Bacteria respond to chemical cues by performing a biased random walk that enables them to migrate towards attractants and away from repellents. Bias is achieved by regulating the duration of the bacterial runs as a function of the environment, inferred from the history of chemoattractant detections experienced by the bacterium. This time-signal is processed using a time convolution function that can be assayed measuring the response of the bacterium to short pulses of chemoattractant. The convolution constitutes an elementary form of memory, which is encoded at the molecular level by the processes of
(de-)methylation and (de-)phosphorilation of the underlying biochemical network. While the latter is being characterized in increasing detail, the evolutionary pressures and the functional reasons shaping the bacterial chemotactic response have largely eluded previous efforts to quantify them. We shall show that the chemotactic response observed experimentally emerges from evolution in hostile natural environments as the game-theoretical MaxiMin strategy. In other words, the observed chemotactic behavior is the response that ensures the highest minimal uptake of chemoattractant for individual bacteria in any profile of chemoattractant.