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Politecnico di Milano Facoltà di Ingegneria dell’Informazione AUTONOMOUS AGENTS AND MULTIAGENT SYSTEMS February 24th, 2010 LAST NAME AND FIRST NAME ROW COLUMN ID NUMBER (MATRICOLA) • The exam is composed of three stapled sheets printed on both sides. • This front page must be filled with last name, first name, ID number, position (row and column communicated by the instructor), and signature. • Exams without a completely filled front page or with missing sheets will not be considered. • Answers can be written only on these sheets. If you need more space, please write on the last page. • Exam is closed books (i.e., no books, notebooks, notes, … are allowed). Cell phones, bags, cases, and wallets are not allowed on the desk during the exam. • All the answers must be justified. SIGNATURE Question 1 (8 points). Consider the following extended form game between two agents, 1 and 2: Ovals are decision nodes for the agents reported in the ovals themselves, rectangles report the payoffs for agent 1 and 2, respectively. 1) Transform the above extended form game in an equivalent normal form game. 2) Find the pure strategy Nash equilibria of the normal form game obtained in 1). 3) Which of the equilibria found in 2) are also subgame perfect equilibria? Why? 4) Assume that there is a third agent and that the extended form game is Answer questions 2) and 3) considering this new situation. 1) The corresponding normal form game is represented by the following payoff matrix: left right Down-Down 2, 1 2, 1 Down-Up 2, 1 2, 1 Up-Down 1, 2 3, 1 Up-Up 1, 2 4, 3 2) Nash equ ilibria can be easi ly f ound by in spection: ( Up-Up, righ t), ( Down-Up, lef t), ( Down-Down, left). 3) The only subgame perfect equilibrium is (Up-Up, right). (Down-Up, left) is not an equilibrium in the subgame…