Lecture 7_Auctions 2010

# Lecture 7_Auctions 2010 - ECON 4109H LECTURE 7 Auctions...

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ECON 4109H LECTURE 7 Auctions September 30, 2010 ECON 4109H LECTURE 7

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Auctions: Ascending Auctions Common Auction Form: People sequentially submit higher bids for an object. When no one wishes to submit a bid higher than the current bid, the person making the current bid wins the object at the price she bid. Imagine each person has a maximal bid v , representing her value, and is willing to raise the bid until it reaches v . ECON 4109H LECTURE 7
Auctions: Second Price Sealed-Bid Auction Imagine that each player instructs a friend to play the ascending auction for him. What information will he have to give his friend? The answer is his value. ECON 4109H LECTURE 7

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Auctions: Second Price Sealed-Bid Auction Suppose that each bidder submits his value, and the bidder with the highest value wins the object for the second highest price. This is equivalent to the ascending auction in this case. Let v i is the value that player i attaches to the object. Assume that the players are numbered so that v 1 > v 2 > ··· > v n > 0. Assume that in case of a tie, the bidder who values the object most (among the winners) gets it. ECON 4109H LECTURE 7
Second Price Sealed Bid Auction Players n bidders Actions Each player may choose any bid (which is any nonnegative number). Preferences For any proﬁle of bids b = ( b 1 , b 2 , . . . , b n ) , if i is the winner (the lowest index bidder among those with the highest bids), his utility is v i - b where b is the highest bid submitted by bidders other than i . Note that if there is a tie, b is equal to i ’s bid. The utility to a loser is 0. ECON 4109H LECTURE 7

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Second Price Sealed Bid Auction: All Equilibria Theorem b = ( b 1 , . . . , b n ) is a Nash equilibrium of the second price auction if and only if 1 Either (i) 1 is the winner and b 1 > v 2 , or (ii) i 6 = 1 is the winner and b i > v 1 . 2 All losing bids are below the winner’s value. ECON 4109H LECTURE 7
Second Price Sealed Bid Auction: All Equilibria Theorem b = ( b 1 , . . . , b n ) is a Nash equilibrium of the second price auction if and only if 1 Either (i) 1 is the winner and b 1 > v 2 , or (ii) i 6 = 1 is the winner and b i > v 1 . 2 All losing bids are below the winner’s value. First argue that if (1) and (2) are satisﬁed, b is an equilibrium. Condition (2) implies winner’s utility nonnegative.

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Lecture 7_Auctions 2010 - ECON 4109H LECTURE 7 Auctions...

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