The main results of the work are new Bayes–Nash equilibria related to the theory of oligopoly. The article is divided into two parts. The first part considers the classical problem of price competition with product differentiation. (That is, each firm produces one product, but the products differ. This is the next most complex class of models after the models where it is assumed that all firms produce a homogeneous product. It is often believed that the goods produced by firms are substitutes.) Each firm knows its marginal costs, but the marginal costs of competing firms are not known, that is, a game with incomplete information is considered. A special feature of this work is that the probabilistic model (for marginal costs) has a very general form. Concerning the joint distribution of random variables, it is only assumed that each random variable has a finite expectation. The second part of the article considers a more complex form of competition according to Bertrand. This part of the article refers to the scientific direction that is sometimes called "price and quality competition". Firms choose not only prices, but also quality levels of the products they produce. At a higher quality level, demand for products increases (at a fixed price), but the firm's costs also increase. Here, a two-stage game is considered. First, firms choose quality levels, and then prices. In the second part of the article, conditions are imposed that random variables have not only finite expectations, but also finite variances. It is assumed that the demand function is linear and that the firms' cost functions are also linear (relative to output volumes). Explicit expressions are given for equilibrium prices and outputs.