Ranking theory delivers an account of iterated contraction; each ranking function induces a specific iterated contraction behavior. The paper shows how to reconstruct a ranking function from its iterated contraction behavior uniquely up to multiplicative constant and thus how to measure ranks on a ratio scale. Thereby, it also shows how to completely axiomatize that behavior. The complete set of laws of iterated contraction it specifies amend the laws hitherto discussed in the literature.
I re-examine Coherence Arguments (Dutch Book Arguments, No Arbitrage Arguments) for diachronic constraints on Bayesian reasoning. I suggest to replace the usual game–theoretic coherence condition with a new decision–theoretic condition ('Diachronic Sure Thing Principle'). The new condition meets a large part of the standard objections against the Coherence Argument and frees it, in particular, from a commitment to additive utilities. It also facilitates the proof of the Converse Dutch Book Theorem. I first apply the improved Coherence Argument to van Fraassen's (...) (1984) Reflection principle. I then point out the failure of a Coherence Argument that is intended to support Conditionalization as a naive, universal, update rule. I also point out that Reflection is incompatible with the universal use of Conditionalization thus interpreted. The Coherence Argument therefore defeats the naive view on Bayesian learning that it was originally designed to justify. (shrink)
All conceptions of equal opportunity draw on some distinction between morally justified and unjustified inequalities. We discuss how this distinction varies across a range of philosophical positions. We find that these positions often advance equality of opportunity in tandem with distributive principles based on merit, desert, consequentialist criteria or individuals' responsibility for outcomes. The result of this amalgam of principles is a festering controversy that unnecessarily diminishes the widespread acceptability of opportunity concerns. We therefore propose to restore the conceptual separation (...) of opportunity principles concerning unjustified inequalities from distributive principles concerning justifiable inequalities. On this view, equal opportunity implies that that morally irrelevant factors should engender no differences in individuals' attainment, while remaining silent on inequalities due to morally relevant factors. We examine this idea by introducing the principle of ‘opportunity dominance' and explore in a simple application to what extent this principle may help us arbitrate between opposing distributive principles. We also compare this principle to the selection rules developed by John Roemer and Dirk Van de Gaer. (shrink)
Consider a group of people whose preferences satisfy the axioms of one of the current versions of utility theory, such as von Neumann-Morgenstern (1944), Savage (1954), or Bolker-Jeﬀrey (1965). There are political and economic contexts in which it is of interest to ﬁnd ways of aggregating these individual preferences into a group preference ranking. The question then arises of whether methods of aggregation exist in which the group’s preferences also satisfy the axioms of the chosen utility theory, and in which (...) at the same time the aggregation process satisﬁes certain plausible conditions (e.g., the Pareto conditions below). (shrink)
Nelson Goodman cast the ‘problem of induction’ as the task of articulating the principles and standards by which to distinguish valid from invalid inductive inferences. This paper explores some logical bounds on the ability of a rational reasoner to accomplish this task. By a simple argument, either an inductive inference method cannot admit its own fallibility, or there exists some non-inferable hypothesis whose non-inferability the method cannot infer (violating the principle of ‘negative introspection’). The paper discusses some implications of this (...) limited self-knowledge for the justifiability of inductive inferences, auto-epistemic logic, and the epistemic foundations of game theory. (shrink)
We characterize seniority rules, also known as lexical dictatorships, under weak consistency constraints on the group’s choice function. These constraints are base triple-acyclicity in the case of binary choices and rationalizability (although not rationality) in the case of choices between an arbitrary number of alternatives. Existing results on these weakened constraints remain silent on the treatment of the group’s most junior individuals and therefore do not yield a complete characterization of seniority rules. We also impose a universal domain, binary strict (...) Pareto optimality, binary Pareto indifference, binary independence of irrelevant alternatives, and the newly introduced condition of conflict resolution. The latter condition requires a social choice rules not to remain indecisive between alternatives for which individuals have conflicting preferences. (shrink)
The scope of Aumann’s (1976) Agreement Theorem is needlessly limited by its restriction to Conditioning as the update rule. Here we prove the theorem in a more comprehensive framework, in which the evolution of probabilities is represented directly, without deriving new probabilities from new certainties. The framework allows arbitrary update rules subject only to Goldstein’s (1983) requirement that current expectations agree with current expectations of future expectations.
This paper works within a model of ungraded belief that characterizes epistemic states as logically closed and consistent sets of sentences. The aim of this paper is to discuss three diachronic coherence conditions for such beliefs. These coherence conditions are formulated in terms of the reasoner's present beliefs about how his present beliefs will evolve in the future, for instance, in response to different pieces of future evidence.
We show that Bayesian ex post aggregation is unstable with respect to refinements. Suppose a group of Bayesians use ex post aggregation. Since it is a joint problem, each agent’s problem is captured by the same model, but probabilities and utilities may vary. If they analyze the same situation in more detail, their refined analysis should preserve their preferences among acts. However, ex post aggregation could bring about a preference reversal on the group level. Ex post aggregation thus depends on (...) how much information is used and may keep oscillating (‘‘flipping’’) as one keeps adding more information. (shrink)
We study the allocation of cadaveric donor kidneys for transplantation based merely on waiting time. This simple allocation rule turns out to possess very attractive ethical and medical properties. Current allocation rules, on the other hand, violate some basic requirements of distributive justice. Perhaps for fear of exacerbating these problems, these rules also fail to consider criteria such as sex, age and race although certain combinations of these criteria are known to affect graft survival rates. We demonstrate that allocation by (...) waiting time automatically protects disadvantaged patient types and puts them in a near to optimal position. The inclusion of sex, age and race will therefore not lead to morally unacceptable allocations. This allows individual patients to improve the expected survival time of their graft relative to the status quo without being penalized by the allocation rule. Moreover, decisions ab out when to start compromising on expected graft survival rates in favour of shorter waiting times are made locally by patients and their medical advisers rather than by a centralized protocol. (shrink)
In “Agreeing to Disagree” , Robert Aumann proves that a group of agents who once agreed about the probability of some proposition for which their current probabilities are common knowledge must still agree, even if those probabilities reflect disparate observations. Perhaps one saw that a card was red and another saw that it was a heart, so that as far as that goes, their common prior probability of 1/52 for its being the Queen of hearts would change in the one (...) case to 1/26, and in the other to 1/13. But if those are indeed their current probabilities, it cannot be the case that both know them, and both know that both know them, etc., etc. (shrink)