## Quantum probability and decision theory, revisited (2002)

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Venue: | IN THE EVERETT INTERPRETATION”, STUDIES IN THE HISTORY AND PHILOSOPHY OF MODERN PHYSICS |

Citations: | 16 - 2 self |

### BibTeX

@INPROCEEDINGS{Wallace02quantumprobability,

author = {David Wallace},

title = {Quantum probability and decision theory, revisited},

booktitle = {IN THE EVERETT INTERPRETATION”, STUDIES IN THE HISTORY AND PHILOSOPHY OF MODERN PHYSICS},

year = {2002},

pages = {415--439},

publisher = {}

}

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### Abstract

An extended analysis is given of the program, originally suggested by Deutsch, of solving the probability problem in the Everett interpretation by means of decision theory. Deutsch’s own proof is discussed, and alternatives are presented which are based upon different decision theories and upon Gleason’s Theorem. It is argued that decision theory gives Everettians most or all of what they need from ‘probability’. Contact is made with Lewis’s Principal Principle linking subjective credence with objective chance: an Everettian Principal Principle is formulated, and shown to be at least as defensible as the usual Principle. Some consequences of (Everettian) quantum mechanics for decision theory itself are also discussed.

### Citations

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(Show Context)
Citation Context ...sult is rather subtle, the Everett interpretation does not introduce any new problems to the analysis; I shall therefore not discuss it further. 20we are physical systems, only imperfectly rational (=-=Dennett 1987-=-, pp. 94–99); certainly we can consider and discuss what ideally rational behaviour is (as shown by the lively debates over (for instance) the sure-thing principle and Newcomb’s problem in the literat... |

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(Show Context)
Citation Context ...tion is correct. It may then be instructive to see how the argument fails in one particular set of non-Everett interpretations: those involving ‘hidden variables’, such as the de Broglie-Bohm theory (=-=Bohm 1952-=-; Holland 1993). Recall that in a hidden-variable theory, the physical state of a system is represented not just by a Hilbert-space vector |ψ〉, but also by some set ω of hidden variables, so that the ... |

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(Show Context)
Citation Context ...rect. It may then be instructive to see how the argument fails in one particular set of non-Everett interpretations: those involving ‘hidden variables’, such as the de Broglie-Bohm theory (Bohm 1952; =-=Holland 1993-=-). Recall that in a hidden-variable theory, the physical state of a system is represented not just by a Hilbert-space vector |ψ〉, but also by some set ω of hidden variables, so that the overall state ... |

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(Show Context)
Citation Context ...y that, if { ̂ P i} is any set of projectors satisfying ∑ i ̂ P i = ̂1, then ∑ i f(̂ P i) = 1. Then there exists a unique density operator ρ on H such that f( ̂ P) = Tr( ̂ Pρ) for all projectors ̂ P.[=-=Gleason 1957-=-] The theorem is remarkable in its generality: notice in particular that no restriction of continuity has been placed on f (although of course the conclusion of the theorem entails that f must in fact... |

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(Show Context)
Citation Context ...r. Whether this is ultimately a sustainable viewpoint depends in part on one’s theory of desires and preferences: Lewis, for instance, regards preferences as wholly determined by dispositions to act (=-=Lewis 1980-=-), in which case preference ordering is necessarily total; Joyce (1999) criticises this view. Further investigation of this controversy lies beyond this paper. 2.4 Defining value functions through add... |

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(Show Context)
Citation Context ...sure. Savage’s axiom system is extremely powerful, and hence inevitably rather complex: it consists of the following (taken directly from Savage 1972, with the tacit axiom S0 made explicit, following =-=Joyce 1999-=-.) S0: Act availability The set A of acts consists of all acts (M, P), for some fixed M ∈ M and for arbitrary functions P : SM → C. (As such, when writing acts we will drop the M and identify acts wit... |

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(Show Context)
Citation Context ... dead. This multiplication comes about not as a consequence of adding extra, world-defining elements to the quantum formalism, but as a consequence of an ontology of macroscopic objects (suggested by =-=Dennett 1991-=-) according to which they are treated as patterns in the underlying microphysics. This account applies to human observers as much as to cats: such an observer, upon measuring an indeterminate event, b... |

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(Show Context)
Citation Context ...ce But clearly neither are applicable to the Everett interpretation, which seems to leave it at a significant disadvantage. In this context it is extremely interesting that David Deutsch has claimed (=-=Deutsch 1999-=-) to derive the quantum probability rule from decision theory: that is, from considerations of pure rationality. It is rather surprising how little attention his work has received in the foundational ... |

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(Show Context)
Citation Context ...is rational not to change my choice of strategy at t ′ . Like all axioms of pure rationality, this seems very obvious (and admittedly, like virtually all of them. it has been challenged; see, e. g., (=-=Elga 2000-=-)). We can use it to deduce the post-measurement strategy very easily, as follows: if you had decided which strategy to adopt before the measurement, you would have opted for maximising expected utili... |

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26 |
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(Show Context)
Citation Context ...before doing an experiment. [Saunders 1998, p. 384; emphasis his.] Whether one accepts such a claim depends upon one’s attitude to the philosophy of probability in general. Many have claimed (e. g. , =-=Mellor 1971-=-) that objective probability is simply another theoretical posit, like charge or mass, in which case presumably all that is required to introduce probability into a theory is a mathematical structure ... |

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23 |
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(Show Context)
Citation Context ...em considered by Savage (1972). The decision problem is specified by the following sets: • A set C of consequences, to be regarded as the “atomic holistic entities that have value to the individual” (=-=Fishburn 1981-=-). Typical consequences might be receiving a thousand-euro cheque, or being hit by a bus. • A set M of chance setups, to be regarded as situations in which a number of possible events might occur, and... |

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15 |
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(Show Context)
Citation Context ... sense of the probability of a world as (for instance) a measure over continuously many identical worlds. Even some proponents of the Many-Minds variant on Everett (notably Albert and Loewer 1988 and =-=Lockwood 1989-=-, 1996), who arguably have no difficulty with the preferred-basis problem, have felt forced to modify quantum mechanics in this way. It is useful to identify two aspects of the problem. The first migh... |

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4 |
Naturalizing Metaphysics
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(Show Context)
Citation Context ...in virtue of Everett’s approach: having an account of quantum mechanics consistent with the last seventy years of physics, not one in which the edifice of particle physics must be constructed afresh (=-=Saunders 1997-=-, p. 44). 1 1 This is by no means universally recognised. Everett-type interpretations can perhaps be divided into three types: (i) Old-style “Many-Worlds” interpretations in which worlds are added ex... |

3 |
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(Show Context)
Citation Context ...ays out the payoffs according to the outcomes. The zero-sum rule is the statement that the most one will pay in the hope of gaining a utility is the least that one will accept for fear of losing it. (=-=Saunders 2002-=-) Deutsch’s postulates can then be axiomatized as follows. D0: Act availability The set of acts is ACQ. 37D1: Transitive preferences (acts) There is a weak order ≻ on ACQ which satisfies measurement ... |

2 | The Structure of the Multiverse. Available online at http://xxx.arXiv.org/abs/quant-ph/0104033 - Deutsch - 2001 |

2 |
Chance and credence: Humean supervenience debugged
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(Show Context)
Citation Context ... frequencies, but this generally leads to circularity (we wish it to be the case that it is very probable that the frequency is close to the objective chance). Lewis has a more sophisticated account (=-=Lewis 1994-=-) in which the objective chances are those given by the laws which best fit the particular facts about the world (hence both frequency considerations, and those based on symmetry or simplicity, get to... |

2 |
The foundations of statistics (2nd edition ed
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- 1972
(Show Context)
Citation Context ...on of ‘more probable than’ into a quantitative probability measure. Savage’s axiom system is extremely powerful, and hence inevitably rather complex: it consists of the following (taken directly from =-=Savage 1972-=-, with the tacit axiom S0 made explicit, following Joyce 1999.) S0: Act availability The set A of acts consists of all acts (M, P), for some fixed M ∈ M and for arbitrary functions P : SM → C. (As suc... |

2 | Why am I me? and why is my world so classical? Available online at http://www.arxiv.org/abs/quant-ph/0011084 - Sudbery - 2000 |

2 | and Structure. Forthcoming in Studies in the History and Philosophy of Modern Physics; available online at http://xxx.arXiv.org/abs/quant-ph/0107144 or from http://philsci-archive.pitt.edu - Wallace |

1 | Why am I me? and why is my world so classical? Available online at http://www.arxiv.org/abs/quant-ph/0011084 - unknown authors - 2000 |