A random-sampling mechanism (RSM) is a
truthful mechanism In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information abou ...
that uses
sampling in order to achieve approximately-optimal gain in
prior-free mechanisms and
prior-independent mechanisms.
Suppose we want to sell some items in an auction and achieve maximum profit. The crucial difficulty is that we do not know how much each buyer is willing to pay for an item. If we know, at least, that the valuations of the buyers are
random variables with some known
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 phenomeno ...
, then we can use a
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 t ...
. But often we do not know the distribution. In this case, random-sampling mechanisms provide an alternative solution.
RSM in large markets
Market-halving scheme
When the market is large, the following general scheme can be used:
# The buyers are asked to reveal their valuations.
# The buyers are split to two sub-markets,
("left") and
("right"), using
simple random sampling
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sam ...
: each buyer goes to one of the sides by tossing a
fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In th ...
.
# In each sub-market
, 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 ...
is calculated.
# The
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 t ...
(Myerson's mechanism) is applied in sub-market
with distribution
, and in
with
.
This scheme is called "Random-Sampling Empirical Myerson" (RSEM).
The declaration of each buyer has no effect on the price he has to pay; the price is determined by the buyers in the other sub-market. Hence, it is a
dominant strategy
In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The o ...
for the buyers to reveal their true valuation. In other words, this is a
truthful mechanism In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information abou ...
.
Intuitively, by the
law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials sho ...
, if the market is sufficiently large then the empirical distributions are sufficiently similar to the real distributions, so we expect the RSEM to attain near-optimal profit. However, this is not necessarily true in all cases. It has been proved to be true in some special cases.
The simplest case is
digital goods auction. There, step 4 is simple and consists only of calculating the optimal price in each sub-market. The optimal price in
is applied to
and vice versa. Hence, the mechanism is called "Random-Sampling Optimal Price" (RSOP). This case is simple because it always calculates feasible allocations. I.e, it is always possible to apply the price calculated in one side to the other side. This is not necessarily the case with physical goods.
Even in a digital goods auction, RSOP does not necessarily converge to the optimal profit. It converges only under the ''bounded valuations'' assumption: for each buyer, the valuation of the item is between 1 and
, where
is some constant. The convergence rate of RSOP to optimality depends on
. The convergence rate also depends on the number of possible "offers" considered by the mechanism.
To understand what an "offer" is, consider a digital goods auction in which the valuations of the buyers, in dollars, are known to be bounded in