We consider a model of price setting duopolists with capacity constraints originated in papers of Bertrand (1883) and Edgeworth (1925). It is well known that this model may not posses a Nash equilibrium. Our aim is to solve original problem in terms of the equilibrium in secure strategies. It coincides with the Nash Equilibrium when Nash Equilibrium exists and takes into account the intention of players to maximize their profit under the condition of security against the actions of other players. This approach reflects the natural logic of behavior of players in this model. We will show that in some cases when Nash equilibrium does not exist there is an Equilibrium in secure strategies. However for some (big enough) capacities equilibrium in secure strategies does not exist either. A criterion for the equilibrium existence is formulated and proved.