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)》之銘言： : 考慮存在於寡占市場中的廠商皆進行價格競爭， : 且每家廠商的邊際成本為0、固定成本為k。 : 現在市場上有兩種類型的消費者， : 第一種是informed consumers：能夠知道最便宜的產品在哪一家store : 第二種是uninformed consumers：只能隨機的選擇store消費 : 而且第一種消費者的數量為I，第二種消費者的數量為2U : 假設每一家廠商在某一期間內確定能招攬到數量U的第二種消費者， : 若想吸引到第一種消費者，產品的定價則必須是市場上的最低價 : 而不論是第一種還是第二種消費者， : 當價格低於保留價格r時，才會購買1單位的產品，否則不會購買。 : 試問，在每一家廠商都採取相同的策略下，此賽局的Nash均衡為何？ --

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