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, $v$ . 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:

w(v):=v-{\frac {1-F(v)}{f(v)}}

Applications

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 $p$ should be determined according to the equation:

w(p)=0

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\cdot (1-F(v))}

Virtual valuations can be used to construct Bayesian-optimal mechanisms also when there are multiple buyers, or different item-types.^[2]

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]
$w(v)=2v-1$ 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.^[3]

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:

w(v):={\frac {f(v)}{1-F(v)}}

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

References

↑ Myerson, Roger B. (1981). "Optimal Auction Design". Mathematics of Operations Research. 6: 58. doi:10.1287/moor.6.1.58.
↑ 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. doi:10.1145/1250910.1250946. ISBN 9781595936530.
↑ See this Desmos graph.

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Virtual valuation

Applications

Examples

Regularity

See also

References