Unique Nash equilibrium

Loch [100] studies a Bertrand duopoly competing for homogeneous customers in a make-to-order environment. Demand arises in a Poisson fashion with an average rate that depends on price and delay. Two firms have independently distributed processing times for a unit of work, and they compete by specifying prices. The author shows that the game has a unique Nash equilibrium when firms are symmetric, that is, their processing time distributions have the same mean and variance. 

In a multiple firm setting, So [138] assumes the overall market size is constant and show that a unique Nash equilibrium solution exists and is easy to compute. Several managerial insights are offered in the paper. For example, he shows that with all other factors being equal, a firm with high capacity should compete with shorter time guarantee, while a firm with lower unit operating cost should compete with lower price. 

Proposition 3. There exists a unique Nash equilibrium pair A )... 

The game has a unique Nash equilibrium (Maintain, Maintain) as whatever one company does, the other prefers maintain to reduce. Table 1. 

Denote the best response function of the retailer by Rr Q) and the best response function of the wholesaler by Ryj A) both defined for zero initial inventory. At this point we have not demonstrated uniqueness or even existence of the equilibrium. It helps, however, to visualize the problem first. We begin by presenting the game graphically (see Figure 14.1, parameters are taken from the example in Section 6 with r = 8). The point A, is a Nash... 

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