1. Auction Theory תכנון מכרזים ומכירות פומביות Class 1 – introduction

2. Administration Instructor: Liad Blumrosen ליעד בלומרוזן – blumrosen@huji.ac.il blumrosen@huji.ac.il – Office hours: Wednesdays 1:30-2:30 – Please send an email before you come. Teaching assistant: Assaf Kovo אסף קובו assafkovo@gmail.com Course web-page: – http://auctiontheorycourse.wordpress.com/ http://auctiontheorycourse.wordpress.com/

3. Requirements Grading: – 40% - home assignments (about 2 exercises). – 60% - home exam. – Academic integrity: You can discuss the home assignments, but please write them on your own. The home exam is done completely on your own without discussing it with anyone. Participation is mandatory.

4. Today Part 1: – Auctions: introduction and examples. – A very brief introduction to game theory. Part 2: – four simple auctions – Modeling – Strategies, truthfulness. – Efficiency.

5. Prices What is the “right” price for objects? Think about the following case: But all the values are private: each value is known only to the potential buyers. You work for the government: how much does the bank worth? The government wants to sell a big bank. Buyer 1: willing to pay 200 Billion. Buyer 2: willing to pay 100 Billion. Buyer 3: does not need a house, but wants to buy the house if he can resale it with a profit.

7. Prices When the preferences of the bidders are known: it is not always clear how to price the objects and how to allocate them. But we don’t know the preferences… Buyers will lie and manipulate to get better prices and better allocation. How can the preferences be revealed?

8. Auctions

9. Some examples 1.Art auctions (Sotheby's) 2.Sponsored search auctions (Google) 3.Spectrum Auctions (FCC) 4.Bond/Stocks issuing

10. Example 1: English Auctions

11. Example2: Auctions for Sponsored Search Real (“organic”) search result Ads: “sponsored search”

12. Example2: Auctions for Sponsored Search

13. An online auction is run for every individual search. Advertising is effective. – Targeted. – Users in search mode. (mostly) Pay-per-click auctions. How ads are sold? We will see later in the course.

14. Market design and sponsored search Google’s revenue from sponsored search: Billions of Dollars each quarter. Every little detail matters. Advertisers are “selfish” agents: will manipulate the auction if possible. Complex software development: hard to experiment  theory to the rescue… Big internet companies (Google, eBay, Microsoft, Yahoo, Facebook, etc.) are hiring well- known economists to design their markets. Use auction theory.

15. Example 3: Spectrum Auctions

16. Federal Communication Commission (FCC) runs auctions for available spectrum since 1994. – Multi-billion dollar auctions. – Also in Europe, Canada, The Pacific, India, etc… Bidders have complex preferences: – Hard to communicate. – Hard to determine. – Hard to compute the outcome. One of the main triggers to recent developments in auction theory.

17. Example 4: bond issues, IPO’s 17

18. Why should one learn auction theory? 1.A widely-used sale tool: – Bonds, rights for natural resources, privatization, procurement, houses, agricultures, equipment, transportation, art, etc… 2.Popularity grew considerably with the Internet: – Gigantic e-commerce platforms: eBay, amazon. – B2B – Online advertising (search engines, social networks, display advertising). 3.Simple, well-defined economic environment. – An applicative branch of game theory, information economics. – Popularity  testable ground for theory. 4.Some beautiful economic theory – implications in different area in economics.

19. (Estimated) Course outline Part 1: selling a single item 1.Basic auction formats and concepts. 2.Efficient auctions, optimal auctions. Revenue equivalence, the revelation principle, Myerson’s auction. 3.Extensions to the basic model: Affiliated types, interdependent types, risk aversion. Common values, winner’s course. Part 2: multi-unit auctions 1.Vickrey-Clarke-Groves mechanisms, matching (without money). 2.Ascending-price auctions Auctions for unit demand/ substitutes valuations (Demange-Gale- Sotomayor). Ausubel and Milgrom Auctions 3.Online advertising and sponsored search. 4.Digital goods.

20. Today Part 1: – Auctions: introduction and examples. – A very brief introduction to game theory. Part 2: – four simple auctions – Modeling – Strategies, truthfulness. – Efficiency.

21. Example: Prisoner’s Dilemma Two suspects for a crime can: – Cooperate (stay silent, deny crime). If both cooperate, 1 year in jail. – Defect (blame the other). If both defect, 3 years (reduced since they confessed). – If A defects (blames the other), and B cooperate (silent) then A is free, and B serves a long sentence. CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3

22. 22 Notation We will denote a game G between two players (A and B) by G[ S A, S B, U A (a,b), U B (a,b)] where S A = strategies available for player A (a in S A ) S B = strategies available for player B (b in S B ) U A = utility obtained by player A when particular strategies are chosen U B = utility obtained by player B when particular strategies are chosen

23. Normal-form game: Example Example : – Actions: S A = {“C”,”D”} S B = {“C”,”D} – Payoffs: u A (C,C) = -1, u A (C,D) = -5, u A (D,C) = 0, u A (D,D) = -3 CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3

24. A best response: intuition Can we predict how players behave in a game? First step, what will players do when they know the strategy of the other players? Intuitively: players will best-respond to the strategies of their opponents.

25. 25 A best response: Definition When player B plays b. A strategy a* is a best response to b if U A (a*,b)  U A (a’,b) for all a’ in S A (given that B plays b, no strategy gains A a higher payoff than a*)

26. A best response: example Example: When row player plays Up, what is the best response of the column player? LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1 LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1

27. Dominant Strategies ( אסטרטגיות שולטות / דומיננטיות ) Definition: action a* is a dominant strategy for player A if it is a best response to every action b of B. Namely, for every strategy b of B we have: U A (a*,b)  U A (a’,b) for all a’ in S A

28. Dominant Strategies: in the prisoner’s dilemma CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3 For each player: “Defect” is a best response to both “Cooperate” and “Defect. Here, “Defect” is a dominant strategy for both players…

29. In the prisoner’s dilemma: (Defect, Defect) is a dominant-strategy equilibrium. Dominant Strategy equilibrium שווי משקל באסטרטגיות שולטות Definition: (a,b) is a dominant-strategy equilibrium if a is dominant for A and b is dominant for B. – (similar definition for more players) CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3

30. 30 Stability in games What is the dominant-strategy equilibrium in this game? – None…. So what would be a “stable” outcome in this game? LeftRight Left 1,11,10,00,0 Right 0,00,01,11,1

31. 31 Nash Equilibrium How will players play when dominant-strategy equilibrium does not exist? – We will define a weaker equilibrium concept: Nash equilibrium A pair of strategies (a*,b*) is defined to be a Nash equilibrium if: a* is player A’s best response to b*, and b* is player B’s best response to a*.

32. (Pure) Nash Equilibrium Examples: LeftRight Left 1,11,10,00,0 Right 0,00,01,11,1 SwerveStraight Swerve 0, 0-1, 1 Straight 1, -1-10,-10 Note: when column player plays “straight”, then “straight” is no longer a best response to the row player. Here, communication between players help.

33. Today Part 1: – Auctions: introduction and examples. – A very brief introduction to game theory. Part 2: – Four simple auctions – Modeling – Strategies, truthfulness. – Efficiency.

34. Experiment שיטה שניה : " שלם את הבא אחריך " ההצעה הגבוהה ביותר זוכה, והתשלום הוא ההצעה השניה הכי גבוהה. - כל תלמיד צריך לכתוב לי שתי הצעות מחיר, אחת לכל שיטת מכירה. - ההצעות יכולות, אך לא חייבות, להיות שונות זו מזו. - אפשר להציע 0 אגורות אם לא מעוניינים. - לאחר קבלת הצעות המחיר, אני אטיל מטבע ואבחר באיזו שיטה אני בוחר. שיטה ראשונה : " שלם את הצעתך " ההצעה הגבוהה ביותר זוכה, והתשלום הוא גובה ההצעה. אם המכירה תהיה " שלם את הצעתך ", הצעתי היא 4.31 שקלים אם המכירה תהיה " שלם את הבא אחריך " הצעתי היא 5.11 שלקים לדוגמא:

35. Why Auction We have an item for sale. Problem: how much bidders are willing to pay? We can ask them… They will probably lie. Auction design: motivate the buyers to reveal their values.

36. Mechanism design Auction theory is a sub-field of Mechanism Design. We design the market. “Economists as engineers” Design an auction such that in equilibrium we get the results we want.

37. Goals A seller (“auctioneer”) may have several goals. Most common goals: 1.Maximize revenue (profit) 2.Maximize social welfare (efficiency) – Give the item to the buyer that wants it the most. (regardless of payments.) 3.Fairness: for example, give items to the poor. This is our focus today.

38. Four auctions We will now present the following auctions. 1.English Auctions 2.Dutch Auctions 3.1 st -price/”pay-your-bid” auctions 4.2 nd -price/”Vickrey” auctions “Open Cry” auctions “Sealed bid” auctions

39. English Auctions

40. English Auctions at ebayebay

41. English auction - rules Price p is announced each time. – At the beginning, p=0. Raising hand by a buyer: Agreeing to buy the item for p + $1. If no bidder raised his hand for 1 minute, the item is sold. – To the bidder who made the last offer. – pays his last offer. p=0 p=1p=2p=3 bid=1bid=2bid=3 at $3

42. Dutch Auctions Dutch Flower Market

43. Today

44. Dutch auction - rules Price p is announced each time. – At the beginning, p = maximum price. Seller lowers the price by $1 at each period. First buyer to raise his hand, wins the items. – Pays current price. p=100 p=99p=98p=97 Me! at $97

45. Dutch auctions - trivia 1.One advantage: quick. – Only requires one bid! 2.US department of treasury sells bonds using Dutch auctions. 3.The IPO for Google’s stock was done using a variant of a Dutch auction.

46. Four auctions We will now present the following auctions. 1.English Auctions 2.Dutch Auctions 3.1 st -price/”pay-your-bid” auctions 4.2 nd -price/Vickrey auctions “Open Cry” auctions “Sealed bid” auctions

47. 1 st -price auctions Each bidder writes his bid in a sealed envelope. The seller: – Collects bids – Open envelopes. Winner: bidder with the highest bid. Payment: winner pays his bid. Note: bidders do not see the bids of the other bidders. $5$3$8 at $8 $5

48. 2 nd -price auctions Each bidder writes his bid in a sealed envelope. The seller: – Collects bids – Open envelopes. Winner: bidder with the highest bid. Payment: winner pays the 2 nd highest bid. Note: bidders do not see the bids of the other bidders. $2$3$8 at $5 $5

49. 2 nd -price=Vickrey Second-price auctions are also known as Vickrey auctions. Auction defined by William Vickrey in 1961. Won the Nobel prize in economics in 1996. Died shortly before the ceremony… (we will see his name again later in the course…)

50. Relations between auctions English Auction Dutch auction 1 st -price auction 2 nd -price auction How do they relate to each other?

51. Equivalent auctions 1 1 st -price auctions are strategically equivalent to Dutch auctions. Strategies: 1 st -price: given that no one has a higher bid, what is the maximum I am willing to pay? Dutch: Given that no body has raised their hand, when should I raise mine? No new information is revealed during the auction! $30$100$55$70

52. Equivalent auctions 2 2 st -price auctions are equivalent* to English auctions. Given that bidders bid truthfully, the outcomes in the two auctions are the same. * Actually, in English auctions bidders observe additional information: bids of other players. (possible effect: herd phenomena) But do bidders bid truthfully? $30$100$55$70

53. Modeling n bidders Each bidder has value v i for the item – “willingness to pay” – Known only to him – “private value” If Bidder i wins and pays p i, his utility is v i – p i – Her utility is 0 when she loses. Note: bidders prefer losing than paying more than their value.

54. Auctions scheme v1v1 v2v2 v3v3 v4v4 b1b1 b2b2 b3b3 b4b4 valuesbids winner payments $$$

55. Strategy A strategy for each bidder: how to bid given your value? Examples for strategies: – b i (v i ) = v i (truthful) – b i (v i ) = v i /2 – b i (v i ) = v i /n – If v<50, b i (v i ) = v i otherwise, b i (v i ) = v i +17 Can be modeled as normal form game, where these strategies are the pure strategies. Example for a game with incomplete information. B(v)=vB(v)=v/2B(v)=v/n…. B(v)=v …

56. Strategies and equilibrium An equilibrium in the auction is a profile of strategies B 1,B 2,…,B n such that: – Dominant strategy equilibrium: each strategy is optimal whatever the other strategies are. – Nash equilibrium: each strategy is a best response to the other strategies. Again: a strategy here is a function, a plan for the game. Not just a bid. B(v)=vB(v)=v/2B(v)=v/n…. B(v)=v …

57. Equilibrium behavior in 2 nd -price auctions That is, no matter what the others are doing, I will never gain anything from lying. – Bidding is easy, independent from our beliefs on the value of the others. Conclusion: 2 nd price auctions are efficient (maximize social welfare). – Selling to bidder with highest bid is actually selling to the bidder with the highest value. Theorem: In 2 nd -price auctions truth-telling is a dominant strategy. – in English auctions too (with private values)

58. Truthfulness: proof Case 1: Bidder 1 wins when bidding v 1. – v 1 is the highest bid, b 2 is the 2 nd highest. – His utility is v 1 - b 2 > 0. – Bidding above b 2 will not change anything (no gain from lying). – Bidding less than b 2 will turn him into a loser - from positive utility to zero (no gain from lying). v1v1 b2b2 Let’s prove now that truthfulness is a dominant strategy. We will show that Bidder 1 will never benefit from bidding a bid that is not v 1.

59. Truthfulness: proof Case 2: Bidder 1 loses when bidding v 1. – Let b 2 be the 2 nd highest bid now. – His utility 0 (losing). – Any bid below b 2 will gain him zero utility (no gain from lying). – Any bid above b 2 will gain him a utility of v 1 -b 2 < 0 - losing is better (no gain from lying). v1v1 b2b2 Let’s prove now that truthfulness is a dominant strategy. We will show that Bidder 1 will never benefit from bidding a bid that is not v 1.

60. Efficiency in 2 nd -price auctions Since 2 nd -price is truthful, we can conclude it is efficient: That is, in equilibrium, the auction allocates the item to the bidder with the highest value. With the actual highest value, not just the highest bid. Without assuming anything on the values – (For every profile of values).

61. Remark: Efficiency We saw that 2 nd –price auctions are efficient. What is efficiency (social welfare)? The total utility of the participants in the game (including the seller). For each bidder: v i – p i For the seller: (assuming it has 0 value for the item) Summing: 61

62. What we saw so far… 2 nd price and English auctions are: – Equivalent* – have a truthful dominant-strategy equilibrium. – Efficient in equilibrium. 1 st -price and Dutch auctions are: – Equivalent. – Truthful??? – Efficient???

63. 1 st price auctions Truthful? $30$100 $31 NO!

64. Bayesian analysis There is not dominant strategy in 1 st price auctions. How do people behave? They have beliefs on the preferences of the other players! Beliefs are modeled with probability distributions.

65. Bayes-Nash equilibrium Definition: A set of bidding strategies is a Nash equilibrium if each bidder’s strategy maximizes his payoff given the strategies of the others. – In auctions: bidders do not know their opponent’s values, i.e., there is incomplete information.  Each bidder’s strategy must maximize her expected payoff accounting for the uncertainty about opponent values.

66. Continuous distributions A very brief reminder of basic notions in statistics/probability.

67. Continuous distributions Reminder: Let V be a random variable that takes values from [0,t]. Cumulative distribution function F:[0,t]  [0,1] F(x) = {Probability that V<x} = Pr{V<x} The density of F is the density distribution f(x)=F’(x). The expectation of V:

68. Example: The Uniform Distribution What is the probability that V<x? F(x)=x. Density: f(x)=1 Expectation: 00.250.50.751 01 01 Area = 1

69. Auctions with uniform distributions A simple Bayesian auction model: – 2 buyers – Values are between 0 and 1. – Values are distributed uniformly on [0,1] What is the equilibrium in this game of incomplete information? Are 1 st -price auctions efficient?

70. Equilibrium in 1 st -price auctions Proof: Assume that Bidder 2’s strategy is b 2 (v)=v 2 /2. We show: b 1 (v)=v 1 /2 is a best response to Bidder 1. – (clearly, no need to bid above 1). Bidder 1’s utility is: Prob[ b 1 > b 2 ] × (v 1 -b 1 ) = Prob[ b 1 > v 2 /2 ] × (v 1 -b 1 ) = 2b 1 * (v 1 -b 1 ) [ 2b 1 * (v 1 -b 1 ) ]’ = 2v 1 -4b 1 = 0 (maximize for b 1 )  b 1 = v 1 /2 (  it is a best response for b 2 =v 2 /2) 01 b 1 =1/3 2/3 If v 2 < 2/3 then b 1 wins. Claim: bidding b(v)=v/2 is an equilibrium – 2 bidders, uniform distribution.

71. Equilibrium in 1 st -price auctions We proved: bidding b(v)=v/2 is an equilibrium – 2 bidders, uniform distribution. For n players: bidding b i (v i ) = by all players is a Nash equilibrium. (with more competition, you will bid closer to your true value) Conclusion: 1 st -price auctions maximize social welfare! (efficient) (not truthful, but in equilibrium the bidder with the highest bid wins).

72. Equilibrium in 1 st -price auctions We proved: 1 st -price auction is efficient for the uniform distribution. What about general distributions? Notation: let v 1,…,v n-1 be n-1 draws from a distribution F. Let max [n-1] = max{v 1,…,v n-1 } (highest-order statistic)

73. Equilibrium in 1 st -price auctions – That is, each bidder will bid the expected winning bid of the other players, given that v i wins. – The above b i (v i ) is strictly monotone in v i  1 st -price auction is efficient. – Example: – Proof: next week (it will be a corollary of another result). When v 1,…,v n are distributed i.i.d. from F F is strictly increasing Claim: the following is a symmetric Nash equilibrium

74. What we saw so far… 2 nd price and English auctions are: – Equivalent* – have a truthful dominant-strategy equilibrium. – Efficient in equilibrium. (“efficient”) 1 st -price and Dutch auctions are: – Equivalent. – Truthful??? – Efficient??? Actually true for all distributions, not just the uniform distribution. No! Yes!

75. Model and real life We discussed a simplified model. Real auctions are more complicated. Bidders know their values? – If so many people are willing to pay more than $100, it possibly worth it. (English auctions may help discover the value.) Auctions are (usually) repeated, and not stand-alone. Budgets and wealth effects. – I think that this TV is worth $1000, but my wife will divorce me if I pay more than $100. Manipulation is not only with bids: – collusion, false name bids, etc. Bidder has accurate probabilities? Bidder behave rationally?