Nested identification of subjective probabilities
The theory of games against nature relies on complete preferences among all conceivable acts, i.e. among all potential assignments of consequences to states of nature (case 1). Yet most decision problems call for choosing an element from a limited set of acts. And in games of strategy, the set of strategies available to a player is given and not amenable to artificial extensions. In "Assessing Strategic Risk" (CORE DP 2005/20), R.J Aumann and J.H. Dreze extend the basic result of decision theory (maximisation of subjectively expected utility) to situations where preferences are defined only for a given set of acts, and for lotteries among these and sure consequences (case 2). In this paper, we provide a similar extension for two other situations: those where only the set of optimal elements from a given set of acts is known (case 3); and those where only a single optimal act is known (case 4). To these four cases correspond four nested sets of admissible subjective probabilities over the states or the opponent's strategies, namely a singleton in case 1 and increasing sets in cases 2-4. The results for cases 3 and 4 also define the extent to which subjective probabilities must be specified in order to solve a given decision problem or play a given game.
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Volume (Year): 3 (2012)
Issue (Month): 1 (March)
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- R. J. Aumann & J. H. Dreze, 2004.
"Assessing Strategic Risk,"
Discussion Paper Series
dp361, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
- AUMANN, Robert J. & DREZE, Jacques H., 2005. "Assessing strategic risk," CORE Discussion Papers 2005020, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- AUMANN, Robert J. & DREZE, Jacques H., "undated". "Assessing strategic risk," CORE Discussion Papers RP 2089, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- R.J., AUMANN & Jacques-Henri, DREZE, 2005. "Assessing Strategic Risk," Discussion Papers (ECON - Département des Sciences Economiques) 2005020, Université catholique de Louvain, Département des Sciences Economiques.
- Dubra, Juan & Maccheroni, Fabio & Ok, Efe A., 2004.
"Expected utility theory without the completeness axiom,"
Journal of Economic Theory,
Elsevier, vol. 115(1), pages 118-133, March.
- Juan Dubra & Fabio Maacheroni & Efe A. Ok, 2001. "Expected Utility Theory without the Completeness Axiom," Cowles Foundation Discussion Papers 1294, Cowles Foundation for Research in Economics, Yale University.
- Juan Dubra & Fabio Maccheroni & Efe Oki, 2001. "Expected utility theory without the completeness axiom," ICER Working Papers - Applied Mathematics Series 11-2001, ICER - International Centre for Economic Research.
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