I propose the following game: I toss a coin and while it is in the air, you say whether it will come up heads or tails. If you get it right, you get the coin. Suppose you know that it is always expensive. What is the best strategy for you?

(a) Suppose the coin is unbalanced and it comes out heads 80% of the time and tails 20% of the time. What is your best strategy now?

(b) What happens if heads 50% of the time and tails 50%? What is your best strategy?

(c) Suppose now that I am able to choose the type of coin to toss (where the type of coin is the probability that it will come up heads) and that you know what I have chosen. What type of currency should I choose to minimize my losses?

(d) What is the mixed strategy Nash equilibrium of this game? It might help you to know that there is a lot of symmetry in this game.