Lecture Note on Auctions

Transcription

1 Lecture Note on Auctions Takashi Kunimoto Department of Economics McGill University First Version: December 26 This Version: September 26, 28 Abstract. There has been a tremendous growth in both the number of internet auction websites, where individuals can put up items for sale under common auction rules, and the value of goods sold there. The objective of auction theory is to rigorously understand the properties and implications of auctions used in the real world. I mainly investigate four auction forms: (1) The open ascending price or English auction; (2) the open descending price or Dutch auction; (3) the sealed-bid first-price auction; and (4) the sealed-bid second-price auction. The topics which will be covered include games with incomplete information, the revenue equivalence principle, mechanism design, efficient and optimal mechanisms, auctions with interdependent values, the revenue ranking (linkage) principle, and multiple objects auctions. I am thankful to the students for their comments, questions, and suggestions. Of course, all remaining errors are my own. Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec, H3A2T7, CANADA, takashi.kunimoto@mcgill.ca URL:

2 Syllabus Econ 577: Mathematical Economics I Fall 27, McGill University Tuesdays and Thursdays, 4:5pm - 5:25pm; at Leacock 21 Instructor: Takashi Kunimoto takashi.kunimoto@mcgill.ca 3 Class Web: the WebCT 4 Office: Leacock 438 COURSE DESCRIPTION: The theme of Econ 577 is auction theory. There has been a tremendous growth in both the number of internet auction websites, where individuals can put up items for sale under common auction rules, and the value of goods sold there. The objective of this course is to rigorously understand the properties and implications of auctions used in the real world. This course investigates three main auction forms: (1) The open ascending price or English auction; (2) the sealed-bid first-price auction; and (3) the sealed-bid second-price auction. The topics which will be covered include games with incomplete information, the revenue equivalence principle, mechanism design, efficient and optimal mechanisms, interdependent values auctions, and multiple object auctions. At the same time, I will touch on experimental and empirical results of auctions with special emphasis on practical relevances of auction theory. This course is designed to be self-contained. Besides, the level of mathematics will be adjusted according to the students understanding during the course. READING: The first three books listed below are available at the McGill bookstore and reserved in the library. 1. Lecture Note on Auctions, by Takashi Kunimoto (Main course material; This note is largely based on Krishna s Auction Theory and is available at the WebCT.) 2. Auction Theory, by Vijay Krishna, Academic Press, 22 (Main Textbook) 5 3. Putting Auction Theory to Work, by Paul Milgrom, Cambridge University Press, 24 (Supplementary) 3 In the semester, I might send s to all students through the WebCT. But, do not (or reply to) me through the WebCT. You should directly use takashi.kunimoto@mcgill.ca to contact me. 4 I will not answer how to use the WebCT. 5 In this course, the contents of the book will be simplified to some extent. 1

3 4. Auctions: Theory and Practice, by Paul Klemperer, Princeton University Press, 24. (Supplementary) 5. Auctions: A Survey of Experimental Research, by John H. Kagel in The Handbook of Experimental Economics, Princeton University Press, (Supplementary; this article is available upon request.) 6. Empirical Models of Auctions, by Susan Athey and Philip A. Haile in Advances in Economics and Econometrics, Theory and Applications: The Ninth World Congress of the Econometric Society, Cambridge University Press (Supplementary; I make this paper available at the WebCT). 7. An Empirical Perspective on Auctions, by Ken Hendricks and Robert H. Porter forthcoming in Handbook of Industrial Organization, vol 3 (Supplementary; I make this paper available at the WebCT). 8. Lecture Note on Mathematics for Economists, by Takashi Kunimoto (Supplementary; If you don t have good mathematical background, you might find this helpful; I make this note available at the WebCT). REFERENCES ON GAME THEORY: Auction theory is exclusively based on the language of game theory. Therefore, I want to introduce you to some references on game theory. There are two books which are, based on my choices (or bias?), considered introductory. You can find these two books in the reserve desk at the (McLennan? I think) library. As far as this course is concerned, you can completely skip any argument on mixed strategy. An Introduction to Game Theory by Martin J. Osborne, Oxford University Press, (24); Consult only Chapter 1(this is introduction if you wish to read), 2, 3, and 9; This book is regarded as a (significantly) less mathematical version of A Course in Game Theory by Osborne and Rubinstein; Exercises in the book can be helpful for your understanding game theory and there are answers for some of exercises at Osborne s website at Department of Economics, University of Toronto. Game Theory for Applied Economists by Robert Gibbons, Princeton University Press, (1992); Consult only Chapter 1 and 3. If you want to skip lots of definitions and theorems and immediately go to applications of game theory, this book might be for you. I picked out two advanced textbooks for this course. These two books are also reserved in the library. A Course in Game Theory by Martin J. Osborne and Ariel Rubinstein, the MIT press, (1994); Consult only Chapter 1 (this is again an introduction, which 2

4 is very interesting, though), and 2. You can completely ignore the section on strictly competitive (i.e., zero-sum two person) games. If you go to, again, Martin J. Osborne s website, you will find the solution manual for exercises in the book. Nevertheless, it is not an easy job to understand what the solution manual says. Don t blame yourself if you don t understand it. Note that this book takes preferences as a primitive and deduce payoff functions as a representation for preferences. Game Theory: Analysis of Conflict by Roger B. Myerson, Harvard University Press, (1991); Consult only Chapter 1 (A very good introduction to game theory and a very detailed discussion on the single person decision theory), 2 (you can ignore 2.9), and 3 (you can ignore 3.5 throughout up to the end of the chapter except 3.11, which is about auctions). Exercises are very tough. You don t have to try them. Do you want to know more? If so, please don t hesitate to talk to me. OFFICE HOURS: Tuesdays and Thursdays, 5:3pm - 6:3pm ASSESSMENT: There are three ingredients which determine your final grade: (1) Problem Sets 1%; (2) midterm exam 3% and (3) final exam 6%. 6 Only if he/she has a serious reason why he/she cannot take the midterm exam, the grade of that person will be solely based on the problem sets and the final exam. However, this treatment is very exceptional. You must take both exams. Moreover, both midterm and final exams are given in the take-home style. PROBLEM SETS: There will be approximately 5 problem sets. Problem sets are essential to help you understand the course and to develop your skill to analyze economic problems. Besides, it should be expected that these problem sets are very good proxies for the exams. The solution to each problem set will be reviewed in the class. 6 McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the code of student conduct and disciplinary procedures (See for more information). 3

5 Contents 1 Introduction Game Theory What is an Auction? Some Common Auction Forms The English Auction The Dutch Auction The Sealed-Bid First-Price Auction The Sealed-Bid Second-Price Auction Valuations Equivalent Auctions Revenue versus Efficiency Games with Complete Information Rational Behaviors Strategic Games Common Knowledge Nash Equilibrium Examples Existence of Nash Equilibrium Sealed-Bid Auctions with Complete Information The First Price Sealed-Bid Auction The Second Price Sealed-Bid Auction Discontinuity of Payoff Functions in First Price Auctions Games with Incomplete Information A Motivating Example Bayesian Games Formalizing Bayesian Games Example: Second-Price Auction Independent Private Values (IPV) Auctions with Symmetric Bidders Continuous Distributions

6 CONTENTS 4.2 Order Statistics Highest Order Statistic Second-Highest Order Statistic The Symmetric Model Second Price Sealed-Bid Auctions The expected payment of Second Price Auctions First Price Sealed-Bid Auctions Symmetric Equilibrium of First Price Auctions Expected Payment in First Price Auctions Multiple Integrals and Stochastic Dominance Multiple Integrals First-Order Stochastic Dominance Second-Order Stochastic Dominance Revenue Equivalence between the First and Second Price Auctions Revenue Comparison between the First and Second Price Auctions Hazard Rates Reserve Price Reserve Prices in Second Price Auctions Reserve Prices in First Price Auctions Revenue Effects of Reserve Prices The Optimal Reserve Price in a Second Price Auction Entry Fees as a Substitute of Reserve Prices The Revenue Equivalence Principle Applications of the Revenue Equivalence Principle Equilibrium of All-Pay Auctions Equilibrium of Third Price Auctions Uncertain Number of Bidders Extensions of Independent Private Values Auctions Auctions with Risk-Averse Bidders Auction with Budget Constrained Bidders Budget Constraints Second Price Auctions First Price Auctions Revenue Comparison Auctions with Asymmetric Bidders Asymmetric First Price Auctions with Two Bidders Hazard Rate Dominance Reverse Hazard Rate Dominance Weakness Leads to Aggressive Behaviors Efficiency Comparison Resale Experimental Analysis for IPV Sealed-Bid Auctions Experimental Procedures

7 CONTENTS Tests of the Second Price Auctions Tests of the Strategic Equivalence of Second Price and English Auctions Tests of the First Price Auctions Tests of the Strategic Equivalence of First Price and Dutch Auctions Tests of Efficiency of Auctions Empirical Analysis for IPV Sealed-Bid Auctions Structural Analysis of Second Price Auctions Structural Analysis of First Price Auctions Mechanism Design The Mechanism Design Approach Mechanisms The Revelation Principle Incentive Compatibility Characterizations of Incentive Compatible Direct Mechanisms Revenue Equivalence Individual Rationality Optimal Mechanisms Setup Solution Discussion and Interpretation of the Optimal Mechanisms Efficient Mechanisms The VCG Mechanism Budget Balance Bilateral Trade Auctions with Interdependent Values Interdependent Values Correlated Signals The Symmetric Model Second Price Auctions with Interdependent Values An Example of Common Value Auctions English Auctions Constructing the Symmetric Equilibrium in an English Auction Characteristics of the Symmetric Equilibrium Ex Post Equilibrium Affiliation Log-Supermodularity Monotone Likelihood Ratio Property Likelihood Ratio Dominance First Price Auctions with Interdependent Values Technical Notes on Differential Equations

8 CONTENTS 7.9 Revenue Ranking English versus Second Price Auctions Second Price versus First Price Auctions An Example Affiliated Signals Revenue Ranking Efficiency The Revenue Ranking ( Linkage ) Principle First Price versus Second Price Auctions The Revenue Equivalence Principle Revisited Value of Public Information When Public Information is not Available When Public Information is Available Public Information in a First Price Auction Existence of Symmetric Equilibria When Public Information is Available Public Information in Second Price and English Auctions The Extended Revenue Ranking Principle Ranking All-Pay Auctions Asymmetries and Other Complications The Symmetry Assumption Second Price Auctions First Price Auctions Failures of the Revenue Ranking Principle Asymmetric Uniform Distributions Second Price Auctions First Price Auctions Revenue in the First Price Auction Failure of the Revenue Ranking between English and Second Price Auctions Equilibrium and Revenues in a Second Price Auction Equilibrium and Revenues in an English Auction Appendix Efficiency and the English Auction The Single Crossing Condition Two-Bidder Auctions The Average Crossing Condition Three or More Bidders Generalized Single Crossing Condition English Auctions with Reentry Appendix for the Proof of Proposition

9 CONTENTS 1 Mechanism Design with Interdependent Values Efficient Mechanisms The Generalized VCG Mechanism The Working of the Generalized VCG Mechanism Optimal Mechanisms Full Surplus Extraction Comments on the CM mechanism

10 Chapter 1 Introduction 1.1 Game Theory Auction theory is exclusively based on the language of game theory. So, let me start by describing what game theory is. Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision makers interact. Game theory is sometimes called interactive decision theory. The basic assumptions that underlie the theory are that decision makers pursue well-defined exogenous objectives (they are rational) and take into account their knowledge or expectations of other decision makers behavior (they reason strategically). A game is a description of strategic interaction that includes the constraints on the actions that the players can take and the players interests, but does not specify the actions that the players do take. A solution is a systematic description of the outcomes that may emerge in a family of games. Game theory suggests reasonable solutions for classes of games and examines their properties. Auction theory models the decision problems collectively facing bidders in an auction as a game of incomplete information. A game with incomplete information is a game in which, at the first point in time when the players can begin to plan their moves in the game, some players already have private information about the game that other players do not know. 1.2 What is an Auction? An auction format is a mechanism to allocate resources among a group of bidders. An auction model includes three major parts: a description of the potential bidders, the set of possible resource allocations (describing the number of goods of each type, whether the goods are divisible, and whether there are legal or other restrictions on how the goods may be allocated), and the values of various resource allocations to each participant. Thus, a wide variety of selling institutions fall under the rubric of an auction. A common aspect of auction-like mechanisms is that they elicit 9

11 CHAPTER 1. INTRODUCTION information, in the form of bids, from potential buyers regarding their willingness to pay and the outcome - that is, who wins what and pays how much - is determined solely on the basis of the received information. An implication of this is that auctions are universal in the sense that they may be used to sell any good. A second important aspect of auction-like mechanisms is that they are anonymous. By this I mean that the identities of the bidders play no role in determining who wins the object and who pays how much. 1.3 Some Common Auction Forms The English Auction The English auction is the open ascending price auction. In one variant of the English auction, the sale is conducted by an auctioneer who begins by calling out a low price and raises it, typically in small increments, as long as there are at least two interested bidders. The auction stops when there is only one interested bidder The Dutch Auction The Dutch auction is the open descending price counterpart of the English auction. Here the auctioneer begins by calling out a price high enough so that presumably no bidder is interested in buying the object at that price. This price is gradually lowered until some bidder indicates her interest The Sealed-Bid First-Price Auction In this auction form, bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object and pays what he bid The Sealed-Bid Second-Price Auction As its name suggests, once again bidders submit bids in sealed envelopes; the person submitting the highest bid wins the object but pays not what he bid, but the second highest bid. 1.4 Valuations Auctions are used precisely because the seller is unsure about the values that bidders attach to the object being sold - the maximum amount each bidder is willing to pay. If each bidder knows the value of the object to himself at the time of bidding, the situation is called one of private values. Implicit in this situation is that no bidder knows with certainty the values attached by other bidders and knowledge of other 1

12 CHAPTER 1. INTRODUCTION bidders values would not affect how much the object is worth to a particular bidder. The assumption of private values is most plausible when the value of the object to a bidder is derived from its consumption or use alone. In many situations, how much the object is worth is unknown at the time of the auction to the bidder himself. He may have only an estimate of some sort or some privately known signal - such as an expert s estimate or a test result - that is correlated with the true value. Indeed, other bidders may possess information, that if known, would affect the value that a particular bidder attaches to the object. Such a specification is called one of interdependent values and is particularly suited for situations in which the object being sold is an asset that can possibly be resold after the auction. 1.5 Equivalent Auctions Open auctions requires that the bidders collect in the same place, whereas sealed bids may be submitted by mail, so a bidder may observe the behavior of other bidders in one format and not in another. For rational decision makers, however, some of these differences are superficial. The Dutch open descending price auction is strategically equivalent to the first-price sealed-bid auction. When values are private, the English open ascending auction is also (not strategically) equivalent to the second-price sealed-bid auction. 1.6 Revenue versus Efficiency The main questions that guide auction theory involve a comparison of the performance of different auction formats as economic institutions. From the perspective of the seller, a natural yardstick in comparing different auction forms is the revenue, or the expected selling price, that they fetch. From the perspective of society as a whole, however, efficiency - that the object end up in the hands of the person who values it the most ex post - may be more important. 11

13 Chapter 2 Games with Complete Information 2.1 Rational Behaviors The models we study assume that each decision maker is rational in the sense that he is aware of his alternatives, forms expectations about any unknown, has clear preferences, and chooses his action deliberately after some process of optimization. In the absence of uncertainty the following elements constitute a model of rational choice. A set A of actions from which the decision maker makes a choice. A set C of possible consequences of these actions. A consequence function g : A C that associates a consequence with each action. A utility function u : C R. Given any set B A of actions that are feasible in some particular case, a rational decision maker chooses an action a B (i.e., feasible) and optimal in the sense that a is the solution to max a B u(g(a)). In the models we study, individuals often have to make decisions under uncertainty. The players may be uncertain about the objective parameters of the environment imperfectly informed about events that happen in the game uncertain about actions of the other players that are not deterministic uncertain about the reasoning of the other players. 12

14 CHAPTER 2. GAMES WITH COMPLETE INFORMATION To model decision making under uncertainty, we adopt the theories of von Neumann and Morgenstern (1944) and of Savage (1954). That is, if the consequence function is stochastic and known to the decision maker (i.e., for each a A, the consequence g(a) is a lottery (probability distribution) on C) then the decision maker is assumed to behave as if he maximizes the expected value of a (von Neumann- Morgenstern utility) function that attaches a number to each consequence. If the stochastic connection between actions and consequences is not given, the decision maker is assumed to behave as if he has in mind a state space Ω, a probability measure over Ω, a function g : A Ω C, and a utility function u : C R; heis assumed to choose an action a that maximizes the expected value of u(g(a, ω)) with respect to the probability measure. 2.2 Strategic Games Here I follow A Course in Game Theory, by Martin J. Osborne and Ariel Rubinstein (1994, the MIT Press). A strategic game is a model of interactive decision making in which each decision-maker chooses his plan of action once and for all, and these choices are made simultaneously. Definition 2.1 A strategic game consists of a finite set N (the set of players) for each player i N a nonempty set A i (the set of actions available to player i) for each player i N, apayoff function u i : A R, where A = A 1 A n. If the set A i of actions of every player i is finite then the game is called finite. When we analyze a game, we say that a player in the game is intelligent if he knows everything that we know about the game and he can make any inferences about the situation that we can make. In game theory, we generally assume that players are intelligent in this sense. 2.3 Common Knowledge Following Aumann (1976), we say that a fact is common knowledge among the players if every player knows it, every players knows every player knows it, and so on ad infinitum. In general, whatever model of a game we may choose to study, the methods of game theory compel us to assume that this model must be common knowledge among the players. To understand why, recall the intelligent assumption. This assumption implies that, whatever model of the game we may study, we must assume that the players know this model, too. Furthermore, because we know that the players all know the model, the intelligent players must also know that they all know 13

15 CHAPTER 2. GAMES WITH COMPLETE INFORMATION the model. Having established this fact, we also recognize that the intelligent players also know that they all know that they all know the model, and so on ad infinitum. 2.4 Nash Equilibrium Nash equilibrium captures a steady state of the play of a strategic game in which each player holds the correct expectation about the other players behavior and acts rationally. It does not attempt to examine the process by which a steady state is reached. Definition 2.2 A Nash equilibrium of a strategic game (N,(A i ) i N, (u i ) i N ) is a profile a A of actions with the property that for every player i N, we have u i (a i,a i) u i (a i,a i) for all a i A i. Thus for a to be a Nash equilibrium it must be that no player i has an action yielding an outcome that he prefers to that generated when he chooses a i, given that every other player j chooses his equilibrium action a j. The following restatement of the definition is sometimes useful. For any a i A i define B i (a i ) to be the set of player i s best actions given a i : { } B i (a i )= a i A i u i (a i,a i ) u i (a i,a i) for all a i A i. We call the set-valued function B i the best-response function of player i. A Nash equilibrium is a profile a of actions for which a i B i (a i) for all i N. Notice that I have not directly argued that intelligent rational players must use equilibrium strategies in a game. When I am asked why players in a game should behave as in some Nash equilibrium, my response is to ask Why not? and let the challenger specify what he thinks the players should do. (See Myerson (1991) for this argument) If this specification is not a Nash equilibrium, then I can show that it would destroy its own validity if the players believed it to be an accurate description of each other s behavior. 2.5 Examples Example 2.1 (Battle of the Sexes) Two people wish to go out together to a concert of music by either Bach or Stravinsky. Their main concern is to go out together, but one person prefers Bach and the other person prefers Stravinsky. Representing the individuals preferences by payoff functions, we have the game described below. The game has two Nash equilibria: (Bach, Bach) and (Stravinsky, Stravinsky). 14

16 CHAPTER 2. GAMES WITH COMPLETE INFORMATION Player 2 Bach Stravinsky Player 1 Bach 2, 1, Stravinsky, 1, 2 Example 2.2 (The Prisoner s Dilemma) Two suspects in a crime are put into separate cells. If they both confess, each will be sentenced to three years in prison. If only one of them confess, he will be freed and used as a witness against the other, who will receive a sentence of four years. If neither confess, they will both be convicted of a minor offense and spend one year in prison. Player 2 Don t confess Confess Player 1 Don t confess 3, 3, 4 Confess 4, 1, 1 Whatever one player does, the other prefers Confess to Don t confess, so that the game has a unique Nash equilibrium (Confess, Confess). Definition 2.3 An action a i is said to (weakly) dominate a i if for all a i A i, u i (a i,a i ) u i (a i,a i ), with a strict inequality for some a i A i. The strategy a i is weakly dominant if it (weakly) dominates every other action a i. If every player has a dominant action a i, then we will refer to a as a dominant strategy equilibrium. It is important to note that the unique Nash equilibrium of the Prisoner s dilemma is a dominant strategy equilibrium. 2.6 Existence of Nash Equilibrium Example 2.3 (Matching Pennies) Each of two people chooses either Head or Tail. If the choices differ, player 1 pays player 2 a dollar; if they are the same, player 2 pays player 1 a dollar. Each player cares only about the amount of money that he receives. A game that models this situation is shown below. The game Matching Pennies has no Nash equilibrium. Player 2 Head Tail Player 1 Head 1, 1 1, 1 Tail 1, 1 1, 1 Not every strategic game has a Nash equilibrium, as the game Matching Pennies shows. I now present some conditions under which the set of Nash equilibria of a 15

17 CHAPTER 2. GAMES WITH COMPLETE INFORMATION game is nonempty. An existence result has two purposes. First, if we have a game that satisfies the hypothesis of the result, then we know that there is some hope that our efforts to find an equilibrium will meet with success. Second, and more important, the existence of an equilibrium shows that the game is consistent with a steady state solution. To show that a game has a Nash equilibrium, it suffices to show that there is a profile a of actions such that a i B i (a i ) for all i N. Define the set valued function B : A A by B(a) = i N B i (a i ). Fixed point theorems give conditions on B under which there indeed exists a value of a for which a B(a ). The fixed point theorem that we use is due to Kakutani (1941). Lemma 2.1 (Kakutani s fixed point theorem (1941)) Let X be a compact convex subset of R n and let f : X X be a set valued function for which for all x X, the set f(x) is nonempty and convex the graph of f is closed (i.e., for all sequences {x n } and {y n } such that y n f(x n ) for all n, x n x and y n y, we have y f(x)). Then, there exists x X such that x f(x ). I omit the proof of Kakutani s fixed point theorem. A function u i : A R is said to be quasi-concave on A i if for any a i A i and any a i,a i A i, we have u i (βa i +(1 β)a i,a i) min{u i (a i,a i ),u i (a i,a i)} for any β [, 1]. Proposition 2.1 The strategic game (N,(A i ) i N, (u i ) i N ) has a (pure strategy) Nash equilibrium if for all i N, the set A i of actions of player i is a nonempty compact convex subset of a Euclidean space and the payoff function u i is continuous quasi-concave on A i. Proof of Proposition 2.1: Define B : A A by B(a) = i N B i (a i ) (where B i is the best response function of player i). For every i N, the set B i (a i )is nonempty since u i is continuous and A i is compact. This follows from Weierstrass s theorem: Any continuous real-valued function whose domain is a compact nonempty set has both the maximum and minimum of it. B i (a i ) is convex because u i is quasiconcave on A i. We shall check this. Let a i,a i B i(a i ). Set βa i +(1 β)a i for β [, 1]. This implies that u i (a i,a i ) u i (ã i,a i ) ã i A i u i (a i,a i ) u i (ã i,a i ) ã i A i 16

18 CHAPTER 2. GAMES WITH COMPLETE INFORMATION Since u i is quasi-concave on A i, we conduct the following computation: u i (βa i +(1 β)a i ) min{u i(a i,a i ),u i (a i,a i) u i (ã i,a i ) ã i A i This implies that βa i +(1 β)a i B i(a i ). Furthermore, B has a closed graph since u i is continuous. To see why, consider the following: we shall prove this by contradiction. Consider sequences {a n } and {b n } such that b n B(a n ) for each n, a n a and b n b as n, but we have b/ B(a). Then, there exists player i for whom b i is not a best response to a i. From this hypothesis, there is no loss of generality to assume that b i is a better response to a i than b i. Then, we have u i (b i,a i) >u i (b i,a i ). On the other hand, the fact that b n B(a n ) for each n implies that u i (b n i,a n i) u i (b i,a n i) n. This implies that u i exhibits a discontinuity at (b i,a i ), which contradicts the hypothesis that u i is continuous. Thus, by Kakutani s theorem, B has a fixed point. We have noted that any fixed point is a Nash equilibrium of the game. Note that this result asserts that a strategic game satisfying certain conditions has at least one Nash equilibrium. 2.7 Sealed-Bid Auctions with Complete Information An object is to be assigned to a player in the set {1,...,n} in exchange for a payment. Player i s valuation of the object is v i, and v 1 >v 2 > >v n >. The mechanism used to assign the object is a sealed-bid auction: the players simultaneously submit bids (nonnegative numbers), and the object is given to the player with the lowest index among those who submit the highest bid, in exchange for a payment The First Price Sealed-Bid Auction In a first-price auction, the payment that the winner makes is the price that he bids. I shall formulate the first price auction as a game with complete information. N = {1,...,n}: The set of players (bidders). B i =[, ) for each i N: The set of possible bids by player i. A generic bid by player i is denoted b i B i. u i (b) =u i (b 1,...,b n )=v i b i if player i is the lowest index among those who submit b i = max j N b j ; and u i (b) = otherwise: player i s payoff function. Claim 2.1 Player 1 obtains the object in all Nash equilibria. 17

19 CHAPTER 2. GAMES WITH COMPLETE INFORMATION Proof of Claim 2.1: Fix a Nash equilibrium b B 1 B n. Suppose, on the contrary, that player 1 does not obtain the object. Assume that player j 1is the winner. Because of rationality of player j, we have b j [,v j]. Set b 1 = b j + ε for ε> small enough so that v 1 > b 1 = b j + ε. Then, we have u 1 ( b 1,b 1) =v 1 b 1 >u 1 (b )=. This contradicts the hypothesis that b is a Nash equilibrium The Second Price Sealed-Bid Auction In a second price auction, the payment that the winner makes is the highest bid among those submitted by the players who do not win (so that if only one player submits the highest bid then the price paid is the second highest bid). I shall formulate the second price auction. N = {1,...,n}: The set of players (bidders). B i =[, ) for each i N: The set of possible bids by player i. A generic bid by player i is denoted b i B i. u i (b) =u i (b 1,...,b n )=v i max j i b j if b i > max j i b j ; and u i (b) = otherwise: player i s payoff function. Claim 2.2 In a second price auction, the bid v i of any player i is a weakly dominant action. Proof of Claim 2.2: Let b i = v i. Let b i be any other bid than b i. Consider two cases: (Case 1) b i >v i and (Case 2) b i <v i. Case 1: u i (b i,b i) = > u i ( b i,b i ) for b i B i with the property that bi > max j i b j >v i. For any other b i, we have u i (b i,b i) u i ( b i,b i ). Case 2: u i (b i,b i) =v i max j i b j > =u i ( b i,b i ) for any b i B i with the property that v i > max j i b j > b i. For any other b i, we have u i (b i,b i) u i ( b i,b i ). With the consideration of cases 1 and 2, we can conclude that b i = v i is a weakly dominant action. Claim 2.3 In a second price auction, there is a(n) ( inefficient ) Nash equilibrium in which the winner is not player 1. Proof of Claim 2.3: We construct the following action profile b : b j >v 1; 18

20 CHAPTER 2. GAMES WITH COMPLETE INFORMATION b 1 <v j; b i = for any i/ {1,j}. It remains to show that b is indeed a Nash equilibrium. It is relatively easy to check that no player has any profitable deviation Discontinuity of Payoff Functions in First Price Auctions Here I will illustrate that in first price auctions, the payoff functions may be discontinuous. This prevents us from using Kakutani s fixed point theorem to guarantee the existence of Nash equilibrium. To make this point, I will check all the conditions for Proposition 2.1. It turns out that all other conditions are satisfied except the continuity of payoff functions. In a first-price auction, the payment that the winner makes is the price that he bids. I shall formulate the first price auction as a game with complete information. Claim 2.4 B i is compact and convex. Proof of Claim 2.4: It is straightforward to see that the set [, b i ] is compact and convex. (Do you see why it is straightforward?) Claim 2.5 u i : B R is quasi-concave on B i. Proof of Claim 2.5: We focus on player i. Fix any other players bids b i B i. Let any two bids by player i be b i,b i B i. We consider the following cases: Case 1: Player i is the winner both when the submitted bids are (b i,b i ) and (b i,b i). Note that b i max j i b j and b i max j i b j. Note also that u i (b i,b i )=v i b i and u i (b i,b i) = v i b i. Set αb i +(1 α)b i for α [, 1]. We know that αb i +(1 α)b i max j i b j. Hence, i is the winner when the submitted bids are (αb i +(1 α)b i,b i). The resulting payoff of player i is examined as follows: u i ( αb i +(1 α)b i,b i ) [ ] = v i αb i +(1 α)b i v i max{b i,b i} = min{v i b i,v i b i } = min{u i (b i,b i ),u i (b i,b i)}. Case 2: Player i is not the winner both when the submitted bids are (b i,b i ) and (b i,b i). 19