Horde Mode. Your fire team defeats enemies that comes in waves, and in the final one you get to fight a boss.
Boss Rush Mode. Your fireteam defeats bosses solely.
Competitive. Your fireteam races another fireteam to seek out which team defeats more enemies than other.
Survival Mode. Like Horde Mode, only endless.
I personally think that the Arena mode will be something like a Horde Mode, but a Boss Rush Mode seems plausible too.
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I think because it's called the prison of elders it will be a boss rush mode. Maybe ultras with randomly generated fallen names