  We consider the fairness in cooperative data exchange (CDE) problem among a
set of wireless clients. In this system, each client initially obtains a subset
of the packets. They exchange packets in order to reconstruct the entire packet
set. We study the problem of how to find a transmission strategy that
distributes the communication load most evenly in all strategies that have the
same sum-rate (the total number of transmissions) and achieve universal
recovery (the situation when all clients recover the packet set). We formulate
this problem by a discrete minimization problem and prove its $M$-convexity. We
show that our results can also be proved by the submodularity of the feasible
region shown in previous works and are closely related to the resource
allocation problems under submodular constraints. To solve this problem, we
propose to use a steepest descent algorithm (SDA) based on $M$-convexity. By
varying the number of clients and packets, we compare SDA with a deterministic
algorithm (DA) based on submodularity in terms of convergence performance and
complexity. The results show that for the problem of finding the fairest and
minimum sum-rate strategy for the CDE problem SDA is more efficient than DA
when the number of clients is up to five.
