Contention-based multiple access is a crucial component of many wireless systems. Multiple-packet reception (MPR) schemes that use interference cancellation techniques to receive and decode multiple packets that arrive simultaneously are known to be very efficient. However, the MPR schemes proposed in the literature require complex receivers capable of performing advanced signal processing over significant amounts of soft undecodable information received over multiple contention steps. In this paper, we show that local channel knowledge and elementary received signal strength measurements, which are available to many receivers today, can actively facilitate multi-packet reception and even simplify the interference canceling receiver's design. We introduce two variants of a simple algorithm called dual power multiple access (DPMA) that use local channel knowledge to limit the receive power levels to two values that facilitate successive interference cancellation. The resulting receiver structure is markedly simpler, as it needs to process only the immediate received signal without having to store and process signals received previously. Remarkably, using a set of three feedback messages, the first variant, DPMA-Lite, achieves a stable throughput of 0.6865 packets per slot. Using four possible feedback messages, the second variant, turbo-DPMA, achieves a stable throughput of 0.793 packets per slot, which is better than all contention algorithms known to date.