Consider the following public goods game. A group has 4 members. Each group member is endowed with $10 and needs to allocate $ ( is an integer between 0 and 10) out of the $10 to a public account and $(10 ) to a private account. The total amount in the public account is the summation of the money allocated by all group members. Each dollar from the public account gives 50 cents as a return to each group member, and each dollar 2 from the private account gives a full return of $1 to the group member who owns it. The payoff for each player is the total payoff from the public account and the private account.

(1) Specify what are the players, strategies, and the payoffs of the game. Write down the payoff as a function of .

(2) Solve all of the pure-strategy Nash equilibria.

(3) What is the efficient outcome of the game? Is it one of the equilibrium outcome(s)? If not, explain why the efficient outcome cannot be achieved.