Alice challenges Bob for the following game:

First Bob takes 2⁹ numbers from the set {0,1,2,3,...,1024}, then Alice takes 2⁸ of the remaining numbers. Then Bob takes 2⁷ numbers and successively, until only two numbers remain. Alice will have to give Bob the difference between these two numbers in euros. What is the largest amount Bob can win, whatever Alice's strategy is?