A necessary condition for third-degree price discrimination to increase social welfare is an increase in total output. As a result, a focal point has been the analysis of the effects of price discrimination on output. It is known from Pigou (1920) that under linear demands price discrimination does not change output. In the general non-linear case, however, the effect of price discrimination on output may be either positive or negative. Shih, Mai and Liu (1988) and Cheung and Wang (1994) obtain more general results and Aguirre (2009), Aguirre, Cowan and Vickers (2010) and Cowan (2016) show that the effect of third-degree price discrimination on total output is intrinsically related to both the shape of demands and inverse demands in strong markets compared to the shape of direct and inverse demands in weak markets. Over the last few decades much research has analyzed price discrimination in oligopolistic markets both under price competition and quantity competition. Here we focus on price discrimination under quantity competition. In this paper, we extend the traditional analysis of the output effect under monopoly third-degree price discrimination to a multimarket Cournot oligopoly. We show that under symmetric Cournot oligopoly (all firms selling in all markets) similar results to those under monopoly are obtained: in order for total output to increase with price discrimination the demand of the strong market (the high price market) should be, as conjectured by Robinson (1933), more concave than the demand of the weak market (the low price one). When competitive pressure (measured by the number of firms) varies across markets the effect of price discrimination on total output crucially depends on what market, the strong or the weak, is more competitive.