1662537687 February1 2022

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GAME THEORY 5 cfu February 1, 2022 Surname: Name: Matricola: The exam, or the single exercise, will not be evaluated if: • The surname (last name) and name are not clearly written in the right place; • The explanations are not clearly written and contain cancellations; Grades: Exercise 1:2+3+3, Exercise 2: 2+2+3+3 Exercise 1 Given the two player game, described by the following bimatrix, where a,b are real parameters and digits represent utilities:   (1, 1) (3 , 2) (2 , 3) (6,b ) (5 , 2) (1 , 6) (3, 5) ( a, 5) (2 , 2)   1. Find a,b (if any) such that the game is a potential game; 2. find q such that the third row is the first player’s best reaction to the strategy (q, 0, 1−q) of the second player; 3. find a,b (if any) such that there is a NEp with support the second and third row for the first player and first and second column for the second player. Write all NEp in such cases. Answer 1 The game is never a potential game. Answer 2 0≤q≤ 1 4 Answer 3 a> 5,b = 2 the NEp are [(0,p, 1−p), ( a−5 a−2 )] with 0<p< 3 7 Explanations 1. A possible potential, built starting putting 1 in the top left, writing the first row then the columns, is: 1 2 3 6 4 2 3 a-1 3 However in the second row we have 6− 2̸= 2− 4; 2. The following system must be solved:    q + 2− 2q ≤ 3q + 2− 2q 6q + 1−q ≤ 3q + 2− 2q 0<q < 1 3. The following systems must be solved:    6q + 5− 5q = 3 q +a−aq 6q + 5− 5q ≥ q + 3− 3q 0<q < 1    pb + 5− 5p = 2 p + 5− 5p 2p + 5− 5p ≥ 6p + 2− 2p 0<p< 1 1 Exercise 2 Consider the following TU game (N,v ): v(S) = {|S| + 2, if{1, 2}⊆ S |S|, otherwise. 1. For n = 3 find the Shapley and Banzhaf values; 2. for n = 3 find the core and draw a picture; 3. for generic n find the Shapley and Banzhaf values; 4. for generic n find the core. Answer 1 σ(v) = β(v) = (2, 2, 1) Answer 2…