In the expected utility case, the risk-aversion measure is given by the Arrow-Pratt index. Three proposals of a risk-aversion measure for the nonexpected utility case are examined. The first one sets "the second derivative of the acceptance frontier as a measure of local risk aversion." The second one takes into account the concavity in the consequences of the partial derivatives of the preference function with respect to probabilities. The third one measures risk aversion through the ratio between the risk premium and the standard deviation of the lottery. The third proposal catches the main feature of risk aversion, while the other two proposals are not always in accordance with the same crude definition of risk aversion, by which there is risk aversion when an agent prefers to get the expected value of a lottery rather than to participate in it.
Risk aversion exists when an agent prefers to get the expected value of a lottery rather than to participate in it. Risk attraction corresponds to the opposite behavior. This crude definition (Varian, 1984, p. 158) is the basis of the risk-aversion measure, which can be referred to as the amount an agent is willing to pay in order to avoid risk.
In the expected utility theory, the risk aversion measure is given by the Arrow-Pratt index. However, this index is meaningless outside the realm of expected utility, for at least two reasons: (1) the Arrow-Pratt index requires the yon Neumann-Morgenstern utility function, which generally does not exist in the nonexpected utility case; (2) the expected utility theory excludes a source of risk aversion (or attraction) that is stronger than the one considered by the Arrow-Pratt index.
Several proposals of a risk-aversion measure for the nonexpected utility case have been proposed. Three of them will be considered here, all of which are relevant for understanding the greater complexity of analyzing risk aversion in the nonexpected rather than in the expected utility case. The first one, introduced by Yaari (1969, pp. 317-318), sets "the second derivative of the acceptance frontier.., as a measure of local risk aversion." The second one, introduced by many authors, takes into account the concavity in the consequences of the partial derivatives with respect to probabilities of the preference function (for instance, Machina, 1987, pp. 134-135). The last one, which was proposed by this author in several articles (Montesano