Table 1.3: Erica and Fred's game. Fred Testify Remain silent Testify (-14, –14) (0, – 15) Erica Remain silent (-15, 0) (-1,–1) What is the best global outcome, and how many years total prison time does it lead to? Both confess, 14 years in prison Both confess, 28 years in prison Both silent, 15 years in prison Both silent, 2 years in prison
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- 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.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?
- Splitting Pizza: You and a friend are in an Italian restaurant, and the owner offers both of you a free eight-slice pizza under the following condition. Each of you must simultaneously announce how many slices you would like; that is, each player i ∈ 1, 2 names his desired amount of pizza, 0 ≤ si ≤ 8. If s1 + s2 ≤ 8 then the players get their demands (and the owner eats any leftover slices). If s1 + s2 > 8, then the players get nothing. Assume that you each care only about how much pizza you individually consume, and the more the better.What outcomes can be supported as pure-strategy Nash equilibria?Consider the following Bayesian game. There are two players 1 and 2. Both players choose whether to play A or B. Two states are possible, L and R. In the former, players play a stag-hunt game, and in the latter, players play a matching pennies game. Suppose that Player 2 knows the state, while Player 1 thinks that the state is L with probability q and R with probability 1 ! q. Payo§s in each state respectively satisfy: Player 1 is the row player, and their payo§ is the first to appear in each entry. Player 2 is thecolumn player and their payo§ is the second to appear in each entry. (a) What is the set of possible strategies for the two players in this game? (b) Find all the pure strategy Bayes Nash equilibria for any value of q 2 (0, 1).Imagine that two firms in two different countries want to bring a new product to market. Due to economies of scale, if both firms do this, they will both lose £50 million. But if only one firm does this, it will gain £300 million. (a) What is the best strategy for firm A, if firm B has not yet entered the market, and why? (b) Illustrate this with a game theory diagram, showing appropriate payouts. (c) What is the welfare-maximising strategy for a government, and why?
- John and Paul are walking in the woods one day when suddenly an angry bear emerges from the underbrush. They each can do one of two things: run away or stand and fight. If one of them runs away and the other fights, then the one who ran will get away unharmed (payoff of 0) while the one who fights will be killed (payoff -200). If they both run, then the bear will chase down one of them and eat them to death but the other one will get away unharmed. Assuming they don't know which one will escape we will call this a payoff of -100 for both. If they BOTH fight, then they will successfully drive off the bear but they may be injured in the process (payoff -20). Construct a payoff matrix for this game and identify the pure strategy Nash equilibrium. (Indicate it with words not with a circle!)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…12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s <w. If an army attacks but its rival does not, no costs are borne by either side. Identify all 12.7 Exercises • 267 the pure-strategy Bayesian Nash equilibria of this game for the following two cases, and briefly describe…
- Keith and Blake play a simultaneous one-shot game. Keith chooses between Top and Bottom, whereas Blake chooses between left and right. Payoffs are given by (Keith's payoffs are listed first and Blake's payoffs second): Top+left: 3,-6. Top+right:0,-7 Bottom+left:4,0. Bottom+right:-2,6. In a mixed strategy equilibria, Keith will play Top with what probability? 1. 6/7 2. 1/4 3. 1/2 4. 1/7 5. 0 6. 3/4UNIT 9 CHAPTER 5 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. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. 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? 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.Imagine that two firms in two different countries want to bring a new product tomarket. Due to economies of scale, if both firms do this, they will both lose £50million. But if only one firm does this, it will gain £300 million.(a) What is the best strategy for firm A, if firm B has not yet entered the market, andwhy?(b) Illustrate this with a game theory diagram, showing appropriate payouts.(c) What is the welfare-maximising strategy for a government, and why?
