A wireless sensor network is a network of compact micro-sensors equipped with wireless communication capability. In a wireless sensor network deployed to collect and update data, sensor nodes are usually battery-powered. Moreover, they are hardly able to support complex computation. Such unique features give rise to a critical issue; how to devise a simple energy-aware MAC scheme. Intending to resolve the issue, we consider a MAC scheme based on p-persistence slotted ALOHA. To determine the value of the attempt probability p, we construct the p-persistence slotted ALOHA as a simple non-cooperative game which involves generalized payoffs reflecting energy saving and throughput. Then, we derive a Nash equilibrium in a closed form. A Nash equilibrium is known to possibly incur a protocol failure and network instability. Using the Nash equilibrium, we thus investigate the effect of the payoffs on the stability of a wireless sensor network. For the stability of the wireless sensor network, numerical examples suggest an appropriate reflection of the traffic load on the payoffs.