# The Effects of Payout and Probability Magnitude on the Allais Paradox

By Weber, Bethany J. | Memory & Cognition, July 2008 | Go to article overview

# The Effects of Payout and Probability Magnitude on the Allais Paradox

Weber, Bethany J., Memory & Cognition

The Allais paradox decision bias was first offered as a challenge to the expected utility theory over 60 years ago. Although the Allais paradox is a standard challenge for normative theories of risky choice, its causes are not well understood. The present experiment uses two manipulations of the Allais paradox to investigate the commonly proposed probability-weighting explanation of the paradox. Reducing the magnitude of the outcomes did not affect the size of the Allais paradox, contradicting previous literature and supporting the probability weighting hypothesis. Reducing the probability of the nonzero outcomes to eliminate certainty reduced, but did not eliminate, the Allais paradox, a result inconsistent with probability weighting and other theories of the Allais paradox. The results suggest that the certainty effect alone cannot explain the Allais paradox.

The goal of most decisions is to find the option that leads to the best possible outcome, but the outcomes of decisions are often not known in advance. One proposed axiom of rational choice under conditions of uncertainty is the independence axiom, which states that any outcome common to two sets of options should not influence the choice between the two options (Savage, 1954).

Although highly intuitive, the independence axiom is the subject of a famous challenge proposed by Allais in 1953. When presented with the pairs of gambles shown in Table 1, decision makers tend to choose the safe common consequence (CC)-high gamble and the risky CC-low gamble: They prefer a 100% chance of \$1 million to the 3-outcome risky gamble, but prefer the 10% chance of \$5 million to the 11% chance of \$1 million (Allais, 1953).

To see why these preferences violate the independence axiom, suppose these gambles are to be resolved by drawing a ball from an urn with 100 balls numbered 1 to 100, as shown in Table 2. When making a choice between the first pair of gambles, the only question should be your preferences when drawing Balls 1-11. Because winning \$1 million on a draw of Ball 12-100 is common to both options-a common consequence-haw you feel about it is irrelevant to the decision. Moreover, changing the common consequence from \$1 million to \$0 (as in the second pair of gambles) should not change the decision, since the outcome of drawing Balls 1-11 has not changed (Savage, 1954).

This violation of the independence axiom is commonly known as the Allais paradox, a robust and widely demonstrated phenomenon (e.g., Birnbaum, 2004; Camerer, 1989; Conlisk, 1989; Kahneman & Tversky, 1979; Slovic & Tversky, 1974; Wu & Gonzalez, 1996).

The most commonly proposed explanations for the Allais paradox involve probability-weighting functions. Examples of theories which explain the Allais paradox (and many other decision biases) via the shape of a probability-weighting function include prospect theory (PT) (Kahneman & Tversky, 1979); cumulative prospect theory (Tversky & Kahneman, 1992); and rank-dependent expected utility (Quiggin, 1982).

Under PT, a widely used probability-weighting theory, the value of a gamble with probability p of winning outcome X is π(p)U(X). Here π is a probability-weighting function that overweights small values of p and underweights large values of p, and U is a utility function assigning a utility to the monetary outcome of the gamble. Because π(0) = O and π(1) = 1, the fact that the π overweights small probabilities and underweights large ones means that the π function changes very rapidly (and is possibly discontinuous) near 0 and 1.

According to PT, the Allais paradox occurs because of the steepness of the π function near the endpoints. When decision makers look at the risky option in the CC-high gamble pair, they overweight the 1% chance of winning \$0, treating it as if it were much more likely than it actually is. This makes the risky option seem unacceptably high, compared with the certain win of \$1 million in the safe CC-high option. …

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