Consensus estimate explained

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

R _{max} - the maximum profit possible given the valuations.

1. Calculate a price that guarantees that we get a profit of

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 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.

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.

See also

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

Notes and References

1. Competitiveness via Consensus . 14 March 2016 . Andrew V. Goldberg, Jason D. Hartline . Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms . 2003 . SODA 03.
2. 10.1145/2465769.2465773. Mechanism Design via Consensus Estimates, Cross Checking, and Profit Extraction. ACM Transactions on Economics and Computation. 1. 2. 1. 2013. Ha. Bach Q.. Hartline. Jason D.. 1108.4744.