Risk-averse Expected Utility (EU) decision makers with wealth-dependent utility functions may find themselves indifferent between accepting and rejecting an indivisible risky prospect. Bell (1988) showed that under these circumstances it is EU-enhancing for the decision maker to engage in a pre-decision side bet, accepting the indivisible risky prospect conditional upon winning the side bet. The side bet places the decision maker on the convex hull between the initial-wealth utility function and the utility function with risky-prospect-augmented wealth. We show that decision makers restricted to actuarially unfair side bets engage in a sequence of individually EU-enhancing side bets. This occurs because optimal stake size is modest for actuarially unfair side bets, whereby wealth remains within the interval of interim convexity upon losing the side bet. As optimal stake size falls strongly with each successive side-bet round, wealth remains within the interval of interim convexity. The EU enhancement conferred
by each successive round is also strongly diminishing. Hence the side-bet sequence is eventually truncated when no further EU enhancement is available.