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?
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.