I thought Bernoulli’s approach was about diminishing marginal utility, not risk aversion. And while Bernoulli’s choice of a logarithmic utility function does not generalise to all possible games of chance, his approach does generalise if we choose a different function. Nor do we have to choose something silly like a constant function. Consider the following simple utility function (where X is the amount of money you have and negative values of X are not allowed):
This is a strictly monotonously increasing function (satisfying the assumption that extra money is always at least a little bit useful) with an upper bound of 1, so no matter the payout structure, the expected utility can’t possibly exceed 1. This is also a more realistic utility function than the one offered by Bernoulli since it seems intuitively obvious that money alone can’t make you arbitrarily happy or satisfied.