关于同质产品Bertrand Competition推广到N家企业的均衡问题

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收藏 2011-09-14

正在准备学校的考试，有一题很有疑惑，拿出来请教一下各位高人:

Consider Bertrand's oligopoly game (the case of homogenous goods) when the cost and demand functions are as discussed in the class and there are n firms, with n>=3. Show tat the set of Nash Equlibria is the set of profiles (p1,...,pn) of prices for which pi>=c for all i and ar least two prices are equal to c.

目前已知的是：
对于Homogenous good其Betrand函数为
Demand i (pi,pj)= Demand (pi) = a - pi       if pi<pj
                        Demand (pi)/2                if pi=pj
                           0                                     if pi>pj
请问如何求证题目中得：至少存在两个企业的价格等于c？
谢谢！

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vesperw

2011-9-16 03:52:16

1. obviously any pi<c cannot be optimal, so it must be pi>=c for all i
2. suppose pi>c for all i, any i can unilateral deviate to pi=min pj (j=/=i) -epsilon , for small epsilon >0, i will be able to capture all the profit in the market, hence pi>c for all i is not a NE . Because of 2, in any NE, there must be an i with pi=c.
3. if pj>c for all j=/=i, and pi=c, i can unilaterally deviate to set pi=min pj (j=/=i) -epsilon , for small epsilon >0, to increase his profit from 0 to strictly positive amount. Deviation is profitable, so this is not a NE. Because of this, in any NE, there must be at least two firms charging c, and pi>=c for any other i
4. lastly prove the stated strategy profile is a NE....

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caifeifei

2011-9-20 06:45:10

vesperw 发表于 2011-9-15 20:52
1. obviously any pi=c for all i
2. suppose pi>c for all i, any i can unilateral deviate to pi=min p ...

多謝解答！

很詳細，很有邏輯！

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