A guessing game…

I was playing this guessing game with a bunch of friends couple of days back, where each of us had a note pasted on our foreheads with a name of a famous person written on it. The goal was to guess the name, asking only polar questions. I was stuck trying to think of sports which doesn’t involve a ball (turns out I had Usain Bolt) and couldn’t help contemplating on the optimality of the questions asked. I found it kinda interesting, and hope you will too.

Here’s a simplified model of the game.
There are 2 players. S is a set having N (finite) elements, known to both players. Each player picks one element from S at random.

Goal:

• Each player has to guess the element the other player has picked.

Rules:

• A player, in his turn can ask yes-no questions.
• If the answer is ‘no’, the turn passes to the other player.
• If the answer is ‘yes’ the player retains his turn, up to M number of consecutive questions.(i.e. after M questions, the other player gets the turn)

Q1. What is the optimal strategy (best questions to ask) for a player?
Optimal = the expected number of turns taken to correctly guess is minimized

Q2. What is the maximum number of turns a player would need to arrive at the correct element?

p.s.
Hint:
From an information theory perspective, the idea is to maximize the information from each turn, with different probabilities of yes/no.

p.p.s.
Another important question on a different note.
Q3.What is the probability that someone thinking of optimal strategies in party-games would be invited again to one? (given that the crowd is just as geeky) :p

p.p.p.s Answer is here.