Reddit reviews Two-Sided Matching (Econometric Society Monographs)

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.

Ah, the bonus chapter to his book.

The matching with women proposing would be: m1/w3, m2/w1, m3/w4, m4/w2

This is because in the first round, women will choose different men than vice-versa. You're totally correct that there will also be only one round.

Honestly, this example is really shitty/construed. There are no further rounds/rejections because

• There's no outside option. Every man is acceptable to every woman and vice-versa.
  
• In the first round, every proposer chooses a different man/woman.
  
  An interesting fact of stable two-sided matchings is the following: When men (women) propose, from all possible stable matchings, the one that is best for men (women) materializes. So the interests of men and women are exactly opposing (i.e. what's good for one men is also good for all other men, and bad for all women).
  
  Armed with this fact, we can conclude something else about this problem: Since m4/w2 are matched in both matchings (i.e. best for men and best for women), they are matched in every stable matching. You could also deduce this from the fact that they are each other's first choice, but that statement would hold even if they weren't.
  
  If you want to study this in detail, this is the book to read: https://www.amazon.com/Two-Sided-Matching-Econometric-Society-Monographs/dp/0521437881

There is some material in MWG that is considered mechanism design. Chapters 13, 14, 21, 23, maybe some elements in others.

For Contract Theory, this is a great introduction and this is a slightly more advanced followup.

For Matching, this is a good introduction to the basics.

I don't have a good idea for a text for Voting. It's mentioned in MWG to some extent.

If you want seminal papers, these books provide citations of those, as well as many, many others.