# How much will you bid if three are 4 players?

Suppose you run a 1st price auction and players valuations are private and drawn independently from Show more Suppose you run a 1st price auction and players valuations are private and drawn independently from U. Assume every player shades his bid by the same fraction and maximizes his utility. a. Suppose your valuation is 40. How much will you bid if three are 4 players? 5 players 6 players? What do you observe? b. What is the expected revenue for the seller when there are 4 players? 5 players? 6 players? [dont use simulations here]. What do you observe? c. Provide another way to calculate the expected revenues in part b for the case of 4 players [hint: use the revenue equivalence result between 1st and 2nd price auctions]. The seller is considering setting a single posted price instead of running an auction. In a posted price mechanism the seller just sets a price p and if there is at least one player with valuation above p the seller will sell the item in price p to one of these players in which case his revenue will be p but if there is no player with a valuation above p no player will want the item and his revenue will be 0. d. Suppose there are 5 bidders. Find a formula for the optimal price the seller can use in a posted price mechanism (one that maximizes his revenue). [Idea: write down the expected revenue as a function of p and take the derivative]. e. Compare the expected revenues the seller will get in an optimal posted price vs a 1st price auction (with no reserve price) when there are (i) 2 players (ii) 5 players (iii) 10 players. I. What do you find? [You can use simulations or exact calculations to to find the expected revenues even if you didnt calculate formally the price and revenue in part c]. II. What will happen to the difference between the revenues under the auction and the posted price mechanism if the number of bidders will grow very large (say more than 1000)? Show less

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