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, . 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
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
1. The buyer's valuation has a continuous uniform distribution in . So:
- in [0,1]
- in [0,1]
- in [0,1]
- , 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
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.
See also
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.