6. Consider the two-player Rock-Paper-Scissors game The payoff table is as follows. Player 2 Paper -1 0 1 Player 1 Rock Paper Scissors Rock 0 1 -1 Scissors 1 -1 0 (a) Follow the approach described in Sec 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. Please give the linear programming problem for each player. (b) Use Excel Solver (the simplex method) to find the optimal mixed strategies for each player. Also find the value of the game. Is this a fair game? If not, is this game in favor of player 1 or 2?

6. Consider the two-player Rock-Paper-Scissors game The payoff table is as follows. Player 2 Paper -1 0 1 Player 1 Rock Paper Scissors Rock 0 1 -1 Scissors 1 -1 0 (a) Follow the approach described in Sec 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. Please give the linear programming problem for each player. (b) Use Excel Solver (the simplex method) to find the optimal mixed strategies for each player. Also find the value of the game. Is this a fair game? If not, is this game in favor of player 1 or 2?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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1. Consider the zero-sum game given by 3 1 2 A = 4 0 2 1 2 -2 3 1 (a) Calculate the gain floor and loss ceiling for A. (b) Without using dominance, use the Equilibrium Theorem to determine whether it is possible for A to have a mixed Nash Equilibrium where player two's strategy is of the form (a, b, c, 0) with a, b, c > 0.
1. One-Shot, Pricing Game. You are a pricing manager at Argyle Inc.-a medium-sized firm that recently introduced a new product into the market. Argyle's only competitor is Baker Company, which is significantly smaller than Argyle. The management of Argyle has decided to pursue a short-term strategy of maximizing this quarter's revenues, and you are in-charge of formulating a strategy that will permit the firm to do so. After talking with an employee who was recently hired from the Baker Company, you are confident that a) Baker is constrained to charge $10 or $20 for its product, b) Baker's goal is to maximize this quarter's profit, and c) Baker's relevant unit costs are identical to yours. You have been authorized to price the product at two possible levels ($5 or $10) and know that your relevant costs are $2 per unit. The marketing department has provided the following information about the expected number of units sold (in millions) this quarter at various prices to help you…
2. Consider the two-player strategic form game represented in the following payoff table: t 1 m b 7,2 1,2 3,7 2 M 2,4 3,4 1,6 R 1,3 7,1 4,2 Player 1 chooses between rows, S₁ = {t, m, b), and receives the first listed payoff in each cell. Player 2 chooses between columns, S₂ = {L, M, R), and receives the second listed payoff in each cell. a. Does either player have a strictly dominant strategy? Explain. b. Explaining your steps carefully, identify the set of rationalizable strategies in this game. c. Making sure to explain your solution method, solve for all pure strategy Nash equilibria in this game.
4. Consider a normal form game with N= {1,2} and S= {A,B,C,D} x {W,X,Y,Z}. Assume that L1 allows to eliminate D and X, L2 allows to eliminate B and C, and L3 allows to eliminate Z. Provide payoffs that lead to this outcome. State the rationality assumptions explicitly.
Suppose that, in an ultimatum game, the proposer may not propose less than $1 nor fractional amounts, and therefore must propose $1, $2, …, or $10 (see image attached below). The responder must Accept (A) or Reject (R). Suppose, first, that this game is played by two egoists, for whom u(x,y)=x. Find all subgame-perfect equilibria in this game. Suppose, second, that this game is played by two altruists, for whom u(x,y) = ⅔(x)1/2 + ⅓(y)1/2. Find all subgame-perfect equilibria in this game.
Prove that in the variation on the centipede game given in figure 14.5(b) the unique sequential equilibrium described is, in fact, the unique Nash equilibrium. (Hint: Take some presumed Nash equilibrium and suppose information set 2n+ 1 [for player 2] is the first unreached information set. Derive an immediate contradiction. Then suppose that node (2n) t is the first unreached information set and derive a contradiction that is one degree removed from immediate.)
2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.
7. Solving for dominant strategies and the Nash equilibrium Suppose Antonio and Caroline are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Antonio chooses Right and Caroline chooses Right, Antonio will receive a payoff of 9 and Caroline will receive a payoff of 8. Caroline Left Right Left 8, 5 8. 7 Antonio Right 3, 6 9, 8 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Antonio chooses and Caroline chooses
c) Consider the centipede game which illustrates the situation where it is mutually beneficial for all players to stay in a relationship while a player would like to exit the relationship if they know that the other player will exit in the next day. Note that the fourth day is the choice for Player 2 to move and choose either o or B. Use the logic of backwards induction to find the subgame perfect Nash equilibrium of the game in Figure 1. Figure 1: The Centipede Game (1 (64,16) (2,8) O (4,2) (16,4) (8,32)
Problem 3 Consider the following simultaneous normal form game: P.2 R 0,0 -1, 1 1, -1 P.1 P 1, -1 0,0 -1, 1 S -1, 1 1, -1 0,0 Now assume the game is not simultaneous and player 1 gets to play first. Create a game tree that represents the sequential game. Also, name each node, define the set of information sets for each player, show which nodes are in which information set and add define action sets at each info set.
Define a new game as a modified version of the game in Problem 1, which we will call the Matching Two Pennies game: each player choose heads (H) or tails (T) for two pennies. In this game, Player 1 wins if both coins show the same combination of heads and tails, while Player 2 wins if they do not. (a) Write the strategy sets S1 and S2 for this game. (b) Give the payoffs for each player in this Matching Two Pennies game, by defining T;(81, 82) for each player and for each strategy pair, either as a list or in matrix form.
3. Repeated Game Consider the game in which the stage game depicted below is infinitely repeated and in which both players discount future payoffs with discount factor 6 € [0, 1]. Player 1 E S Player 2 E 3,3 4, -1 S -1.4 1,1 (a) Suppose that both players play the tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player copies the action chosen by his opponent in the previous period (i.e if Player 2 shirks (plays S) in period 2, then Player 1 plays S in period 3). Then, for what values of d can tit-for-tat be supported on the equilibrium path of play? (b) Suppose that both players play the perfect tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player plays E unless the actions are disagreed in the previous period. Then, for what values of d can perfect tit-for-tat be supported on the equilibrium path of play?