(b) Suppose there are two roommates John and Bill; each has a wealth of $1000. They are trying to decide whether or not to buy a TV that costs $800, when each of them values it at $600. They agree on the following procedure to decide whether or not to buy the TV. Each person will write on a piece of paper whether or not he thinks the TV should be purchased. If both say yes, they buy the TV and split the cost evenly. If both say no, they don't buy the TV. If one says yes, and the other says no, the person who says yes is obligated to buy the TV on his own. In this context, answer the following: (i) Is it Pareto efficient for the two roommates to buy the TV? and What is the dominant strategy of each player in this game? (ii)

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…

Consider 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…

Table 22-21 Three longtime friends-Linda, Stella, and Lydia are deciding how they will spend their Sunday afternoon. They all agree that they should do one of the three things: go to a comedy club, play soccer, or go to a tennis tournament. They also agree that they will have two pairwise votes to determine how to spend their afteroon, with the majority determining the outcome on each vote. The first, second, and third choices for each person are as indicated in the following table. First choice Second choice Third choice Linda tennis tournament soccer comedy club Stella soccer comedy club tennis tournament Lydia comedy club tennis tournament soccer Refer to Table 23-2. If (1) the first vote pits "tennis tournament" against "comedy club," and (2) the second vote pits "soccer against the winner of the first vote, then the outcome is as follows: ⒸA "Tennis tournament" wins the first vote and "tennis tournament" wins the second vote, so they go to a tennis tournament OB. "Tennis…

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).

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…

Consider the following game played by four individuals, players 1, 2, 3, and 4. Each individual has $10,000. Each player can donate between $0 and $10,000 to build a public park that costs $20,000. If they collect enough money, they construct the park, which is worth $9,000 to each of them. However, if they collect less than $20,000, they cannot build a park. Furthermore, regardless of whether the park is built or not, individuals lose any donations that they make. a) Describe the Nash equilibria for a simultaneous game. What makes them equilibria? Hint: There are many equilibria, so you may want to use a mathematical expression! b) Suppose that players 1, 2, and 3, each donate $4,000 for the park. How much will player 4 donate and why. What are the resulting payoffs for the players? c) Suppose instead that player 1 donated first, player 2 second, player 3 third, and player 4 last. Furthermore, players could only donate in intervals of 1,000 (0, $1,000, $2,000, etc.). How much will…

4 In a gambling game, Player A and Player B both have a $5 and a $10 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. (a) Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. Player B Player A $5 $10 $5 (b) Is there a pure strategy? Why or why not? ---Select--- . Since the maximum of the row minimums is --Select--- (c) Determine the optimal strategies and the value of this game. probability Player A selects $5 probability Player A selects $10 probability Player B selects $5 probability Player B selects $10 = = $10 = = Does the game favor one player over the other? Yes O No and the minimum of the column maximums is (d) Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the…

Consider the following sequential-move game: This game involves three players, each making sequential decisions. The game proceeds as follows: • Player 1 initiates the game by choosing between two actions: Left and Right. • Depending on Player 1's choice, Player 2 then decides, choosing between two actions, left (I) and right (`r`). • Finally, Player 3 makes the last move in the sequence, choosing between actions `a` and `b`. The payoffs are determined by the sequence of choices made by all three players. Each payoff is represented by the triplet (x,y,z), where x, y, and z denote the payoffs for Player 1, Player 2, and Player 3, respectively. For example, if Player 1 chooses 'Left', Player 2 chooses 'I', and Player 3 chooses `a` the resulting payoff would be (3,1,2). This indicates that Player 1 receives a payoff of $3, Player 2 receives a payoff of $1, and Player 3 receives a payoff of $2 for this particular sequence of actions. Left Player 2 Player 1 Right Player 2 Player 3 Player 3…