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- 1. Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1 Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1 Exercise 1: Solve the game provided using backwards induction. (Hint: Notice that at the lower history where Player 1 moves again there is a “tie.” This indicates that there will be more than one solution by backward induction.) Describe each backward induction solution by giving, in each solution, each player’s complete plan for playing the game. Since Player 1 is indifferent between choosing A or B in the last branch of the game tree, we can outline two possible SPNEs in the two game trees below. DYNAMIC OPTIMIZATION & ECONOMIC Our online Tutors are available 24*7 to provide Help with Dynamic Optimization & Economic Homework/Assignment or a long term Graduate/Undergraduate Dynamic Optimization & Economic Project. Our Tutors being experienced and proficient in Dynamic Optimization & Economic sensure to provide high quality Dynamic Optimization & Economic Homework Help. Upload your Dynamic Optimization & Economic Assignment at ‘Submit Your Assignment’ button or email it to info@assignmentpedia.com. You can use our ‘Live Chat’ option to schedule an Online Tutoring session with our Dynamic Optimization & Economic Tutors.
- 2. Exercise 2: For each backward induction solution in question 1, what is the path of play that will be realized in the game? What payoffs will result. As shown on the slides for the game of 5 stones, “Path of Play” always consists of what the first mover does, followed by what the second mover does in that event, and so on until you get to a terminal node. Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1 Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1 In this Sub-game Perfect Nash Equilibrium (SPNE), Player 1 actually plays L and Player 2 Plays X resulting in payoffs of (4,3). In this Sub-game Perfect Nash Equilibrium (SPNE), Player actually 1 plays R and Player 2 Plays Y resulting in payoffs of (3,3). In this Sub-game Perfect Nash Equilibrium (SPNE), Player 1 plays LA and Player 2 Plays XY resulting in payoffs of (4,3). In this Sub-game Perfect Nash Equilibrium (SPNE), Player 1 plays RB and Player 2 Plays WY resulting in payoffs of (3,3).
- 3. Exercise 3: Using the following steps, give and solve the strategic form of this game. a. What are the possible strategies for Player 1? Player one can choose, 1.) LA; 2.) LB; 3.) RA; 4.) RB. b. What are the possible strategies for Player 2? Player one can choose, 1.) WY; 2.) WZ; 3.) XY; 4.) XZ. c. What Nash Equilibrium does this game have? See the “*” boxes in the strategic form below. Note that there are 5. WY WZ XY XZ LA LB RA RB 2 1 1 2 4 3 4 3 2 5 5 2 4 3 4 3 3 3 2 1 3 3 1 2 3 3 2 1 3 3 1 2 * * * * * * SPNE * NE only
- 4. Exercise 4: For Nash equilibrium that are not consistent with backward induction, what “non- credible threat is involved – that is, what move would not be rational if actually required? (Hint: for one of them, you need to assume that a later move specified by the Nash Equilibrium is anticipated by earlier moves in the sequential game. Based on the strategic form game, we first see that there are three Nash Equilibria that are not SPNEs: 1. (LA, XZ); 2. (RA, WY) and 3. (LB, WZ). We can determine what move constitutes the non-credible threat in each NE by viewing the extensive form games on the slides that follow. WY WZ XY XZ LA LB RA RB 2 1 1 2 4 3 4 3 2 5 5 2 4 3 4 3 3 3 2 1 3 3 1 2 3 3 2 1 3 3 1 2 * * * * * * SPNE * NE only
- 5. Exercise 4: Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1 Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1 In this Nash Equilibrium, Player 2 makes the non- credible threat of playing Z if Player 1 moves Right. In this case, the resulting payoffs are identical to the SPNE of (LA, XY)= (4,3) since Player one will actually move Left. 1.) Nash Equilibrium of (LA, XZ) 2.) Nash Equilibrium of (RA, WY) In this Nash Equilibrium, Player 1 plays A if Player 2 plays W (which is in fact a rational move since Player 1 is indifferent between both payoffs of 2 in his last move.) Player 2 threatens to Play W if Player 1 plays Left (resulting in lower payoffs for both players of (2,1) instead of (4,3) if the game reaches this node. In response to Player 2’s non- credible threat, Player 1 plays Right resulting in payoffs of (3,3) which are identical to a SPNE at (RB, WY).
- 6. Exercise 4: In this Nash Equilibrium, Player 1 plays B if Player 2 plays W (which is in fact a rational move since Player 1 is indifferent between both payoffs of 2 in his last move.) Player 2 threatens to Play Z if Player 1 plays Right. This would result in a lower payoff of (1,2) for each player. Based on this non-credible threat, Player 1 would choose to Play Left instead of Right to at least get a payoff of 2 rather than 1. This will enable Player 2 to realize his highest possible payoff of 5. 3.) Nash Equilibrium of (LB, WZ) Player 1 Y Z 3 3 1 2 L X 4 3A B R W 2 5 2 1 Player 2 Player 2 Player 1