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: 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 $$R_{max}$$ by bidding strategically. To solve this problem, we can replace the exact $$R_{max}$$ with an approximation - $$R_{app}$$ - that, with high probability, cannot be influenced by a single agent.
 * 1) Ask the buyers to tell their valuations.
 * 2) Calculate $$R_{max}$$ - the maximum profit possible given the valuations.
 * 3) Calculate a price that guarantees that we get a profit of $$R_{max}$$.

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 $$R_{app} = \lfloor R_{max} \rfloor$$ = the value of $$R_{max}$$ rounded to the nearest integer below it. Intuitively, in "most cases", a single agent cannot influence the value of $$R_{app}$$ (e.g., if with true reports $$R_{max}=56.7$$, then a single agent can only change it to between $$R_{max}=56.6$$ and $$R_{max}=56.8$$, but in all cases $$R_{app}=56$$).

To make the notion of "most cases" more accurate, define: $$R_{app} = \lfloor R_{max} + U \rfloor$$, where $$U$$ is a random variable drawn uniformly from $$[0,1]$$. This makes $$R_{app}$$ a random variable too. With probability at least 90%, $$R_{app}$$ cannot be influenced by any single agent, so a mechanism that uses $$R_{app}$$ is truthful with high probability.

Such random variable $$R_{app}$$ is called a consensus estimate:
 * "Consensus" means that, with high probability, a single agent cannot influence the outcome, so that there is an agreement between the outcomes with or without the agent.
 * "Estimate" means that the random variable is near the real variable that we are interested in - the variable $$R_{max}$$.

The disadvantages of using a consensus estimate are:
 * It does not give us the optimal profit - but it gives us an approximately-optimal profit.
 * It is not entirely truthful - it is only "truthful with high probability" (the probability that an agent can gain from deviating goes to 0 when the number of winning agents grows).

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.