  We consider five different peer-to-peer file sharing systems with two chunks,
with the aim of finding chunk selection algorithms that have provably stable
performance with any input rate and assuming non-altruistic peers who leave the
system immediately after downloading the second chunk. We show that many
algorithms that first looked promising lead to unstable or oscillating
behavior. However, we end up with a system with desirable properties. Most of
our rigorous results concern the corresponding deterministic large system
limits, but in two simplest cases we provide proofs for the stochastic systems
also.
