Sensing extracellular changes initiates signal transduction and is the first stage of cellular decision-making. Yet relatively little is known about why one form of sensing biochemistry has been selected over another. To gain insight into this question, we studied the sensing characteristics of one of the biochemically simplest of sensors: the allosteric transcription factor. Such proteins, common in microbes, directly transduce the detection of a sensed molecule to changes in gene regulation. Using the Monod-Wyman-Changeux model, we determined six sensing characteristics--the dynamic range, the Hill number, the intrinsic noise, the information transfer capacity, the static gain, and the mean response time--as a function of the biochemical parameters of individual sensors and of the number of sensors. We found that specifying one characteristic strongly constrains others. For example, a high dynamic range implies a high Hill number and a high capacity, and vice versa. Perhaps surprisingly, these constraints are so strong that most of the space of characteristics is inaccessible given biophysically plausible ranges of parameter values. Within our approximations, we can calculate the probability distribution of the numbers of input molecules that maximizes information transfer and show that a population of one hundred allosteric transcription factors can in principle distinguish between more than four bands of input concentrations. Our results imply that allosteric sensors are unlikely to have been selected for high performance in one sensing characteristic but for a compromise in the performance of many.