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

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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   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.[2]

Examples

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1. The buyer's valuation has a continuous uniform distribution in  . So:

  •  
  •  
  •  
  •  , 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.   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.[3]

Regularity

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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.

The proof is simple: the monotone hazard rate implies   is weakly increasing in   and therefore the virtual valuation   is strictly increasing in  .

See also

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References

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  1. ^ Myerson, Roger B. (1981). "Optimal Auction Design". Mathematics of Operations Research. 6: 58. doi:10.1287/moor.6.1.58.
  2. ^ Chawla, Shuchi; Hartline, Jason D.; Kleinberg, Robert (2007). "Algorithmic pricing via virtual valuations". Proceedings of the 8th ACM conference on Electronic commerce – EC '07. p. 243. arXiv:0808.1671. doi:10.1145/1250910.1250946. ISBN 9781595936530.
  3. ^ See this Desmos graph.