Reddit reviews Auction Theory

If you're interested in all of that, you should start by reading up on mechanism design, which you can find in any good microeconomics or game theory textbook. I like Fudenberg and Tirole.

• Fudeberg and Tirole - Game Theory
  
  I mainly see Krishna's book used for auctions. Menezes and Monteiro is also good. There are also books by Paul Klemperer and Paul Milgrom which are less textbook like.
  
  
• Auction Theory - Vijay Krishna
  
  
• Menezes and Monteiro - An Introduction to Auction Theory
  
  For matching, the guide to the "classical" theory is Roth & Sotomayor, but it's getting pretty old. There's a new "Handbook of Market Design" but I haven't read it and can't vouch for it.
  
  
• Roth and Sotomayor - Two-Sided Matching
  
  For bargaining, you can't go wrong with Osbourne and Rubinstein.
  
  
• Osbourne and Rubinstein - Bargaining and Markets
  
  Sadly, I don't know of a good voting textbook. Much of the recent work in voting has moved out of economics into computer science, political science, etc., so I would look there for current work on voting.

1. If you are an undergrad and your adviser isn't publishing Mech Design papers, then you can't do a Mech Design dissertation.
   
   
2. If you haven't taken a PhD level micro theory course and a PhD level game theory course, then you don't know enough to even start writing a Mech Design paper.
   
   
3. If you are a PhD student and asking this question, then you have no business doing Mech Design.
   
   
4. Here's a book https://www.amazon.com/Auction-Theory-Vijay-Krishna/dp/0123745071

As all-pay auctions are standard, you can derive the answer from the revenue equivalence principle.

Under all the usual assumptions and given that a bidder values the item under auction at x, the symmetric equilibrium is

β^AP (x) = ∫(0 to x) y g(y) dy

where g is the probability density of the bidder winning the auction with a bid of y.

I recommend Krishna - "Auction Theory" for more details

Edit: I should make clear what the assumptions for this equilibrium are:

You have n bidders where bidder i values the item at X_i, the X_i are drawn IID from a distribution F. Each bidder only knows n and F, but has no additional information about the X_i (except their own, which they of course know). All bidders are assumed to be risk-neutral, that is, they only want to maximise the expected value they gain out of this auction (value of the item minus price paid)