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
v
v
F(v)
f(v):=F'(v)
The virtual valuation of the agent is defined as:
r(v):=v-
| 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
p
r(p)=0
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}vv ⋅ (1-F(v))
Virtual valuations can be used to construct Bayesian-optimal mechanisms also when there are multiple buyers, or different item-types.[2]
1. The buyer's valuation has a continuous uniform distribution in
[0,1]
F(v)=vin[0,1]
f(v)=1in[0,1]
r(v)=2v-1in[0,1]
r-1(0)=1/2
2. The buyer's valuation has a normal distribution with mean 0 and standard deviation 1.
w(v)
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):=
| f(v) | |
| 1-F(v) |
The proof is simple: the monotone hazard rate implies
| - | 1 |
| r(v) |
v
| v- | 1 |
| r(v) |
v