First-Price and Second-Price Auctions - Consider an auction of a single indi- visible object with 5 bidders, 1, 2, 3, 4 and 5, whose personal valuations (willingness to pay) for the object are v₁ 10, V₂ = 8, V3 = 7, V4 5 and v5 3. The bidders simultaneously submit their bids and the winner is the one with the highest bid. = Suppose that, when the highest bidder is not unique, the bidder with the smallest number (highest valuation) wins. (a) Suppose that this is a second-price auction, where the winner pays the highest bid among those from her opponents, determine whether each of the following ilibrium Explain bidding profil in a Noch =

First-Price and Second-Price Auctions - Consider an auction of a single indi- visible object with 5 bidders, 1, 2, 3, 4 and 5, whose personal valuations (willingness to pay) for the object are v₁ 10, V₂ = 8, V3 = 7, V4 5 and v5 3. The bidders simultaneously submit their bids and the winner is the one with the highest bid. = Suppose that, when the highest bidder is not unique, the bidder with the smallest number (highest valuation) wins. (a) Suppose that this is a second-price auction, where the winner pays the highest bid among those from her opponents, determine whether each of the following ilibrium Explain bidding profil in a Noch =

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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1. Consider a sequential continuum game with two players, in which Player 1 first chooses a continuous variable £₁, and then Player 2 selects £2, with payoff functions ₁ and 7₂. Find the best response for P2, as the function î2 = BR₂(x1), for each given #₂: (a) (b) (c) T₂(x1, x₂) ) = (1400 — x1 — x2)x2 − 13x² T₂(x1, x₂) T₂(X1, X2₂) = : 100x2 + 3x1x2 − 14x1 – 30x1x² = 15x2 x² + x² +1
(The All-Pay Auction). The seller has an item for sale. The valuations of the bidders are independently and identically distributed on R+ with a c.d.f. F. Find the symmetric equilibrium of an auction with two bidders in which both bidders pay their own bids but only the highest bidder wins the object. Show that each bidder’s expected payment is the same in this auction and in the first-price auction.
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P1| P2 Left Middle Right Left 4,2 3,3 1,2 Middle 3,3 5,5 2,6 Right 2,1 6,2 3,3 Consider the simultaneous move game represented by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted, or the discount factor is equal to 1. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. a) (Left, Left) is played in both periods. b) (Right, Right) is played in both periods. c) (Middle, Middle) is played in both periods. d) (Middle, Middle) is played in the first period, followed by (Left, Left). e) (Middle, Middle) is played in the first period, followed by (Right, Right).
Question 4 Consider allocating an object to one of two players when each player's pref- erences are her private information. Player 1's value for the object, denoted by vị is drawn from a continuous distribution with (1. 2] as its support. Player 2's value for the object, denoted by v2, is likewise drawn from the interval 0, 1]. 1. What is the equilibrium in dominant strategies if a second price auction is used to allocate the object. Is the outcome ex-post efficient? 2. For what pairs (v1, v2) should Player 1 and Player 2 respectively be allocated the object in a mechanism that maximizes expected revenue for the seller? 3. Comment on the differing allocations you obtained in the previous two parts.
Why do sellers generally prefer a Vickrey auction to a regular sealed bid if sellers don’t receive the highest bid in the Vickrey auction? Sellers only have to sell their item if the bid is the highest-price bid. The second-highest bid in a Vickrey auction is generally higher than the highest bid in a regular sealed-bid auction. The second-highest bid is about the same in both auctions. Sellers prefer the final price is not revealed to all bidders. Sellers would never prefer Vickrey auctions.
Consider the version of the Beauty Contest game from the class, in which the players are trying to guess the average guess minus one. More precisely, the utilities now are: ui (si; sj ) = -|si-(-1)| where sj/#l forj belong tI (a) Which strategies are strictly dominated in this game, and by which alternative strategies? And so, what is the set of strategies BR11 (b) What is the set BR21 ? What is the set BRni for any n > 0?
Consider the following payoff matrix B R U 7, 7 4,17 A 17,4 6,6 Players A and B repeatedly play this game with an infinite horizon. Future payoffs are discounted with 1/(1+r), where r is the long-term interest rate. Determine the interval of r for which the Carrot-and-Stick strategy does form a Nash-equilibrium when the opponent can discover any deviation immediately. Remember the following relation TA+Lt-1 (14+t) TB TB = TA + O r>0.1 O r0.2 Or-1 O r1