Determine whether each of the following statements is true or false and justify your answer. (1) The set of rationalizable action profiles of a strategic game must contain all of its Nash equilibria. (2) In a first-price auction, bidding one’s valuation weakly dominates bidding any higher real number.
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Determine whether each of the following statements is true or false and justify your answer.
(1) The set of rationalizable action profiles of a strategic game must contain all of its
Nash equilibria.
(2) In a first-price auction, bidding one’s valuation weakly dominates bidding any higher
real number.
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- Consider a 4-bidder auction model. The auction is second price sealed bid. However, now, the bidders know each-others valuation. Assume that v1 > v2 > v3 > v4. Find a Nash Equilibrium of the 2nd price sealed-bid auction where bidder 4 obtains the object.Consider the two period Repeated Prisoner's Dilemma Game where each player is interested in the SUM of the payoffs she gets in each period. Players see the outcome after the play in each period. (The period payoffs are 10,5,1,0.) (i) Write out this game in its strategic form. (ii) Find all Nash equilibria and all Subgame Perfect Nash Equilibria.Consider the following Guessing Game. There are n = 10 players simultaneously choosing a number in {1, 2, 3}. The winners are those closest to 1/2 the average guess (they evenly split the prize between the winners if there is more than one). Find the set of rationalizable strategy profiles. Justify your answer. please no handwriting and this course about game theory (topic Rationalizability) answer with all steps, please
- Consider a Common Value auction with two bidders who both receive a signal X that is uniformly distributed between 0 and 1. The (common) value V of the good the players are bidding for is the average of the two signals, i.e. V = (X1+X2)/2. the symmetric Nash equilibrium bidding strategy for the second-price sealed-bid auction assuming that players are risk-neutral and have standard selfish preferences. Furthermore, you may assume that the other bidder is following a linear bidding strategy. Make sure to explain your notation and the steps you take to derive the result.First-Price Auction with Two Bidders - Consider a first-price sealed-bid auction = with only two bidders, 1 and 2. Assume that V₁ = V2 : 10 and player 1 is chosen as the winner whenever b₁ = b₂. Find all the Nash equilibria of this game and justify your answer. (Note that the game here does not fit the framework we introduced in lecture, because the two players have the same valuation. However, the rationale behind the necessary conditions for an action profile to be a Nash equilibrium still applies.)Suppose that you are a manager. You are considering whether or not to monitor employees with the payoffs in the normal-form accompanying game. Worker Work Shirk Manager Monitor -1,1 1,-1 Don't Monitor 1,-1 -1,1 Which of the following pairs of strategies constitutes a Nash equilibrium? Multiple Choice Manager monitors and worker shirks. Manager does not monitor and worker works. Manager monitors and worker works. O None of the answers is correct.
- Consider the game in Table 1 and answer the following questions. (1) Write down each player's best response functions and find all the Nash equilibria. (2) For each action of each player, determine whether it is strictly, weakly, or not domi- nated by any other actions. Justify your answer. (3) Find the set of rationalizable action profiles by iteratedly eliminate strictly dominated actions. Clearly show the steps. TABLE 1. A Strategic Game L CR 2,2 1,2 3,3 2,3 1,2 0,3 0,3 1,2 2,1 2,2 M BTen commuters must decide simultaneously in the morning to use route A or route B to go from home (same place for all) to work (ditto). If a of them use route A, each of them will travel for 10a + 40 minutes; if b of them use route B, each of them will travel for 10b minutes. Everyone wishes to minimize his/her commuting time. Your tasks: Describe the pure Nash equilibrium (or Nash equilibria) of this ten-person game. Compute the corresponding profile of commuting times. Explicitly list all equilibrium conditions that are satisfied. à. b. What is the traffic pattern (strategies) minimizing the total travel time of all commuters (the sum of their travel times)? Describe the corresponding profile of commuting times (individual payoffs/cost). What does this mean about the Price of Anarchy of this game (assuming that the objective function is the total travel time)? с.Suppose two bidders compete for a single indivisible item (e.g., a used car, a piece of art, etc.). We assume that bidder 1 values the item at $v1, and bidder 2 values the item at $v2. We assume that v1 > v2. In this problem we study a second price auction, which proceeds as follows. Each player i = 1, 2 simultaneously chooses a bid bi ≥ 0. The higher of the two bidders wins, and pays the second highest bid (in this case, the other player’s bid). In case of a tie, suppose the item goes to bidder 1. If a bidder does not win, their payoff is zero; if the bidder wins, their payoff is their value minus the second highest bid. a) Now suppose that player 1 bids b1 = v2 and player 2 bids b2 = v1, i.e., they both bid the value of the other player. (Note that in this case, player 2 is bidding above their value!) Show that this is a pure NE of the second price auction. (Note that in this pure NE the player with the lower value wins, while in the weak dominant strategy equilibrium where both…
- Consider the game with the payoffs below. Which of the possible outcomes are MORE efficient than the Nash Equilibrium (NE)? Note, they do NOT need to be Nash equilibria themselves, they just need to be more efficient than the NE. Multiple answers are possible, but not necessary. You need to check ALL correct answers for full credit. JILL High Medium LowMAGGIE Left 3,4 2,3 2,2Center 4,8 9,7 8,7Right 7,6 8,5 9,4Group of answer choices (Left, Low) There is no strategy combination that is more efficient than the Nash equilibrium for this game. (Right, Medium) (Left, High) (Center, Medium) (Center, High) (Center, Low) (Left, Medium) (Right, Low) (Right, High)Suppose t = 1 and c = 3/5, what is the payoff matrix for the game and state all Nash equilibria in pure strategies. Also find the unique Nash equilibirum in mixed strategies.Consider a Common Value auction with two bidders who both receive a signal X that is uniformly distributed between 0 and 1. The (common) value V of the good the players are bidding for is the average of the two signals, i.e. V = (X1+X2)/2. Compute the symmetric Nash equilibrium bidding strategy for the second-price sealed-bid auction assuming that players are risk-neutral and have standard selfish preferences. Furthermore, you may assume that the other bidder is following a linear bidding strategy. Make sure to explain your notation and the steps you take to derive the result.