<p><strong>Problem statement:</strong>
A seller is considering how to auction off an item of value. He knows bidders have valuations drawn from a uniform distribution [0:30]. At the moment, he has 4 prospective bidders. he is approached by an Auction Success Consultant, who promises that, for a fixed fee, he will be able to bring 2 more bidders to the auctions who also have valuations drawn from the same uniform distribution [0:30]. What is the maximal fee that the seller would agree to, if he is interested only in maximizing his expected revenue from the auction?</p>
<p><strong>Reasoning:</strong>
Expected valuations drawn from uniform distribution can be calculated as follows: If N valuations are drawn randomly from a uniform distribution, the expected <span class="math-container">$i^{th}$</span> valuation is</p>
<p><span class="math-container">$$\dfrac{N - i +1}{N +1}$$</span></p>
<p>we know that, by definition, the winner pays an amount equal to th second highest bid, so we can restate this where <span class="math-container">$i=2$</span>:
<span class="math-container">$$\left( \dfrac{N - 1}{N + 1}\right)30$$</span></p>
<p>in scenario 1 there are 4 bidders, so the expected revenue is:
<span class="math-container">$$\dfrac{3}{5} \times 30 = 18$$</span></p>
<p>In scenario 2, there are 6 bidders, so the expected revenue is:
<span class="math-container">$$\dfrac{5}{7} \times 30 = 21.4268$$</span>
obviously this is rounded to 4 decimal places/</p>
<p>Finally, i think the maximum that the seller would be willing to pay is the difference between the two, as it leaves him no worse off with the consultant's services or without. Therefore at this price he is indifferent:
<span class="math-container">$$21.4268 - 18 = 3.4286$$</span></p>
<p>So i think the answer is 3.4286</p>
<p>However, whats bothering me is that &quot;<em>in equilibrium, bidders bid only half their values, so the expected revenue of the seller is <span class="math-container">$\dfrac{1}{2}$</span> of the expected value of the higher bidder value</em>&quot;. Does this mean that i should divide each result by 2 in order to get the correct answer?</p>
<p>Thanks in advance for any guidance or help.</p>

CC BY-SA 4.0
2021-11-18T04:00:21.373Z
Hi! Where does this bothersome quote come from?
Giskard
CC BY-SA 4.0
2021-11-19T21:44:30.463Z
HI, it came from the course notes in the subject I'm currently doing. I'm finding it hard to reconcile that quote with what i calculate. I think maybe I'm taking the quote out of context?
Michael Kingston
CC BY-SA 4.0
2021-11-20T06:15:57.560Z
Likely, but to be sure you should probably ask the TA or a peer.
Giskard
CC BY-SA 4.0
2021-11-21T22:33:46.713Z
This is a second price auction, so the quote isn't even true in this scenario.
Walrasian Auctioneer
<game-theory>
expected revenue at private auction
Michael Kingston
# expected revenue at private auction

(Asked by: username_0 on 2021-11-18)

**Problem statement:**
A seller is considering how to auction off an item of value. He knows bidders have valuations drawn from a uniform distribution [0:30]. At the moment, he has 4 prospective bidders. he is approached by an Auction Success Consultant, who promises that, for a fixed fee, he will be able to bring 2 more bidders to the auctions who also have valuations drawn from the same uniform distribution [0:30]. What is the maximal fee that the seller would agree to, if he is interested only in maximizing his expected revenue from the auction?

**Reasoning:**
Expected valuations drawn from uniform distribution can be calculated as follows: If N valuations are drawn randomly from a uniform distribution, the expected $i^{th}$ valuation is

$$\dfrac{N - i +1}{N +1}$$

we know that, by definition, the winner pays an amount equal to th second highest bid, so we can restate this where $i=2$:
$$\left( \dfrac{N - 1}{N + 1}\right)30$$

in scenario 1 there are 4 bidders, so the expected revenue is:
$$\dfrac{3}{5} \times 30 = 18$$

In scenario 2, there are 6 bidders, so the expected revenue is:
$$\dfrac{5}{7} \times 30 = 21.4268$$
obviously this is rounded to 4 decimal places/

Finally, i think the maximum that the seller would be willing to pay is the difference between the two, as it leaves him no worse off with the consultant's services or without. Therefore at this price he is indifferent:
$$21.4268 - 18 = 3.4286$$

So i think the answer is 3.4286

However, whats bothering me is that "*in equilibrium, bidders bid only half their values, so the expected revenue of the seller is $\dfrac{1}{2}$ of the expected value of the higher bidder value*". Does this mean that i should divide each result by 2 in order to get the correct answer?

Thanks in advance for any guidance or help.

**Tags:** game-theory

> **Comments:**
> **username_1**: Hi! Where does this bothersome quote come from?  
> **username_0**: HI, it came from the course notes in the subject I'm currently doing. I'm finding it hard to reconcile that quote with what i calculate. I think maybe I'm taking the quote out of context?  
> **username_1**: Likely, but to be sure you should probably ask the TA or a peer.  
> **username_2**: This is a second price auction, so the quote isn't even true in this scenario.  

---

## 0 answers

