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(1) 20 points] Consider the following coalition game with four students. Each student happens to have a shoe (not a pair of shoes), and the students are trying to form pairs of shoes. Specifically, student 1 has a left-shoe in red, student 2 a right-shoe in red, student 3 a left-shoe in blue, and student 4 a right-shoe in blue. Once shoes are collected, they can sell a pair of shoes. If they collectively have a pair of blue shoes, the shoes are sold at $30, a pair of red shoes at $20, and a pair of red-blue or blue-red shoes (mixed-color shoes) at $10. If there are multiple ways to sell the shoes, they will sell the highest-price pair of shoes first (that is, in order of blue, red, and mixed). For example, if all of them coordinate, then they would first sell the blue shoes at $30, and then the red shoes at $20 to make a total profit of $50. (a)11 points] Formally describe this as a coalitional game. N- 11,2,3,4l, and you are to describe the characteristic function v : 2N -> [$0, $10, $20, $30, $50). Fill out the table below to specify v. Only specify values for v(S) where v(S) $0 (to avoid tedious work). Use the table below as a reference. 1,2 $20 2,3) $10 (b) [5 points] Compute the Shapley value, ?(N,U), for each student i E N. (c) [1 point] True or false: this game is additive. (d) [1 point] True or false: this game is superadditive. (e) [1 point] True or false: this game is constant-sum (f) [1 point] True or false: the core of this game is empty. (1) 20 points] Consider the following coalition game with four students. Each student happens to have a shoe (not a pair of shoes), and the students are trying to form pairs of shoes. Specifically, student 1 has a left-shoe in red, student 2 a right-shoe in red, student 3 a left-shoe in blue, and student 4 a right-shoe in blue. Once shoes are collected, they can sell a pair of shoes. If they collectively have a pair of blue shoes, the shoes are sold at $30, a pair of red shoes at $20, and a pair of red-blue or blue-red shoes (mixed-color shoes) at $10. If there are multiple ways to sell the shoes, they will sell the highest-price pair of shoes first (that is, in order of blue, red, and mixed). For example, if all of them coordinate, then they would first sell the blue shoes at $30, and then the red shoes at $20 to make a total profit of $50. (a)11 points] Formally describe this as a coalitional game. N- 11,2,3,4l, and you are to describe the characteristic function v : 2N -> [$0, $10, $20, $30, $50). Fill out the table below to specify v. Only specify values for v(S) where v(S) $0 (to avoid tedious work). Use the table below as a reference. 1,2 $20 2,3) $10 (b) [5 points] Compute the Shapley value, ?(N,U), for each student i E N. (c) [1 point] True or false: this game is additive. (d) [1 point] True or false: this game is superadditive. (e) [1 point] True or false: this game is constant-sum (f) [1 point] True or false: the core of this game is empty