We study in this paper two competing AIMD flows that share a common bottleneck link. When congestion occurs, one (or both) flows will suffer a loss that will cause its throughput to decrease by a multiplicative factor. The identity of the flow that will suffer a loss is determined by a randomized "loss strategy" that may depend on the throughputs of the flows at the congestion instant. We analyze several loss strategies: the one in which the identity of the flow experiencing the loss is independent of the current throughput and the one in which the flow with the largest throughput is to suffer the loss; this is compared with the strategy that assigns loss probabilities proportionally to the throughputs (thus a flow with a larger throughput has a larger loss probability). After deriving some results for the general asymmetric case, we focus in particular on the symmetric case and study the influence of the strategy on the average throughput and average utilization of the link. As the intuition says, a strategy that assigns a loss to a flow with a higher throughput is expected to give worse performance since the total instantaneous throughput after a loss is expected to be lower with such a strategy. Surprisingly, we show that this is not the case. We show that the average throughput and average link utilizations are invariant: they are the same under any possible strategy; the link utilization is 6/7 of the link capacity. We show, in contrast, that the second moment of the throughput does depend on the strategy.