Consider two companies bidding to be at the top of a search engine's results for a given keyword. Company 1 values the top position at v₁ = 8. Both company 1 and company 2 know company 1's value. However, only company 2 knows its own valuation for the top position, which can take two values: v2 = 6 or v2 = 10. Company 1 believes that company 2 has a valuation of v2 = 6 with probability and a valuation of v₂ = 10 with probability. Each company chooses simultaneously whether to submit a bid of b = 6 or a bid of b = 8. The company which submitted the highest bid wins the auction and obtains the top position in the search engine. If both firms submit the same bid, then firm 1 wins the auction. A company's payoff is therefore: V₁ = vi 0 bi if it wins the auction if it loses the auction
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- Each of the two players independently (and simultaneously with the other) decides whether to go to a play or a concert. Each would rather go with the other to a concert than with them to a play, but prefers this to not being together, in which case they don't care where they go alone. Additionally, each is indifferent between attending the play together and participating in a lottery where both go to the concert with a probability of ¾ and to different events with a probability of ¼. Describe the game in matrix form and find all its equilibria under the assumption that the players have von Neumann-Morgenstern preferences.The mixed stratergy nash equalibrium consists of : the probability of firm A selecting October is 0.692 and probability of firm A selecting December is 0.309. The probability of firm B selecting October is 0.5 and probability of firm selecting December is 0.5. In the equilibrium you calculated above, what is the probability that both consoles are released in October? In December? What are the expected payoffs of firm A and of firm B in equilibrium?Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will beequal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their incomewill be y − L. The chance of a safe return is 50%. Now suppose that there are two identical merchants, A and B, who are both risk averse expected utilitymaximisers with utility of income given by u(y) = ln y. The income of each merchant will be 8 if theirown ship returns and 2 if it sinks. As previously, the probability of a safe return is 50% for each ship.However, with probability p ≤ 1/2 both ships will return safely. With the same probability p both willsink. Finally, with the remaining probability, only one ship will return safely.(iv) Compute the increase in the utility of each merchant that they could achieve from pooling theirincomes (as a function of p). How does the benefit of pooling depend on the probability p? Explainintuitively why this is the case.
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- Two bidders compete in a second price auction (i.e., the winning bidder pays the losing bidder’s bid, and the losing bidder does not pay anything). They submit sealed bids, and the one with the highest bid wins the contract and pays the other bidder’s bid. Each bidder i’s private valuation is vi and is distributed independently and uniformly between 0 and 50. 1. For any given bidder, prove that he has a dominant strategy bid and show what it is. 2. Assuming each bidder bids his dominant strategy noted above, if a bidder with vi = 40 wins, what price does he expect to pay?You and a rival are engaged in a game in which there are three possible outcomes: you win, your rival wins (you lose), or the two of you tie. You get a payoff of 50 if you win, a payoff of 20 if you tie, and a payoff of 0 if you lose. What is your expected payoff in each of the following situations? (a) There is a 50% chance that the game ends in a tie, but only a 10% chance that you win. (There is thus a 40% chance that you lose.) (b) There is a 50–50 chance that you win or lose. There are no ties. (c) There is an 80% chance that you lose, a 10% chance that you win, and a 10% chance that you tie.Portsmouth Bank has foreclosed on a home mortgage and is selling the house at auction. There are three bidders for the house, Emily, Anna, and Olga. Portsmouth Bank does not know the willingness to pay of these three bidders for the house, but on the basis of its previous experience, the bank believes that each of these bidders has a probability of 1/3 of valuing it at $600,000, a probability of 1/3 of valuing at $500,000, and a probability of 1/3 of valuing it at $200,000. Portsmouth Bank believes that these probabilities are independent among buyers. If Portsmouth Bank sells the house by means of a second- bidder, sealed- bid auction (Vicktey auction), what will be the bank's expected revenue from the sale?
- If Firm 1 chooses to release the console in October with probability of 0.692 or December with a probability of 0.308, then Firm 2 is indifferent between choosing a release date. If Firm 2 released the console in October with probability of 0.50 or December with a probability of 0.50, then Firm 1 is indifferent between choosing a release date Suppose now that instead of choosing the release date at the same time, the firms choose sequentially (but still in advance). Firm A chooses its release date first, then firm B observes that date and chooses its own date. Thepayoffs are otherwise the same as above. Represent the game tree corresponding to this dynamic game.Suppose that there are two types of entrepreneur: skilled and unskilled. Skilled entrepreneurs have a probability p = 2/3 of success if they get the loan. Unskilled entrepreneurs have zero chance of being successful. Despite that, assume that unskilled entrepreneurs want to take up the loan, because it is cool to say you have a startup. The bank does not observe skill. The share of skilled entrepreneurs is s. Question 1: 1A). TRUE OR FALSE: If L = 2, R = 6, and s = 0.5, then the bank would have zero expected profits, but entrepreneurs would never take up the loan. 1B. ) TRUE OR FALSE: If the loan amount is L = 2, the payback amount is R = 3, and the share of skilled entrepreneurs is s = 0.9, then the bank will have positive expected profits.You are considering a $500,000 investment in the fast-food industry and have narrowed your choice to either a McDonald’s or a Penn Station East Coast Subs franchise. McDonald’s indicates that, based on the location where you are proposing to open a new restaurant, there is a 25 percent probability that aggregate 10-year profits (net of the initial investment) will be $16 million, a 50 percent probability that profits will be $8 million, and a 25 percent probability that profits will be −$1.6 million. The aggregate 10-year profit projections (net of the initial investment) for a Penn Station East Coast Subs franchise is $48 million with a 2.5 percent probability, $8 million with a 95 percent probability, and −$48 million with a 2.5 percent probability. Considering both the risk and expected profitability of these two investment opportunities, which is the better investment? Explain carefully.