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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.

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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 $p$.

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  • $\begingroup$ Is there a reason for auctions we think in terms of the 'virtual valuation' instead of the change in revenue? $\endgroup$
    – Sentient
    Nov 7 '18 at 0:15
  • $\begingroup$ @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. $\endgroup$
    – Ubiquitous
    Nov 7 '18 at 7:28
  • $\begingroup$ Ah, thank you for the pointer to their papers! $\endgroup$
    – Sentient
    Nov 7 '18 at 9:07
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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.

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  • $\begingroup$ Could you elaborate on what is stochastic dominance and hazard rate? I have never heard of those terms before :o $\endgroup$
    – Sentient
    Nov 7 '18 at 0:06
  • $\begingroup$ You can refer to Vijay Krishna's Auction Theory for a detailed analysis of the optimal auction problem. $\endgroup$
    – superhulk
    Nov 7 '18 at 17:27

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