Beyond Risk: How Projective Expected Utility Reshapes Decision Theory

The expected utility hypothesis, developed by John von Neumann and Oskar Morgenstern, suggests that individuals make rational decisions by maximizing their expected utility. This is calculated based on probabilities and outcomes of different choices. Its influence stems from its simplicity and tractability, making it widely applicable in economics, game theory, finance, and political science. However, it doesn't fully capture the complexities of human decision-making, particularly in uncertain situations where paradoxes like the Allais and Ellsberg paradoxes emerge, revealing deviations from purely rational choices. These deviations highlight the need for models like Projective Expected Utility that incorporate subjective perceptions and context-dependent preferences.

The Allais paradox demonstrates that people's preferences change when the same risk is present in all options, challenging the independence axiom of expected utility theory. The Ellsberg paradox highlights ambiguity aversion, where individuals prefer options with known probabilities over those with unknown probabilities, even if the expected value of the ambiguous option is higher. These paradoxes reveal that individuals consider more than just objective probabilities when making decisions; they also factor in subjective beliefs, attitudes toward risk, and the decision's context. This is where Projective Expected Utility comes in, offering a framework that allows for non-linear preferences and context-dependent evaluations.

Projective Expected Utility differs from traditional expected utility theory by incorporating insights from quantum mechanics to provide a more flexible framework that accounts for subjective perceptions and context-dependent preferences. Traditional expected utility theory assumes preferences are linear in probabilities, which the Allais and Ellsberg paradoxes show do not always hold true. Projective Expected Utility addresses these limitations by representing preferences as projections in a Hilbert space, allowing for a more nuanced representation of subjective beliefs and attitudes toward risk. This approach seeks to provide a more realistic model of decision-making under risk and uncertainty, whereas traditional expected utility is based on purely rational choices based on probabilities.

Using concepts from quantum mechanics, like representing preferences as projections in a Hilbert space, allows Projective Expected Utility to model the complexities of subjective beliefs and attitudes towards risk more effectively than traditional models. Hilbert space provides a mathematical structure that can capture the non-linear and context-dependent aspects of preferences that are often observed in human decision-making. It is important to note, however, that Projective Expected Utility uses the mathematical structure of quantum mechanics as an analogy without implying that decision-making processes are literally quantum mechanical phenomena.

Projective Expected Utility offers a promising avenue for understanding decision-making in complex and uncertain environments. By incorporating insights from quantum mechanics and allowing for subjective perceptions, this theory provides a more realistic and flexible framework than traditional expected utility. This could lead to improved decision-making models in various domains such as finance, economics, and public policy. Further research and applications of this approach could lead to a deeper understanding of human behavior. The ability to model and predict deviations from purely rational behavior could have significant implications for designing more effective policies and interventions in situations where individuals' choices are influenced by biases, emotions, or cognitive limitations.