  We find that existing multi-party quantum key agreement (MQKA) protocols
designed for fairness of the key are, in fact, unfair. Our analysis shows that
these protocols are sensitive to collusive attacks; that is, dishonest
participants can collaborate to predetermine the key without being detected. In
fact, the transmission structures of the quantum particles in those unfair MQKA
protocols, three of which have already been analyzed, have much in common. We
call these unfair MQKA protocols circle-type MQKA protocols. Likewise, the
transmission structures of the quantum particles in MQKA protocols that can
resist collusive attacks are also similar. We call such protocols
complete-graph-type MQKA protocols. A MQKA protocol also exists that can resist
the above attacks but is still not fair, and we call it the tree-type MQKA
protocol. We first point out a common, easily missed loophole that severely
compromises the fairness of present circle-type MQKA protocols. Then we show
that two dishonest participants at special positions can totally predetermine
the key generated by circle-type MQKA protocols. We anticipate that our
observations will contribute to secure and fair MQKA protocols, especially
circle-type protocols.
