Question 4 Consider a group of 15 players, each of whom has to choose between joining a club (action labeled IN) and staying out (action labeled OUT). The larger the club, the more benecial it is for any one player to join, and the more costly to stay out. The players are numbered from 1 to 15 in decreasing order of their desire to join. Specically, the payoffs are as follows: When 11 others have chosen IN, for any player k, if player k chooses IN, his payoff is n k if player k chooses OUT, his payoff is 3* (k r1) 6 (For example, if 10 players are already in (n=10), the payoff for player 7 (k=7) is either 10-7=3, if he chooses to enter and be the 11th member, or 3*(7-10)6=-15 if he stays out.) Observe that both these payoffs are functions of k and n, where k ranges from 1 to 15, and n ranges from 0 to 14. When n = 0, the choice of \"join\" by any one player is to be interpreted as founding or forming the club. (a) Show that player k will choose IN if k n + 3/2. In parts (b) and (c), suppose that the game is played with simultaneous moves. (b) Show that for player 1, joining is the dominant strategy. (0) Show that the game is dominance solvable, and that in equilibrium all players choose IN. In parts (d) and (e), suppose that the game is played with sequential moves. (d) If the order of moves is 1, 2, 3, 15, show that the outcome from the rollback equilibrium is the same as that of the simultaneous-move game above. (Do not try to draw a tree; you can do the rollback reasoning without drawing the game tree.) (e) Give a brief verbal argument to show that the same outcome will result irrespective of the order of moves. In parts (f)(h), the equilibrium outcome refers to the common outcome of all the cases (c)-(e) above (everyone has chosen 1N, so for any one player, n : 14). The status quo is the situation where the club does not exist at all (n : 0 and everyone has chosen OUT)