We introduce a state-based distributed prioritization mechanism for a sensor associated with a dynamical system to access a wireless network, when multiple such systems share the same network link. The priorities, designated Attention Factors, are assigned by each sensor to its data packets, based on measurements of the system state. The Attention Factor represents a quantized value of the minimum risk in not transmitting a given measurement. The Attention Factors from different sensors are evaluated and allotted slots, in a distributed manner, using a dominance-based protocol called tournaments. Packets with the same Attention Factor in a tournament collide, and are lost. We analytically evaluate the probability of a successful transmission using this access mechanism. We also find an upper bound for the estimation and control performance of a system using tournament access, which shows the benefits of using state-based priorities. The proposed tournament mechanism is implemented on the IEEE 802.15.4 standard protocol stack, and evaluated in a hardware-in-the-loop experimental setup.