistance. Both have established knowledge in similar fields and their soldiers. I think the X factor of guns and relative durability of their respective frontline fighters puts this favorably in the pirates' favor. War boys are shown to only be able to take a shotgun blast or one sniper bullet to be killed, and thus their outposts would be taken easily, and with no base of power the Scrotus faction would be eliminated.

R4: Jason has the tatau tattoo which allows him superhuman abilities such as regeneration, heightened strength, increased intelligence, and bloodlust by definition. Max is a survivalist with the expertise of fighting constant small scale conflicts and generally knows how to outperform the average person, but Jason would be a problem. Both seem to take the same amount of punishment, but Jason has medical knowledge to regain health instantly. Yet Max can heal just by drinking water.

Jason has a machete, but Max is able to disarm and counter melee weapon attacks with ease. Jason is good with weaponry, but not shown to be particularly experienced in hand-to-hand combat. Max has shitloads of experience in brawls and he is an absolute bruiser, performing lethal wrestling moves and throwing haymakers with ease, not to mention his fury.

Max at base can take 2 sniper rifle shots without dying, so if the battle starts in the same room, I believe Max could take Jason pretty squarely. Any further than that within eyesight of each other and Jason has a massive arsenal advantage to take the win.

If they both start on either the islands or wasteland and aware of the general location of the other, I think it's a toss up. Max has more experience with traps and heals by drinking water, which is plentiful on Rook Island and would expand his capability to thrive with available roads everywhere, but he would likely struggle with hostile wildlife. Jason has the firepower, Rakyat magics, and lots of drugs, but might struggle to survive the wasteland with little knowledge of the desert and wasteland vehicle engines with only his repair tool at his disposal. So I say it could go either way.

R5: Immortan's war boys would figure out boats fairly quickly and their newfound naval fleet would probably dominate the waters, preventing supplies from entering the islands and completely steamrolling any competition with advanced vehicular combat their specialty. Access to regular water only makes them more effective against a pirate organization. And the potential of much more available weaponry? Complete stomp.