I want to first say that the shotguns do NOT need a nerf.
In all of Bungie's past games, they had a similar and relative damage output and they were great......no one was complaining then. But why?
The answer is because, in Halo, you had to find and pick up special weapons at specific spawn points and the ammo was very limited. There would be a scramble to get the ammunition/weapon and then a couple minutes where those people rained hell on others. The 'Special Ammo' actually felt special, when you finally got a chance to use it and had a limited batch.
To avoid everyone simply/exclusively using a shotgun and, inevitably, taking away from the "tactics" of PvP, all that would need to be done is a minor limitation of the ammo.
Now hear me out: imagine in the first 2 minutes of each match (rather than spawning with enough to get you to the first 'Special Ammo Drop') no one started off with special ammo at all. The only 'Special Ammo' would be found in a crate, in a neutral location on the map. In simpler terms: no initial spawning with special ammo. Immediately, people would have to use a different gun, value the ammo location/drops (instead of the 40 that show up every minute) and it would actually be a 'Special Weapon.'
It would make guns like the Invective, the Ice Breaker, etc, TRULY exotic. It would add a whole 'hot zone' to the early stages of each game.....but most of all, it would put some shotguns away until the people actually find ammo.
All that has to be done is a limitation on ammo, and less people will be running around recklessly.....essentially playing Tag.
Edit: for those who love it the way it is, I would suggest a new game mode entirely: All players spawn with ammo in each category, supers/grenades/melee recharge much faster, ammo crates are everywhere and I give you...........(wait for it) "Chaos."
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How bout NERF movement? Make it so everyone is on rails which only allows side to side movement so nobody can even get close to eachother.