Virtual valuation
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, <math>v</math>. The seller does not know <math>v</math> exactly, but he assumes that <math>v</math> is a random variable, with some cumulative distribution function <math>F(v)</math> and probability distribution function <math>f(v) := F'(v)</math>.
The virtual valuation of the agent is defined as:
::<math>r(v) := v - \frac{1-F(v)}{f(v)}</math>
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 <math>p</math> should be determined according to the equation:
::<math>r(p) = 0</math> 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: :<math>p = \operatorname{argmax}_v v\cdot (1-F(v)) </math>
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 <math>[0,1]</math>. So: * <math>F(v) = v \text{ in } [0,1] </math> * <math>f(v) = 1 \text{ in } [0,1] </math> * <math>r(v) = 2v-1 \text{ in } [0,1] </math> * <math>r^{-1}(0) = 1/2</math>, 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. <math>w(v)</math> 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:
::<math>r(v) := \frac{f(v)}{1-F(v)}</math> Monotone-hazard-rate implies regularity, but the opposite is not true.
The proof is simple: the monotone hazard rate implies <math>-\frac{1}{r(v)} </math> is weakly increasing in <math>v</math> and therefore the virtual valuation <math>v-\frac{1}{r(v)}</math> is strictly increasing in <math>v</math>.