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:
# Ask the buyers to tell their valuations.
# Calculate $R_$ - the maximum profit possible given the valuations.
# Calculate a price that guarantees that we get a profit of $R_$.
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 $R_$ by bidding strategically. To solve this problem, we can replace the exact $R_$ with an approximation - $R_$ - 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
$R_ = \lfloor R_ \rfloor$ = the value of $R_$ rounded to the nearest integer below it. Intuitively, in "most cases", a single agent cannot influence the value of $R_$ (e.g., if with true reports $R_=56.7$, then a single agent can only change it to between $R_=56.6$ and $R_=56.8$, but in all cases $R_=56$).

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

Such random variable $R_$ 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 $R_$.

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.

See also

* Random-sampling mechanism - an alternative approach to prior-free mechanism design.

References

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Mechanism design