We present a simple formalism for parton-shower Markov chains. As a first step towards more complete "uncertainty bands," we incorporate a comprehensive exploration of the ambiguities inherent in such calculations. To reduce this uncertainty, we then introduce a matching formalism which allows a generated event sample to simultaneously reproduce any infrared-safe distribution calculated at leading or next-to-leading order in perturbation theory, up to subleading corrections. To enable a more universal definition of perturbative calculations, we also propose a more general definition of the hadronization cutoff. Finally, we present an implementation of some of these ideas for final-state gluon showers, in a code dubbed Vincia.