<p>The idea is simple: the seller wants to target that individuals who's ready to pay him the highest amount, thus targets the person with the highest virtual valuation. </p>

<p>To target the individual who's ready to pay the most, we appeal to the concept of stochastic dominance(specifically, we are talking about first order stochastic dominance). The term <span class="math-container">$\lambda_i =\frac{ f_i(v_i)}{1-F_i(v_i)}$</span> is known as the hazard rate. Reverse of hazard rate, <span class="math-container">$\frac{1}{\lambda_i}$</span> is used to check the hazard rate dominance(which will imply stochastic dominance as well) of one random variable over the other. The random variable which dominates all is given the object by the seller. </p>

CC BY-SA 4.0
2018-11-07T00:06:32.630Z
Could you elaborate on what is stochastic dominance and hazard rate? I have never heard of those terms before :o
Sentient
CC BY-SA 4.0
2018-11-07T17:27:54.883Z
You can refer to Vijay Krishna's Auction Theory for a detailed analysis of the optimal auction problem.
superhulk
CC BY-SA 4.0
2018-11-06T12:01:59.773Z
2018-11-06T17:33:00.297Z
2018-11-06T17:33:00.297Z
superhulk
<p>Suppose you face a single buyer whose willingness to pay, <span class="math-container">$v$</span>, is distributed according to <span class="math-container">$F(v)$</span>. If you charge a price <span class="math-container">$p$</span>, he will buy if and only if <span class="math-container">$v&gt;p$</span>, leaving you with expected revenue of <span class="math-container">$$r(p)=\Pr(v&gt;p)p=[1-F(p)]p.$$</span></p>

<p>Let's maximise revenue by computing an FOC:</p>

<p><span class="math-container">$$r'(p)=1-F(p)-F'(p)p=0.$$</span></p>

<p>We can rearrange this as</p>

<p><span class="math-container">$$\phi(p)\equiv p-\frac{1-F(p)}{F'(p)}=0$$</span>
i.e., the 'virtual valuation' should be zero.</p>

<p>If we return to <span class="math-container">$r'(p)$</span> and think about what the individal terms mean, we can see where the "cost" you ask about comes from: a unit increase in price causes an extra unit of revenue in the fraciton <span class="math-container">$1-F$</span> of the time that the buyer is willing to buy, but reduces the liklihood of him buying by <span class="math-container">$F'$</span>. This is the fundamental trade-off that a seller who doesn't know the buyer's willingness to pay must make.</p>

<hr>

<p>If you know anything about standard monopoly theory then this setup should be quite familiar. Usually, when we look at a profit-maximising monopolist with demand <span class="math-container">$D(p)$</span> and zero marginal cost we solve
<span class="math-container">$$\max_p D(p)p\iff \underbrace{D'(p)p-D(p)}_{\text{marginal revenue}}=0.$$</span>
In an auction-like setting, the 'demand' <span class="math-container">$1-F(p)$</span> is simply the probability of the buyer being willing to pay <span class="math-container">$p$</span>.</p>

CC BY-SA 4.0
2018-11-07T00:15:09.870Z
Is there a reason for auctions we think in terms of the 'virtual valuation' instead of the change in revenue?
Sentient
CC BY-SA 4.0
2018-11-07T07:28:54.393Z
@Sentient the term virtual valuation comes from Myerson's landmark 1981 paper on optimal auction design. The realisation that the problem is equivalent to a monopolist's profit maximisation (with the corresponding notion of marginal revenue) did not come until Bulow & Klemperer, 1989. By then, the virtual value terminology had stuck.
Ubiquitous
CC BY-SA 4.0
2018-11-07T09:07:34.260Z
Ah, thank you for the pointer to their papers!
Sentient
CC BY-SA 4.0
2018-11-06T17:25:34.827Z
2018-11-06T17:25:34.827Z
Ubiquitous
<p>I’m reading through a textbook in auction design when it describes a term, virtual valuation</p>

<p><span class="math-container">$$\phi_i(v_i) = v_i - \frac{1 - F(v_i)}{f(v_i)}$$</span></p>

<p>where <span class="math-container">$f$</span> is the pdf of a bidders valuation and <span class="math-container">$F$</span> is the corresponding cdf.</p>

<p>It is described as the price you want to charge minus the cost of not knowing <span class="math-container">$v_i$</span>. Can someone explain how the cost was derived from the probability distributions? Note I’m not an economics student by training.</p>

CC BY-SA 4.0
2018-11-06T02:34:04.853Z
Does the Wikipedia entry (https://en.wikipedia.org/wiki/Virtual_valuation) help?
Herr K.
CC BY-SA 4.0
2018-11-06T09:48:30.390Z
No, it didn't have an intuitive explanation :(
Sentient
<auctions><mechanism-design>
What is virtual valuation?
Sentient
# What is virtual valuation?

(Asked by: username_0 on 2018-11-06)

I’m reading through a textbook in auction design when it describes a term, virtual valuation

$$\phi\_i(v\_i) = v\_i - \frac{1 - F(v\_i)}{f(v\_i)}$$

where $f$ is the pdf of a bidders valuation and $F$ is the corresponding cdf.

It is described as the price you want to charge minus the cost of not knowing $v\_i$. Can someone explain how the cost was derived from the probability distributions? Note I’m not an economics student by training.

**Tags:** auctions, mechanism-design
> **Comments:**
> **username_1**: Does the Wikipedia entry (https://en.wikipedia.org/wiki/Virtual_valuation) help?  
> **username_0**: No, it didn't have an intuitive explanation :(  

---

## 2 answers

### Answer by username_3 on 2018-11-06. Score: 10 | Accepted answer

Suppose you face a single buyer whose willingness to pay, $v$, is distributed according to $F(v)$. If you charge a price $p$, he will buy if and only if $v>p$, leaving you with expected revenue of $$r(p)=\Pr(v>p)p=[1-F(p)]p.$$

Let's maximise revenue by computing an FOC:

$$r'(p)=1-F(p)-F'(p)p=0.$$

We can rearrange this as

$$\phi(p)\equiv p-\frac{1-F(p)}{F'(p)}=0$$
i.e., the 'virtual valuation' should be zero.

If we return to $r'(p)$ and think about what the individal terms mean, we can see where the "cost" you ask about comes from: a unit increase in price causes an extra unit of revenue in the fraciton $1-F$ of the time that the buyer is willing to buy, but reduces the liklihood of him buying by $F'$. This is the fundamental trade-off that a seller who doesn't know the buyer's willingness to pay must make.

---

If you know anything about standard monopoly theory then this setup should be quite familiar. Usually, when we look at a profit-maximising monopolist with demand $D(p)$ and zero marginal cost we solve
$$\max\_p D(p)p\iff \underbrace{D'(p)p-D(p)}\_{\text{marginal revenue}}=0.$$
In an auction-like setting, the 'demand' $1-F(p)$ is simply the probability of the buyer being willing to pay $p$.

> **Comments:**
> **username_0**: Is there a reason for auctions we think in terms of the 'virtual valuation' instead of the change in revenue?  
> **username_3**: @Sentient the term virtual valuation comes from Myerson's landmark 1981 paper on optimal auction design. The realisation that the problem is equivalent to a monopolist's profit maximisation (with the corresponding notion of marginal revenue) did not come until Bulow & Klemperer, 1989. By then, the virtual value terminology had stuck.  
> **username_0**: Ah, thank you for the pointer to their papers!  

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### Answer by username_2 on 2018-11-06. Score: 6

The idea is simple: the seller wants to target that individuals who's ready to pay him the highest amount, thus targets the person with the highest virtual valuation.

To target the individual who's ready to pay the most, we appeal to the concept of stochastic dominance(specifically, we are talking about first order stochastic dominance). The term $\lambda\_i =\frac{ f\_i(v\_i)}{1-F\_i(v\_i)}$ is known as the hazard rate. Reverse of hazard rate, $\frac{1}{\lambda\_i}$ is used to check the hazard rate dominance(which will imply stochastic dominance as well) of one random variable over the other. The random variable which dominates all is given the object by the seller.

> **Comments:**
> **username_0**: Could you elaborate on what is stochastic dominance and hazard rate? I have never heard of those terms before :o  
> **username_2**: You can refer to Vijay Krishna's Auction Theory for a detailed analysis of the optimal auction problem.  

---

