The continuity axiom employed by von Neumann and Morgenstern...

The continuity axiom employed by von Neumann and Morgenstern holds that if $A>B>C$ then there exist some probabilities $p$ and $q$ such that $A p C>$ $B>A q C$. Let $$A=\$ 10,000,001$$ and $$B=\$ 10,000,000$$, and $C=50$ years in prison. (a) Do you think it is really true that there are any values of $p$ and $q$ such that $A p C>B>A q C$ truly describes your preferences? (b) Psychological studies suggest that most people cannot distinguish between very small probabilities, i.e. that their preferences over lotteries in which there is a small probability of a very bad outcome is unaffected by exactly how small the probability is. Does this show that there is something wrong with von Neumann and Morgenstern's theory?

Key Concepts

Preferences Over Lotteries

Preferences over lotteries involve the ranking or ordering of different probabilistic scenarios (or lotteries) based on anticipated outcomes. This concept is central to analyzing decision-making under risk, as it reflects the idea that individuals compare and choose among varied mixtures of outcomes rather than deterministic results.

Sensitivity to Small Probabilities

Sensitivity to small probabilities refers to the empirical observation that individuals often do not distinguish finely between very small probability differences, especially when these probabilities are linked to extreme outcomes. This discrepancy between theoretical models and actual behavior highlights a limitation of normative theories like expected utility theory, suggesting that human perception and psychological factors can modify how probabilities influence decision-making.

Continuity Axiom

The continuity axiom is a key component in the expected utility framework that posits if one outcome is preferred over a second, which in turn is preferred over a third, then there exists a probability mix of the best and worst outcomes that is equally desirable as the middle outcome. This axiom ensures that utility can be measured continuously and that indifference can be interpolated between extreme outcomes.

Expected Utility Theory

Expected utility theory is a foundational concept in decision-making under risk, where outcomes are evaluated based on the sum of the utilities of possible results weighted by their probabilities. This theory provides a normative model that helps explain how rational individuals should make choices when facing uncertainty, by maximizing expected utility.

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