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Random-access mechanisms play an important role in wireless networks, and have been extensively studied in recent years. Although many previous studies have proposed enhanced algorithms, each one has only considered either throughput or fairness. In this paper, we propose an efficient random-access mechanism called Multi-round Collision Avoidance (MrCA) that considers throughput and fairness together. The key idea in MrCA is to avoid collisions through multiple contentions, each with a smaller sized contention window. With this simple modification, we can significantly reduce the collision probability as well as the access delay, in addition to increasing fairness index. We find the collision probability and throughput analytically. Through simulation, we validate our analytical model and find appropriate parameters for achieving good performance. We also demonstrate that, compared to the IEEE 802.11 DCF, MrCA makes the collision probability extremely low, so that it increases throughput by 25% as well as short-term fairness by 50% with 50 contending nodes. When MrCA and 802.11 DCF schemes are combined with the auto rate fallback scheme, the performance gain of MrCA over 802.11 DCF increases because MrCA lowers the collision probability significantly, which makes channel error estimation more accurate. We also discuss the issues of implementation and backward compatibility.