The two commands affect the same account balance, so they don't commute, so these commands conflict. Every EPaxos worker is required to be able to determine whether any two commands are conflicting, in this case it would be something like:

def do_commands_conflict(c1): return len(write(c1) & read(c2)) > 0 or len(write(c2) & read(c1)) > 0 or len(write(c1) & write(c2)) > 0

Whenever an EPaxos node learns about a new command, it compares it to the commands that it already knows about. If it conflicts with any current commands, then it gains a dependency on them (see Figure 3, "received PreAccept"). So the commands race; the first node to learn about both of them is going to determine the dependency order [in some cases, two nodes will disagree on the order that the conflicting commands were received -- this is what the "Slow Path" is for].

The clients don't coordinate this; the EPaxos nodes choose the order. The cluster as a whole guarantees linearity. This just means that there's at least one possible ordering of client requests that would produce the observed behavior; if two clients send requests concurrently, there's no guarantee of who goes first.

(in particular, the committed dependency graph is durable, even though it's arbitrary, so in the event of a failure/restart, all of the nodes will agree on the dependency graph, which means that they'll always apply non-commuting commands in the same order)