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Expected Utility from a Constructive Viewpoint

Kislaya Prasad

Abstract

This paper introduces a space of variable lotteries and proves a constructive version of the expected utility theorem. The word ``constructive'' is used here in two senses. First, as in constructive mathematics, the logic underlying proofs is intuitionistic. In a second sense of the word, ``constructive'' is taken to mean ``built up from smaller components.'' Lotteries as well as preferences vary continuously over some topological space. The topology encodes observability or verifiability restrictions -- the open sets of the topology serve as the possible truth values of assertions about preference and reflect constraints on the ability to measure, deduce, or observe. Replacing an open set by a covering of smaller open sets serves as a notion of refinement of information. Within this framework, inability to compare arises as a phenomenon distinct from indifference and this gives rise to the constructive failure of the classical expected utility theorem. A constructive version of the theorem is then proved, and accomplishes several things. First, the representation theorem uses continuous real-valued functions as indicators of preference for variable lotteries and these functions reflect the inability to compare phenomenon. Second, conditions are provided whereby local representations of preference over open sets can be collated to provide a global representation. Third, the proofs are constructive and do not use the law of the excluded middle, which may not hold for variable lotteries. Fourth, a version of the classical theorem is obtained by imposing a condition on the collection of open sets of the topology which has the effect of making the logic classical.

Expected Utility from a Constructive Viewpoint

Abstract

This paper introduces a space of variable lotteries and proves a constructive version of the expected utility theorem. The word ``constructive'' is used here in two senses. First, as in constructive mathematics, the logic underlying proofs is intuitionistic. In a second sense of the word, ``constructive'' is taken to mean ``built up from smaller components.'' Lotteries as well as preferences vary continuously over some topological space. The topology encodes observability or verifiability restrictions -- the open sets of the topology serve as the possible truth values of assertions about preference and reflect constraints on the ability to measure, deduce, or observe. Replacing an open set by a covering of smaller open sets serves as a notion of refinement of information. Within this framework, inability to compare arises as a phenomenon distinct from indifference and this gives rise to the constructive failure of the classical expected utility theorem. A constructive version of the theorem is then proved, and accomplishes several things. First, the representation theorem uses continuous real-valued functions as indicators of preference for variable lotteries and these functions reflect the inability to compare phenomenon. Second, conditions are provided whereby local representations of preference over open sets can be collated to provide a global representation. Third, the proofs are constructive and do not use the law of the excluded middle, which may not hold for variable lotteries. Fourth, a version of the classical theorem is obtained by imposing a condition on the collection of open sets of the topology which has the effect of making the logic classical.
Paper Structure (8 sections, 10 theorems, 49 equations, 7 figures)

This paper contains 8 sections, 10 theorems, 49 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Expected Utility Theorems
  3. The Framework of Variable Lotteries
  4. Constructive Failure of the Expected Utility Theorem
  5. The Constructive Expected Utility Theorem
  6. The Classical Expected Utility Theorem
  7. Example 4
  8. Conclusion

Key Result

Theorem 1

Suppose $X$ is a topological space, with $\mathcal{O}(X)$ its collection of open sets and $P$ a sheaf of variable lotteries. A binary relation $\prec$ is defined on $P$ and satisfies Assumptions 1--4. Then there is an open covering $\{W_i\}$ of $X$ (with $\bigcup_i W_i=X$), and there exist mappings where $\Phi(W_i) \subseteq \{1, 2, \ldots n\}$. Moreover, the $u_{W_i}$ are unique up to positive l

Figures (7)

  • Figure 1: Ordering continuous functions
  • Figure 2: Negative transitivity for continuous functions
  • Figure 3: Failure of minimal comparability
  • Figure 4: Commutative diagram summarizing the results in Theorem 1 and Corollary 1 when $P(W)$ is equipped is with a binary relation $\prec$ satisfying Assumptions 1-4.
  • Figure 5: The uniqueness result. $P(W)$ is equipped is with a binary relation $\prec$ satisfying Assumptions 1-4.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Theorem 1: Constructive Expected Utility Theorem
  • Corollary 1
  • Theorem 2: Expected Utility Theorem with Classical Logic
  • Corollary 2
  • Lemma 1: Monotonicity
  • Lemma 2: Solvability
  • Lemma 3: Independence for $\precsim$ and $\sim$
  • Lemma 4: Local representation by affine mapping
  • Lemma 5: Local expected utility representation
  • Lemma 6: Uniqueness
  • ...and 4 more