Market design is an interdisciplinary, engineering-driven approach to economics and a practical methodology for creation of markets of certain properties, which is partially based on mechanism design. In market design, the focus is on the rules of exchange, meaning who gets allocated what and by what procedure. Market design is concerned with the workings of particular markets in order to fix them when they are broken or to build markets when they are missing. Practical applications of market design theory has included labor market matching (e.g. the national residency match program), organ transplantation, school choice, university admissions, and more.

Early research on auctions focused on two special cases: common value auctions in which buyers have private signals of an items true value and private value auctions in which values are identically and independently distributed. Milgrom and Weber (1982) present a much more general theory of auctions with positively related values. Each of n buyers receives a private signal ${\displaystyle {{x}_{i}}}$ . Buyer i’s value ${\displaystyle \phi ({{x}_{i}},{{x}_{-i}})}$ is strictly increasing in ${\displaystyle {{x}_{i}}}$ and is an increasing symmetric function of ${\displaystyle {{x}_{-i}}}$. If signals are independently and identically distributed, then buyer i’s expected value ${\displaystyle {{v}_{i}}={{E}_{{x}_{-i}}}\{\phi ({{x}_{i}},{{x}_{-i}})\}}$ is independent of the other buyers’ signals. Thus, the buyers’ expected values are independently and identically distributed. This is the standard private value auction. For such auctions the revenue equivalence theorem holds. That is, expected revenue is the same in the sealed first-price and second-price auctions.

Milgrom and Weber assumed instead that the private signals are “affiliated”. With two buyers, the random variables ${\displaystyle {{v}_{1}}}$ and ${\displaystyle {{v}_{2}}}$ with probability density function ${\displaystyle f({{v}_{1}},{{v}_{2}})}$ are affiliated if

Applying Bayes’ Rule it follows that ${\displaystyle f({{v}_{2}}^{\prime }|{{v}_{1}}^{\prime })f({{v}_{2}}|{{v}_{1}})\geq f({{v}_{2}}|{{v}_{1}}^{\prime })f({{v}_{2}}^{\prime }|{{v}_{1}})}$, for all ${\displaystyle v}$ and all ${\displaystyle {v}'<v}$.

Rearranging this inequality and integrating with respect to ${\displaystyle {{v}_{2}}^{\prime }}$ it follows that

It is this implication of affiliation that is critical in the discussion below.

For more than two symmetrically distributed random variables, let ${\displaystyle V=\{{{v}_{1}},...,{{v}_{n}}\}}$ be a set of random variables that are continuously distributed with joint probability density function f(v) . The n random variables are affiliated if

Suppose each of n buyers receives a private signal ${\displaystyle {{x}_{i}}}$. Buyer i’s value ${\displaystyle \phi ({{x}_{i}},{{x}_{-i}})}$ is strictly increasing in ${\displaystyle {{x}_{i}}}$ and is an increasing symmetric function of ${\displaystyle {{x}_{-i}}}$. If signals are affiliated, the equilibrium bid function in a sealed first-price auction ${\displaystyle {{b}_{i}}=B({{x}_{i}})}$ is smaller than the equilibrium expected payment in the sealed second price auction.