混合賽局 - 經濟

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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