4. Consider a game. Player 1 decides up or down (U, D). Player 2 simultaneously and independently decides whether to go left or right (L, R). The payoffs from each action combination are given by (Player 1's payoff is the first entry in each cell). Player 2 L R Player 1 U 1, 1 3, 3 D 2, 4 4, 2 a) Find all the Nash equilibria of this game. Which player, if any, has a dominant strategy? b) Now suppose that Player 1 moves first by choosing either U or D. Player 2 observes Player 1's action and then chooses L or R. For every action combination, the players' payoffs are the same as in the above payoff matrix. Draw a tree of this new game. How many strategies does player I have and what are they? How many strategies does player 2 have and what are they? Find all the sub-game perfect equilibria of this game. c) Now find all the Nash equilibria of the game in part b). Are there any Nash equilibria that are not sub-game perfect? Which Nash equilibrium is more believable? Explain. Does Player 1 receive a higher payoff in a static game of part a) or in a dynamic game of part b)? Why do you think this difference arises? d) Now suppose there are new payoffs. How many Pure Strategy Nash Equilibriums are there? Find all Mixed Strategy Nash Equilibrium. Player 2 L R Player 1 2, -3 1, 2 D 1, 1 4, -1