ECN 614 An Introduction to Game Theory Problem Set 2 (Due on April 6 (Tuesday), 11:59 pm) You need to submit your homework by email. Please make sure that the scanned version of your homework assignment is i) in good quality and ii) in SINGLE PDF file. I will not accept photos of homework assignments. Problem 1. Consider the following game. Notice that player 2 has two information sets. L M 1 b 1, 2 a 4, 1 2 d 3, 1 c 2, 2 2 (a) Find all subgame perfect Nash equilibria (SPNE) by using backward induction. (b) Consider mixed strategies σ ∗ 1 = ( 1 2 , 1 2 ) and σ ∗ 2 = ( 1 3 , 1 3 , 0, 1 3 ). Notice that σ ∗ 2 (bc) = 0. Find behavioural strategies b ∗ 1 and b ∗ 2 such that u1(σ ∗ 1 , σ∗ 2 ) = u1(b ∗ 1 , b∗ 2 ) and u2(σ ∗ 1 , σ∗ 2 ) = u2(b ∗ 1 , b∗ 2 ). Problem 2. Consider the following three-player game. First, player 1 chooses a real number x1 between 0 and 1 (i.e. x1 ∈ [0, 1]). After observing player 1’s choice of x1, player 2 chooses x2 between 0 and 1 (i.e. x2 ∈ [0, 1]). Then, after observing players 1 and 2’s choices of x1 and x2, player 3 chooses x3 between 0 and 1 (i.e. x3 ∈ [0, 1]). If the sum of x1, x2 and x3 does not exceed 1, players 1, 2 and 3 receive x1, x2 and x3 respectively; otherwise they all receive 0. In other words, if x1, x2 and x3 are chosen, ui(x1, x2, x3) = ( xi if x1 + x2 + x3 ≤ 1 0 if x1 + x2 + x3 > 1 (a) Describe the strategy spaces of the players. (b) Find a SPNE (in pure strategies) of this game. (b) Find a Nash equilibrium s ∗ that is not subgame perfect and satisfies u1(s ∗ )