As synthetic nanocatalysis strives to create and apply well-defined catalytic centers containing as few as a handful of active metal atoms, it becomes particularly important to understand the structures, compositions, and reactivity of small metal clusters as a function of size and chemical environment. As a part of our effort to better understand the oxidation chemistry of Pt clusters, we present here a comprehensive set of density functional theory simulations combined with thermodynamic modeling that allow us to map out the T- pO(2) phase diagrams and predict the oxygen affinity of PtxOy clusters, x = 1-3. We find that the Pt clusters have a much stronger tendency to form oxides than does the bulk metal, that these oxides persist over a wide range of oxygen chemical potentials, and that the most stable cluster stoichiometry varies with size and may differ from the stoichiometry of the stable bulk oxide in the same environment. Further, the facility with which the clusters are reduced depends both on size and on composition. These models provide a systematic framework for understanding the compositions and energies of redox reactions of discrete metal clusters of interest in supported and gas-phase nanocatalysis.