Consider the following two-player game: In the first round, Kamal picks an integer x between 1 and 10. In the next round, Sana picks another integer y such that x < y < x + 11. Then, Kamal picks integer z such that y < z < y + 11, etc. In short: Kamal and Sana must add to the previous number at least one but no more than 10. The first player who picks 100 or more is the
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- Suppose Edison and Hilary 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 Edison chooses Right and Hilary chooses Right, Edison will receive a payoff of 3 and Hilary will receive a payoff of 7. Hilary Left Right Left 4, 6 6, 8 Edison Right 7, 5 3, 7 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Edison chooses and Hilary choosesSuppose Antonio and Trinity 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 Antonio chooses Up and Trinity chooses Down, Antonio will receive a payoff of 7 and Trinity will receive a payoff of 5. Trinity Up Down Up 4,8 7,5 Antonio Down 3,2 5,6 In this game, the only dominant strategy is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Antonio chooses, and Trinity chooses Grade It Now Save & Continue Continue without saving @ 2 F2 #3 80 Q F3 MacBook Air 44 F7 Dll F8 44 F10 74 $ 4 05 Λ & % 5 6 7 8 * 0 Q W E R T Y U 1 A N S X 9 0 -O O D F G H J K L on را H command C > B N M Λ - - P [ H Λ command optiSuppose 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 6 and Deborah will receive a payoff of 5. Deborah Left Right Carlos Left 8, 4 4, 5 Right 5, 4 6, 5 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 .
- $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.Suppose Yakov and Ana 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 Yakov chooses Right and Ana chooses Right, Yakov will receive a payoff of 6 and Ana will receive a payoff of 6. Yakov Left Left 2,3 Right 3,7 Ana Right 4,4 6,6 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Yakov chooses and Ana choosesLee and Cody are playing a game in which Lee has the first move at A in the accompanying decision tree: Once Lee has chosen either aggression or cooperation, Cody, who can see what Lee has chosen, must choose either aggression or cooperation at B or C. Both players know the payoffs at the end of each branch. Lee chooses A Aggression Cooperation Cody chooses 8 Multiple Choice C Cody chooses Aggression aggression: aggression Cooperation Aggression In the equilibrium of this game, Lee chooses Cooperation aggression; cooperation -10 for Lee -10 for Cody 40 for Lee O for Cody O for Leo 40 for Cody 25 for Lee 25 for Cody and then Cody chooses.
- Suppose Charles and Dina 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 Charles chooses Right and Dina chooses Right, Charles will receive a payoff of 5 and Dina will receive a payoff of 3. Dina Left Right Left 4, 4 6, 3 Charles Right 5, 4 5, 3 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Charles chooses and Dina choosesConsider 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.6. Two players each pick a positive integer between 1 and 100. If the numbers are identical, noone wins. If the numbers differ by 1, the one with the lower number pays 1 to the opponent. If the difference is at least 2, the one with the higher number pays 2 to the opponent. What is the NE? Hint: Draw a sketch of the game matrix (while you can't consider all 100 strategies, at least consider the lowest 5-6 numbers) and think about dominated strategies.
- Game 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…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.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.