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Threshold Models for Comparative Probability on Finite Sets.

Nakamura1

  • 1University of Tsukuba, Tsukuba, Japan

Journal of Mathematical Psychology
|September 7, 2000
PubMed
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This study introduces threshold models for comparative probability relations, defining when one event succeeds another based on probability and a bivariate function. These models provide a framework for understanding probabilistic comparisons with specific axiomatic conditions.

Area of Science:

  • Decision Sciences
  • Probability Theory
  • Mathematical Economics

Background:

  • Comparative probability relations are fundamental in decision theory.
  • Existing models often lack a comprehensive axiomatic foundation for specific comparison structures.

Purpose of the Study:

  • To present and discuss necessary and sufficient axioms for threshold models of comparative probability.
  • To establish a general representational form for these models.

Main Methods:

  • Defining a comparative probability relation 'succeeds' on subsets of a finite state space.
  • Developing a general representational form involving a probability measure P and a bivariate set function Omega.
  • Imposing conditions like skew-monotonicity and additive separability on Omega.

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Main Results:

  • Established a general form: A succeeds B if and only if P(A) > P(B) + Omega(A, B).
  • Identified necessary and sufficient axioms for several threshold models.
  • Demonstrated the impact of imposing conditions on the functional form of Omega.

Conclusions:

  • The proposed threshold models offer a structured approach to comparative probability.
  • The axiomatic framework provides a rigorous foundation for these probabilistic comparisons.
  • Further research can explore applications of these models in various decision-making contexts.