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In economics, oligopoly is a market structure in which a small number of
firms producing substitutable goods compete by setting prices strategically
in an uncertain demand environment (Bertrand model). For instance, Pepsi
and Coca-Cola in the soft drink industry, Apple and Samsung in the smart
phone industry. Firms choose prices to maximize profit in the sense of
Nash Equilibrium. The Bertrand Oligopoly can be modelled as nonzero sum
differential games. The model can be solved by the dynamic programming
principle, which yields a system of Hamilton-Jacobi-Bellman (HJB) equations.
More precisely, let 
be the value functions, which represent the
expected discount lifetime profit. Then for the special case where there
are only two players (duopoly), the unknowns satisfy:
