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, formula_1. The seller does not know formula_1 exactly, but he assumes that formula_1 is a random variable, with some cumulative distribution function formula_4 and probability distribution function formula_5. 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 formula_7 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 formula_10. So: 2. The buyer's valuation has a normal distribution with mean 0 and standard deviation 1. formula_15 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.