The Sure-Thing Principle

In 1954, Jim Savage introduced the Sure Thing Principle to demonstrate that preferences among actions could constitute an axiomatic basis for a Bayesian foundation of statistical inference. Here, we trace the history of the principle, discuss some of its nuances, and evaluate its significance in the light of modern understanding of causal reasoning.

A Different Way to Look at Random Variables

A classical problem in Decision Theory is to represent a preference preorder among random variables. The fundamental Debreu's Theorem states that, in the discrete case, a preference satisfies the so-called Sure Thing Principle if and only if it can be represented by means of a function that can be additively decomposed along the states of the world where the random variables are defined. Such a representation suggests that every discrete random variable may be seen as a “histogram” (union of rectangles), i.e., a set. This approach leads to several fruitful consequences, both from a theoretical and an interpretative point of view. Moreover, an immediate link can be found with another alternative approach, according to which a decision maker sorts random variables depending on their probability of outperforming a given benchmark. This way, a unified approach for different points of view may be achieved.

Costs of abandoning the Sure-Thing Principle

Risk-weighted expected utility theory (REU theory for short) permits preferences which violate the Sure-Thing Principle (STP for short). But preferences that violate the STP can lead to bad decisions in sequential choice problems. In particular, they can lead decision-makers to adopt a strategy that is dominated – i.e. a strategy such that some available alternative leads to a better outcome in every possible state of the world.

MATHEMATICAL STRUCTURE OF QUANTUM DECISION THEORY

One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a "Quantum Decision Theory" of decision-making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to explain a variety of interesting fallacies and anomalies that have been reported to particularize the decision-making of real human beings. The theory describes entangled decision-making, non-commutativity of subsequent decisions, and intention interference of composite prospects. We demonstrate how the violation of the Savage's sure-thing principle (disjunction effect) can be explained as a result of the interference of intentions, when making decisions under uncertainty. The conjunction fallacy is also explained by the presence of the interference terms. We demonstrate that all known anomalies and paradoxes, documented in the context of classical decision theory, are reducible to just a few mathematical archetypes, all of which allow the finding of straightforward explanations in the frame of the developed quantum approach.