Assume you are one of the two bidders in a second-price sealed bid auction for a preserved grilled cheese sandwich which purportedly bore the portrait of the Virgin Mary: • This sandwich is worth $8000 to you. • You know that your rival bidder’s valuation is uniformly distributed between $0 and $12,000. a) What is the probability that your bid exceeds your rival’s bid given that you know your rival’s valuation is uniformly distributed between $0 and $12,000? (In other words, what is the probability that your rival’s bid is lower than your valuation?) Remember the equilibrium bidding strategies for a SPA in weakly dominant strategies ! b) What is your rival’s average bid, if you win the auction (considering that he must have bid below your bid)? c) What is your expected payoff from this auction, if you win the bidding ? (Hint: Combine b) and c), i.e. compute the payoff from winning giving the probability of winning)

Assume you are one of the two bidders in a second-price sealed bid auction for a preserved grilled cheese sandwich which purportedly bore the portrait of the Virgin Mary: • This sandwich is worth $8000 to you. • You know that your rival bidder’s valuation is uniformly distributed between $0 and $12,000. a) What is the probability that your bid exceeds your rival’s bid given that you know your rival’s valuation is uniformly distributed between $0 and $12,000? (In other words, what is the probability that your rival’s bid is lower than your valuation?) Remember the equilibrium bidding strategies for a SPA in weakly dominant strategies ! b) What is your rival’s average bid, if you win the auction (considering that he must have bid below your bid)? c) What is your expected payoff from this auction, if you win the bidding ? (Hint: Combine b) and c), i.e. compute the payoff from winning giving the probability of winning)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.10P

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6. Consider a partnership game between two people, in which each player simultaneously chooses an effort e; ≥ 0 to expend on a joint project, and receives in return a payoff π₁1 = == R(e1, €2) - e³ where the revenue R shared between the players is described by the function R(e₁, €2) = 100 (e₁ + ₂ + €502) €₂ 1 = R(e₁, 0₂) - π2 = (a) Find the best response functions BR₁(e₂) and BR₂(e₁), and the Nash Equilibrium (ei, e) for this game. (b) Find the payoff amounts ₁ and 7₂ at the NE. Note: recall that for all duopoly games, the payoff function for each player is of the form Tį = P(q1, 92)qi – Ci(qi) uch that the game payoff is equal to the profit (= Revenue - Cost).
You are evaluating the possibility that your company bids $150,000 for a particular construction job. (a) If a bid of $150,000 corresponds to a relative bid of 1.20, what is the dollar profit that your company would make from winning the job with this bid? Show your work. (b) Calculate an estimate of the expected profit of the bid of $150,000 for this job. Assume that, historically, 55 percent of the bids of an average bidder for this type of job would exceed the bid ratio of 1.20. Assume also that you are bidding against three other construction companies. Show your work.
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3) Consider the following two gambles: Option A: you win $50 with probability 0.75 and lose $100 with probability 0.25. Option B: You win $100 with probability 0.25 and lose $50 with probability 0.75. Which option is preferred according to expected utility if i) You use an exponential utility function with R=5 ii) You use an exponential utility function with R=10 iii) You use an exponential utility function with R=100 iv) You use an exponential utility function with R=1000 v) You use an exponential utility function with R=10,000
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5. You are a risk-averse decision maker with a utility function U(1) = VI, where I denotes your income. Your income is $100,000 (thus, I=100). However, there is a 0.2 chance that you will have an accident that results in a loss of $10,000. Now, suppose you have the opportunity to purchase an insurance policy that fully insures you against this loss (i.e., that pays you $10,000 in the event that you incur the loss). What is the highest premium that you would be willing to pay for this insurance policy?
Question 5 You negotiate with a retailer over a contract according to which the retailer would buy a large fraction of your current production for next year. The retailer is perfectly informed about consumer demand, but you do not know whether demand is high or low. You only know that the probability for high demand is 80%. If demand is high, the retailer's profit is £5 million minus what he pays to you according to your contract. If demand is low, the retailer's profit is £3 million minus what he pays to you. Your costs of producing the output specified in the contract are £1 million. You can make sequential offers for the retailer's total payment for you to deliver a fixed quantity of your production. As you know that your competitor is also seeking a similar contract with this retailer, and the retailer can only supply one firm due to limited shelf space, you know that you can only make at most two offers. If your first offer is rejected, the retailer will strike the deal with your…
1. Equilibria in an auction Suppose a single item will be sold to one of two bidders. Each bidder's value is drawn uniformly at random from the interval [0, 1]. Each bidder will simultaneously name a price and the item will be sold to the higher bidder for the price that bidder named. If this price is p and the winning bidder has a value of v then the utility of the winning bidder is v-p and the utility of the losing bidder is 0. This situation is more complex than the normal form games we studied in class for two reasons. First, the set of strategies is infinite. Second, rather than known utilities there is uncertainty over them, so we need a Bayesian model and in general the equilibrium strategy will depend on (i.e. be a function of) v. This assignment walks you through the derivation of an equilibrium in this setting. (a) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. If you bid p, what is your expected utility? (b) Suppose your value is…
4. Consider a first-price, sealed-bid auction in which the bidders’ valuations are independently and uniformly distributed on [0,1]. Suppose that each bidder uses a strategy of b(vi) = avi. What is the symmetric Bayesian Nash equilibrium of this game when there are n bidders?
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1. Consider the odds and evens game introduced in Sec. 15.1 (Week 7, second video) and whose payoff table is shown as below. Strategy Player 1 (even player) 1 (finger) 2 (fingers) 1 (finger) Player 2 (odd player) 2 (fingers) 3 -4 −4 3 (a) Show that this game does not have a saddle point. (b) Write an expression for the expected payoff for player 1 (the evens player) in terms of the probabilities of the two players using their respective pure strategies. (c) Show what this expression in part (b) reduces to for the following three cases: (i) Player 2 definitely uses his first strategy, (ii) player 2 definitely uses his second strategy, (iii) player 2 assigns equal probabilities to using his two strategies. (Your answer should be like the bottom table on sec 15.3's notes Page 7: each reduced expression only has x₁.)