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

The virtual valuation of the agent is defined as:

${\displaystyle r(v):=v-{\frac {1-F(v)}{f(v)}}}$

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 ${\displaystyle p}$ should be determined according to the equation:

${\displaystyle r(p)=0}$

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:

${\displaystyle p=\operatorname {argmax} _{v}v\cdot (1-F(v))}$

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 ${\displaystyle [0,1]}$. So:

• ${\displaystyle F(v)=v{\text{ in }}[0,1]}$
• ${\displaystyle f(v)=1{\text{ in }}[0,1]}$
• ${\displaystyle r(v)=2v-1{\text{ in }}[0,1]}$
• ${\displaystyle r^{-1}(0)=1/2}$, 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. ${\displaystyle w(v)}$ 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:

${\displaystyle r(v):={\frac {f(v)}{1-F(v)}}}$

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

The proof is simple: the monotone hazard rate implies ${\displaystyle -{\frac {1}{r(v)}}}$ is weakly increasing in ${\displaystyle v}$ and therefore the virtual valuation ${\displaystyle v-{\frac {1}{r(v)}}}$ is strictly increasing in ${\displaystyle v}$.

See also

• Myerson ironing
• Algorithmic pricing

References

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.