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,. The seller does not know exactly, but he assumes that is a random variable, with some cumulative distribution function and probability distribution function.
The virtual valuation of the agent is defined as:

Applications

A key theorem of Myerson says that:
In the case of a single buyer, this implies that the price should be determined according to the equation:
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:
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. So:
2. 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

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