Two drinkers alternately taking half the glass

Two people share a pint. The first drinks half of it, then the second drinks half of what is left, then the first drinks half of the remainder, and they keep alternating forever. What fraction of the pint does each person drink in total?

The first drinks $\tfrac23$, the second $\tfrac13$. At every stage the second person drinks exactly half as much as the first just did, so their totals are in ratio $2:1$. Equivalently, $\tfrac12+\tfrac18+\tfrac1{32}+\dots=\tfrac{1/2}{1-1/4}=\tfrac23$.