Virtual valuation

In auction theory, particularly Bayesian-optimal mechanism design, a virtual 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 the valuation of the buyer to the item, $$v$$. The seller does not know $$v$$ exactly, but he assumes that $$v$$ is a random variable, with some cumulative distribution function $$F(v)$$ and probability distribution function $$f(v) := F'(v)$$.

The virtual valuation of the agent is defined as:


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

Applications
A key theorem of Myerson 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 $$p$$ should be determined according to the equation:


 * $$r(p) = 0$$

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 the expected value of the seller's profit, given the distribution of valuations:
 * $$p = \operatorname{argmax}_v v\cdot (1-F(v)) $$

Virtual valuations can be used to construct Bayesian-optimal mechanisms also when there are multiple buyers, or different item-types.

Examples
1. The buyer's valuation has a continuous uniform distribution in $$[0,1]$$. So:
 * $$F(v) = v \text{ in } [0,1] $$
 * $$f(v) = 1 \text{ in } [0,1] $$
 * $$r(v) = 2v-1 \text{ in } [0,1] $$
 * $$r^{-1}(0) = 1/2$$, so the optimal single-item price is 1/2.

2. The buyer's valuation has a normal distribution with mean 0 and standard deviation 1. $$w(v)$$ is monotonically increasing, and crosses the x-axis in about 0.75, so this is the optimal price. The crossing point moves right when the standard deviation is larger.

Regularity
A probability distribution function is called regular if its virtual-valuation function is weakly-increasing. Regularity is important because it implies that the virtual-surplus can be maximized by a truthful mechanism.

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


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

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

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