You're likely referring to Ghost Mode, the 1v1 competitive multiplayer in Hitman 2. It is not called "Dome Games."
How Ghost Mode Works
Asynchronous Competition: Both players hunt the same targets on the same map but in separate "realities"—you see a "ghost" of your opponent's position/actions but cannot interact directly.
Scoring: Points for kills, specific methods (accidents, poison), disguises, and avoiding detection. Bonus challenges rotate each match.
Time Limit: Typically 10–15 minutes per round.
Win Condition: First to 5 points (or most when time expires) wins.
Variants
Standard Match: Classic 1v1.
Escalation Contracts: Progressive difficulty with modifiers.