To be exact, assume that all firms are risk-netural. Assume first that there are two existing firms. We will check this assumption later. Denote them as 1 and 2. Since we are interested in symmetric strategies, it is sufficient to consider firm 1's strategy only. Simple obsveration: there is no pure-strategy Nash equilibrium. (why?) Define F(x)=prob.{firm 2's price <= x} to be firm 2's equilibrium mixed strategy, and assume that F(x) is differentiable with support [a,b], where 0 <= a < b <= r. Given F(．), if firm 1's price is x, his expected profit is π(x)= x{U+[1-F(x)]I}-k , if x>0; 0 , if x=0. (1) (x=0 corresponds to no production) To justify the symmetric mixed-strategy F(x), we must have π'(x)=0 for all x belongs to [a,b]. (2) Moreover, to justify that no other firm wants to enter the market, we must have π(a)=π(b)=π(0)=0 (3) Note that equations (1) and (2) together generate the following differential equation [xF(x)]'=1+ U/I. (4) From (3), one should be able to find a and b, which provide the boundary condtions of (4).* The rest of the problem follows. *Here I try to endogenize the number of existing firms, and find it is related to the fixed cost k and reserve price r. The condtion for exactly two firms existing in the market is rU-k=0, provided by b=r. (why b should be r?) On the other hand, if the number of firms is exogenous, then (3) can be modified to provide a lower bound of a (b=r still), so the boundary condition becomes π(b)=rU-k. (5) For those who are interested in more complete results of the model, please refer to Varian (1980), A Model of Sales. ※ 引述《esail (我)》之铭言： : 考虑存在於寡占市场中的厂商皆进行价格竞争， : 且每家厂商的边际成本为0、固定成本为k。 : 现在市场上有两种类型的消费者， : 第一种是informed consumers：能够知道最便宜的产品在哪一家store : 第二种是uninformed consumers：只能随机的选择store消费 : 而且第一种消费者的数量为I，第二种消费者的数量为2U : 假设每一家厂商在某一期间内确定能招揽到数量U的第二种消费者， : 若想吸引到第一种消费者，产品的定价则必须是市场上的最低价 : 而不论是第一种还是第二种消费者， : 当价格低於保留价格r时，才会购买1单位的产品，否则不会购买。 : 试问，在每一家厂商都采取相同的策略下，此赛局的Nash均衡为何？ --

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