Milgrom and Weber (1982) present a much more general theory of auctions with positively related values.
be a set of random variables that are continuously distributed with joint probability density function f(v) .
The intuition for this result is as follows: In the sealed second-price auction the expected payment of a winning bidder with value v is based on their own information.
In the sealed high-bid auction such low value buyers therefore bid lower than they would if they had the same beliefs.
Thus the informational effect lowers the equilibrium payment of the winning bidder in the sealed first-price auction.
In the sealed second-price (or Vickrey auction), it is a dominant strategy for each buyer to bid his value.
Appealing to the revenue equivalence theorem, if all buyers have values that are independent draws from the same distribution then the expected payment of the winner is the same in the two auctions.
The ascending proxy auction may be viewed either as a compact representation of a dynamic combinatorial auction or as a practical direct mechanism, the first example of what Milgrom would later call a “core selecting auction.” They prove that, with respect to any reported set of values, the ascending proxy auction always generates a core outcome, i.e. an outcome that is feasible and unblocked.
Moreover, if bidders’ values satisfy the substitutes condition, then truthful bidding is a Nash equilibrium of the ascending proxy auction and yields the same outcome as the Vickrey–Clarke–Groves (VCG) mechanism.
The first of these articles, entitled "The Lovely but Lonely Vickrey Auction", made an important point in market design.
The VCG mechanism, while highly attractive in theory, suffers from a number of possible weaknesses when the substitutes condition is violated, making it a poor candidate for empirical applications.
Additional work in this area by Milgrom together with Larry Ausubel and Peter Cramton has been particularly influential in practical market design.
At the 2008 Nemmers Prize conference, Penn State University economist Vijay Krishna[6] and Larry Ausubel[7] highlighted Milgrom's contributions to auction theory and their subsequent impact on auction design.
This is where market designers try to create interactive platforms with specific rules and constraints to achieve optimal situations.
Matching refers to the idea of establishing a proper relationship between the two sides of the market, the demanders of a good or service and its suppliers.
[9] The idea for the matching emerged in the form of theoretical efforts by mathematicians such as Shapley and Gale.
It matured with the efforts of economists such as Roth, and now market design and matching are of the most important branches of microeconomics and game theory.
They show that a suitable generalization of the deferred acceptance algorithm of David Gale and Lloyd Shapley finds a stable matching in their setting; moreover, the set of stable matchings forms a lattice, and similar vacancy chain dynamics are present.
Hatfield and Milgrom observed that the accumulated offers and rejections formed a lattice, and that the bidding process in an auction and the deferred acceptance algorithm were examples of a cumulative offer process that was an increasing function in this lattice.
Their generalization also shows that certain package auctions (see also: Paul Milgrom: Policy) can be thought of as a special case of matching with contracts, where there is only one agent (the auctioneer) on one side of the market and contracts include both the items to be transferred and the total transfer price as terms.
Thus, two of market design's great success stories, the deferred acceptance algorithm as applied to the medical match, and the simultaneous ascending auction as applied to the FCC spectrum auctions, have a deep mathematical connection.
In addition, this work (in particular, the "cumulative offer" variation of the deferred acceptance algorithm) has formed the basis of recently proposed redesigns of the mechanisms used to match residents to hospitals in Japan[10] and cadets to branches in the US Army.
Usually, employers or firms do not reduce the offered wage to such an extent that supply and demand in the labor market are equal.
Two general types of communication between kidney applicants and donors are chain and cyclical systems of exchanges.
[15] Milgrom has contributed to the understanding of the effect of simplifying the message space in practical market design.
An example of conflation arises in Gale and Shapley's deferred acceptance algorithm for hospital and doctors matching when hospitals are allowed to submit only responsive preferences (i.e., the ranking of doctors and capacities) even though they could be conceivably asked to submit general substitutes preferences.
In the Internet sponsored-search auctions, advertisers are allowed to submit a single per-click bid, regardless of which ad positions they win.
In assignment messages, an agent can encode certain nonlinear preferences involving various substitution possibilities into linear objectives by allowing agents to describe multiple “roles” that objects can play in generating utility, with utility thus generated being added up.
The valuation over a set of objects is the maximum value that can be achieved by optimally assigning them to various roles.
In doing so, the paper has provided a generalization of the Birkhoff-von Neumann Theorem (a mathematical property about Doubly Stochastic Matrices) and applied it to analyze when a given random assignment can be "implemented" as a lottery over feasible deterministic outcomes.