Game Theory and Social Sciences — DU B.A. (Hons.) Economics Previous Year Question Papers

1. (a) Consider the following normal-forma game: Explain how dominance solvability helps in reducing the complexity of strategic games. What conclusion can you draw about the equilibrium outcome of this game after applying IESDS? (6 marks) (b) Two firms, Firm 1 and Firm 2, compete in quantities in a homogeneous product market à la Cournot Competition. The inverse market demand function is given by: 120 where , and , denote outputs of Firm 1 and Firm 2 respectively. The cost functions of the firms are: 10 , 20 (i) (ii) Derive the profit functions for both firms. (2 marks) Derive the best response functions of Firm 1 and Firm 2. (4 marks) (iii) Solve for the equilibrium quantities ∗ and ∗ . (2 marks) (iv) (v) Determine the equilibrium price and profits of both firms.(3 marks) Explain how cost asymmetry affects equilibrium output and profits. (1 marks) 2. (a) In a second-price auction with bidders having valuations ₹150, ₹175, and ₹190, find: (i) Each bidder’s equilibrium bid. (3) (ii) Winner and price paid. (2) (iii) When would an auction designer prefer first price auction over second price auction in such a situation? (4 marks) (b) Solve for the mixed strategy Nash equilibrium and expected payoff for each player in the following game. The first entry in a pay-off vector is row player’s pay-off. (9 3. (a) What is a strictly competitive game? Consider the following normal form game and computer the mixed strategy Nash equilibrium. (9 marks) (b) Consider the following normal-form game: Define the concepts of strictly dominated and weakly dominated strategies and identify dominated strategies, if any, for both the players. (9 marks) 4. Consider the following normal-form game: Player 1 (Aarav) and Player 2 (Bhavna) each choose between two actions: Invest (I) or Not Invest (N). Payoffs are given as (Aarav, Bhavna): Invest Not Invest Not Invest (a) For what values of y does this game have a unique Nash equilibrium? Identify the equilibrium for those values. (b) For what values of y does this game have a mixed-strategy Nash equilibrium? Compute the probability (in terms of y) with which each player chooses Invest (I) in the mixed-strategy equilibrium. (8 marks) (c) Let y=5. Derive and graph the best response functions of both players, and use them to identify all Nash equilibria (pure and mixed). (6 marks) Consider a two-player sequential game with symmetric information in which Player 1 moves first by choosing an action ∈ , , after which Player 2 observes Player 1’s choice and selects an action ∈ , . The resulting payoffs (Player 1, Player 2) are as follows: if Player 1 chooses A, then Player 2 choosing X yields (5, 1) and choosing Y yields (2, 4); if Player 1 chooses B, then Player 2 choosing X yields (3, 3) and choosing Y yields (4, 2). (a) Represent the game in extensive form. (6 marks) (b) Determine Player 2’s optimal strategy using backward induction, identify Player 1’s optimal choice anticipating Player 2’s response, and find the Subgame Perfect