Abstract
The Machina thought experiments pose to major non-expected utility models challenges that are similar to those posed by the Ellsberg thought experiments to subjective expected utility theory (SEUT). We test human choices in the ‘Ellsberg three-color example’, confirming typical ambiguity aversion patterns, and the ‘Machina 50/51 and reflection examples’, partially confirming the preferences hypothesized by Machina. Then, we show that a quantum-theoretic framework for decision-making under uncertainty recently elaborated by some of us allows faithful modeling of the collected data. In the quantum-theoretic framework, subjective probabilities are represented by quantum probabilities, while quantum state transformations enable representations of ambiguity aversion and subjective attitudes toward it.
Highlights
Preprint submitted to Journal of Mathematical Economics designate uncertainty situations that can be described by objective probabilities, while the term ‘ambiguity’ to designate uncertainty situations that cannot be described by objective probabilities (Knight 1921)
A significant difference in the choices f1f2 and f3f4 was observed (p-value=1.91 · 10−35). This pattern agrees with the Ellsberg preferences and points toward ambiguity aversion, while it cannot be reproduced within subjective expected utility theory’ (SEUT)
The Ellsberg threecolor experiment confirmed the typical ambiguity aversion pattern that is known in the literature, and the Machina 50/51 experiment confirmed the typical preferences suggested by Machina, against the predictions of SEUT and its major non-expected utility extensions
Summary
Probability theorists distinguish between probabilities that are known or knowable in principle (‘objective probabilities’) and probabilities that are not known or knowable. The ‘Ellsberg paradox’ led many scholars to rethink the foundations of SEUT and to formulate more general decision-making models which could represent subjective probabilities, ambiguity and subjective attitudes toward it In this respect, the major ‘non-expected utility models’ include, but are not limited to, ‘Choquet expected utility’ (Schmeidler 1989), ‘cumulative prospect theory’ (Tversky and Kahneman 1992), ‘maxmin expected utility’ (Gilboa and Schmeidler 2008), ‘α-maxmin expected utility’ (Ghirardato et al 1995), ‘variational preferences’ (Maccheroni et al 2006a) and ‘smooth ambiguity preferences’ (Klibanoff et al 2005) (see, e.g., (Gilboa and Marinacci 2013) and (Machina and Siniscalchi 2014) for extensive reviews). A decision-making experiment performed by L’Haridon and Placido (2010) confirmed the ‘Machina preferences’ f1 f2 and f4 f3, consistently with an informational symmetry scenario These authors found that the ‘tailseparability property’ is violated in non-expected utility models with rank-dependent utility, like Choquet expected utility and cumulative prospect theory. The immediate consequence is that representing Machina preferences needs a completely new theoretical approach to ambiguity and subjective attitudes toward it (L’Haridon and Placido 2010)
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