Integration of F(v)^(n-1)

[waluma]

Hi all,

maybe it is a dumb question but hopefully somebody can help me.

What is the integral of: $$\int_{a}^{b} F(x)^{(n-1)}\,dx$$

Tank you very much.
Chris

 

[Deleted member 4993]

Hi all,

maybe it is a dumb question but hopefully somebody can help me.

What is the integral of: $$\int_{a}^{b} F(x)^{(n-1)}\,dx$$

Tank you very much.
Chris

What do you think it should be?

You do understand that the answer depends on the nature of F(x).

 

[waluma]

Is it possible to have a general solution without knowing F(x) in the first place. This is part of a general proof. The idea is to solve it for "every" distribution F(x).

 

[Deleted member 4993]

Is it possible to have a general solution without knowing F(x) in the first place. This is part of a general proof. The idea is to solve it for "every" distribution F(x).

I do not think a general solution is available - with the information given in the post.

 

[waluma]

ok, thank you.

 

[blamocur]

Hi all,

maybe it is a dumb question but hopefully somebody can help me.

What is the integral of: $$\int_{a}^{b} F(x)^{(n-1)}\,dx$$

Tank you very much.
Chris

Does $F(x)^{(n-1)}$ by chance stand for the $(n-1)$-th derivative of $F(x)$ ?

 

[Dr.Peterson]

Is it possible to have a general solution without knowing F(x) in the first place. This is part of a general proof. The idea is to solve it for "every" distribution F(x).

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?

 

[waluma]

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:
$$b(x) = x -\int_{a}^{b} F(x)^{(n-1)} \,dx \cdot \dfrac{1}{F(x)^{n-1}}+........$$
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]

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:
$$b(x) = x -\int_{a}^{b} F(x)^{(n-1)} \,dx \cdot \dfrac{1}{F(x)^{n-1}}+........$$
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.

F(x) must follow a particular probability distribution...

 

[Deleted member 4993]

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:
$$b(x) = x -\int_{a}^{b} F(x)^{(n-1)} \,dx \cdot \dfrac{1}{F(x)^{n-1}}+........$$
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.

Can you provide a reference for the expression quoted above?

Can you please answer the question posed in response #6?

 

[waluma]

Does $F(x)^{(n-1)}$ by chance stand for the $(n-1)$-th derivative of $F(x)$ ?

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).

 

[waluma]

Can you provide a reference for the expression quoted above?

Can you please answer the question posed in response #6?

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]

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).

So what does that notation mean? $(n-1)$-th power of $F$, or just a superscript used for indexing? Also, what is $F$? Is there a formula for it?

 

[BigBeachBanana]

"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.