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,
The virtual valuation of the agent is defined as: A key theorem of Myerson[1] says that: In the case of a single buyer, this implies that the price
This exactly equals the optimal sale price – the price that maximizes the expected value of the seller's profit, given the distribution of valuations: Virtual valuations can be used to construct Bayesian-optimal mechanisms also when there are multiple buyers, or different item-types.
The buyer's valuation has a normal distribution with mean 0 and standard deviation 1.
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 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: Monotone-hazard-rate implies regularity, but the opposite is not true.
The proof is simple: the monotone hazard rate implies