Quantum Probability and Decision Theory, Revisited

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.

[1]  J. Neumann,et al.  The Theory of Games and Economic Behaviour , 1944 .

[2]  Michael Lockwood,et al.  Mind, Brain, and the Quantum: The Compound 'i' , 1991 .

[3]  Simon Saunders,et al.  Comment on Lockwood , 1996 .

[4]  Simon Saunders.  Time, Quantum Mechanics, and Probability , 2004, Synthese.

[5]  H. Bandemer.  Savage, L. J.: Foundations of Statistics. Dover Publ., Inc,. New York 1972. 310 S. , 1974 .

[6]  David Deutsch,et al.  The structure of the multiverse , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  On Many-Minds Interpretations of Quantum Theory , 1997, quant-ph/9703008.

[8]  A. Gleason.  Measures on the Closed Subspaces of a Hilbert Space , 1957 .

[9]  James M. Joyce.  The Foundations of Causal Decision Theory , 1999 .

[10]  M. Redhead,et al.  Incompleteness, Nonlocality, and Realism: A Prolegomenon to thePhilosophy of Quantum Mechanics , 1989 .

[11]  B. M. Hill,et al.  Theory of Probability , 1990 .

[12]  David Deutsch,et al.  Quantum theory as a universal physical theory , 1985 .

[13]  Michael Lockwood,et al.  'Many Minds' Interpretations of Quantum Mechanics , 1996 .

[14]  David Wallace,et al.  Everett and structure , 2001 .

[15]  R. Schack,et al.  Quantum probability from decision theory? , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  Simon Saunders.  Derivation of the Born rule from operational assumptions , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  Peter Urbach,et al.  Scientific Reasoning: The Bayesian Approach , 1989 .

[18]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[19]  D. Lewis.  A Subjectivist’s Guide to Objective Chance , 1980 .

[20]  W. Zurek.  Decoherence, einselection, and the quantum origins of the classical , 2001, quant-ph/0105127.

[21]  Mark Kaplan.  Decision Theory as Philosophy , 1983, Philosophy of Science.

[22]  P. Holland.  The Quantum Theory of Motion , 1993 .

[23]  David Deutsch.  Quantum theory of probability and decisions , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  Barry Loewer,et al.  Interpreting the many worlds interpretation , 1988, Synthese.

[25]  Simon French.  Decision, Probability and Utility: Selected Readings , 1989 .

[26]  J. Colquhoun,et al.  A matter of chance. , 1992, Australian family physician.

[27]  Adam Elga,et al.  Self‐locating belief and the Sleeping Beauty problem , 2000 .

[28]  Henry Rutgers Marshall.  Belief and Will , 1899, The International Journal of Ethics.

[29]  W. Zurek.  II. Quantum mechanics and measurement theoryEnvironment-induced decoherence and the transition from quantum to classical , 1993 .

[30]  Why am I me? and why is my world so classical? , 2000, quant-ph/0011084.

[31]  Van Fraassen,et al.  Belief and the Will , 1984 .

[32]  Virendra Singh,et al.  Quantum Mechanics and Reality , 2004, quant-ph/0412148.

[33]  Willard Van Orman Quine,et al.  Word and Object , 1960 .

[34]  H. Everett.  "Relative State" Formulation of Quantum Mechanics , 1957 .

[35]  Richard C. Jeffrey,et al.  Studies in inductive logic and probability , 1971 .

[36]  S. Brison.  The Intentional Stance , 1989 .

[37]  J. Neumann,et al.  Theory of Games and Economic Behavior. , 1945 .

[38]  D. Lewis,et al.  Symposium: Chance and CredenceHumean Supervenience Debugged , 1994 .

[39]  D. Papineau,et al.  Many Minds are No Worse than One , 1996 .

[40]  David Wallace,et al.  Worlds in the Everett interpretation , 2001, quant-ph/0103092.

[41]  P. Fishburn.  Subjective expected utility: A review of normative theories , 1981 .

[42]  Wojciech H. Zurek,et al.  Decoherence, einselection and the existential interpretation (the rough guide) , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[43]  L. Vaidman.  On schizophrenic experiences of the neutron or why we should believe in the many‐worlds interpretation of quantum theory , 1996, quant-ph/9609006.

[44]  A. Baier,et al.  Reasons and Persons , 1984 .

[45]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.