Lecture 10: Auction Mechanisms

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1 Microeconomics I: Game Theory Lecture 10: Auction Mechanisms (see Osborne, 2009, Sect 3.5) Dr. Michael Trost Department of Applied Microeconomics January 17, 2014 Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 1 / 23

2 Auctions An auction is a mechanism for the selling of objects by means of bids submitted by the potential buyers. Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 2 / 23

3 Auctions as allocation mechanisms Auctions are used to allocate economic resources, e.g., work of arts (e.g., The Scream of Edvard Munch for approx. $ 120 mio. in May 2, 2012) Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 3 / 23

4 Auctions as allocation mechanisms Auctions are used to allocate economic resources, e.g., government bonds (e.g., weekly auctions of US Treasury Bills) offshore oil drilling rights (e.g., auction of offshore oil drilling rights in the central Gulf of Mexico in June 20, 2012 ). radio spectrum licenses (e.g., auction of UMTS licenses in 2000 in Germany) merchandise (e.g., online auctions of merchandise in ebay) Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 4 / 23

5 Sealed-bid single-object auctions In this lecture, we restrict ourselves to auctions in which a single object is offered by the auctioneer (so called single-object auctions). the potential buyers submit once and simultaneously a monetary bid for the offered object (so called sealed-bid auctions). Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 5 / 23

6 Auction mechanism An auction mechanism is specific set of rules that determines the course of the auction. These rules stipulate the set of participants of auctions, the set of possible bids the participants are allowed to submit, how a winner is determined by the submitted bids, how much the participants of the auctions must pay for their submitted bids. Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 6 / 23

7 Bidders and bids An auction mechanism is a set of rules which specifies 1 a finite non-empty set I of potential buyers (called bidders) 2 for each bidder i I, a set of possible non-negative real-valued bids b i B i [0, + [. A profile b := (b i ) i I i I B i of bids is a list summarizing the bids submitted by the bidders. Henceforth, the set of all bid profiles is denoted by B := i I B i. Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 7 / 23

8 Winning function An auction mechanism is a set of rules which specifies 3 a winning function w : B (I ) that assigns to each bid profile b := (b i ) i I B a probability vector on I according to which the winner of the auction (i.e., the buyer who receives the object) is chosen. Function w i : B [0, 1] denotes the winning function of bidder i I and gives the probability that i wins the auction if bid profile b B is realized. Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 8 / 23

9 Cost function An auction mechanism is a set of rules which specifies 4 for each bidder i I, a cost function C i : B [0, + [ of bidder i that determines for every bid profile b := (b i ) i I B the payment bidder i must make. Dr. Michael Trost Microeconomics I: Game Theory Lecture 10 9 / 23

10 Auction mechanism Definition 10.1 An auction mechanism A := (I, (B i ) i I, w, (C i ) i I ) consists of the components a finite non-empty set I of bidders, for each bidder i I, a non-empty set B i [0, + [ of possible bids. a winner function w : B (I ) specifying the winner of the auction, for each bidder i I, a cost function C i : B [0, + [ specifying the payment i must settle. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

11 Auctions with free zero-bids An auction mechanism A is said to be an auction with free zero-bids if C i (b) = 0 holds for every bidder i I and for every bid profile b B satisfying b i = 0. This condition says the submission of a bid of zero does not induce any payments for bidders. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

12 Standard auction An auction mechanism A is termed a standard auction whenever the bidder with highest bid is the winner of the auction, and in case there are several bidders who have submitted the highest bid a random device is applied according to which each of them is chosen with the same probability to be the winner of the auction. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

13 Winner set Consider some auction A := (I, (B i ) i I, w, (C i ) i I ). Denote by W (b) the set of possible winners of auction A if bid profile b has been submitted, i.e. W (b) := {i I : w i (b) > 0}. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

14 Standard auction A standard auction is an auction in which 1 the winner set is given by W (b) = {i I : b i b j for every j I } for every bid profile b B. 2 the winning function of every player i I is given by w i (b) := { 1 #W (b) if i W (b), 0 if i / W (b). Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

15 First-price sealed-bid auction A first-price sealed-bid auction is an auction in which every bidder is allowed to submit any non-negative real-valued bid, and the bidder who has submitted the highest bid wins and pays the amount of money she has bid. Bidders who do not win the auction pay nothing. In the case several bidders have submitted the highest bid, a random device is applied according to which each of them is selected with equal probability to be the winner of the auction. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

16 First-price auction EXERCISE: Determine the components of the first-price auction! set of I := bidders set of B bids i := winning w : function cost C function i : Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

17 Second-price sealed-bid auction A second-price sealed-bid auction is an auction in which every bidder is allowed to submit any non-negative realvalued bid, and the bidder who has submitted the highest bid wins and pays the amount of money equal to the second highest bid. Bidder who do not win the auction pay nothing. In case several bidders have submitted the highest bid, a random device is applied according to which each of them is selected with equal probability to be the winner of the auction. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

18 Vickrey auction Second-price sealed bid auctions are also known as Vickrey auctions. Economist William Vickrey was the first who studied properties of this auction form. William Vickrey (June 21, October 11, 1996) received in 1996 the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for fundamental contributions to the economic theory of incentives under asymmetric information. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

19 Vickrey auction EXERCISE: Determine the components of the Vickrey auction! set of I := bidders set of B bids i := winning w : function cost C function i : Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

20 All-pay sealed-bid auction An all-pay sealed-bid auction is an auction in which every bidder is allowed to submit any non-negative real-valued bid, and the bidder who has submitted the highest bid wins. Each bidder must pay the amount of money equal to her submitted bid regardless if she has won the auction, or not. In case several bidders have submitted the highest bid, a random device is applied according to which each of them is selected with equal probability to be the winner of the auction. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

21 All-pay sealed-bid auction EXERCISE: Determine the components of the all-pay sealed-bid auction! set of I := bidders set of B bids i := winning w : function cost C function i : Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

22 Sealed-bid lottery auction A sealed-bid lottery auction is an auction in which every bidder is allowed to submit any non-negative real- valued bid. The winner is chosen by a random device according to which each bidder is selected with the probability equal to the fraction of her bid on the total sum of the bids. In case all bidders have submitted a bid of zero each of them is selected with the same probability to be the winner of the auction. The winner of the auction pays the amount he submitted while the other bidders pay nothing. Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23

23 Sealed-bid lottery auction EXERCISE: Determine the components of a sealed-bid lottery auction! Is this auction mechanism a standard auction? set of I := bidders set of B bids i := winning w : function cost C function i : Dr. Michael Trost Microeconomics I: Game Theory Lecture / 23