Two firms are developing competing products for a market of fixed size. The longer a firm spends on development, the better its product. But the first firm to release its product has an advantage: the customers it obtains will not subsequently switch to its rival. (Once a person starts using a product, the cost of switching to an alternative, even one significantly better, is too high to make a switch worthwhile.) A firm that releases its product first, at time t ,captures the share h(t) of the market, where h is a function that increases from time 0 to time T , with h( 0) = 0 and h(T) = 1. The remaining market share is left for the other _rm. If the _rms release their products at the same time, each obtains half of the market. Each firm wishes to obtain the highest possible market share. Model this situation as a strategic game and and its Nash equilibrium (equilibria?).
(When finding firm i 's best response to firm j 's release time tj , there are three cases: that in which h(tj) < 1/2 (firm j gets less than half of the market if it is the first to release its product), that in which h(tj) = 1/2 , and that in which h(tj) > 1/2 .) Also assume h(t)=t and T=1.
(a) Formulate the game (players, actions, payo_s).
(b) Find the best response functions.
(c) Find all Nash equilibria.