Generalized Mechanism Design, Stanley Reiter diagram and Vickrey Auction I am trying to learn this new topic, Mechanism Design and stumbled upon the "Stanley Reiter" diagram (see the top-right side on the page). 
I have also learned that the Second Price Sealed Bid auction (or Vickrey Auction) implements such design. My question is, in the case for Vickrey acution, how do we interpret the $\Theta$, $\theta$, $f(\theta)$, $X$, $\xi(\cdot)$, $M$ and $g$? Please also note that I am a total noob (Econ/Math is not my major).
What I understood -- 


*

*$\Theta$: All possible valuation values in a particular bidding.

*$\theta$: Bidders' true valuation 

*$M$: Agents' report, that can distort $\theta$. if so, then why we don't say like $M = \phi(\theta)$? where $\phi(\cdot)$ distorts $\theta$?


Now, how do we interpret $f(\cdot) \in X$, $\xi(\cdot)$ and $g$? Strictly in terms of Vickrey auction?
 A: I will answer as informally as possible. What you want to achieve with the auction is allocate a good (say, a painting) to one of $n$ bidders. 
In auctions (or economics more generally) we are interested in efficient allocations. This basically means you want to give the painting to the guy who values it the most. The valuation of the good for bidder $i$ is denoted by $\theta_i$ (this is his actual valuation), but since only the bidder himself knows his valuation of the painting, all we (as auctioneers) know is the range of possible values for $\theta_i$, which is $\Theta_i$ as you correctly stated. Since we have many bidders, all possible combinations of $\theta$ are $\Theta=\Theta_1\times \Theta_2\times\ldots\times\Theta_n$, while the vector of actual valuations is $\theta=(\theta_1,\theta_2,\ldots)$.
Now the $f(\theta)$ function basically is a rule that tells you who of the $n$ bidders receives the painting, depending on their valuations. That is, $x$ could be a vector $(1,0,0,0\ldots)$, which indicates that the first bidder receives the painting (1), and the others don't (0). This was the two upper squares in the diagram and their connecting line.
The problem is that we don't know the actual $\theta_i$'s of the $n$ bidders, and if you ask "Who has the highest valuation?" ("Who wants the painting?") then everybody shouts "ME!". In other words, these bidders may lie about their valuations. This is why we bother with the auction: lying here means you might have to pay more, thus discouraging lying. The auction specifies rules (who can bid, when, who receives the painting in the end..). The bids are the messages $M$ in the diagram - they communicate your willingness to pay (valuation). The auction rules are $g$ - they specify who, as a function of all bids, receives the painting (the highest bidder), and what he has to pay for it. 
$\xi(M,g,\theta)$ now means that, given the auction rules $g$, the valuations of all bidders $\theta$ and the bids $M$ of all bidders, the auction outcome (Bayesian Nash equilibrium) is as if we asked "Who has the highest valuation?", and everybody answered truthfully. That is, in the auction outcome we learn $\theta$ through the bids, because lying in the auction means you bid more than what your valuation is, but this implies you might have to pay more than you are willing to pay --- hence you don't do it. Consequently, the auction with its rules implements the allocation $x$ that $f(\theta)$ would have implemented ($f(\theta)=x$) if everybody could have been relied upon not to lie.
In short, the auction is the mechanism that extracts $\theta_i$ from all bidders and allows us to implement the social choice $f(\theta$). Without the auction, that may not work because bidders may lie about $\theta_i$. With the auction, it may work, because overstating one's true valuation is discouraged, since it may imply higher payments if you are allocated the good. [You cannot just specify some $\phi(\theta_i)$, because you have no idea how they are going to lie as a function of their valuation (double the true valuation, three times the true valuation?). Granted, if you knew how they lied, you could infer the true $\theta_i$ from their lies, but then bidders would anticipate this and lie differently. The only situation where this works is if bidders have no incentive to lie differently once you use some function $\phi(\theta_i)$ to infer $\theta_i$. This would be an equilibrium, and $\phi$ would be the equilibrium bid strategy, and so here we see why the auction helps to reveal the true $\theta_i$ for all bidders.]
