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[1] and later extended to more general settings.[2]

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 ${\displaystyle R_{max}}$ - the maximum profit possible given the valuations.
3. Calculate a price that guarantees that we get a profit of ${\displaystyle R_{max}}$.

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 ${\displaystyle R_{max}}$ by bidding strategically. To solve this problem, we can replace the exact ${\displaystyle R_{max}}$ with an approximation - ${\displaystyle R_{app}}$ - that, with high probability, cannot be influenced by a single agent.[3]^{: 349–350}

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

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

Such random variable ${\displaystyle R_{app}}$ 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 ${\displaystyle R_{max}}$.

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).[3]^: 349

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.[3]^: 350 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.