players choose different integers, the player with the largest integer gets $100, and the two other players get $0. If two players choose the same integer Q, and the third player chooses a different integer Q', then the player playing Q' gets $0, and the two players who played Q get $50 each. Find 2 Nash equilibriums in this game.
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- GAME UUU B1 Player B B2 A1 7,13 5, 10 A2 3,8 9,16 Player A A3 5,8 4,7 In Game UUU (see table above), assuming players move simultaneously, Player A choosing A1 and Player B choosing B3 is a Nash equilibrium. Player A choosing A3 and Player B choosing B2 is a Nash equilibrium. Both Player A choosing A1 and Player B choosing B1 and Player A choosing A2 and player B choosing B2 are Nash equilibria in pure strategies Player A choosing A1 and Player B choosing B2 is a Nash equilibrium.$11. Anne and Bruce would like to rent a movie, but they can't decide what kind of movie to get: Anne wants to rent a comedy, and Bruce wants to watch a drama. They decide to choose randomly by playing "Evens or Odds." On the count of three, each of them shows one or two fingers. If the sum is even, Anne wins and they rent the comedy; if the sum is odd, Bruce wins and they rent the drama. Each of them earns a payoff of 1 for winning and 0 for losing "Evens or Odds." (a) Draw the game table for "Evens or Odds." (b) Demonstrate that this game has no Nash equilibrium in pure strategies.Consider an extensive game with 3 players. Player 1 moves first and chooses an action, A or B. Player 2 observes this and then chooses an action herself. Her actions are C or D, regardless of the past history. Finally, player 3 observes the actions of players 1 and 2 before making his own choice. His possible actions are E or F if player 1 chose A, regardless of what player 2 chose. If player 1 chose B and player 2 chose C, then player 3's possible actions are G or H. If player 1 chose B and player 2 chose D, then player 3's possible actions are I or J. Draw the game tree (without payoffs). Which of the following is a strategy profile [strategies are discussed in the second recorded lecture]? Check all that apply. (A,D,F) is a strategy profile (A,D,EE) is a strategy profile (B,DD,FI) is a strategy profile None of the above.
- Let there be two players in a game, Player 1 and Player 2. Consider a jar containing 5 snakes. 3 of the snakes in the Kjar are venomous, while the remaining 2 are non-venomous. In the game, both the players have to put their hand in the jar one after the other and pick a snake out. Each snake, if picked out of the jar, will bite the player's hand. The event of picking a venomous snake, or equivalently, a venomous snake's bite will earn the player zero points. On the other hand, the event of picking a non-venomous snake, or equivalently, a non-venomous snake's bite will earn the player one point. Let X denote Player 1's pick and let Y denote Player 2's pick. Suppose Player 1 is the first to pick out a snake. The expected value of Player 1's pick is: E(X)= (Express your answer as a fraction or round your answer to two decimal places)Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.Consider the following sequential game: Player 1 plays first and can choose between L and R. Player 2 plays second and can choose between U and D. • If player 1 chooses L, player 2 does not get to play and both players get £1. If player 1 plays R, player 2 gets to play. If player 2 plays U then both players get £2. If player 2 plays D then player 2 gets £3 and player 1 gets £0. What the the outcome of the subgame perfect equilibrium? O a. £3-£0 O b. £1-£1 O c. This game does not have a subgame-perfect Nash-equilibrium. O d. £0-£3 O e. £2-£2
- The Stag Hunt Game - A group of 10 hunters are hunting, where the games include a stag and several (>10) hares. Each hunter can choose whether to hunt the stag or go after a hare. It takes the joint effort of all 10 players to catch the stag, but each hunter on their own is able to catch a hare. Suppose that a stag is worth $100 and a hare is worth $9. Suppose also that when all 10 players together catch the stag, they each get $10 after the game is sold. (a) Formulate this situation as a strategic game and find the Nash equilibria. (b) Now suppose that there are still 10 hunters, who need to choose to either hunt the stag or catch a hare. However, suppose that only 6 hunters are needed to catch a stag. Assume further that after the stag is caught, it can still be sold for $100 and the money shall be divided evenly among the hunters who have chosen to join the stag hunt only. On the other hand, each hunter can always catch a hare on their own and sell it for $9. Formulate this…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?Suppose Carlos and Deborah 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 Carlos chooses Right and Deborah chooses Right, Carlos will receive a payoff of 7 and Deborah will receive a payoff of 6. The only dominant strategy in this game is for ____ to choose ____ . The outcome reflecting the unique Nash equilibrium in this game is as follows: Carlos chooses ____ and Deborah chooses ____ .
- In a gambling game, Player A and Player B both have a $1 and a $5 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. b. Is there a pure strategy? Why or why not? Determine the optimal strategies and the value of this game. Does the game favor one player over the other? d. Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy. с.Question 5 ave Answer Player 1 has two options: L and R. If Player 1 chooses L, then Player 2 has two options; she can choose A or B. If Player 2 chooses A payoffs are (2; 1) and if she chooses B payoffs are (1; 2). If Player 1 chooses R, then Player 2 has two options; she can choose Y or N. If Player 2 shows Y payoffs are (0; 3) and if she chooses N payoffs are (1; 2). Which of the following statements is true? O A. (L; B,Y ) is a subgame perfect equilibrium of this game O B. (R; B,N ) is a subgame perfect equilibrium of this game O C.(L; A,Y) is a subgame perfect equilibrium of this game O D. (R; B,Y ) is a subgame perfect equilibrium of this gameGame Theory Question A non-profit firm is on a local community online donation platform for a community event it wants to hold (only community members can donate via the website). The event will be held only if the non-profit firm collects $20,000 total from members of the community. Each member values the event at $500. Suppose that there are 100 community members. Community members can only donate by purchasing a lottery ticket from the firm. Each ticket costs $200 and only one ticket can be purchased per member. The proceeds will be collected by the firm. The lottery winner gets a premier meal at a local restaurant that's worth $100. Remember, the firm keeps all the donated money. If the amount of donations is less than $20,000, then the firm returns the donated money to the community members (since there'll be no event held but the lottery winner still gets to eat that fancy meal). If the donations sum up to $20,000, the community event will take place. What are the Nash…