Yesterday I argued that in a treasure-based XP system, megadungeon play is about treasure more than exploration.
That is, you explore as needed to get treasure. You fight as needed to get treasure, too. I've often heard combat described as failure, or to be avoided at all costs, but honestly, that doesn't match with D&D and D&D-based systems as I've played them. Combat that nets you less benefit than it costs is a failure. Combat that nets you something you want and costs less than that's worth to you is a net benefit. It costs resources, sure, but so does everything else - light, healing, food, time, even logistical ability to bring out treasure is a limiting factor.
But it's treasure you want.
In other words, combat for combat's sake is a failure.
Exploration for exploration's sake is wasted.
Treasure found and recovered for any reason is success, regardless of if it is coupled with combat or exploration or not. 10,000 gp is 10,000 gp, whether it's in a heap in a corner of room 1 or hidden behind a series of secret doors down on a deep level guarded by demons.
But does this all change if there are other players in the dungeon, competing for access to that treasure? Does that fundamentally change the game to an exploration-based game where treasure is the reward for exploring?
I think no.
Rivals and competition certainly change the dynamic of a dungeon.
In a non-competitive game or single-party game, you can be a lot more relaxed about finding new sources of treasure. You can spend time exploiting loot until you need to move on to look for more.
In a competitive game, though, you could potentially miss out if you don't explore. Your main goal is still loot. But you're more driven to find things first - something not an issue where you're the only group doing so.
You might order a non-competitive game's priorities as:
1. Treasure. Find it and loot it, you need it to advance.
2. Combat. Kill things that are in the way of #1 or #3.
3. Exploration. Discover new areas to loot.
In a competitive game, my own experience of competition puts the priorities as:
1. Treasure. It's still the main goal.
2. Exploration. Being first has potentially great value.
3. Combat. It's to your benefit if your competition has monsters in their way.
Ultimately, then, I think competition just reframes the value of exploration. But only because the pressure to find and exploit treasure first and most makes it so. You can't be leisurely about exploiting finds - or exploring for finds - if you know another group may be more aggressively exploring. You won't spend time thinking, we need to wipe out those orcs/demons/dragons/eyes of death/whatever because they are a long-term risk. You'll be thinking, great, those orcs/demons/dragons/eyes of death/whatever make a great barrier to other groups, let's avoid them and find new places no one has been before "they" get their ahead of us.
Fear of missing out on new areas with new treasures seems like it would drive exploration's value up in a competitive game. Your goal is still identical, but this would become a more of a priority and more of a means you feel you must maximize.