  The price-anticipating Kelly mechanism (PAKM) is one of the most extensively
used strategies to allocate divisible resources for strategic users in
communication networks and computing systems. The users are deemed as selfish
and also benign, each of which maximizes his individual utility of the
allocated resources minus his payment to the network operator. However, in many
applications a user can use his payment to reduce the utilities of his
opponents, thus playing a misbehaving role. It remains mysterious to what
extent the misbehaving user can damage or influence the performance of benign
users and the network operator. In this work, we formulate a non-cooperative
game consisting of a finite amount of benign users and one misbehaving user.
The maliciousness of this misbehaving user is captured by his willingness to
pay to trade for unit degradation in the utilities of benign users. The network
operator allocates resources to all the users via the price-anticipating Kelly
mechanism. We present six important performance metrics with regard to the
total utility and the total net utility of benign users, and the revenue of
network operator under three different scenarios: with and without the
misbehaving user, and the maximum. We quantify the robustness of PAKM against
the misbehaving actions by deriving the upper and lower bounds of these
metrics. With new approaches, all the theoretical bounds are applicable to an
arbitrary population of benign users. Our study reveals two important insights:
i) the performance bounds are very sensitive to the misbehaving user's
willingness to pay at certain ranges; ii) the network operator acquires more
revenues in the presence of the misbehaving user which might disincentivize his
countermeasures against the misbehaving actions.
