A Generalized Mechanism for Gale-Shapley

Question

I am working on some problems in my applied graph theory course, and we have just gotten to matching problems. We are currently working on a graph problem where instead of there being two types of agents in Gale-Shapley, all of the $n$ agents can establish preferences on the other $n-1.$ I am asked to find a mechanism to ensure a stable matching or if I cannot ensure a stable matching, attempt to find a property that biconditionally prevents a stable matching from existing. I was able to pretty clearly figure out that the latter is true, since we can easily see from the preference profile $$1: 2 \succ 3 \succ 1$$ $$2: 3 \succ 1 \succ 2$$ $$3: 1 \succ 2 \succ 3$$ that we cannot form a stable matching since we will have a blocking pair in any of the three matchings. However, I am having problems figuring out what is occurring that prevents stable matches from happening. Does it involve the number of agents? Is it a property of the preference profile? Any assistance will be greatly appreciated.