John and Dave are roommates. When they cook and eat dinner together, they gain 20 utils each. When John cooks dinner alone for both of them, John gains 9 utils, and Dave gains 18 utils. When Dave cooks dinner alone for both of them, John gains 18 utils, and Dave gains 9 utils. If neither decides to cook dinner, they both go hungry and earn 5 utils each.
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- N=2 video broadcasting websites, You and Twi, must decide the number of minutes of ads to be displayed for every video that the user elects to watch. Let tY be the number of ad-minutes per video set by You, and tT the number of ad-minutes per video set by Twi. Streaming one video costs You cY=0.02, while it costs Twi cT=0.03. There are 100 million potential users, and each watches videos according to the following demand curves: qY((tY,tT) =10-2tY+tT=10-2tT+tY a- What is the cross-price elasticity between You and Twi? b- Suppose, for now, that You and Twi enter an (illegal) agreement by which they set tY=tT=t Derive the total number of users in the market as a function of t. Derive the profits for each website as a function of t. c- Now let the two platforms compete by each setting their number of ad-minutes: i. What is the best reply of You? What is the best reply of Twi? ii. Find the Nash Equilibrium of the game. iii. How many total users choose You and how many total users choose…Exercise 4.1 Amy and Bill simultaneously write a bid on a piece of paper. The bid can only be either 2 or 3. A referee then looks at the bids, announces the amount of the lowest bid (without revealing who submitted it) and invites Amy to either pass or double her initial bid. - The outcome is determined by comparing Amy's final bid to Bill's bid: if one is greater than the other then the higher bidder gets the object and pays his/her own bid; if they are equal then Bill gets the object and pays his bid. Represent this situation by means of two alternative extensive frames. Note: (1) when there are simultaneous moves we have a choice as to which player we select as moving first: the important thing is that the second player does not know what the first player did; (2) when representing, by means of information sets, what a player is uncertain about, we typically assume that a player is smart enough to deduce relevant information, even if that information is not explicitly given to…Andy, Brad, and Carly are playing a new online video game: Zombie Civil War. Each has an army of 100 zombies and must decide how to allocate them to battle each of the other two players’ armies. Three simultaneous battles are occurring: one between Andy and Brad, one between Andy and Carly, and one between Brad and Carly. Let Ab denote how many zombie soldiers Andy allocates to his battle with Brad, with the remaining 100 - Ab soldiers in Andy’s zombie army assigned to the battle with Carly. Bc denotes the number of zombie soldiers that Brad allocates to his battle with Carly, and 100 - Bc zombies go to his battle with Andy. Ca is the the number of zombie soldiers that Carly allocates to the battle with Andy, and 100 - Ca in her battle with Brad. To see how payoffs are determined, consider Andy. If Ab > 100 - Bc, so that Andy has more zombies than Brad in the Andy-Brad battle, then Andy wins the battle and receives w points where w > 2. If Ab = 100 - Bc, so that Andy and Brad…
- In the game shown below, Player 1 can move Up or Down, and Player 2 can move Left or Right. The players must move at the same time without knowledge of the other player’s move. The first payoff is for the row player (Player 1) and the second payoff is for the column player (Player 2). Solve each game using game theoretic logic. Player 2 Player 1 Left Right Up 7, 7 2, 9 Down 9, 2 1, 3Consider the following situation. Maipo and Pisco need to decide how to divide a cake between the two of them. Both like cake and want to get as much cake as they can. They decide to let Maipo cut the cake first and then Pisco gets to pick which piece he wants. For simplicity, assume that Maipo can only cut the cake in two ways: He can either divide it into two pieces that are equal size (i.e., both will get half the cake) or he can divide the cake into two pieces where one piece is twice the size of the other (i.e., one will get a piece that is two-thirds of the cake and the other will get a piece that is one-third of the cake). Set up this game as a sequential game and draw the game tree that represents it Note: You can either draw the game tree by hand and then photograph/scan the tree and paste it into the assignment or use the drawing tool in Word to draw the tree. Find the sub-game perfect Nash Equilibria to this game. Underline the strategies or highlight the…Cameron and Luke are playing a game called ”Race to 10”. Cameron goes first, and the players take turns choosing either 1 or 2. In each turn, they add the new number to a running total. The player who brings the total to exactly 10 wins the game. a) If both Cameron and Luke play optimally, who will win the game? Does the game have a first-mover advantage or a second-mover advantage? b) Suppose the game is modified to ”Race to 11” (i.e, the player who reaches 11 first wins). Who will win the game if both players play their optimal strategies? What if the game is ”Race to 12”? Does the result change? c) Consider the general version of the game called ”Race to n,” where n is a positive integer greater than 0. What are the conditions on n such that the game has a first mover advantage? What are the conditions on n such that the game has a second mover advantage?
- Rita is playing a game of chance in which she tosses a dart into a rotating dartboard with 8 equal-sized slices numbered 1 through 8. The dart lands on a numbered slice at random. This game is this: Rita tosses the dart once. She wins $1 if the dart lands in slice 1, $2 if the dart lands in slice 2, $5 if the dart lands in slice 3, and $8 if the dart lands in slice 4. She loses $3 if the dart lands in slices 5, 6, 7, or 8. (If necessary, consult a list of formulas.) (a) Find the expected value of playing the game. | dollars (b) What can Rita expect in the long run, after playing the game many times? O Rita can expect to gain money. She can expect to win dollars per toss. Rita can expect to lose money. She can expect to lose dollars per toss. O Rita can expect to break even (neither gain nor lose money).Subgame Perfect Equilibrium. Stephen J. Seagull must decide whether or not to start a new movie project. If he decides not to, he and Clod VandeCamp both get a utility of 10. If he decides to begin the project, both he and Clod must simultaneously decide who the director should be: George Spellbinder, or Ed Tree. If they disagree on the director, the movie isn't made, but both have wasted time, so they get only a utility of 0. If they agree on George Spellbinder, the movie will be a roaring success, and each gets a utility of 20. If they agree on Ed Tree, the movie will be terrible. and they will only get a utility of 5. (a). Draw the extensive form of this game. (b). Write down the normal form. (c). Find all Nash equilibria. (d). Apply the theory of iterated elimination of weakly dominated strategies and state its prediction. (e). Find all the subgame perfect equilibria.Mr. and Mrs. Ward typically vote oppositely in elections and so their votes “cancel each other out.” They each gain 6 units of utility from a vote for their positions (and lose 6 units of utility from a vote against their positions). However, the bother of actually voting costs each 3 units of utility. The following matrix summarizes the strategies for both Mr. Ward and Mrs. Ward. Mrs. Ward Vote Don't Vote Mr. Ward Vote Mr. Ward: -3, Mrs. Ward: -3 Mr. Ward: 3, Mrs. Ward: -6 Don't Vote Mr. Ward: -6, Mrs. Ward: 3 Mr. Ward: 0, Mrs. Ward: 0 The Nash equilibrium for this game is for Mr. Ward to and for Mrs. Ward to . Under this outcome, Mr. Ward receives a payoff of units of utility and Mrs. Ward receives a payoff of units of utility. Suppose Mr. and Mrs. Ward agreed not to vote in tomorrow's election. True or False: This agreement would decrease utility for each spouse, compared to the Nash equilibrium from the previous part of the question. True…
- Taylor, Faith, Jay, and Jessica are college roommates. They're trying to decide where the four of them should go for spring break: Orlando or Las Vegas. If they order the tickets by 11:00 PM on February 1, the cost will be just $500 per person. If they miss that deadline, the cost rises to $1,200 per person. The following table shows the benefit (in dollar terms) that each roommate would get from the two trips. Roommate Taylor Benefit from Orlando Benefit from Las Vegas Faith $1,250 $800 $550 $800 Jay Jessica $650 $600 $850 $1,050 The roommates tend to put off making decisions. So, when February 1 rolls around and they still haven't made a decision, they schedule a vote for 10:00 PM that night. In case of a tie, they will flip a coin between the two vacation destinations. The roommates will get the most total benefit if they choose to go to Given the individual benefits each roommate receives from the two trips, which trip will each roommate vote for? Fill in the table with each…6. Eventually, Aron and Nora who we met in the previous problem were able to get together and have now been an item for some time. So long, in fact, that there are issues. Aron is a little insecure, which makes him jealous. Nora is none too pleased about it. When she arranges a girls' night out with some of her friends, Aron is very upset and threatens to break up. So Nora has to decide to go out with her friends or stay home with Aron. If she stays home, she gets a payoff of 0, whereas Aron gets what he wants and a payoff of 2 (perhaps he should be a little more concerned with Nora's happiness). On the other hand, if Nora decides to go out, then Aron must make a choice between breaking up or not. If he breaks up, both he and Nora are miserable and each receives a payoff of -1. If he does not, he gets a payoff of 1 while Nora gets what she wants and a payoff of 2. a) Draw the extensive form of this game. b) Find all Nash equilibria. Explain your answer carefully. c) Find the backward…Rosencrantz and Guildenstern play a game in which they simultaneously put down some number of coins with either a head or a tail showing on each coin. Rosencrantz puts down one coin and Guildenstern puts down two coins. Rosencrantz pays Guildenstern one dollar for each coin that shows the side that Rosencrantz played; for example, if Rosencrantz played a head and Guildenstern played a head and a tail, Rosencrantz would pay Guildenstern two dollars, since two heads were displayed among the three coins. a. Formulate a strategic game that represents this situation. b. Find all Nash equilibria of this game (including any mixed strategy equilibria). c. For each of the Nash equilibria in (b), give Guildenstern’s expected payoff.
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