Does a multi-agent system have to have a synchronous component in order for the agents to cooperate?
You're asking about a problem known as leader election.  There are various computational models one can consider, based on what we assume about how the network works and what agents are allowed to do.

The Wikipedia article claims there are no deterministic algorithms for leader election in a setting where all agents start from the same state (have no identity) and are situation in a ring (so connectivity is symmetrical); see https://en.wikipedia.org/wiki/Leader_election#Anonymous_rings.

At the same time, there are randomized algorithms for leader election that work if we make some assumptions about network delay (e.g., that messages won't be delayed forever), and if you're willing to accept algorithms with a finite expected running time.  For instance, a very simple starting point is for each node to pick a random number and broadcast it, and then select the agent with the lowest number as the leader.  In case of ties, repeat the entire procedure.  There are additional details one must work out, depending on the model of computation and the assumptions about communication.

There's lots of research on this topic in the distributed systems literature.  You might want to spend some time reviewing the papers that have been published in that field, to see what you can find.