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This paper deals with duopoly models with vertical product differentiation. These are two-stage games. In the first stage, the quality game, the two firms choose their quality. In the second stage, the price game, they choose their prices. When the market is covered, the properties of this game are well known, even when the consumers are not uniformly distributed. When the market is uncovered the literature is not always clear about these models. In the present paper it is shown that the price game can be solved, when the consumers are distributed according to a distribution function with a log-concave density function.

Non-cooperative game theory has in recent years become one of the main tools in the social sciences and economics. It deals with decision problems of more than one (rational) player. The actions of those players result in outcomes that are the ‘best’ that may have happened to a player, given the actions of the other players. This is precisely the content of the 'Nash equilibrium' concept. It would be nice when each game had a unique Nash equilibrium. Unfortunately, both non-existence and multiple equilibria occur. An important role in game theory is played by the ‘information‘ available to the players. Among the economic applications are: the ‘Theory of Industrial Organisation,’ ‘Bargaining’ and the ‘Economic Theory of Information.’

“Solving Games by Differential Equations”: was the title of a paper by Brown and von Neumann (1950). The idea of solving bargaining games by differential equations goes back to Raiffa (1953). The present paper starts with a solution ϕ of a bargaining game and constructs, by means of solving a set of differential equations, a new solution Cϕ, the continuation of ϕ. A property is called hereditary when it holds for Cϕ whenever it holds for ϕ. Hereditary properties and properties of Cϕ in relation to properties of ϕ are studied. The paper includes a number of open problems.