“Any gain by the winner must harm the loser.” Is this statement true or false? Explain your reasoning in one or two sentences. Alice, Bob, and Confucius are bored during recess, so they decide to play...

“Any gain by the winner must harm the loser.” Is this statement true or false? Explain your reasoning in one or two sentences.

Alice, Bob, and Confucius are bored during recess, so they decide to play a new game. Each of them puts a dollar in the pot, and each tosses a quarter. Alice wins if the coins land all heads or all tails. Bob wins if two heads and one tail land, and Confucius wins if one head and two tails land. The quarters are fair, and the winner receives a net payment of $2 ($3 2 $1 5 $2), and the losers lose their $1.

(a) What is the probability that Alice will win and the probability that she will lose?

(b) What is Alice’s expected payoff?

(c) What is the probability that Confucius will win and the probability that he will lose?

(d) What is Confucius’ expected payoff?

(e) Is this a zero-sum game? Please explain your answer.