Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $1 and Player B gets $2. If Player A plays UP and Player B plays RIGHT then Player A gets $4 and Player B gets $3. If Player A plays DOWN and Player B plays LEFT then Player A gets $2 and Player B gets $4. If Player A plays DOWN and Player B plays RIGHT then Player A gets $3 and Player B gets $1. What is the probability that Player A will play UP and Player B will play LEFT?
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- Consider the following game of ’divide the dollar.’ There is a dollar to be split between two players. Player 1 can make any offer to player 2 in increments of 25 cents; that is, player 1 can make offers of 0 cents, 25 cents, 50 cents, 75 cents, and $1. An offer is the amount of the original dollar that player 1 would like player 2 to have. After player 2 gets an offer, she has the option of either accepting or rejecting the offer. If she accepts, she gets the offered amount and player 1 keeps the remainder. If she rejects, neither player gets anything. Draw the game tree of the modified version of the ’divide the dollar’ game in which player 2 can make a counteroffer if she does not accept player 1’s offer. After player 2 makes her counteroffer - if she does - player 1 can accept or reject the counteroffer. As before, if there is no agreement after the two rounds of offers, neither player gets anything. If there is an agreement in either round then each player gets the amount agreed…The following table contains the possible actions and payoffs of players 1 and 2. Player 2 Cooperate Not Cooperate Player Cooperate 15 , 15 -20 , 20 1 Not Cooperate 20 , -10 10 , 10 This game is infinitely repeated, and in each period both players must choose their actions simultaneously. If both players follow a tit-for-tat strategy, then they can Cooperate in equilibrium if the interest rate r is . At an interest rate of r=0.5, . If instead of playing an infinite number of times, the players play the game only 10 times, then in the first period player 1 receives a payoff ofConsider the following game. There are two payers, Player 1 and Player 2. Player 1 chooses a row (10, 20, or 30), and Player 2 chooses a column (10/20/30). Payoffs are in the cells of the table, with those on the left going to Player 1 and those on the right going to player 2. Suppose that Player 1 chooses his strategy (10, 20 or 30), first, and subsequently, and after observing Player 1’s choice, Player 2 chooses his own strategy (of 10, 20 or 30). Which of the following statements is true regarding this modified game? I. It is a simultaneous move game, because the timing of moves is irrelevant in classifying games.II. It is a sequential move game, because Player 2 observes Player 1’s choice before he chooses his own strategy.III. This modification gives Player 1 a ‘first mover advantage’. A) I and IIB) II and IIIC) I and IIID) I onlyE) II only
- 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…Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $2 and Player B gets $4. If Player A plays UP and Player B plays RIGHT then Player A gets $3 and Player B gets $6. If Player A plays DOWN and Player B plays LEFT then Player A gets $5 and Player B gets $2. If Player A plays DOWN and Player B plays RIGHT then Player A gets $1 and Player B gets $1. What is the mixed strategy expected payout for Player B? 1 40/15 39/15 11/2Try the following variant of the Let’s Make a Deal game. Again one of the three boxes contains a prize, but now there are two players, 1 and 2. Assume Player 1 picks Box A and Player 2 picks Box B. The host (who again has perfect knowledge) opens Box B, which contains junk, and Player 2 leaves the show. Player 1 can either stay with Box A or switch to Box C. Using Bayes’ theorem, show that now it does not pay Player 1 to switch, that is, the probability Player 1 will win with Box C is 1/2, the same as the probability Player 1 will win by staying with Box A.
- Consider a game with two players A and B and two strategies X and Z. If both players play strategy X, A will earn $300 and B will earn $700. If both players play strategy Z, A will earn $1,000 and B will earn $600. If Player A plays strategy X and player B plays strategy Z, A will earn $200 and B will earn $300. If Player A plays strategy Z and player B plays strategy X, A will earn $500 and B will earn $400. Player B finds that: a) strategy Z is a dominant strategy. b) strategy X is a dominant strategy. c) he has no dominant strategy. d) strategy X is a dominated strategy. e) strategy Z is a dominated strategy.Consider the following dynamic game. There are two players (P1, P2). Player 2 tries to rob Player 1. If Player 1 pays $100, the game is over with Player 1 (victim) paying $100 to Player 2 (robber) (P1: -$100, P2: +$100). If Player 1 refuses to pay $100, then Player 2 has two choices: one is to hurt Player 1 (P1: -$5,000, P2: -$1,000) and the other is to walk away (P1: 0, P2: 0). Explain how to find an equilibrium in this game.Suppose you are playing a game in which you and one other person each picks a number between 1 and 100, with the person closest to some randomly selected number between 1 and 100 winning the jackpot. (Ask your instructor to fund the jackpot.) Your opponent picks first. What number do you expect her to choose? Why? What number would you then pick? Why are the two numbers so close? How might this example relate to why Home Depot and Lowes, Walgreens and Rite-Aid, McDonald’s and Burger King, and other major pairs of rivals locate so close to each other in many well-defined geographical markets that are large enough for both firms to be profitable?
- We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does this game have a Nash Equilibrium? (If not, why not? If yes, what is the Nash Equilibrium?)We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does Jerry have any strictly dominant strategy? Does Elaine?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.