On The Robustness of Price-Anticipating Kelly Mechanism

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: 1) the performance bounds are very sensitive to the misbehaving user's willingness to pay at certain ranges and 2) the network operator acquires more revenues in the presence of the misbehaving user which might disincentivize his countermeasures against the misbehaving actions.