Solving Steve Ballmer's Interview Riddle with Game Theory

The quest Imagine that one day, billionaire Steve Ballmer invites you to join his mission. Before you can sign on, he poses a challenge: “I am thinking of an integer between 1 and 100. You must guess the number. After each guess, I will tell you if my number is higher or lower. Each guess costs you $1. You must keep playing until you get it right.” 1 The question is: How much money should Steve…