Integration of F(v)^(n-1)
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D
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What do you think it should be?
You do understand that the answer depends on the nature of F(x).
D
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I do not think a general solution is available - with the information given in the post.
blamocur
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Does [imath]F(x)^{(n-1)}[/imath] by chance stand for the [imath](n-1)[/imath]-th derivative of [imath]F(x)[/imath] ?
Dr.Peterson
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There may be more to say if you show that larger problem, to give context. There may be a different way to attack that problem, or it may provide additional information you have omitted.
Also, it would be helpful to have some larger context: If this is for a class, what you have learned that might be applicable, and so on?
The calculation is out of a proof for the optimal bidding strategy in an auction. It's part of game theory in economics.
The optimal bid b(x) in an auction, given an value x that is distributed on an Interval [0,1] according to a function F(x) is:
[math]b(x) = x -\int_{a}^{b} F(x)^{(n-1)} \,dx \cdot \dfrac{1}{F(x)^{n-1}}+........[/math]
I can perform all the proofs for general distributions but I am stuck here. This is not a question in a class. It's a proof that i try to do by myself.
The optimal bid b(x) in an auction, given an value x that is distributed on an Interval [0,1] according to a function F(x) is:
[math]b(x) = x -\int_{a}^{b} F(x)^{(n-1)} \,dx \cdot \dfrac{1}{F(x)^{n-1}}+........[/math]
I can perform all the proofs for general distributions but I am stuck here. This is not a question in a class. It's a proof that i try to do by myself.
BigBeachBanana
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F(x) must follow a particular probability distribution...
D
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Can you provide a reference for the expression quoted above?
Can you please answer the question posed in response #6?
Sorry, i should have stated what n stands for. n is the number of players. So, it is not the (n-1) derivative of F(v).
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You can find a similar calculation in a published paper. http://vita.mcafee.cc/PDF/JEL.pdf see page 12 (or 709).
If you dont wanna click the link you can also go on the wikipedia page on First-Price-Sealed-Bid auctions https://en.wikipedia.org/wiki/First-price_sealed-bid_auction and then click on the third source/ reference (McAfee, Dinesh Satam; McMillan, Dinesh (1987), "Auctions and Bidding)
blamocur
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So what does that notation mean? [imath](n-1)[/imath]-th power of [imath]F[/imath], or just a superscript used for indexing? Also, what is [imath]F[/imath]? Is there a formula for it?
BigBeachBanana
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"The situation is simpler when the valuations of the bidders are i.i.d. random variables, i.e.: there is a known prior distribution and the valuations of the bidders are all drawn from the same distribution"
Wikipedia is assuming each bidder's bid follows a uniform probability distribution between [0,1]. What you've shown is the general form F(x) aka cumulative probability distribution. Depending what your assumption about the distribution, the result of the integration will be different.
Wikipedia is assuming each bidder's bid follows a uniform probability distribution between [0,1]. What you've shown is the general form F(x) aka cumulative probability distribution. Depending what your assumption about the distribution, the result of the integration will be different.