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2 min if y g 0.5 1.5 +3; min if y > 0.5 D 1min if1yO.7 0.3+(1y) min if1y>0.7 Numbers on the edges show the travel time (in minutes) on each road; :3, 1 x, y, 1 y are fractions of all drivers on roads L, l, H and h, respectively. All drivers start at A and nish at B. The difference from the assignment problem in (a)(f) is that we will assume that there are only two drivers (hence, when one driver takes a road, then 0.5 fraction of all drivers take this road). (a) [4pt] Identify the pure strategies of each player. Compute the payoffs to each of the players from Sim strategy proles. If you compute more than six payoffs, the rst siX will be checked by a marker. You can answer this question by drawing a table with strategies and payoffs. Note that c is not given a numeric value; do not substitute any value for 0. Do not eliminate any strictly dominated strategies, but assume that no driver drives away from B or uses the same road twice (that is, do not consider such strategies as leL . . . ) For (b)(e), we set c = 1. (b) [3pt] Find best response(s) for one of the players for each strategy you have identied in (a). (c) [3pt] Identify all strictly dominated strategies and prove that they are. Explain why other strategies are not strictly dominated. (d) [3pt] Identify all strictly dominant strategies and prove that they are. Explain why other strategies are not strictly dominant. (e) [4pt] Find all pure strategy Nash equilibria of this game. Explain why they are Nash equilibria and why there are no other pure strategy Nash equilibria. (f) [4pt] Suppose now that c = 0.3. Find all pure and mixed strategy Nash equilibria of that game. We now return to the model with innitely many drivers (the road network is exactly the same). Suppose that driving time on s is different for different drivers and given by the function C(i) = i, where 72 6 [0,1] is the index of a driver. Hence, for any x, the fraction 3: of all drivers have the driving time less than w and the fraction (1 :17) of drivers have the driving time more than CI}. The minimum driving time, for driver 2' = 0, is 0 and the maximum driving time, for driver 1' = 1, is 1. (g) [8pt] Find a pure-strategy Nash equilibrium of this game. We now return to the problem with two drivers. Suppose that the one driver moves rst (we will call her the rst driver) and the other driver moves next (the second driver). Suppose further that, when deciding upon his strategy, the second driver sees whether the rst driver took road L or 1, but does not know what happens when the rst driver reaches points C or D, respectively. Note that the second driver decides upon a strategy (that is, a complete route) when he starts driving and does not change the route en route. (h) [5pt] Represent this game as a sequential game (draw a tree). As in (a), you need to identify only six payoffs. (i) [4pt] Find one subgame perfect Nash equilibrium (SPNE) and explain why it is an SPNE