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Jacob Ross - One of the best experts on this subject based on the ideXlab platform.
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Sleeping Beauty, Countable Additivity, and Rational Dilemmas
2014Co-Authors: Jacob Ross, Frank Arntzenius, Cian Dorr, Adam Elga, En Fitelson, Matthew Kotzen, Chris Meacham, Sarah Moss, Mark Schroeder, Teddy SeidenfeldAbstract:Currently, the most popular views about how to update de se or self-locating beliefs entail the one-third solution to the Sleeping Beauty prob-lem.1 Another widely held view is that an agent’s credences should be countably additive.2 In what follows, I will argue that there is a deep ten-sion between these two positions. For the assumptions that underlie the one-third solution to the Sleeping Beauty problem entail a more general principle, which I call the Generalized Thirder Principle, and there are situations in which the latter principle and the principle of Countable Additivity cannot be jointly satisfied. The most plausible response to this tension, I argue, is to accept both of these principles and tomaintain that when an agent cannot satisfy them both, he or she is faced with a rational dilemma. In writing this essay, I benefited enormously from comments from, and discussions with
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All roads lead to violations of Countable Additivity
Philosophical Studies, 2012Co-Authors: Jacob RossAbstract:This paper defends the claim that there is a deep tension between the principle of Countable Additivity and the one-third solution to the Sleeping Beauty problem. The claim that such a tension exists has recently been challenged by Brian Weatherson, who has attempted to provide a Countable Additivity-friendly argument for the one-third solution. This attempt is shown to be unsuccessful. And it is argued that the failure of this attempt sheds light on the status of the principle of indifference that underlies the tension between Countable Additivity and the one-third solution.
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sleeping beauty Countable Additivity and rational dilemmas
The Philosophical Review, 2010Co-Authors: Jacob RossAbstract:Currently, the most popular views about how to update de se or selflocating beliefs entail the one-third solution to the Sleeping Beauty problem.1 Another widely held view is that an agent’s credences should be countably additive.2 In what follows, I will argue that there is a deep tension between these two positions. For the assumptions that underlie the one-third solution to the Sleeping Beauty problem entail a more general principle, which I call the Generalized Thirder Principle, and there are situations in which the latter principle and the principle of Countable Additivity cannot be jointly satisfied. The most plausible response to this tension, I argue, is to accept both of these principles and tomaintain that when an agent cannot satisfy them both, he or she is faced with a rational dilemma.
C. Elliot - One of the best experts on this subject based on the ideXlab platform.
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e t jaynes s solution to the problem of Countable Additivity
Erkenntnis, 2020Co-Authors: C. ElliotAbstract:Philosophers cannot agree on whether the rule of Countable Additivity should be an axiom of probability. Edwin T. Jaynes attacks the problem in a way which is original to him and passed over in the current debate about the principle: he says the debate only arises because of an erroneous use of mathematical infinity. I argue that this solution fails, but I construct a different argument which, I argue, salvages the spirit of the more general point Jaynes makes. I argue that in Jaynes’s objective Bayesianism we might have good reasons to adopt Countable Additivity, and some of the major problems this adoption is known to entail need not worry us. In particular, I propose to adopt this new angle on Countable Additivity in Jon Williamson’s version of objective Bayesianism.
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E.T. Jaynes’s Solution to the Problem of Countable Additivity
Erkenntnis, 2020Co-Authors: C. ElliotAbstract:Philosophers cannot agree on whether the rule of Countable Additivity should be an axiom of probability. Edwin T. Jaynes attacks the problem in a way which is original to him and passed over in the current debate about the principle: he says the debate only arises because of an erroneous use of mathematical infinity. I argue that this solution fails, but I construct a different argument which, I argue, salvages the spirit of the more general point Jaynes makes. I argue that in Jaynes’s objective Bayesianism we might have good reasons to adopt Countable Additivity, and some of the major problems this adoption is known to entail need not worry us. In particular, I propose to adopt this new angle on Countable Additivity in Jon Williamson’s version of objective Bayesianism.
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Countable Additivity in the Philosophical Foundations of Probability
2014Co-Authors: C. ElliotAbstract:In this thesis, I study an open problem in the current philosophy of science: should Countable Additivity be an axiom of probability? We say that a probability function is finitely additive if the probability of a union of two (or any finite number of) events is equal to the sum of the single probabilities of each event. This accepted by all schools of thought on probability. Countable Additivity just extends this property to countably infinite unions of events. That this should hold is a hotly debated issue. In mathematics, probability is defined axiomatically, its properties prescribed without need of justification. Countable Additivity is used in almost all modern mathematical probability, because of the powerful integration technique, and convergence theorems it makes possible. Many philosophers object that it is hard to justify this adoption, and that the principle makes it impossible, amongst other things, to model a Humean scepticism towards induction, and impossible to follow some very basic intuitions which regard uniform distribution of probability over all possible events. Having examined the available philosophical arguments, I reach the conclusion that they all, on both sides of the debate, sometimes openly but sometimes not, crucially rely on two deep intuitions which are simply incompatible: one regards Additivity, the other regards (the possibility of) uniformity between probability values. Given that any argument for or against the principle of Countable Additivity must contrast one of these two intuitions, this explains why the debate is still open, and will most likely stay that way. Finally, I examine a recent attempt at solving the deadlock, which makes use of non-standard analysis, at the price of losing real-valued probabilities and our usual idea of sum.
Michael Nielsen - One of the best experts on this subject based on the ideXlab platform.
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Convergence to the Truth Without Countable Additivity
Journal of Philosophical Logic, 2020Co-Authors: Michael NielsenAbstract:Must probabilities be countably additive? On the one hand, arguably, requiring Countable Additivity is too restrictive. As de Finetti pointed out, there are situations in which it is reasonable to use merely finitely additive probabilities. On the other hand, Countable Additivity is fruitful. It can be used to prove deep mathematical theorems that do not follow from finite Additivity alone. One of the most philosophically important examples of such a result is the Bayesian convergence to the truth theorem, which says that conditional probabilities converge to 1 for true hypotheses and to 0 for false hypotheses. In view of the long-standing debate about Countable Additivity, it is natural to ask in what circumstances finitely additive theories deliver the same results as the countably additive theory. This paper addresses that question and initiates a systematic study of convergence to the truth in a finitely additive setting. There is also some discussion of how the formal results can be applied to ongoing debates in epistemology and the philosophy of science.
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the strength of de finetti s coherence theorem
Synthese, 2020Co-Authors: Michael NielsenAbstract:I show that de Finetti’s coherence theorem is equivalent to the Hahn-Banach theorem and discuss some consequences of this result. First, the result unites two aspects of de Finetti’s thought in a nice way: a corollary of the result is that the coherence theorem implies the existence of a fair Countable lottery, which de Finetti appealed to in his arguments against Countable Additivity. Another corollary of the result is the existence of sets that are not Lebesgue measurable. I offer a subjectivist interpretation of this corollary that is concordant with de Finetti’s views. I conclude by pointing out that my result shows that there is a sense in which de Finetti’s theory of subjective probability is necessarily nonconstructive. This raises questions about whether the coherence theorem can underwrite a legitimate theory of rational belief.
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The strength of de Finetti’s coherence theorem
Synthese, 2020Co-Authors: Michael NielsenAbstract:I show that de Finetti’s coherence theorem is equivalent to the Hahn-Banach theorem and discuss some consequences of this result. First, the result unites two aspects of de Finetti’s thought in a nice way: a corollary of the result is that the coherence theorem implies the existence of a fair Countable lottery, which de Finetti appealed to in his arguments against Countable Additivity. Another corollary of the result is the existence of sets that are not Lebesgue measurable. I offer a subjectivist interpretation of this corollary that is concordant with de Finetti’s views. I conclude by pointing out that my result shows that there is a sense in which de Finetti’s theory of subjective probability is necessarily nonconstructive. This raises questions about whether the coherence theorem can underwrite a legitimate theory of rational belief.
Kevin X. D. Huang - One of the best experts on this subject based on the ideXlab platform.
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Valuation in infinite-horizon sequential markets with portfolio constraints
Economic Theory, 2002Co-Authors: Kevin X. D. HuangAbstract:We develop a theory of valuation of assets in sequential markets over an infinite horizon and discuss implications of this theory for equilibrium under various portfolio constraints. We characterize a class of constraints under which sublinear valuation and a modified present value rule hold on the set of non-negative payoff streams in the absence of feasible arbitrage. We provide an example in which valuation is non-linear and the standard present value rule fails in incomplete markets. We show that linearity and Countable Additivity of valuation hold when markets are complete. We present a transversality constraint under which valuation is linear and countably additive on the set of all payoff streams regardless of whether markets are complete or incomplete.
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Asset price bubbles in Arrow-Debreu and sequential equilibrium*
Economic Theory, 2000Co-Authors: Kevin X. D. Huang, Jan WernerAbstract:Price bubbles in an Arrow-Debreu equilibrium in an infinite-time economy are a manifestation of lack of Countable Additivity of valuation of assets. In contrast, the known examples of price bubbles in a sequential equilibrium in infinite time cannot be attributed to the lack of Countable Additivity of valuation. In this paper we develop a theory of valuation of assets in sequential markets (with no uncertainty) and study the nature of price bubbles in light of this theory. We define a payoff pricing operator that maps a sequence of payoffs to the minimum cost of an asset holding strategy that generates it. We show that the payoff pricing functional is linear and countably additive on the set of positive payoffs if and only if there is no Ponzi scheme, provided that there is no restriction on long positions in the assets. In the known examples of equilibrium price bubbles in sequential markets valuation is linear and countably additive. The presence of a price bubble means that the dividends of an asset can be purchased in sequential markets at a cost lower than the asset's price. We present further examples of equilibrium price bubbles in which valuation is nonlinear, or linear but not countably additive.
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Valuation bubbles and sequential bubbles
1997Co-Authors: Kevin X. D. Huang, Jan WernerAbstract:Price bubbles in an Arrow-Debreu valuation equilibrium in infinite-time economy are a manifestation of lack of Countable Additivity of valuation of assets. In contrast, known examples of price bubbles in sequential equilibrium in infinite time cannot be attributed to the lack of Countable Additivity of valuation. In this paper we develop a theory of valuation of assets in sequential markets (with no uncertainty) and study the nature of price bubbles in light of this theory. We consider an operator, called payoff pricing functional, that maps a sequence of payoffs to the minimum cost of an asset holding strategy that generates it. We show that the payoff pricing functional is linear and countably additive on the set of positive payoffs if and only if there is no Ponzi scheme, and provided that there is no restriction on long positions in the assets. In the known examples of equilibrium price bubbles in sequential markets valuation is linear and countably additive. The presence of a price bubble indicates that the asset's dividends can be purchased in sequential markers at a cost lower than the asset's price. We also present examples of equilibrium price bubbles in which valuation is nonlinear but not countably additive.
J Maitland D Wright - One of the best experts on this subject based on the ideXlab platform.
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decoherence functionals for von neumann quantum histories boundedness and Countable Additivity
Communications in Mathematical Physics, 1998Co-Authors: J Maitland D WrightAbstract:Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I2 direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of Countable Additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give:
