Question

Billy and Cam are playing the following game: each player has a coin and decides whether to leave it as heads or tails before showdown (both player reveals their coin simultaneously). If both coins are heads, Billy wins $2. If both are tails, Billy wins $0.50. Otherwise, Cam wins $1. Find the optimal strategy for Billy.

Answer 1

means . Xe 3 To find the optimal strategy for Billya to find profit for Billy strategy which has the highest expected Let x be a random variable denoting the profit Billy, Loss is considered as negatire profit. Billy has hoo possible strategy leave it to as heads o leave it as tails For strategy, leaving it as heads $2 f lam has to shows heads { Son - $1 :f cam shows tails PC cam shows tails) = 1 / 2 = P(cam shows heads) since Billy doesn't know what Cam will show taking heads & tails to have equal prob. makes sense. So, E (x)= 2.1/2 - 1 1/2 = $ 1 / for strategy, leaving it as tails, $0.5 of cam shows tacts -$1 of Cam shows heads So, E(X)= 0.5.1 / -$14 so strategy ③ has a les expected loss of $0.25 so strategy of leaving it has heads is optimal for Billy - X-