5. Consider the payoff matrix we saw in the run-pass game: Payoff Matrix for Football Game Run defense Pass Defense Offense Runs - 5 Offense Passes 10 If the offense decides to play 60% of the time pass and 40% of the time run, while the defense chooses to defend the pass and the run 50% of the time each, what is the expected yards gained from the offense? а. 3 b. О 10

5. Consider the payoff matrix we saw in the run-pass game: Payoff Matrix for Football Game Run defense Pass Defense Offense Runs - 5 Offense Passes 10 If the offense decides to play 60% of the time pass and 40% of the time run, while the defense chooses to defend the pass and the run 50% of the time each, what is the expected yards gained from the offense? а. 3 b. О 10

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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7. Solving for dominant strategies and the Nash equilibrium Suppose Russell and Aaliyah are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Russell chooses Up and Aaliyah chooses Down, Russell will receive a payoff of 6 and Aaliyah will receive a payoff of 4. Russell Up Down Aaliyah Up 6,3 3,3 Down 6,4 7,4 In this game, the only dominant strategy is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Russell chooses and Aaliyah chooses
12. Consider a game where each player picks a number from 0 to 60. The guess that is closest to half ofthe average of the chosen numbers wins a prize. If several peopleare equally close, then they share theprize. The game theory implies that (A) all players have dominant strategies to choose 0 (B) all players have dominant strategies to choose 30 (C) there is a Nash equilibrium where all players pick 0 (D) there is a Nash equilibrium where all players pick positive numbers 13. Behavioral data in such games suggests that (A) most subjects choose 0; (B) most subjects choose 30; (C) common answers include 30, 15, 7.5, and 0; (D) most subjects use randomization. Can you help me answer number 13 please?
4. Consider a three-player bargaining, where the players are negotiating over a surplus of one unit of utility. The game begins with player 1 proposing a three-way split of the surplus. Then player 2 must decide whether to accept the proposal or to substitute for player 1's proposal his own alternative proposal. Finally, player 3 must decide whether to accept or reject current proposal (it is player 1's if player 2 accepts or player 2's if player 2 offer a new one). If he accepts, then the players obtain the specified shares of the surplus. If player 3 rejects, then the players each get 0. (a) Draw the extensive form game of this perfect-information game. (b) Determine the subgame perfect NE.
4. Consider a two player game with Fred and Barney, who tal turns removing matchsticks from a pile. They start with 33 matchsticks, and Fred goes first. On each turn, ecach player may remove either one, two, three, four, or five matchsticks. The player to remove the last matchstick wins the game. What are the optimal strategies for each player? Who will win? b. Suppose now that they can remove up to six matchsticks, how will the optimal strategies change for- each player? a.
10. Game theory terminology Select the term that best describes each definition listed in the following table. Definition Nash Equilibrium Dominant Strategy Collusion Tit-for-tat Strategy Payoff Matrix/Table Prisoners' Dilemma Game A set of strategies (one for each player) in which each player's strategy is the best option for that player, given the chosen strategy of the player's opponents A strategy in which a player cooperates until the other player defects and then defects until the other player cooperates again A case in which individually rational behaviour leads to a jointly inefficient outcome A player's best choice, if it exists, regardless of his or her opponent's strategy
Karen goes to the casino with 1000$ and plays the following strategy: In her first game, she putsin 1$. If she wins, she finishes playing and takes her profit home. If she looses a game, she willplay another game and put in twice as much money as she did in the previous game. If she wins,she will finish playing and take her profits home. She will keep going with that strategy, untilshe either wins a game or runs out of money.(a) Create a table for n = 1, 2, ..., 10 which lists: (a) The amount of money she puts in in game n;(b) the sum of the amount of money she has put in in all games up to this game n; and (c)her overall profit if the game n is the first game she wins.(b) What is the likelihood of her loosing 8 games in a row?(c) How many games can she loose in a row before she cannot afford to continue with herstrategy of doubling her bet?(d) Given the low likelihood of loosing, is this a smart strategy for Karen to pursue?
4 Consider an extensive game where player 1 starts with choosing of two actions, A or B. Player 2 observes player 1’s move and makes her move; if the move by player 1 is A, then player 2 can take three actions, X, Y or Z, if the move by player 1 is B, then player 2 can take of of two actions, U or V. Write down all teminal histories, proper subhistories, the player function and strategies of players in this game.
3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W x y Z 2,1 0,2 2,1 a 3,3 b 1,1 1,2 1,0 1,4 c 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 Find all the Nash equilibria in the game in pure strategies.
3. Consider a two-player, sequential-move game where each player can choose to play right or left. Player 1 moves first. Player 2 observes player 1's actual move and then decides to move right or left. If player 1 moves right, player 1 receives £20 and player 2 receives £45. If both move left, player 1 receives £15 and player 2 receives £30. If player 1 moves left and player 2 moves right, player 1 receives £40 and player 2 receives £40. a. Draw the above situation in the form of an extensive form game. b. Find the sub-game perfect Nash equilibrium of the extensive form game.
5. Instead of choosing Game A or Game B in the first stage, player 1 chooses either game B or directly which action to play in game A. Game U D L (10,5) (5,10) R (0,0) (0,0) :Game B: L R U (6,6) (0,0) D (0,0) (0,0) (a) Write the game in extensive form. (b) Solve for all SPNEs of the modified game. (c) Do all the SPNEs give intuitively reasonable predictions? Why or why not? (Hint: think about what the modification of the payoff to game A entail for player 2 in that game and how the predictions contrast with the case where the game is chosen first, then the action.)
when selecting the lineup for relays, teams start with the slowest runner and finishwith the fastest. In Beijing 2008, Jamaica were the hot favorites to win the 4 X 100 since theUK and USA were disqualified in the semis. Moreover, Usain Bolt came into the lineup. Thisleft the Jamaicans to make a key decision: Which position should they run Bolt? Look at thesplits both Bolt and Asafa Powell had and mention which position each runner should havecome in in race time.
Consider the following scenario to understand the relationship between marginal and average values. Suppose Sam is a professional basketball player, and his game log for free throws can be summarized in the following table. Fill in the columns with Sam's free-throw percentage for each game and his overall free-throw average after each game. Game Result Total Game Free-Throw Percentage Average Free-Throw Percentage Game 1 8/10 80 12/20 14/28 16/32 22/40 2 3 4 5 8/10 4/10 2/8 2/4 6/8 80 On the following graph, use the orange points (square symbol) to plot Sam's free-throw percentage for each game individually, and use the green points (triangle symbol) to plot his overall average free-throw percentage after each game. Note: Plot your points in the order in which you would like them connected. Line segments will connect the points automatically.