(d) Suppose we now make this game sequential so that Player 2 moves first. Player 1 observes Player 2's action and moves second. What are the (pure strategy) Nash equilibria and subgame perfect equilibria of this new game? (e) Give one example of how the players or a third party could prevent a suboptimal outcome from occurring in this game.
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- Question 4. Zeynep and Mehmet will eventually play the following game. Mehmet L R U 3,1 0,0 Zeynep D 0,3 1,3 In a preliminary stage, Zeynep has already asked Mehmet to allow her to move first and proposed to pay him 1 unit of her own payoff in exchange. So Mehmet has to options: • If he accepts Zeynep's offer: they will play the sequential move version of the above game in which Zeynep moves first. Mehmet will receive 1 unit of utility more, and Zeynep will receive 1 unit of utility less (in any outcome of the game) compared to the payoffs given in the bimatrix. 2 • If he rejects Zeynep's offer: they will play the simultaneous move game. a. Represent this strategic interaction in a game tree. b. How many information sets does each player have? c. Characterize the set of pure strategies for both players. d. Present this game as a normal-form game, and characterize the set of pure strategy Nash equi- libria.2. Consider the following game. Two criminals are thinking about pulling off a bank robbery. The take from the bank would be $20,000 each, but the job requires two people (one to rob the bank and one to drive the getaway car. Each criminal could instead rob a liquor store. The take from robing a liquor store is only $1000 but can be done with one person acting alone. Write the payoff matrix of this game Player A Bank job Liqour store Player B Bank job Liquor store a. What are the Nash equilibria in this game? b. Explain why there can be multiple equilibria in this game. c. How the game will be played if mixed strategies are allowed? Discuss.4. Consider a two player game with Fred and Barney, who tal turns removing matchsticks from a pile. They start with 33 matchsticks, and Fred goes first. On each turn, ecach player may remove either one, two, three, four, or five matchsticks. The player to remove the last matchstick wins the game. What are the optimal strategies for each player? Who will win? b. Suppose now that they can remove up to six matchsticks, how will the optimal strategies change for- each player? a.
- The next 3 questions involve the following game. There are two players, a husband and wife. They can either be selfish (S) or selfless (U) in their marriage. If they choose to be selfish, then there is a negative ʻguilt’ payoff of g. The payoff matrix is below. Figure 1: The Marriage Game Wife S U S 10-g, 10-g 15-g, 2 U 2, 15-g 12, 12 Number left (right) of comma refers to H's (W's) payoff. 23. Suppose that g = 0. What is the Nash equilibrium (or equilibria)? (A) (S, S). (B) (U, S) and (S, U). (C) (S, S) and (U, U) (D) (U, U). 24. Suppose that g= 5. What is the Nash equilibrium (or equilibria)? (A) (S, S). (B) (U, S) and (S, U). (C) (S, S) and (U, U) (D) (U, U). 25. Suppose that g = 10. What is the Nash equilibrium (or equilibria)? (A) (S, S). (B) (U, S) and (S, U). (C) (S, S) and (U, U) (D) (U, U). Husbanda. The president of a Japanese electronics company, Takashi Hashiyama, famously chose the auction housefor his company’s art collection by having representatives from two auction houses, Christie’s and Sotheby’s,play a round of rock-paper-scissors. At stake were millions of dollars in commissions. Show the payoff matrixfor a game where the payoff is 1 if you win, 0 if you tie, and -1 if you lose. I’m sure you know this already, butjust in case: rock breaks scissors, scissors cut paper, and paper smothers rock. Here’s an article about the actualgame if you’re interested. b. If you were in a high stakes game…2 Consider Anna and Joe, who value a certain good by the same amount v and can choose to either contribute (C) e to get the good or not (N) where v > e> 0. Obtaining the good only requires c from one person. The game is summarized in the payoff table below: Joe V-c,V-c V-c,V v,V-c 0,0 Find a pure strategy equilibrium and a mixed strategy equilibrium. Anna O Z
- . 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? c. 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.5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:A writes down either number 1 or number 2, and B must guess which one.If the number that A has written down is i and B has guessed correctly, B receives i units from A.If B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.Suppose B guesses 1 with probability p and 2 with probability 1 −p.Let X1 denote B’s gain (or loss) in a game where A has written down 1.Let X2 denote B’s gain (or loss) in a game where A has written down 2.(a) Find the pmf of X1 and X2(b) Find B’s expected gain for these two cases, E[X1] and E[X2].(c) What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player ASuppose that A writes down 1 with probability q and 2 with probability 1 −q.Let Y1 be A’s loss (or gain) if B chooses number 1.Let Y2 be A’s loss (or gain) if B…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…
- Consider the following price game: Firm 1 Firm 2 High Low High 20, 20 12, 24 Low 24, 12 14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?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-£22. Consider a game that game theory people refer to as the “ultimatum game.”We will refer to our two players as the “offerer” and the “decider”. How the gameworks is that the offerer proposes a way to split $1000 between the two players.While this could be done in a variety of ways, we will assume that the offerersonly has two possible proposals: Either a 50-50 split, or she offers the decider$50 and keeps the rest. The decider can either accept or reject the offer. If the offer is accepted, the money is split as proposed. If the offer is rejected, themoney spontaneously combusts and nobody gets anything. a) List the strategies for each player and write an extensive form version of thegame with payouts. b) List all the Nash equilibria of this game. c) Explain which, if any of the Nash equilibrium are not sub-game perfect. d) Write the game out in normal form and find the pure strategy Nashequilibrium. Explain how this matches with your answers to (b) and (c) . Alsoexplain why there…