A prior-free mechanism (PFM) is a
mechanism
Mechanism may refer to:
*Mechanism (engineering), rigid bodies connected by joints in order to accomplish a desired force and/or motion transmission
*Mechanism (biology), explaining how a feature is created
*Mechanism (philosophy), a theory that a ...
in which the designer does not have any information on the agents' valuations, not even that they are random variables from some unknown probability distribution.
A typical application is a seller who wants to sell some items to potential buyers. The seller wants to price the items in a way that will maximize his profit. The optimal prices depend on the amount that each buyer is willing to pay for each item. The seller does not know these amounts, and cannot even assume that the amounts are drawn from a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. The seller's goal is to design an auction that will produce a reasonable profit even in worst-case scenarios.
PFMs should be contrasted with two other mechanism types:
*
Bayesian-optimal mechanism A Bayesian-optimal mechanism (BOM) is a mechanism in which the designer does not know the valuations of the agents for whom the mechanism is designed, but the designer knows that they are random variables and knows the probability distribution of th ...
s (BOM) assume that the agents' valuations are drawn from a known probability distribution. The mechanism is tailored to the parameters of this distribution (e.g, its median or mean value).
*
Prior-independent mechanism A Prior-independent mechanism (PIM) is a mechanism in which the designer knows that the agents' valuations are drawn from some probability distribution, but does not know the distribution.
A typical application is a seller who wants to sell some i ...
s (PIM) assume that the agents' valuations are drawn from an unknown probability distribution. They sample from this distribution in order to estimate the distribution parameters.
From the point-of-view of the designer, BOM is the easiest, then PIM, then PFM. The approximation guarantees of BOM and PIM are in expectation, while those of PFM are in worst-case.
What can we do without a prior? A naive approach is to use
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
: ask the potential buyers what their valuations are and use their replies to calculate an
empirical distribution function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function ...
. Then, apply the methods of
Bayesian-optimal mechanism design A Bayesian-optimal mechanism (BOM) is a mechanism in which the designer does not know the valuations of the agents for whom the mechanism is designed, but the designer knows that they are random variables and knows the probability distribution of ...
to the empirical distribution function.
The problem with this naive approach is that the buyers may behave strategically. Since the buyers' answers affect the prices that they are going to pay, they may be incentivized to report false valuations in order to push the price down. The challenge in PFMD is to design
truthful mechanisms. In truthful mechanisms, the agents cannot affect the prices they pay, so they have no incentive to report untruthfully.
Several approaches for designing truthful prior-free mechanisms are described below.
Deterministic empirical distribution
For every agent
, let
be the empirical distribution function calculated based on the valuations of all agents except
. Use the Bayesian-optimal mechanism with
to calculate price and allocation for agent
.
Obviously, the bid of agent
affects only the prices paid by other agents and not his own price; therefore, the mechanism is truthful.
This "empirical
Myerson mechanism" works in some cases but not in others.
Here is a case in which it works quite well. Suppose we are in a
digital goods auction
In auction theory, a digital goods auction is an auction in which a seller has an unlimited supply of a certain item.
A typical example is when a company sells a digital good, such as a movie. The company can create an unlimited number of copies ...
. We ask the buyers for their valuation of the good, and get the following replies:
# 51 buyers bid "$1"
# 50 buyers bid "$3".
For each of the buyers in group 1, the empirical distribution is 50 $1-buyers and 50 $3-buyers, so the empirical distribution function is "0.5 chance of $1 and 0.5 chance of $3". For each of the buyers in group 2, the empirical distribution is 51 $1-buyers and 49 $3-buyers, so the empirical distribution function is "0.51 chance of $1 and 0.49 chance of $3". The Bayesian-optimal price in both cases is $3. So in this case, the price given to all buyers will be $3. Only the 50 buyers in group 2 agree to that price, so our profit is $150. This is an optimal profit (a price of $1, for example, would give us a profit of only $101).
In general, the empirical-Myerson mechanism works if the following are true:
* There are no feasibility constraints (no issues of incompatibility between allocations to different agents), like in a
digital goods auction
In auction theory, a digital goods auction is an auction in which a seller has an unlimited supply of a certain item.
A typical example is when a company sells a digital good, such as a movie. The company can create an unlimited number of copies ...
;
* The valuations of all agents are drawn independently from the same unknown distribution;
* The number of the agents is large.
Then, the profit of the empirical Myerson mechanism approaches the optimum.
If some of these conditions are not true, then the empirical-Myerson mechanism might have poor performance. Here is an example. Suppose that:
[
# 10 buyers bid "$10";
# 91 buyers bid "$1".
For each buyer in group 1, the empirical distribution function is "0.09 chance of $10 and 0.91 chance of $1" so the Bayesian-optimal price is $1. For each buyer in group 2, the empirical distribution function is "0.1 chance of $10 and 0.9 chance of $1" so the Bayesian-optimal price is $10. The buyers in group 1 pay $1 and the buyers in group 2 do not want to pay $10, so we end up with a profit of $10. In contrast, a price of $1 for everyone would have given us a profit of $101. Our profit is less than %10 of the optimum. This example can be made arbitrarily bad.
Moreover, this example can be generalized to prove that:][
::There do not exist constants and a symmetric deterministic truthful auction, that attains a profit of at least in all cases in which the agents' valuations are in ]