Answer the following questions?

3. A simple game on N is a family if? of so-eaJled winning subsets of N. For example, simple majority with a voters is a simple game with winning subsets {D : |D| :=- 11,!2}. A game is griaioekea if there is a subset of N such that neither it nor its complement is winning. Thus, simple majority is gridlocked if and only if n is even. Individual i is pieotai if it belongs to every winning subset, that is, i E D for all D E 5?. A game is free if it has no pivotal individuals, and dictatorial if it has only one. For example, the game 9p = {S C N : P C S]- is free if and only if P is empty and dictatorial if and only if |P| = 1. Prove that if a simple game is neither free nor gridlocked then it is dictatorial. 2. Consider the following game. There are N players. Each player i has two choices: A or B. The state of the world is also either A or B, and each player gains one util from matching her action to the state, no utils from mismatching. All players share a common prior belief about the probability of A, denoted 11' = INA) E (U, 1). Each player 1' observes a private signal 35, conditionally IID across players. The signal is either a or b, with probability 113(3, = cpl) = lP'(s,- = MB) = q E g, 1). Players take turns making their choices, and observe the choices made by their pre- decessors. The order of moves coincides with the natural ordering of players' names, thus, player 1 chooses rst, player 2 second, etc. (a) Find player i's posterior beliefs and optimal choice given her signal predecessors' actions for i = 1, 2, 3. (b) Assume that, when a player is indifferent between two choices, she ips a fair coin to decide. Prove that player 3 relies on her information to make her own decision with probability q{1 q}, and that, with probability 1 q(1 q}, every subsequent player starting from player 3 will disregard her own signal when making a decision