Between Liberalism and Democracy

We study a class of voting rules that bridge between majoritarianism and liberalism. An outcome of the vote specifies who among the voters are eligible to a certain right or qualification. Each outcome serves also as a permissible ballot. We characterize axiomatically a family of rules parameterized by the weight each individual has in determining his or her qualification. In one extreme case, the Liberal Rule, each individual's qualification is determined by her. In the other, an individual's qualification is determined by a majority. We also propose a formalization of self-determination, and apply it in a characterization of the Liberal Rule.liberalism, voting rules, social choice

Inductive Inference: An Axiomatic Approach

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of finitely many past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventuality-case pairs, as follows. Given a memory, each eventuality is ranked according to the sum of the numbers in its row, over cases in memory. The number attached to an eventuality-case pair can be interpreted as the degree of support that the past case lends to the plausibility of the eventuality. Special instances of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classification problems. Interpreting the same result for rankings of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with rankings by the likelihood functions.Inductive inference, case-based reasoning,case-based decision theory, maximum likelihood

Cognitive Foundations of Probability

Prediction is based on past cases. We assume that a predictor can rank eventualities according to their plausibility given any memory that consists of repetitions of past cases. In a companion paper, we show that under mild consistency requirements, these rankings can be represented by numerical functions, such that the function corresponding to each eventuality is linear in the number of case repetitions. In this paper we extend the analysis to rankings of events. Our main result is that a cancellation condition a la de Finetti implies that these functions are additive with respect to union of disjoint sets. If the set of past cases coincides with the set of possible eventualities, natural conditions are equivalent to ranking events by their empirical frequencies. More generally, our results may describe how individuals form probabilistic beliefs given cases that are only partially pertinent to the prediction problem at hand, and how this subjective measure of pertinence can be derived from likelihood rankings.Bayesian prior, case-based decision theory, qualitative probabilities

Empirical Similarity

An agent is asked to assess a real-valued variable Y_{p} based on certain characteristics X_{p} = (X_{p}^{1},...,X_{p}^{m}), and on a database consisting (X_{i}^{1},...,X_{i}^{m},Y_{i}) for i = 1,...,n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Y_{p}^{s}, be the weighted average of all previously observed values Y_{i}, where the weight of Y_{i}, for every i =1,...,n, is the similarity between the vector X_{p}^{1},...,X_{p}^{m}, associated with Y_{p}, and the previously observed vector, X_{i}^{1},...,X_{i}^{m}. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular functional form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations. We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction.Similarity, estimation

Rationality of Belief Or: Why Savage's axioms are neither necessary nor sufficient for rationality, Second Version

Economic theory reduces the concept of rationality to internal consistency. As far as beliefs are concerned, rationality is equated with having a prior belief over a “Grand State Space”, describing all possible sources of uncertainties. We argue that this notion is too weak in some senses and too strong in others. It is too weak because it does not distinguish between rational and irrational beliefs. Relatedly, the Bayesian approach, when applied to the Grand State Space, is inherently incapable of describing the formation of prior beliefs. On the other hand, this notion of rationality is too strong because there are many situations in which there is not sufficient information for an individual to generate a Bayesian prior. It follows that the Bayesian approach is neither sufficient not necessary for the rationality of beliefs.Decision making, Bayesian, Behavioral Economics

Utilitarian Aggregation of Beliefs and Tastes

Several authors have indicated a contradiction between consistent aggregation of subjective beliefs and tastes, and a Pareto condition. We argue that the Pareto condition that implies the contradiction is not compelling. Society should not necessarily endorse a unanimous choice when it is based on contradictory beliefs. Restricting the Pareto condition to choices that only involve identical beliefs allows a utilitarian aggregation: both society's utility function and its probability measure are linear combinations of those of the individuals.Utilitarianism, Pareto