Subjective Expected Utility With Incomplete Preferences

Abstract

This paper extends the subjective expected utility model of decision making under uncertainty to include incomplete beliefs and tastes. The main results are two axiomatizations of the multiprior expected multiutility representations of preference relations under uncertainty. The paper also introduces new axiomatizations of Knightian uncertainty and the expected multiutility model with complete beliefs.

Irresolute choice behavior

“…By Theorem 1 of Galaabaatar and Karni (2013), the stochastic choice relations $${\succ }^{\alpha },\alpha \in [0,1]$$ are strict partial orders (i.e., transitive and irreflexive) satisfying the Archimedean (i.e., for all $$f,g,h\in H$$, if $$f{\succ }^{\alpha }g$$ and $$g{\succ }^{\alpha }h$$ then there exist $$\beta ,\gamma \in (0,1)$$ such that $$\beta f+(1-\beta )h{\succ }^{\alpha }g$$ and $$g{\succ }^{\alpha }\gamma f+(1-\gamma )h$$), independence (i.e., for all $$f,g,h\in H$$ and $$\alpha \in (0,1],f{\succ }^{\alpha }g$$ if and only if $$\alpha f+(1-\alpha )h{\succ }^{\alpha }\alpha g+(1-\alpha )h)$$, dominance (i.e., for all $$h,g\in H$$, and $$\alpha \in [0,1]$$…”

Variational Free Energy and Economics Optimizing With Biases and Bounded Rationality

“…We can, however, measure a number attached to utility, but only if there exists an equivalency between optimal behavior, utility maximizing behavior, and mathematical optimization, naturally fostering a discussion about the degree to which optimal behavior is representative of actual human behavior. While there have been many attempts over the years to augment expected utility theory by removing or altering various axioms (Rényi, 1955;Aumann, 1962;Dubins, 1975;Giles, 1976;Giron and Rios, 1980;Fishburn, 1982;Blume et al, 1991;Galaabaatar and Karni, 2013;Zaffalon and Miranda, 2017), all of these alternative approaches to expected/subjective utility theory rely on a general axiomatic foundation for utility nonetheless, either directly or by approximation (i.e., lexicographic preferences/ordering). This paper, far from criticizing the work of von Neumann and Morgenstern (1947), Savage (1972) and others, will, however, try to take a different approach to the concept of optimal behavior.…”

Reasons and Means to Model Preferences as Incomplete

“…This can be traced back at least to Aumann [1962] whose axioms do not require uniqueness of utilities: x ≻ y ⇒ u(x) > u(y), without requiring the reverse. More recently, Galaabaatar and Karni [2013] are interested in the Savage-like context where probabilities are unknown and represent an incomplete preference relation in uncertaintly using a set of pairs of probabilities and utilities.…”