We study a Bertrand duopoly game in which firms adopt a gradient-based mechanism to update their prices. In this competition, one of the firms knows somehow the price adopted by the other firm next time step. Such asymmetric information of the market price possessed by one firm gives interesting results about its stability in the market. Under such information, we use the bounded rationality mechanism to build the model describing the game at hand. We calculate the equilibrium points of the game and study their stabilities. Using different sets of parameter values, we show that the interior equilibrium point can be destabilized through flip and Neimark-Sacker bifurcations. We compare the region of stability of the proposed model with a classical Bertrand model without asymmetric information. The results show that the proposed game's map is noninvertible with type Z(0)-Z(2) or Z(1)-Z(3), while the classical model is of type Z(0)-Z(2) only. This explains the quite complicated basins of attraction given for the proposed map.
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