question on Bertrand and Cournot Equilibriums

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Bertrand and Cournot Equilibria

How does a company choose between the two? What are the deciding factors?
In a two-market Bertrand duopoly, each of two firms chooses one of two markets and a price in that market. All four choices are made simultaneously. In a two-market Cournot duopoly, the firms choose quantities rather than prices. It is well known that in the one-market case the threat of price undercutting means that Bertrand equilibrium prices and profits will be lower and quantities higher than Cournot equilibrium prices, profits and quantities. Companies find a quite different consequence of price undercutting in two-market duopoly. In the two-market case the threat of price undercutting means that Bertrand equilibria are in continuous mixed strategies, while every Cournot duopoly has equilibrium in pure strategies, or in strategies that are pure in each market.

Since Bertrand's famous criticism on Cournot's homogeneous duopoly model, there has been a widely held conjecture, if not a belief, that price competition results in lower prices and higher outputs than does quantity competition. A comparison of these two benchmark oligopoly models has been widely undertaken by companies. Examples include duopolies that examined a duopoly market with general demand and cost functions. They found that at least one firm's price is higher in Cournot equilibrium than in Bertrand equilibrium. Under duopoly companies found that both firms' prices are higher and outputs are lower in quantity competition than in price competition, if each firm can make a profit when the other's price or output is zero. Some companies have obtained similar results with a geometric approach.
Extending the analysis to a general oligopoly model, US companies have obtained a nice conclusion that under certain conditions, prices are lower in Bertrand equilibrium.
These papers significantly improved our understanding of the two models.
However, some questions remain unanswered. First, are prices (outputs) always lower (higher) in price competition? Other companies gave a counter example in
which prices (outputs) are higher (lower) in Bertrand equilibrium than in Cournot equilibrium. However, his demand function xi = 1 - pi - 3pj, i, j = 1, 2, cannot be generated from a concave utility function. It has not been known whether this can happen if the demand function is derived from a concave utility function. It is seen that even if demand is linear and derived from a concave utility function, marginal costs are constant, a firm's price need not be higher in Cournot equilibrium when goods are complements, and output need not be lower when goods are substitutes. Then if both complement and substitute goods exist, neither a price must be higher nor an output must be lower in Cournot equilibrium. We actually can say little whether price competition is more competitive.
Secondly, as a Bertrand equilibrium does not necessarily imply lower prices or higher outputs with a mixture of ...