In the first regime, where we calculate at the population level, we stop and average the wealth of the whole population after every coin toss. So if E[X_t] is the expected sum of wins and losses from the coin toss,
W_t = exp(k E[X_t])
Whereas the wealth of any individual grows like
W_t = E[k exp(X_t)]
There's a result called Jensen's inequality that says that
f(E[X]) > E[f(X)]
for any random variable X and any convex function f (exp is one such function). In a sense, I think this all just falls out of Jensen's inequality.
W_t = exp(k E[X_t])
Whereas the wealth of any individual grows like
W_t = E[k exp(X_t)]
There's a result called Jensen's inequality that says that
f(E[X]) > E[f(X)]
for any random variable X and any convex function f (exp is one such function). In a sense, I think this all just falls out of Jensen's inequality.