/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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

A central issue in school choice is the design of a student assignment mechanism. Education literature provides guidance for the design of such mechanisms but does not offer specific mechanisms. The flaws in the existing school choice plans result in appeals by unsatisfied parents. We formulate the school choice problem as a mechanism design problem and analyze some of the existing school choice plans including those in Boston, Columbus, Minneapolis, and Seattle. We show that these existing plans have serious shortcomings, and offer two alternative mechanisms each of which may provide a practical solution to some critical school choice issues.

/pdf/school-choice-a-mechanism-design-approach-11t5vq18ed.pdf

School Choice: A Mechanism Design Approach

We propose and axiomatize a model of preferences over acts such that the decision maker
prefers act f to act g if and only if Eμφ (Eπu ◦ f) ≥ Eμφ (Eπu ◦ g), where E is the
expectation operator, u is a vN-M utility function, φ is an increasing transformation,
and μ is a subjective probability over the set Π of probability measures π that the
decision maker thinks are relevant given his subjective information. A key feature of our
model is that it achieves a separation between ambiguity, identified as a characteristic
of the decision maker’s subjective information, and ambiguity attitude, a characteristic
of the decision maker’s tastes. We show that attitudes towards risk are characterized
by the shape of u, as usual, while attitudes towards ambiguity are characterized by the
shape of φ. We also derive φ (x) = −1
α
e−αx as the special case of constant ambiguity
aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by
properties of the subjective set of measures Π. This characterization of ambiguity is
formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang
(2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the welldeveloped
machinery for dealing with risk attitudes can be applied as well to ambiguity
attitudes. The model is also distinct from many in the literature on ambiguity in that
allows smooth, rather than kinked, indifference curves. This leads to different behavior
and improved tractability, while still sharing the main features (e.g., Ellsberg’s Paradox,
etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of
measures may be seen as an extreme case of our model with infinite ambiguity aversion.
Two illustrative applications to portfolio choice are offered.

/pdf/a-smooth-model-of-decision-making-under-ambiguity-yoi825dq9u.pdf

A Smooth Model of Decision Making Under Ambiguity

We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

/pdf/a-smooth-model-of-decision-making-under-ambiguity-2qvm7ggyao.pdf

A Smooth Model of Decision Making under Ambiguity

http://www.tayfunsonmez.net/wp-content/uploads/2013/10/ErginSonmez-JPubE2006.pdf

Games of school choice under the Boston mechanism

A theory of subjective compound lotteries

https://eml.berkeley.edu/~hie/research/matching-with-indifferences.pdf

Two-sided matching with indifferences

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show the equivalence of aversion to this risk with issue preference, the Ellsberg paradox, and uncertainty aversion

A subjective theory of compound lotteries

Liver exchange has been practiced in small numbers, mainly to overcome blood-type incom- patibility between patients and their living donors. A donor can donate either his smaller left lobe or the larger right lobe, although the former option is safer. Despite its elevated risk, right-lobe transplantation is often utilized due to size-compatibility requirement with the patient. We model liver exchange as a market-design problem, focusing on logistically simpler two-way exchanges. First, with two patient-donor sizes, we introduce an algorithm when only the safer left-lobe transplantation is feasible. We then introduce an individually rational, Pareto-efficient, and incentive-compatible mechanism that truthfully elicits the right- lobe-donation willingness of donors, and finally extend these results to a general model with any number of patient/donor sizes. The generalization requires new technical tools regarding bilateral exchanges under partial-order-induced preferences. Through simulations we show that not only liver exchange can increase the number of transplants by more than 30%, it can also increase the share of the safer left-lobe transplants.

/pdf/efficient-and-incentive-compatible-liver-exchange-nkx66gbwxx.pdf

Efficient and Incentive-Compatible Liver Exchange

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