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- Solve parts 1(d) and 1(e) ONLY(a) Find all the Nash Equilibria, if there is any. (no explanation needed for this part (b) Does player 1 (choosing rows) have any dominant action? If yes, which action(s)? Any dominated action(s)? If yes, which ones? Answer the same questions for player 2, too. (c) If player 1 moves first (and player 2 moves next), what would be the sequentially rational equilibrium (draw the game tree and use backward induction)?What if player 2 moves first (and then player 1 moves next)? (d) Looking at your findings in (c), would player 1 want to move first or second or is she indifferent (the order doesn’t matter)?1. Suppose Company A is about to play a game with Company B. The following facts are known about the two players. The first Company (Company A) is a row player and uses three different strategies i.e. (Strategy X, Strategy Y and Strategy Z). Whereas the column player (Company B) has two different strategies i.e. (M and N) that can be used accordingly. The payoff matrix is given in the table below. Player B Player A M N X -5 3 Y 3 -7 Z -6 5 Answer the following questions based on the information given above a. Determine the strategies of each firm using graphical technique b. Compute the value of the game.
- 8) Find the mixed strategy Nash equilibrium of the following normal form game. Player 2 T1 T2 T3 2, 3 3, 5 1, 1 Player 1 S2 1, 4 4, 3 0, 5 Player 1 attaches probability (S1, S2) = () and Player 2 attaches probability (T1, T2, T3) = ( ) Player 1 attaches probability (S1, S2) = (.) and Player 2 attaches probability (T1, T2, T3) = (qi, 42, 1 – q1 – 92) where q1 , and 0 < q2 S %3D Player 1 attaches probability (S1, S2) = (G,;) and Player 2 attaches probability (T1, T2, T1) = (qı.42, 1 – q1 – 42) where 0 < qi <, and q2 = 3. Player 1 attaches probability (S1, S) = (;, -) and player 2 attaches probability (T1, T2, T3) = (1.42, 1- q1- 42) where 0 s qı s and q2 =Refer to Table Left-Right-ZigZag. How many pure strategy Nash equilibria does this game have? Table: Left-Right-ZigZag Player 1 Drive left Driver right Zigzag 1) 1 2) 2 3) 3 4) 4 5) None of the above. Drive left (1,1) (-1,-1) (0,0) Player 2 Driver right (-1,-1) (1,1) (0,0) Zigzag (0,0) (0,0) (0,0)2. Consider the following 2 → 2 normal form games: In both games Player 1 (the row player) chooses either strategy A or strategy B. Player 2 (the column player) chooses simultaneously strategy A or B. The outcomes are defined by the following matrices. The first number in each cell indicates the payoff of Player 1, the second number (after the comma) indicates the payoff of Player 2. A B Game I A B 50X, 30X 10X, 10X 10X, 10X 30X, 50X Game II A A 110X, 30X B 10X, 10X B 10X, 10X 30X, 50X where X > 0 is a payoff scale parameter. (a) For both games: find all Nash equilibria in pure strategies, if any, and find the Nash equilibrium in mixed strategies. How would you describe this strategic interaction in the two games?
- 2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.4Question 1 Consider the following game. Player 1 has 3 actions (Top, middle,Bottom) and player 2 has three actions (Left, Middle, Right). Each player chooses their action simultaneously. The game is played only once. The first element of the payoff vector is player 1’s payoff. Note that one of the payoffs to player 2 has been omitted (denoted by x). A) Suppose that the value of x is such that player 2 has a strictly dominant strategy. Find the solution to the game. What solution concept did you use to solve the game? B) Suppose that the value of x is such the player 2 does NOT have a strictly dominant strategy. Find the solution to the game. What solution concept did you use to solve the game?
- 1. Contract Enforcement Between Buyer and Seller in a Supply Chain. Suppose that two firms, 1 and 2, must undergo a certain amount of investment so as to coordinate their efforts in order to enable the supplier to produce a good to be used as an input in the buyer's production process. Such investment may involve the costs incurred by both parties in the initial planning and design phase of the relationship and/or involve the firms in having to incur the associated costs of purchasing specialized equipment and developing certain production processes in order to satisfy the needs and specific requirements of both parties. In any case, suppose that the technology of the relationship and the structure of the underlying game are given by the following normal form representation of the game: Firm 1 Invest (I) Not Invest (N) Firm 2 Invest (I) Z₁, Z₂ (41, 35) X1, Y2 (92,-42) Not Invest (N) Y1, X2 (-32, 21) (0, 0) Further suppose that the players contract in a setting of court-imposed breach…1. Consider the following game in which each of the two players has two possible actions, cooperate with each other (C) and defect (D, i.e., Don't cooperate with each other). The monetary payoffs to each action profile are given in the following table. Player1 / Player 2 Defect (D) Cooperate (C) 3, 0 Defect (D) 1, 1 Cooperate (C) 0,3 2, 2 However, the above table does not represent the true preferences of the players, as they care about what the other player earns as well as their own earnings. In particular, if mi is the amount of money earned by Player 1 and m2 is the amount of money earned by Player 2, then the payoff to Player 1 is mi + am2 ; where a 2 0. (Of course, when a = 0; the payoffs coincide with the ones in the above table). For example, Player 1's payoff to action profile (C,C) is u1 (C;C) = 2 + 2a; and Player 1's payoff to action profile (C,D) is u1 (C;D ) = 0+ 3a = 3a a) Write down the strategic form of this game for a= 1. Is this game a Prisoners' Dilemma game? b) Find…3. DON'T MESS WITH ME GAME TREE. Suppose Player #1 (P1) anticipates the possibility of being challenged by rival Player #2 (P2). To discourage P2 from challenging P1 tomorrow, P1 could invest in mixed martial arts training today, thereby making P1 a more formidable rival in case it has to battle tomorrow. Suppose the game tree below accurately models this dynamic game. TIP: Review the dynamic games slides to refresh your memory about analyzing games trees and finding SPNE. Fight (F) (100, -50) Challenge (C) P1 ММА Training (M) P2 Give Up (G) (-40, 100) Do Not (D) (60, 0) P1 fight (f) (-20, 150) challenge (c) P1 No Training (N) give up (g) P2 (0, 100) do not (d) (100, 0) а) Find all Subgame Perfect Nash Equilibria (SPNE) of this game. TIP: What is the definition of SPNE? b) In the subgames in the game tree directly following P1's choice to invest in MMA Training, identify ALL of the noncredible threats and explain why each is noncredible.