In a game with eight players, each player has x > 10 coins. Each player also chooses the number of coins to contribute to a 'common fund'. The remaining coins are deposited into a 'bank'. Each coin contributed to the common fund relieves 23 - 0.25a in return (where 'a' is the total number of coins contributed to the fund). Each coins contributed to the 'bank' gets a return of 5. The utility function of each player is the total return from all x tokens.
Formulate this as a simultaneous-move game, and find the total amount contributed to the common fund in the unique symmetric (same contribution for each player) pure strategy Nash equilibrium of the game. Why is this outcome not socially optimal?