Experimental studies of microbial communities routinely reveal that they have multiple stable states. While each of these states is generally resilient, certain perturbations such as antibiotics, probiotics, and diet shifts, result in transitions to other states. Can we reliably both predict such stable states as well as direct and control transitions between them? Here we present a new conceptual model-inspired by the stable marriage problem in game theory and economics-in which microbial communities naturally exhibit multiple stable states, each state with a different species' abundance profile. Our model's core ingredient is that microbes utilize nutrients one at a time while competing with each other. Using only two ranked tables, one with microbes' nutrient preferences and one with their competitive abilities, we can determine all possible stable states as well as predict inter-state transitions, triggered by the removal or addition of a specific nutrient or microbe. Further, using an example of seven Bacteroides species common to the human gut utilizing nine polysaccharides, we predict that mutual complementarity in nutrient preferences enables these species to coexist at high abundances.