PROBLEM (5) (Behavioral Game Theory - Centipede game) Each of the two players A and B has $10 dollars. First, A can choose to (C)ontinue or (S)top, if she chooses to continue then it is B's turn and she can choose to continue or stop, and if she chooses to continue it is A's turn again to choose to continue or stop,... until one of them chooses to stop or each had 3 turns to play (after B's 3* round choice to continue or stop, the game ends). Each time a player chooses to continue, her money decreases by $1 but the other player's money increases by $2. When a player stops, the game ends with each player getting whatever dollar she currently has as her utility. (a) What is the sequentially rational (backward induction) Nash Equilibrium of this game? Describe what each player does at each of his/her decision nodes and identify the outcome of the game when this equilibrium is played. (b) Clearly there are outcomes (terminal payoffs) in this game that would have made each player strictly better off compared to the outcome in (b). If both players had the other regarding utility functions us(a,b) =a+ abandug(a,b) =b + aa (when they end up with $a and $b respectively, when the game ends) instead; what should at least be, so that each player plays C on all the 3 rounds they get to play, in the sequentially rational Nash Equilibrium of the game with these new utility functions? (c) Instead of the utilities in (b), assume they have Fehr-Schmidt preferences us(a,b) = a amax{ab,0}and ug(a,b) = b a max{b a, 0} where a measures the degree of embarrassment (guilt) each feels about getting more than their opponent. Take an extreme value a = 2, (where a player would be willing to give up $2 of her own dollars not to be getting $1 more than her opponent). What would be the sequentially rational Nash Equilibrium of the game with these modified payoffs? Would they still continue till the end? (d) In (c), what is the possible range of values for a for which the sequentially rational Nash Equilibrium play of the game calls for Continue till the end for both players