<p><a href="https://stats.stackexchange.com/a/214514">Discrete distributions have no pdf</a>.</p>

<p>And since $r(v)$ is defined via $f(v)$ virtual valuations also do not exist in this case. Even if one were to approximate the discrete distribution $X$ with continuous distributions $X_n$ in such a manner that
$$
\forall v: \lim_{n \to \infty} F_n(v) = F(v),
$$
the limit $\lim_{n \to \infty} r_n(v)$ would not exist where $\lim_{n \to \infty} f_n(v) = 0$.</p>

CC BY-SA 4.0
2018-06-01T13:06:03.930Z
So, is there a way to use the optimal auction theory when buyers' valuations come from discrete distributions?
Erel Segal-Halevi
CC BY-SA 4.0
2018-06-01T13:15:28.003Z
It depends on the distribution. But that was not your question, was it?
Giskard
CC BY-SA 4.0
2018-06-01T11:12:45.197Z
2018-06-01T11:18:05.013Z
2018-06-01T11:18:05.013Z
Giskard
<p>Virtual valuation is the derivative of the expected revenue function with respect to the tail probability <span class="math-container">$q$</span> that is then evaluated at value <span class="math-container">$v$</span>. The Revenue function is</p>
<p><span class="math-container">$$R(q) = q\cdot v(q),\;\;\; q = 1-F(v),\;\;\; v(q) = F^{-1}(1-q) \tag{1}$$</span></p>
<p>and</p>
<p><span class="math-container">$$r(v(q)): = \frac{dR(q)}{dq}=  \frac{d}{dq}\big[q\cdot F^{-1}(1-q)\big] \tag{2}$$</span></p>
<p>written, <em>after</em> the calculations, in terms of <span class="math-container">$v$</span>, so</p>
<p><span class="math-container">$$r(v)=v - \frac{1-F(v)}{f(v)} \tag{3}$$</span></p>
<p>If <span class="math-container">$v$</span> is discrete we have</p>
<p><span class="math-container">$$q_j = 1-F(v\leq v_j),\;\;\; j=1,...,k \tag{4}$$</span></p>
<p>where now <span class="math-container">$F$</span> is a distribution function of a discrete rv, an where <span class="math-container">$j$</span> counts the ordered discrete values that <span class="math-container">$v$</span> may take, <span class="math-container">$v \in \{v_1,...,v_k\}$</span>.</p>
<p>Discretizing the relationship we could define</p>
<p><span class="math-container">$$r(v(q_j, q_{j+1})):= \frac{R(q_{j+1})-R(q_j)}{q_{j+1}-q_j} \tag{5}$$</span></p>
<p>which in terms of <span class="math-container">$v$</span> becomes</p>
<p><span class="math-container">$$r(v(q_j, q_{j+1})):= \frac{v_{j+1}\cdot [1- F(v\leq v_{j+1})] -v_{j}\cdot [1- F(v\leq v_{j})]}{1- F(v\leq v_{j+1})-1+F(v\leq v_{j})} $$</span></p>
<p><span class="math-container">$$=\frac{v_{j+1}\cdot Pr(v &gt; v_{j+1}) -v_{j}\cdot Pr(v &gt; v_{j})}{-Pr(v=v_{j+1})}$$</span></p>
<p>Using</p>
<p><span class="math-container">$$Pr(v &gt; v_{j}) = Pr(v=v_{j+1}) + Pr(v&gt;v_{j+1})$$</span></p>
<p>we get</p>
<p><span class="math-container">$$r(v_j,v_{j+1}) = v_{j} - (v_{j+1}-v_j)\cdot \frac{Pr(v&gt;v_{j+1})}{Pr(v=v_{j+1})}$$</span></p>
<p>or</p>
<p><span class="math-container">$$r(v_j,v_{j+1}) = v_{j} - (v_{j+1}-v_j)\cdot \frac{1-F(v_{j+1})}{Pr(v=v_{j+1})} \tag{6}$$</span></p>
<p>Since to a first approximation, if <span class="math-container">$v$</span> is continuous, and <span class="math-container">$v_j,v_{j+1}$</span> are &quot;close enough&quot; we have</p>
<p><span class="math-container">$$Pr[v \in (v_j,v_{j+1})] \approx f(v_{j+1}) \cdot (v_{j+1}-v_{j})$$</span></p>
<p>It is clear how the obtained relation for the discrete case can be seen as the analogue of the continuous case.</p>
<p>For practical applications, one could think also of using a &quot;continuity correction&quot; for the probability in the denominator of <span class="math-container">$(6)$</span>.</p>

CC BY-SA 4.0
2018-06-01T13:48:12.377Z
2021-12-12T13:38:17.690Z
2021-12-12T13:38:17.690Z
Alecos Papadopoulos
<p>The <a href="https://en.wikipedia.org/wiki/Virtual_valuation" rel="nofollow noreferrer">virtual valuation</a>  of a buyer in an auctin is a function used to calculate the expeced revenue of a seller from that buyer. When the buyer's value comes from a continuous distribution with pdf $f$ and cdf $F$, the virtual valuation is:
$$
r(v) := v - \frac{1-F(v)}{f(v)}
$$
How exactly should the virtual valuation be calculated when the distribution is discrete?</p>

<p>For example, suppose the valuation of a buyer can take three values: 1 USD - with probability 0.3, 2  USD - with probability 0.3, 3  USD - with probability 0.4. What exactly are $F$ and $f$ in this case?</p>

<auctions>
Virtual valuation when the distribution is discrete
Erel Segal-Halevi
# Virtual valuation when the distribution is discrete

(Asked by: username_0 on 2018-06-01)

The [virtual valuation](https://en.wikipedia.org/wiki/Virtual_valuation) of a buyer in an auctin is a function used to calculate the expeced revenue of a seller from that buyer. When the buyer's value comes from a continuous distribution with pdf $f$ and cdf $F$, the virtual valuation is:
$$
r(v) := v - \frac{1-F(v)}{f(v)}
$$
How exactly should the virtual valuation be calculated when the distribution is discrete?

For example, suppose the valuation of a buyer can take three values: 1 USD - with probability 0.3, 2 USD - with probability 0.3, 3 USD - with probability 0.4. What exactly are $F$ and $f$ in this case?

**Tags:** auctions

---

## 2 answers

### Answer by username_2 on 2018-06-01. Score: 5 | Accepted answer

Virtual valuation is the derivative of the expected revenue function with respect to the tail probability $q$ that is then evaluated at value $v$. The Revenue function is

$$R(q) = q\cdot v(q),\;\;\; q = 1-F(v),\;\;\; v(q) = F^{-1}(1-q) \tag{1}$$

and

$$r(v(q)): = \frac{dR(q)}{dq}= \frac{d}{dq}\big[q\cdot F^{-1}(1-q)\big] \tag{2}$$

written, *after* the calculations, in terms of $v$, so

$$r(v)=v - \frac{1-F(v)}{f(v)} \tag{3}$$

If $v$ is discrete we have

$$q\_j = 1-F(v\leq v\_j),\;\;\; j=1,...,k \tag{4}$$

where now $F$ is a distribution function of a discrete rv, an where $j$ counts the ordered discrete values that $v$ may take, $v \in \{v\_1,...,v\_k\}$.

Discretizing the relationship we could define

$$r(v(q\_j, q\_{j+1})):= \frac{R(q\_{j+1})-R(q\_j)}{q\_{j+1}-q\_j} \tag{5}$$

which in terms of $v$ becomes

$$r(v(q\_j, q\_{j+1})):= \frac{v\_{j+1}\cdot [1- F(v\leq v\_{j+1})] -v\_{j}\cdot [1- F(v\leq v\_{j})]}{1- F(v\leq v\_{j+1})-1+F(v\leq v\_{j})} $$

$$=\frac{v\_{j+1}\cdot Pr(v > v\_{j+1}) -v\_{j}\cdot Pr(v > v\_{j})}{-Pr(v=v\_{j+1})}$$

Using

$$Pr(v > v\_{j}) = Pr(v=v\_{j+1}) + Pr(v>v\_{j+1})$$

we get

$$r(v\_j,v\_{j+1}) = v\_{j} - (v\_{j+1}-v\_j)\cdot \frac{Pr(v>v\_{j+1})}{Pr(v=v\_{j+1})}$$

or

$$r(v\_j,v\_{j+1}) = v\_{j} - (v\_{j+1}-v\_j)\cdot \frac{1-F(v\_{j+1})}{Pr(v=v\_{j+1})} \tag{6}$$

Since to a first approximation, if $v$ is continuous, and $v\_j,v\_{j+1}$ are "close enough" we have

$$Pr[v \in (v\_j,v\_{j+1})] \approx f(v\_{j+1}) \cdot (v\_{j+1}-v\_{j})$$

It is clear how the obtained relation for the discrete case can be seen as the analogue of the continuous case.

For practical applications, one could think also of using a "continuity correction" for the probability in the denominator of $(6)$.


---

### Answer by username_1 on 2018-06-01. Score: 2

[Discrete distributions have no pdf](https://stats.stackexchange.com/a/214514).

And since $r(v)$ is defined via $f(v)$ virtual valuations also do not exist in this case. Even if one were to approximate the discrete distribution $X$ with continuous distributions $X\_n$ in such a manner that
$$
\forall v: \lim\_{n \to \infty} F\_n(v) = F(v),
$$
the limit $\lim\_{n \to \infty} r\_n(v)$ would not exist where $\lim\_{n \to \infty} f\_n(v) = 0$.

> **Comments:**
> **username_0**: So, is there a way to use the optimal auction theory when buyers' valuations come from discrete distributions?  
> **username_1**: It depends on the distribution. But that was not your question, was it?  

---

