Inauction theory, particularlyBayesian-optimal mechanism design, avirtual valuation of an agent is a function that measures the surplus that can be extracted from that agent.

A typical application is a seller who wants to sell an item to a potential buyer and wants to decide on the optimal price. The optimal price depends on thevaluation of the buyer to the item,${\displaystyle v}$. The seller does not know${\displaystyle v}$ exactly, but he assumes that${\displaystyle v}$ is a random variable, with somecumulative distribution function${\displaystyle F(v)}$ andprobability distribution function${\displaystyle f(v):=F^{\prime }(v)}$.

Thevirtual valuation of the agent is defined as:

${\displaystyle r(v):=v-\frac{1-F(v)}{f(v)}}$

A key theorem of Myerson^[1] says that:

The expected profit of any truthful mechanism is equal to its expected virtual surplus.

In the case of a single buyer, this implies that the price${\displaystyle p}$ should be determined according to the equation:

${\displaystyle r(p)=0}$

This guarantees that the buyer will buy the item, if and only if his virtual-valuation is weakly-positive, so the seller will have a weakly-positive expected profit.

This exactly equals the optimal sale price – the price that maximizes theexpected value of the seller's profit, given the distribution of valuations:

${\displaystyle p={\mathrm{argmax}}_{v}v\cdot (1-F(v))}$

Virtual valuations can be used to constructBayesian-optimal mechanisms also when there are multiple buyers, or different item-types.^[2]

1. The buyer's valuation has acontinuous uniform distribution in${\displaystyle [0,1]}$. So:

• ${\displaystyle F(v)=v\text{ in }[0,1]}$
• ${\displaystyle f(v)=1\text{ in }[0,1]}$
• ${\displaystyle r(v)=2v-1\text{ in }[0,1]}$
• ${\displaystyle r^{-1}(0)=1/2}$, so the optimal single-item price is 1/2.

2. The buyer's valuation has anormal distribution with mean 0 and standard deviation 1.${\displaystyle w(v)}$ is monotonically increasing, and crosses thex-axis in about 0.75, so this is the optimal price. The crossing point moves right when the standard deviation is larger.^[3]

Aprobability distribution function is calledregular if its virtual-valuation function is weakly-increasing. Regularity is important because it implies that the virtual-surplus can be maximized by atruthful mechanism.

A sufficient condition for regularity is monotone hazard rate, which means that the following function is weakly-increasing:

${\displaystyle r(v):=\frac{f(v)}{1-F(v)}}$

Monotone-hazard-rate implies regularity, but the opposite is not true.

The proof is simple: the monotone hazard rate implies${\displaystyle -\frac{1}{r(v)}}$ is weakly increasing in${\displaystyle v}$ and therefore the virtual valuation${\displaystyle v-\frac{1}{r(v)}}$ is strictly increasing in${\displaystyle v}$.

• Myerson ironing
• Algorithmic pricing

• Mechanism design

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