You could play a gambit in chess. It is an opening where one side sacrifices material for an attack. It would often be a proper noun, but wouldn’t have to be. Morphy occasionally played the King’s Gambit. He enjoys playing gambits. Some people think that declining gambits gives you a psychological edge. Otherwise, I would not use “gambit” with “play” and definitely not with “run”. Their gambit is to contact the sellers immediately and offer to put down a deposit while pretending to be very interested in buying. But they are just fishing for information they can use. ‘Gauntlets’ can be thrown down and picked up (one meaning) or run (a different meaning). I wouldn’t use. “Gamut” I would use. But more likely as “the whole gamut”. “Run the gamut” sounds too annoyingly cute for my taste, but not wrong.