Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

ONE PARTY TO AN EXCHANGE often knows something relevant to the transaction that the other party does not know. Such asymmetries of information are pervasive in economic activity: for example, in the relationship between employer and employee when the employee's effort cannot be monitored perfectly; between the stockholders and the manager of a firm; between insurer and insured; between a regulated firm and the regulatory agency; between the supplier and the consumers of a public good; between a socialist firm and the central planner; or (as is the subject of this paper) between buyer and seller when the value of the item is uncertain. Forty years ago, F. A. Hayek criticized theories that purport to describe the price system but start from the assumption that individuals have symmetric information: The peculiar character of the problem of a rational economic order is determined precisely by the fact that the knowledge of the circumstances of which we must make use never exists in concentrated or integrated form but solely as the dispersed bits of incomplete and frequently contradictory knowledge which all the separate individuals possess. The economic problem of society is thus not merely a problem of how to allocate "given" resources-if "given" is taken to mean given to a single mind which deliberately solves the problem set by these "data." It is rather a problem of how to secure the best use of resources known to any of the members of society, for ends whose relative importance only these individuals know. Or, to put it briefly, it is a problem of the utilization of knowledge which is not given to anyone in its totality. (Hayek 1945, p. 519)

Auctions and Bidding

Economics has much to do with incentives--not least, incentives to work hard, to produce quality products, to study, to invest, and to save. Although Adam Smith amply confirmed this more than two hundred years ago in his analysis of sharecropping contracts, only in recent decades has a theory begun to emerge to place the topic at the heart of economic thinking. In this book, Jean-Jacques Laffont and David Martimort present the most thorough yet accessible introduction to incentives theory to date. Central to this theory is a simple question as pivotal to modern-day management as it is to economics research: What makes people act in a particular way in an economic or business situation? In seeking an answer, the authors provide the methodological tools to design institutions that can ensure good incentives for economic agents. This book focuses on the principal-agent model, the "simple" situation where a principal, or company, delegates a task to a single agent through a contract--the essence of management and contract theory. How does the owner or manager of a firm align the objectives of its various members to maximize profits? Following a brief historical overview showing how the problem of incentives has come to the fore in the past two centuries, the authors devote the bulk of their work to exploring principal-agent models and various extensions thereof in light of three types of information problems: adverse selection, moral hazard, and non-verifiability. Offering an unprecedented look at a subject vital to industrial organization, labor economics, and behavioral economics, this book is set to become the definitive resource for students, researchers, and others who might find themselves pondering what contracts, and the incentives they embody, are really all about.

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The Theory of Incentives: The Principal-Agent Model

Foundations Of Statistics

We consider auctions for a single indivisible object, in the case where the bidders have information about each other which is not available to the seller. We show that the seller can use this information to his own benefit, and we completely characterize the environ- ments in which a well chosen auction gives him the same expected payoff as that obtainable were he able to sell the object with full information about each bidder's willingness to pay. We provide this characterization for auctions in which the bidders have dominant strate- gies, and for those where the relevant equilibrium concept is Bayesian Nash. In both set-ups, the existence of these auctions hinges on the possibility of constructing lotteries with the correct properties. WE CONSIDER the situation in which an agent, the seller, possesses one indivisible unit of a good to which he attaches no value. But the good has value to a number of potential buyers, and its transfer to one of them would increase social welfare. In particular, the transfer to the buyer with the highest valuation maximizes social welfare. In this paper, we completely characterize environments in which the seller can design an auction that will enable him to capture for himself the full increase in social welfare induced by the transfer of the good to the bidder with the highest willingness to pay. If the seller had full information about the reservation prices of potential buyers, his optimal selling strategy would be very simple. He would announce a price equal or very close to the highest reservation value. The optimal strategy for the bidder with the highest evaluation would be to accept the offer. (Note that we are treating a situation in which the seller can commit himself to a price.) As a result of the exchange, the utility of the seller increases by the full amount of the increase in social welfare, and he has been able to fully extract the surplus. In many circumstances, however, a seller has only imperfect knowledge of the buyers' willingnesses to pay. In this case, he must find some mechanism, or auction, which will enable him to maximize his benefit from the sale of the object. The auction literature starts with this observation and shows how the seller can, by an astute choice of auction, extract the largest possible fraction of the surplus. In general, the literature has shown that this proportion is strictly less than one. In some circumstances, the bidders will have information about each other which is not available to the seller. For instance, in auctions for petroleum drilling rights, bidders know the results of geological tests which they have

Full extraction of the surplus in bayesian and dominant strategy auctions

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This paper deals with the optimal design of resource allocation mechanisms in the presence of asymmetric information. A buyer's valuation function is allowed to depend on the characteristics of other buyers as well as his own and sufficient conditions are provided under which the seller can extract the full surplus from the buyers in an "ex post Nash" equilibrium. The result is then applied to the important problem of optimal auction design.

Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist when Demands are Interdependent

We examine a general equilibrium model with asymetrically informed agents. The presence of asymmetric information generally presents a conflict between incentive compatibility and Pareto efficiency. We present a notion of informational size and show that the conflict between incentive compatibility and efficiency can be made arbitrarily small if agents are sufficiently small informationally.

Informational Size and Incentive Compatibility

We examine a general equilibrium model with asymmetrically informed agents. The presence of asymmetric information generally presents a conflict between incentive com- patibility and Pareto efficiency. We present a notion of informational size and show that the conflict between incentive compatibility and efficiency can be made arbitrarily small if agents are of sufficiently small informational size. The incompatibility of Pareto efficiency and incentive compatibility is a central theme in economics and game theory. The issues associated with this incompatibility are particularly important in the design of resource allocation mechanisms in the presence of asymmetrically informed agents where the need to acquire information from agents in order to compute efficient outcomes and the incentives agents have to misrepresent that information for personal gain come into conflict. Despite a large literature that focuses on these issues, there has been little work aimed at understanding those situations in which informational asymmetries are quantitatively important. Virtually every transaction is characterized by some asymmetry of information: any investor who buys or sells a share of stock generally knows something rel- evant to the value of the share that is not known to the person on the other side of the transaction. In order to focus on more salient aspects of the problem, many models (rightly) ignore the incentive problems associated with informa- tional asymmetries in the belief that, for the problem at hand, agents are "infor- mationally small." However, few researchers have investigated the circumstances under which an analysis that ignores these incentive problems will yield results similar to those obtained when these problems are fully accounted for. In this paper, we study a class of mechanism design problems. Our goal is to formalize informational size in a way that, when agents are informationally small, 1 Early versions of this paper circulated under the titles "Informational Smallness and Pareto Effi- ciency" and "Informational Smallness and Incentive Compatibility." We thank Jacques Cremer, Dean Foster, Andreu Mas-Colell, Stephen Morris, David Schmeidler, and particularly, Faruk Gul for help- ful conversations and Rajiv Vohra for a very careful reading. We also thank the numerous partici- pants in seminars at which this paper was presented for helpful comments. We also thank an editor and the referees for very helpful comments, and thank Ichiro Obara for pointing out an error in an earlier version. Postlewaite gratefully acknowledges support from the National Science Foundation.

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Richard P. McLean

Papers

Full extraction of the surplus in bayesian and dominant strategy auctions

Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist when Demands are Interdependent

Informational Size and Incentive Compatibility

Informational size and incentive compatibility

Industry structure with sequential technology choice