The purpose of this paper is to indicate that the opportunity to obtain information regarding the probability distribution of the return on a risky asset, such as a portfolio or a mutual fund, may cause a risk-averse decision maker to accept a single-period actuarially unfair gamble. This behavior is the same as that implied by utility functions that have convex segments, as originally considered by Friedman and Savage [2] and by Markowitz [12], but the utility function derived is not convex on any interval, since it is the envelope of a finite set of strictly increasing, strictly concave functions. Similar utility functions have been obtained, by Fleming [1] because of transactions costs, by Hakansson [4] by imposing a borrowing restriction on an investment-consumption model, and by Masson [14] in the context of an imperfect capital market. In this paper acceptance of single-period actuarially unfair gambles by an individual risk averse with respect to future wealth levels results from the opportunity to acquire information. The acquisition of information creates a set of conditional decisions each of which the individual may treat in an optimal manner, and that set of conditional decisions may induce risk-taking behavior.