Consider an online dating site with four men m1, m2, m3, m4 and four women wi, wz, W3, w4. Suppose the dating site manages to compute a "compatibility" score for each couple as follows. wi Em 8 5 6 3 2 m2 4 m3 8 2 m4 3 4 (a) Suppose that individuals' preferences coincide exactly with the company's com- patibility score. Find a stable matching and show that it is unique. (b) Does this matching maximize the total compatibility of the matched couple? Explain why or why not.
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- 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 following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.Consider two bidders – Alice and Bob who are bidding for a second-hand car. Each of them knows the private value she/he assigns to the car, but does not know the exact value of others. It is common knowledge that the value of other bidders is randomly drawn from a uniform distribution between 0 and $10000. Assume that Alice values the car at $8500 and Bob values the car at $4500. a) If Alice and Bob participated in the second-price sealed bid auction, what would they bid and what would be the result of the auction? Explain your answer. b) If they participate instead in a first-price sealed bid auction, what would they bid and what would be the result of the auction? Explain your answer. c) Calculate and compare the revenue of the seller in the above situations. Which type of auction should the seller use? Explain your answer
- True or false: Discuss if False Homoscedasticity means that E (e^2).Consider the following ‘war of attrition’. Two animals are in a stand off for a prey. Theyindependently decide when to give up. Waiting is costly, but the animal giving up last winsthe prey (they each get nothing if they walk away at the exact same time). Getting the preygives a benefit of 80 while waiting costs 2 per unit of time. Formally payoffs are given asfollows:u1(t1, t2) =(−2t1 if t1 ≤ t280 − 2t2 if t1 > t2u2(t1, t2) =(80 − 2t1 if t1 < t2−2t2 if t1 ≥ t2,where ti is the amount of time animal i decided to wait. Assuming that animals aim tomaximize payoffs (consciously or not), figure out the Nash equilibria of this game by answeringthe following questions (similar to how we proceeded to solve the Bertrand game).(a) Show that there is no Nash equilibrium where both animals wait a strictly positiveamount of time. For this, consider two subcases: (i) both wait the same amount oftime, or (ii) one gives in earlier than the other.(b) Assume now that one animal, say the first one,…(a) Calculate the safety levels of both players.(b) Find the set of all Nash equilibria (pure and mixed).
- Q:1 Jhon have server and many clients or bidders bidders who are bidding with their resources (datasize,bandwidth) and price they want to sell their resources. Jhon is using first price sealed bid auction here. Jhon need to select winners. Please apply Nash equilibrium strategy for clients to maximize the expected profit and expected utility theory to give guidance to the aggregator to obtain the expected resources from clients. Note: related material is attached in pictures.Game theory: Consider a collective action game with thirty individuals (N = 30). When the number of participants in the joint project is n, each individual, including shirkers, receives a benefit of B(n) = 18n and each participant incurs a cost of C(n) = 32 − 2n. 1. Find all of the Nash equilibria, both stable and unstable ones. 2. Find the socially optimal outcome. 3. Check if any of the Nash equilibria is socially optimal. Explain your answer.Find all NE of the stage game.(b) Consider a two-period game without discounting in which the stage game is played ineach period. Find all pure strategy SPNE.(c) What’s the min-max payoff of each player?(c1) Consider pure strategies only.(c2) Consider all strategies, including the mixed ones.(d) Now suppose the stage game is repeated infinitely many times. Use the Fudenberg-Maskin Folk theorem to find all possible values of payoff that can be supported as aSPNE.
- Question 1 Consider a first-price sealed bid auction of a single object with two biddersj = 1,2 and no reservation price. Bidder 1′s valuation is v1 = 2, and bidder 2′s valuation isv1 = 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.(a) Find an equilibrium of this game.(b) Is the allocation of your answer to (a) efficient?Choice under uncertainty. Consider a coin-toss game in which the player gets $30 if they win, and $5 if they lose. The probability of winning is 50%. (a) Alan is (just) willing to pay $15 to play this game. What is Alan’s attitude to risk? Show your work. (b) Assume a market with many identical Alans, who are all forced to pay $15 to play this coin-toss game. An insurer offers an insurance policy to protect the Alans from the risk. What would be the fair (zero profit) premium on this policy? can you help me for par (b) plase?Choice under uncertainty. Consider a coin-toss game in which the player gets $30 if they win, and $5 if they lose. The probability of winning is 50%. (a) Alan is (just) willing to pay $15 to play this game. What is Alan’s attitude to risk? Show your work.(b) Assume a market with many identical Alans, who are all forced to pay $15 to play this coin-toss game. An insurer offers an insurance policy to protect the Alans from the risk. What would be the fair (zero profit) premium on this policy? i need help with question B please.