John W. Pratt is a professor of business administration, emeritus, at Harvard Business School. He was educated at Princeton and Stanford, specializing in mathematics and statistics. Except for two years at the University of Chicago, and a sabbatical in Kyoto on a Guggenheim fellowship, Pratt has been at Harvard for his entire professional career. Editor of the Journal of the American Statistical Association from 1965 to 1970, he is a fellow of five professional societies and has chaired National Academy of Sciences committees on environmental monitoring, census methodology, and the future of statistics. His recent research has been on risk aversion, risk sharing incentives, and the nature and discovery of stochastic laws, statistical relationships that describe the effects of decisions. He is co-author of a book entitled: Introduction to Statistical Decision Theory, published by MIT Press, 1995.
Drawbacks of existing procedures are illustrated and a method of efficient fair division is proposed that avoids them. Given additive participants' utilities, each item is priced at the geometric mean (or some other function) of its two highest valuations. The utilities are scaled so that the market clears with the participants' purchases proportional to their entitlements. The method is generalized to arbitrary bargaining sets and existence is proved. For two or three participants, the expected utilities are unique. For more, under additivity, the geometric mean separates the prices where uniqueness holds and where it fails; it holds for the geometric mean except in one case where refinement is needed.
Pratt, John W. "Efficient Risk Sharing: The Last Frontier." Management Science 46, no. 12 (December 2000): 1545–1553. (Japanese version, translated by Fumiko Seo, in Modeling and Decision Making in Ambiguous Environments, Fumiko Seo and Takao Fukuchi, eds., 2002 (Kyoto U. Press).)
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Conditions one might impose on fair allocation procedures are introduced. Nondiscrimination requires that agents share an item in proportion to their entitlements if they receive nothing else. The "price" procedures of Pratt (2007), including the Nash bargaining procedure, satisfy this. Other prominent efficient procedures do not. In two-agent problems, reducing the feasible set between the solution and one agent's maximum point increases the utility cost to that agent of providing any given utility gain to the other and is equivalent to decreasing the dispersion of the latter's values for the items he does not receive without changing their total. One-agent monotonicity requires that such a change should not hurt the first agent, limited monotonicity that the solution should not change. For prices, the former implies convexity in the smaller of the two valuations, the latter linearity. In either case, the price is at least their average and hence spiteful.
Noncooperative games are examined from the point of view of an outside observer who believes that the players are rational and that they know at least as much as the observer. The observer is assumed to be able to observe many instances of the play of the game; these instances are identical in the sense that the observer cannot distinguish between the settings in which different plays occur. If the observer does not believe that he will be able to offer beneficial advice then he must believe that the players are playing a correlated equilibrium, though he may not initially know which correlated equilibrium. If the observer also believes that, in a certain sense, there is nothing connecting the players in a particular instance of the game then he must believe that the correlated equilibrium they are playing is, in fact, a Nash equilibrium.
