Player Rhas a $2, a $5, and a $10 bill. Player Chas a $1, a $5, and a $10 bill. Each player selects and shows (simultaneously) one of his or her three bills. If the total value of the two bills shown is even, Rwins Cs bill; if the value is odd, C wins Rs bil (Which player would you rather be?) ) Set up the payoff matrix for the game. ) Solve the game using the simplex method discussed in this section. (Remove any recessive rows and columns, if present,
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- (a) Stan and Ollie are two students who share a flat. Both of them prefer to live in a clean flat. However, neither is too fond of housecleaning. Each of them receives a payoff of 12 if they both clean the flat. If neither person cleans the flat, they receive a payoff of 6 each. If one person cleans the flat but the other person does not, then the payoff for the person who does the cleaning is 5 and the payoff for the person who doesn't do any cleaning is 15. (i) Write down the payoff matrix of this game. Derive the dominant strategy equilibrium. Is this also a Nash equilibrium? (ii) Expiain your reasoning. Consider a game with N players. Each player chooses Black or White. If a player (b) chooses Black, she gets 100 if everyone else also chooses Black, and she gets 0 if any of the other players does not choose Black. If a player chooses White, she always gets 50. Show that everyone choosing Black and everyone choosing White are both Nash equilibria of this game.(a) Consider a ROCK PAPER SCISSOR game. Two players indicate either Rock, Paper or Scissor simultaneously. The winner is determined by: Rock crushes Scissors, Paper covers Rock, and Scissor cut Paper. In the case of a tie, there is no payoff. In the case of a win, the winner collects 5 dollars. Write the payoff matrix for this game. (b) Find the optimal row and column strategies and the value of the matrix game. 3 2 4 -2 1 -4 5Consider the payoff matrix listed below: IS |1, -1 3, 0 |2, 1 0, 3 1, 2 |0, 0 3, 1 5, 3 2, 1 What is the Nash Equilibium of the game? a. (B, R) b. (U, Q) c. (B, S) d. (C, Q) M/2/6
- Player 1 Cooperate (C) Defect (D) If the game has a dominant strategy, what is it? There is none. If the game has a Nash equilibrium in pure strategies, what is it? There is none. Cooperate (C) 3,3 8,0 Cooperate (C) is a dominant strategy for both players. Defect (D) is a dominant strategy for both players. Cooperate (C) is a dominant strategy for 1, and Defect (D) is a dominant strategy for 2. C, C is the only Nash equilibrium. D, D is the only Nash equilibrium. C, C and D, D are both Nash equilibria. Player 2 Defect (D) 0,8 1,1Suppose two players, First and Second, take part in a sequential-move game. First moves first, Second moves second, and each player moves only once. (a). Draw a game tree for a game in which First has two possible actions (Up or Down), and Second has three possible actions (Top, Middle, or Bottom) at each node. Show which nodes are terminal/decision and write down all the (pure) strategies of each player. (b). Draw a game tree for a game in which First and Second each have three possible actions (Sit, Stand, or Jump) at each node. Show which nodes are terminal/decision and write down all the (pure) strategies of each player.Please no written by hand and no emage Two classmates A and B are assigned a group project. Each student can choose to Shirk or Work. If one or more students chooses Work, the project is completed and provides each with credit valued at 4 payoff units each. The cost of completing the project is that 6 total units of effort (measured in payoff units) is divided equally among all players who choose to Work and this is subtracted from their payoff. If both Shirk, they do not have to expend any effort but the project is not completed, giving each a payoff of 0. The instructor can only tell whether the project is completed and cannot determine which students contributed to the project. A. Write down (draw) the normal form game, with payoff in the format of (player A payoff, player B payoff). Note that we assume students choose to Shirk or Work simultaneously. B. Does either player have a dominant strategy? C. Find the Nash equilibrium or equilibria. D. Explain ‘Prisoners’ dilemma’ using the…
- Two politicians from the same party decide whether to support or oppose the gov- ernment's environmental policy. Politician A prefers that they both support, but Politician B prefers that they both oppose. Both politicians agree that the party should have a coherent position and the worst outcome is that one supports and the other opposes the policy. (a) Formulate this situation as a strategic game - specify the players, actions, and payoffs (you can set-up a table of the actions and payoffs). (b) What are the Nash equilibria of this game? Explain your answer.1. Draw tree of the game clearly, handwritten is preferable.A cool kid is willing to rename himself for a profit. He decides to auctionoff the naming right. Two bidders show interest. Their valuations for thenaming right are independently and uniformly distributed over [0,100].There are several possible ideas to design the auction.(a) The auction runs as follows. Both bidders are invited to the sameroom; an auctioneer will start the auction with an initial price 0, and increase it by $1 every minute. The bidders are not allowed to say anything during the process, but they can walk out of the room at any moment. If one bidder walks out of the room when the price increases to p (the bidder does not need to pay), the remaining bidder will be awarded the naming right for a price of p. If both walk out when the price reaches p, the naming right is not assigned and the two bidders do not need to pay. What should the bidders do? Explain your answer. (b) Both bidders are invited to submit their bids covertly (bids are non-negative real…
- Roger and Michelle like going out and enjoy each others’ company, but have different taste in entertainment. Michelle would like to go to a professional basketball game, while Roger prefers opera. If they fail to agree to go together to either the game or to the opera this Saturday night they will stay at home and be miserable. Both will be worse off than if they had gone together to either the game or the opera. (a) Construct a pay-off matrix for this game. (You can make up your own numbers for this problem; just make sure the numbers correctly depict the given situation.) (b) Find two Nash equilibria for this game. Explain why they are indeed Nash equilibria.Player 1 Cooperate (C) Defect (D) O Cooperate (C) 3,3 8,0 0 Player 2 Suppose the game is repeated infinitely many times, and 2 plays a grim trigger strategy (cooperates as long as 1 cooperates, but defects forever after 1 defects). What is the value of 1's future payoffs if 1 defects today and then best-responds to 2? Assume that the discount factor is still 0.5 (i.e. payoffs in the next period are worth half as much as payoffs today). 01 02 8 O 9 Defect (D) 0,8 1,1 If the game is repeated infinitely many times, what is the smallest discount factor that would support cooperation in Nash equilibrium? O 1/2 O 5/7 3/8A and B are a couple and they want to go into a concert. A comes from Bonn and loves the music of Ludwig van Beethoven. B comes from Salzburg and loves the music of Wolfgang Amadeus Mozart. There are two concerts in town: one with the music of Beethoven and one with the music of Mozart. Both (A and B) prefer to go to a concert together. If both go to a concert of Beethoven, A has a pay-off of 4 and B has a pay-off of 2. If both go to a concert of Mozart, B has a pay-off of 4 and A has a pay-off of 2. If A goes to a concert of Mozart and B to a concert of Beethoven, they are both miserable and get a pay-off of 0. But, if B goes to a concert of Mozart and A to a concert of Beethoven, they are both a little better off with a pay-off of 1. Both A and B have to make a simultaneous decision and cannot communicate prior to the decision. (a) Construct the pay-off matrix for this game. (b) Identify the Nash equilibrium or equilibria of the game. (c) Which off the allocations are…