For two ideally rational agents, does learning a finite amount of shared evidence necessitate agreement? No. But does it at least guard against belief polarization, the case in which their opinions get further apart? No. OK, but are rational agents guaranteed to avoid polarization if they have access to an infinite, increasing stream of shared evidence? No.

The ProblemEconomic efficiency is naturally thought to be a virtue of social policies and decisions, and cost-benefit (CB) analysis is commonly regarded as a technique for measuring economic efficiency. It is not surprising, then, that CB analysis is so widely used in social policy analysis. However, there is a great deal of controversy about CB analysis, including controversy about its underlying philosophical rationale. The rationales that have been proposed fall into three basic, though not mutually exclusive categories. There aremoralistviews to (...) the effect that an acceptable CB analysis would provide, or contribute to, anethicalappraisal of proposed policies or projects. There arerationalistviews to the effect that an acceptable CB analysis would contribute to the selection of social policies and projects that are “socially rational.” Finally, there are so-calledmanagement scienceviews to the effect that the purpose of CB analysis is to promote the achievement of objectives held by the policy maker, whatever they may be. Different positions are available within each of these categories. But there is also the possibility that CB analysis lacksanyviable rationale. I will examine some of the major rationales for CB analysis in this paper, and I will suggest that the last view is close to the truth. (shrink)

We discuss two general issues concerning diverging sets of Bayesian (conditional) probabilities—divergence of “posteriors”—that can result with increasing evidence. Consider a setof probabilities typically, but not always, based on a set of Bayesian “priors.” Incorporating sets of probabilities, rather than relying on a single probability, is a useful way to provide a rigorous mathematical framework for studying sensitivity and robustness in Classical and Bayesian inference. See: Berger (1984, 1985, 1990); Lavine (1991); Huber and Strassen (1973); Walley (1991); and Wasserman and (...) Kadane (1990). Also, sets of probabilities arise in group decision problems. See: Levi (1982); and Seidenfeld, Kadane, and Schervish (1989). Third, sets of probabilities are one consequence of weakening traditional axioms for uncertainty. See: Good (1952); Smith (1961); Kyburg (1961); Levi (1974); Fishburn (1986); Seidenfeld, Schervish, and Kadane (1990); and Walley (1991). (shrink)

The Problem Economic efficiency is naturally thought to be a virtue of social policies and decisions, and cost-benefit analysis is commonly regarded as a technique for measuring economic efficiency. It is not surprising, then, that CB analysis is so widely used in social policy analysis. However, there is a great deal of controversy about CB analysis, including controversy about its underlying philosophical rationale. The rationales that have been proposed fall into three basic, though not mutually exclusive categories. There are moralist (...) views to the effect that an acceptable CB analysis would provide, or contribute to, an ethical appraisal of proposed policies or projects. There are rationalist views to the effect that an acceptable CB analysis would contribute to the selection of social policies and projects that are “socially rational.” Finally, there are so-called management science views to the effect that the purpose of CB analysis is to promote the achievement of objectives held by the policy maker, whatever they may be. Different positions are available within each of these categories. But there is also the possibility that CB analysis lacks any viable rationale. I will examine some of the major rationales for CB analysis in this paper, and I will suggest that the last view is close to the truth. (shrink)

Bayesians often confuse insistence that probability judgment ought to be indeterminate (which is incompatible with Bayesian ideals) with recognition of the presence of imprecision in the determination or measurement of personal probabilities (which is compatible with these ideals). The confusion is discussed and illustrated by remarks in a recent essay by R. C. Jeffrey.

Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [(1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end (...) by arguing that it would be inconsistent for mathematicians to suspend judgement on the truth of mathematical propositions on the basis of worries about the reliability of randomized algorithms. (shrink)

We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with our (...) describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both. (shrink)

Conditioning can make imprecise probabilities uniformly more imprecise. We call this effect "dilation". In a previous paper (1993), Seidenfeld and Wasserman established some basic results about dilation. In this paper we further investigate dilation on several models. In particular, we consider conditions under which dilation persists under marginalization and we quantify the degree of dilation. We also show that dilation manifests itself asymptotically in certain robust Bayesian models and we characterize the rate at which dilation occurs.

In The Enterprise of Knowledge, I proposed a general theory of rational choice which I intended as a characterization of a prescriptive theory of ideal rationality. A cardinal tenet of this theory is that assessments of expected value or expected utility in the Bayesian sense may not be representable by a numerical indicator or indeed induce an ordering of feasible options in a context of deliberation. My reasons for taking this position are related to my commitment to the inquiry-oriented approach (...) to human knowledge and valuation favored by the American pragmatists, Charles Peirce and John Dewey. A feature of any acceptable view of inquiry ought to be that during an inquiry points under dispute ought to be kept in suspense pending resolution through inquiry. (shrink)

Suppose we have a group of Bayesian agents, and suppose that theywould like for their group as a whole to be a Bayesian agent as well. Moreover, suppose that thoseagents want the probabilities and utilities attached to this group agent to be aggregated from theindividual probabilities and utilities in reasonable ways. Two ways of aggregating their individual data areavailable to them, viz., ex ante aggregation and ex post aggregation. The former aggregatesexpected utilities directly, whereas the latter aggregates probabilities and utilities (...) separately.A number of recent formal results show that both approaches have problematic implications. This studydiscusses the philosophical issues arising from those results. In this process, I hope to convincethe reader that these results about Bayesian aggregation are highly significant to decision theorists, butalso of immense interest to theorists working in areas such as ethics and political philosophy. (shrink)