Maximizing Utility Among Selfish Users in Social Groups

We consider the problem of a social group of users trying to obtain a "universe" of files, first from a server and then via exchange amongst themselves. We consider the selfish file-exchange paradigm of give-and-take, whereby two users can exchange files only if each has something unique to offer the other. We are interested in maximizing the number of users who can obtain the universe through a schedule of file-exchanges. We first present a practical paradigm of file acquisition. We then present an algorithm which ensures that at least half the users obtain the universe with high probability for $n$ files and $m=O(\log n)$ users when $n\rightarrow\infty$, thereby showing an approximation ratio of 2. Extending these ideas, we show a $1+\epsilon_1$ - approximation algorithm for $m=O(n)$, $\epsilon_1>0$ and a $(1+z)/2 +\epsilon_2$ - approximation algorithm for $m=O(n^z)$, $z>1$, $\epsilon_2>0$. Finally, we show that for any $m=O(e^{o(n)})$, there exists a schedule of file exchanges which ensures that at least half the users obtain the universe.