With the emergence of global digital service providers, concerns about digital oligopolies have increased, with a wide range of potentially harmful effects being discussed. One of these relates to cyber security, where it has been argued that market concentration can increase cyber risk. Such a state of affairs could have dire consequences for insurers and reinsurers, who underwrite cyber risk and are already very concerned about accumulation risk. Against this background, the paper develops some theory about how convex cyber risk affects Cournot oligopoly markets of data storage. It is demonstrated that with constant or increasing marginal production cost, the addition of increasing marginal cyber risk cost decreases the differences between the optimal numbers of records stored by the oligopolists, in effect offsetting the advantage of lower marginal production cost. Furthermore, based on the empirical literature on data breach cost, two possibilities are found: (i) that such cyber risk exhibits decreasing marginal cost in the number of records stored and (ii) the opposite possibility that such cyber risk instead exhibits increasing marginal cost in the number of records stored. The article is concluded with a discussion of the findings and some directions for future research. © 2020 by the authors.