Various sets of rationality axioms (vonNeumann-Morgenstern, Savage, and others) imply that people behave so as to maximize expected utility. A differentiable utility function is (by definition) approximately linear over small ranges, so for small bets, maximizing expected utility is the same thing as maximizing expected value. For large bets, this reasoning of course does not apply.
( In case that was unclear---consider unclear---suppose you start with wealth I and consider a bet where you might win the amount x with probability p and lose the amount y with probability 1-p. The theorem says that there exists a function U such that you will take this bet if and only if pU(xpU(I+x) > (1-p)U(y). 1-p)U(I-y). When x and y are sufficiently small, this is essentialy equivalent to px > (1-p)y. )