A Framework for Probabilistic Decision-Making Using Change-of-Probability Measures

Probabilistic decision-making is a fundamental problem considered in many disciplines from engineering to social sciences. In this article, we address decision-making in contexts where the law of large numbers (LLN) does not apply. Non-LLN regimes include almost all high-impact decisions. The rise of artificial intelligence (AI) decision making is further increasing the importance of developing principled approaches for such problems. In this regard, we first introduce a method called bounded expectation (BE) to apply the accepted principle of ignoring negligible probabilities. We show that BE provides some satisfactory results and insights into some decision-making problems. Pointing out some shortcomings of BE, we then turn to a much more general setting, using change-of-probability measures. We show that the proposed approach can be considered a generalization of expected utility theory (EUT) from two different perspectives. First, the approach converges to EUT as the number of repetitions grows. Additionally, when the fundamental distortion parameter, $\epsilon $ , is set to zero, the proposed theory reduces to EUT. We then propose a systematic approach to applying the developed framework to non-LLN decisions. Finally, through a real-world example, we compare the decisions made with the proposed method and the conventional methods. It is speculated that due to the complexity and multidimensionality of decision-making under non-LLN regimes, the presented ideas can potentially lead to considerable further research, some of which is discussed in this article.