Savage's Theorem Under Changing Awareness

Dietrich, Franz (2016): Savage's Theorem Under Changing Awareness.

Abstract

This paper proposes a simple unified framework of changing awareness, addressing both outcome and (nature) state awareness, and both how fine and how exhaustive the awareness is. Six axioms characterize an (essentially unique) expected-utility representation of preferences, in which utilities and probabilities are revised systematically under changes in awareness. Revision is governed by three well-defined rules: (R1) certain utilities are transformed affinely, (R2) certain probabilities are transformed proportionally, and (R3) certain (`objective') probabilities are preserved. Rule R2 parallels Karni and Viero's (2013) 'reverse Bayesianism' and Ahn and Ergin's (2010) 'partition-dependence'. Savage's (1954) theorem emerges in the special case of fixed awareness. The theorem draws mathematically on Kopylov (2007), Niiniluoto (1972) and Wakker (1981).

• Item Type: MPRA Paper
• Language: English
• Keywords: Decision under uncertainty, outcome unawareness versus state unawareness, non-refinement versus non-exhaustiveness, utility revision versus probability revision
• Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D80 - General; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty

References

Ahn, D., Ergin, H. (2010) Framing Contingencies, Econometrica 78: 655--695

Anscombe, F. J., Aumann, R. J. (1963) A Definition of Subjective Probability, Annals of Mathematical Statistics 34 (1): 199--205

Dekel, E., Lipman, B. L., Rustichini, A. (1998) Standard state-space models preclude unawareness, Econometrica 66: 159--73

Halpern, J. Y. (2001) Alternative Semantics for Unawareness, Games and Economic Behavior 37: 321--39

Halpern, J. Y., Rego, L. C. (2008) Interactive Unawareness Revisited, Games and Economic Behavior 62: 232--62

Hill, B. (2010) Awareness Dynamics, Journal of Philosophical Logic 39: 113--37

Karni, E., Schmeidler, D. (1991) Utility Theory with Uncertainty. In: Handbook of Mathematical Economics, Vol. 4, edited by Werner Hildenbrand and Hugo Sonnenschein, 1763--1831, New York: Elsevier Science

Karni, E., Viero, M. (2013) Reverse Bayesianism: a choice-based theory of growing awareness, American Economic Review 103: 2790-2810

Karni, E., Viero, M. (2015) Awareness of unawareness: a theory of decision making in the face of ignorance, working paper, Johns Hopkins University

Kopylov, I. (2007) Subjective probabilities on "small" domains, Journal of Economic Theory 133: 236-265

Niiniluoto, I. (1972) A note on fine and tight qualitative probabilities, Annals of Mathematical Statistics 43: 1581-91

Pivato, M., Vergopoulos, V. (2015) Categorical decision theory, working paper, University Cergy-Pontoise

Savage, L. J. (1954) The Foundations of Statistics, New York: Wiley

Schmeidler, D,, Wakker, P. (1987) Expected Utility and Mathematical Expectation. In: The New Palgrave: A Dictionary of Economics, first edition, edited by J. Eatwell, M. Milgate, and P. Newman, New York: Macmillan Press

Wakker, P. (1981) Agreeing probability measures for comparative probability structures, Annals of Statistics 9: 658-62