I guess I could sub divide this into
professional and personal

professional: It happens the most in my professional career. I am in tech sales. Overall I would say that most arrogant people in that world are in insecure or in a delicate position and it is them pushing back to their situation.

In that idea it is a mix of compassion (for their situation) but having them own whatever situation created the arrogance in fhe first place. But it never anything person/ it just business to me.

In business If it is a personality trait, which is more rare I don't take it personally but note it in terms of negocations and such

On a personal level --I just move around it and don't interact with them anymore than I absolutely have to I don't get mad but or upset because that is a "you problem" not a we problem. I just don't spend time or energy on arrogant people and feel apathy to their existence on a personal level.

I usually laugh and carry on with my life.

I don't. I just trip them as they're walking away.

My face does the talking and I ignore them.

I avoid them.

Quite often they can simply be ignored.

Just keep being me.

I try to stay away from them

I would say I ignore them but they wouldnt know it lol

I ignore them.

Shut them down.

Ignore and wait for an out

Ignore and move on.

Ignore, keep the distance and gradually fade into obscurity

Ignore

I arrogant them right back.

Deny them the adoration, admiration and attention they so desperately crave.
"Hurt me" said the masochist and the sadist said "No"

I simply ignore and avoid them. Otherwise, it could get ugly.

I roll my eyes and walk away