# Bayesian Mechanism vs Prior-free Mechanism

I have a double auction mechanism in which the valuations of the agents for the items are drawn from a known random distribution. To be precise, the valuations are the probability of each agent utilizing the items, i.e. in the range [0,1]. So I guess this is a Bayesian mechanism!

First: Why is it necessary to show that the mechanism is prior free? Second: How could I prove that?

• Perhaps the Wikipedia entry for Prior-free mechanism can clarify things for you? – Herr K. Feb 13 '18 at 18:56

You may need to provide more information for a clear answer. How does the mechanism look like in detail? What is the goal you want to achieve?

Regarding your first question, it is not "necessary" to show that a mechanism is prior-free but it is a nice-to-have feature. Whether it is of first-order importance, as always, depends on the circumstances.

For example, Myerson's optimal (revenue-maximizing) auction requires that the seller allocates her object to the buyer with the highest "virtual valuation", $$v_i - \frac{1-F_i(v_i)}{f_i(v_i)}$$. This construction depends on $$F_i$$ (the cdf of $$i$$'s value distribution) and its derivative. That is, in order to design the optimal auction, the seller needs a prior about $$i$$'s value - and if this belief is way off, the designed auction can be far from optimal.

As another example, consider a bidder in a first-price auction. How should this bidder figure out an optimal bid if they have no clue about how the other bidders' valuations are distributed? Do they even know the mean or even the support of this distribution?

If a seller simply wants to allocate her good efficiently, i.e., to the buyer with the highest value, she can set a second-price auction without a reserve price. It is a (weakly) dominant strategy for each bidder to bid the true value and no knowledge about priors is required (for the bidders or the designer).

A participant in your double auction may know the support $$[0,1]$$, but what is the optimal price to be submitted if they don't know the value distributions? So what goal do you want to achieve with this double auction?

Mechanism designers refer to research weakening assumptions on common priors as following the "Wilson doctrine" (yeah...) because of this quote:

Game theory has a great advantage in explicitly analyzing the consequences of trading rules that presumably are really common knowledge; it is deficient to the extent it assumes other features to be common knowledge, such as one player’s probability assessment about another’s preferences or information.

I foresee the progress of game theory as depending on successive reductions in the base of common knowledge required to conduct useful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality. Wilson (1987)

Bergemann and Morris have done some nice research on this issue. Maybe you want to read this essay or get a copy of their book.

First, I do not see why this is a Bayesian construction. Indeed, if you do mean that the density is known, it cannot possibly be a Bayesian construction. Consider the case where $f(x)=280{x^3}(1-x)^4,x\in[0,1]$,. There is no unknown parameter here. This is both your prior and your posterior as no amount of data will alter anything.

If this were a Bayesian construction then there would have to be some uncertain parameter, but there is no uncertain parameter. This is a beta distribution with $\alpha=5$ and $\beta=4$ The prior is forced to be $\Pr(\alpha=5;\beta=4)=1$.

The question is "what is x?" The only uncertainty is in the valuation. This is a Frequentist problem.

EDIT

In the case where it is drawn from an unknown distribution, you are facing two options, even if the distribution is known with certainty to the actors.

The first is to use Bayesian non-parametric methods, the second is to use Frequentist non-parametric methods. Depending on what I wanted to accomplish, I would choose one or the other.

The Bayesian method will be coherent and so you could place gambles on it. It will also likely be very difficult to implement. There cannot be a Bayesian solution that is free of its prior. Such a thing does not exist. It might be that it is uninformative, but it must exist. The alternative is to use Fisher's failed method of fiducial statistics. The Frequentist method will minimize the maximum loss you could experience from making a choice based on the data by using an incorrect inference. It will also allow you to control for power. It will usually be far simpler to implement.

Bayesian non-parametric methods are potentially infinite dimensional constructions and you would need to do a bit of reading on them. A simple approximation though would be to use the beta distribution because of its incredible flexibility, although you could use any high degree polynomial that stays above the axis since your bounding guarantees that a constant of integration exists. You would then perform model selection.

As long as you believe it is unimodal, the bounding on both sides guarantees that a mean exists. Even though your distribution is unknown, it is guaranteed to have moments. The t-test is probably inappropriate because of the bounding is so tight, but you could use the empirical quantiles to test significance. If you felt you needed the higher moments, the method of moments is always available.

Finally, in either case, you have kernel methods available to you.

You cannot avoid a prior using Bayesian methods, but the greatest advantage of Frequentist statistics is to be able to solve problems when you cannot form a prior.

• Thanks, @dave-harris, So let me rephrase my question according to your answer. The fact that all the agents know that everyone's' valuation is in the range [0,1] does not mean that the mechanism is Bayesian. If so, it is not necessary anymore to proof prior freeness!? My confusion is rooted in the fact that when simulating the market I use random distributions to generate the agents' valuations. And this made me think that I'm facing a Bayesian mechanism. – Nima Afraz Feb 13 '18 at 15:21
• -1 This is a question about mechanism design, not statistics. – Michael Greinecker Feb 14 '18 at 8:30
• @Nima Afraz: Knowing the underlying distribution of a mechanism in a model does not imply bayesian-ness, as Dave quite eloquently pointed out. Take OLS: One assumes ( often anyway ) that the error term is distributed as N(0, $\sigma^2$) but this does not imply that the model is bayesian. – mark leeds Jun 25 '18 at 19:38