Harman Patil (Editor)

Virtual valuation

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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.

Contents

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 . The seller does not know v exactly, but he assumes that v is a random variable, with some cumulative distribution function F ( v ) and probability distribution function f ( v ) := F ′ ( v ) .

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 p 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:

p = arg ⁡ max v v ⋅ ( 1 − F ( v ) )

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 [ 0 , 1 ] . So:

  • F ( v ) = v in [0,1]
  • f ( v ) = 1 in [0,1]
  • w ( v ) = 2 v − 1 in [0,1]
  • w − 1 ( 0 ) = 1 / 2 , 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. w ( v ) 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.

    References

    Virtual valuation Wikipedia
    (Text) CC BY-SA