On the existence of linear equilibria in models of market making

We derive necessary and sufficient conditions for a linear equilibrium in three types of competitive market making models: Kyle type models (when market makers only observe aggregate net order flow), Glosten-Milgrom and Easley-O'Hara type models (when market makers observe and trade one order at a time), and call markets models (individual order models when market makers observe a number of orders before pricing and executing any of them). We study two cases: when privately informed (strategic) traders are symmetrically informed and when they have differential information, We derive necessary and sufficient conditions on the distributions of the random variables for a linear equilibrium We also explore those features of the equilibrium that depend on linearity as opposed to the particular distributional assumptions and we provide a large number of examples of linear equilibria for each of the models.