(a) In the following pair of games, check whether the players' preferences over lotteries on the strategy profiles are identical (i.e. row player's preferences on the left to the row player's preferences on the right and column player's preferences on the left to the column player's preferences on the right). L M R L M R 2.-2 1,1 -3,7 12,-1 | 5,0 -3.2 1,10 0,4 0,4 5.3 3,1 3.1 -2,1 1,7 -1,-5 -1,0 5,2 1,-2 (b) Under Postulates P1-5 of Savage, let D1, D2,...; D, be disjoint non-null events such that Di~Dy...~Dn, where > and ~ are the at least as likely as and as likely as relations between events, derived from betting preferences as in the class. Given any subsets N and N' of {1, 2. .... n}, show that UD.= UD. - IN1 2 IN'I. iEN iEN' 3. Bergson becomes a benevolent dictator. He has n subjects i = 1, ... .n with CARA utilities u1.-.., un, respectively. (Write a, for the absolute risk aversion of i.) The total wealth in the society, Y, is a function of an unknown state w and is normally distributed with mean / and variance o'. Bergson can choose any allocation a = (21, ...,2, ) such that an (w) + ... +2, (w) & In(Is+4). *=1 where , 6 6 (0, 1). She gets return of r > I from her savings so that her wealth at t + 1 is with = " (ut - 21) if her wealth at t is no, and she consumes r, at t. (b) Find a sophisticated-optimal consumption strategy for her in which the self at any given date s consumes yu,. Compute the constant y and briefly verify that this is indeed a subgame-perfect equilibrium of the multi-agent game. (c) For 8