1. Consider the game with payoff matrix (0,0) (0, 2) (2, 0) (3, ) and let (–1, –1) be the NTU disagreement point. Find the corresponding agreement point based on the Nash bargaining that satisfies the Nash's bargaining axioms.
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- Consider a first-price sealed bid auction of a single object with two bidders j = 1,2 and no reserved price. Bidder 1's valuation is v1 = 2. and bidder 2's valuation is v2= 5. Both v1 and v2 are known to both bidders. Bids must be in whole dollar amounts. In the event of a tie, the object is awarded by a flip of a fair coin. Is there a Nash equilibrium? What is it? Is it unique? Is it efficient?A seller would like to sell a painting using second price (Vickrey) auction. The seller knows that there are three buyers and each buyer is equally likely to have High (H), Medium (M), or Low (L) valuation for the painting. The valuations of the buyers are independently distributed. Assuming that the buyers will play the Nash equilibrium in the second price auction, how much revenue will the seller make?Exercice 1: In the following table i) by describing precisely your reasoning, find the balance(s) of the game ii) Justify that the equilibrium(s) found is/are nash equilibria
- 1. Two risk neutral players compete to win a prize, P. The probability of winning depends on effort; the probability that player i wins equals e(e++e2), where ei is the effort level of player i. The cost of effort is ciei. a) Find the Nash equilibrium effort levels. b) What is the probability player 1 wins at equilibrium?2. Two friends are playing poker with one another and need to decide on how to play their next tum. The payoffs in this game are as shown in the following payoff matrix: Player 2 Fold All-in Fold (-20,-30) (-20,100) Player 1 All-in (120,-30) (-80,-150) Determine the mixed-strategy Nash Equilibrium for this game.How does a Nash Equilibrium affect consumers? It makes some customers travel farther than necessary to purchase product. It eliminates the possibility of a socially optimal solution. (c ) It enables customers the ability to compare the products (price, variety, etc.) with ease. (D All of the above
- A roulette wheel has 30 slots, consisting of 2 blues, 8 whites, and 20 red slots. You will receive P100 if the roulette stops spinning on a blue slot, you will receive P50 on a white slot, but you will be penalized P10 if the roulette stops spinning on a red slot. Let X be the amount you will recieve (or pay). The probaility of receiving P50 is 2/30 or 1/15, and of receiving P50 is 8/30 or 4/15. What is the probability of penalty of P10? 2/3 1/3 -2/3In Hotelling's Ice Cream Truck game. Assume there are 9 spots: {-4, −3, −2, −1, 0, 1, 2, 3, 4}Assume there are two customers at each spot (18 in total). In a Nash equilibrium, each truck will locate at 0. How many customers would truck A lose by deviating and moving from position 0 to position 1. Select one: a.none of the other answers. b.3 c.1 d.0 e.24. (Watson 23.2) Consider the following two-country tariff game. There are two countries that are labeled 1 and 2. Let x, be the tariff level of country i (in percent), for i=1,2. If country i picks x, and the other country j selects x,, then country i gets a payoff of 2000+ 60x + xix-x7-90xj. Assume that x₁ and x₂ must be between 0 and 100 and that the countries set tariff levels simultaneously and independently. (a) Compute the Nash equilibrium of the stage game. (b) Find conditions on the discount factor such that zero tariff (x₁= x₂ = 0) can be sustained each period by a subgame perfect equilibrium. Use the grim-trigger strategy profile.
- Little Kona is a small coffee company that is considering entering a market dominated by Big Brew. Each company’s profit depends on whether Little Kona enters and whether Big Brew sets a high price or a low price: a) Does either player in this game have a dominant strategy? b) Does your answer to part (a) help you figure out what the other player should do? What is the Nash equilibrium? Is there only one? c) Big Brew threatens Little Kona by saying, “If you enter, we’re going to set a low price, so you had better stay out.” Do you think Little Kona should believe the threat? Why or why not?2. Consider a simultaneous-move auction in which two players simultaneously choose bids which must be in nonnegative integer multiples of one cent. The higher bidder wins a dollar bill. If the bids are equal, neither player receives the dollar. Each player must pay his/her own bid whether or not he or she wins the dollar. Each player's payoff is simply the net winnings. Construct a symmetric mixed-strategy equilibrium in which every bid less than 1.00 has a positive probability.4. Consider the game below. Find the mixed-strategy Nash equilibrium (MSNE). Consider a perturbation of this game with incomplete information and construct a Bayesian Nash equilibrium (BNE) such that as the perturbed game converges to the original game, the BNE converges to the mixed-strategy NE. R U 1,6 4,0 9, -8 -7,1 Figure 2: Non-descript two by two