Consensus estimate

Consensus estimate is a technique for designing truthful mechanisms in a prior-free mechanism design setting. The technique was introduced for digital goods auctions and later extended to more general settings.

Suppose there is a digital good that we want to sell to a group of buyers with unknown valuations. We want to determine the price that will bring us maximum profit. Suppose we have a function that, given the valuations of the buyers, tells us the maximum profit that we can make. We can use it in the following way:

1. Ask the buyers to tell their valuations.
2. Calculate R m a x - the maximum profit possible given the valuations.
3. Calculate a price that guarantees that we get a profit of R m a x .

Step 3 can be attained by a profit extraction mechanism, which is a truthful mechanism. However, in general the mechanism is not truthful, since the buyers can try to influence R m a x by bidding strategically. To solve this problem, we can replace the exact R m a x with an approximation - R a p p - that, with high probability, cannot be influenced by a single agent.

As an example, suppose that we know that the valuation of each single agent is at most 0.1. As a first attempt of a consensus-estimate, let R a p p = ⌊ R m a x ⌋ = the value of R m a x rounded to the nearest integer below it. Intuitively, in "most cases", a single agent cannot influence the value of R a p p (e.g, if with true reports R m a x = 56.7 , then a single agent can only change it to between R m a x = 56.6 and R m a x = 56.8 , but in all cases R a p p = 56 ).

To make the notion of "most cases" more accurate, define: R a p p = ⌊ R m a x + U ⌋ , where U is a random variable drawn uniformly from [ 0 , 1 ] . This makes R a p p a random variable too. With probability at least 90%, R a p p cannot be influenced by any single agent, so a mechanism that uses R a p p is truthful with high probability.

Such random variable R a p p is called a consensus estimate:

The disadvantages of using a consensus estimate are:

In practice, instead of rounding down to the nearest integer, it is better to use exponential rounding - rounding down to the nearest power of some constant. In the case of digital goods, using this consensus-estimate allows us to attain at least 1/3.39 of the optimal profit, even in worst-case scenarios.