more of one quantity would imply more of another, are found pervasively in economic analysis. Milgrom pioneered in the development of new mathematical methods for understanding monotonic relationships in economics. His work on auctions with Robert Weber introduced the concept of affiliation of random variables, to indicate systems of unknown quantities where learning that any one of them is higher than some given level would cause beliefs about others to be higher. His work with John Roberts and Chris Shannon advanced the use of supermodularity as a property of individuals' preferences that can yield general monotonicity results in economic analysis. The work of Milgrom and Shannon (1994) showed that comparative statics results could often be obtained through more relevant and intuitive ordinal conditions. Indeed, they show that their concept of quasi-supermodularity (a generalization of supermodular function) along with the single-crossing property, is necessary and sufficient for comparative statics to obtain on arbitrary choice sets. Their theory extends earlier work in the Operations Research literature (Topkis, 1968; Veinott, 1989) which already uses lattice theory but focuses on cardinal concepts. Milgrom and John Roberts (1994) extended this to comparative statics on equilibria, while Milgrom (1994) demonstrated its wider applicability in comparing optima. Milgrom and Roberts (1996) also generalized Paul Samuelson's application of Le Chatelier's Principle in economics. In related work, Milgrom and Ilya Segal (2002) reconsidered the Envelope Theorem and its applications in light of the developments in monotone comparative statics. Due to the influence of Milgrom and Shannon's paper and related research by Milgrom and others, these techniques, now often referred to as monotone comparative statics, are widely known and used in economic modelling. The single-crossing property as reformulated by Milgrom and Shannon was subsequently shown by Joshua Gans and Michael Smart not only to resolve Condorcet's Voting paradox in majority voting and social choice theory but also to give rise to a complete characterization of social preferences. Susan Athey extended these results to consider economic problems with uncertainty. Writing in 1994 on the comparative statics and theoretical modeling, Milgrom relates a theorem that would demonstrate when a result with a specific functional form may easily generalize, and notes: These conclusions do not mean that functional form assumptions are either useless or inconsequential for economic analysis. Functional form assumptions may be helpful for deriving explicit formulas for empirical estimation or simulations or simply to lend insight into the problem structure, and