Auction Theory

1. Auction Theory Class 2 – Revenue equivalence

2. This class: revenue • Revenue in auctions • Connection to order statistics • The revelation principle • The revenue equivalence theorem • Example: all-pay auctions.

3. English vs. Vickrey • The English Auction: • Price starts at 0 • Price increases until only one • bidder is left. • Vickrey (2nd price) auction: • Bidders send bids. • Highest bid wins, pays 2nd highest bid. • Private value model: each person has a privately known value for the item. • We saw: the two auctions are equivalent in the private value model. • Auctions are efficient:dominant strategy for each player: truthfulness.

4. Dutch vs. 1st-price • The Dutch Auction: • Price starts at max-price. • Price drops until a bidder agrees to buy. • 1st-price auction: • Bidders send bids. • Highest bid wins, pays his bid. • Dutch auctions and 1st price auctions are strategically equivalent. (asynchronous vs simple & fast) • No dominant strategies. (tradeoff: chance of winning, payment upon winning.) • Analysis in a Bayesian model: • Values are randomly drawn from a probability distribution. • Strategy: a function. “What is my bid given my value?”

5. Bayes-Nash eq. in 1st-price auctions • We considered the simplest Bayesian model: • n bidders. • Values drawn uniformly from [0,1]. Then: In a 1st-price auction, it is a (Bayesian) Nash equilibrium when all bidders bid • An auction is efficient, if in (Bayes) Nash equilibrium the bidder with the highest value always wins. • 1st price is efficient!

6. Optimal auctions • Usually the term optimal auctions stands for revenue maximization. • What is maximal revenue? • We can always charge the winner his value. • Maximal revenue: optimal expected revenue in equilibrium. • Assuming a probability distribution on the values. • Over all the possible mechanisms. • Under individual-rationality constraints (later).

7. Example: Spectrum auctions • One of the main triggers to auction theory. • FCC in the US sells spectrum, mainly for cellular networks. • Improved auctions since the 90’s increased efficiency + revenue considerably. • Complicated (“combinatorial”) auction, in many countries. • (more details further in the course)

8. New Zealand Spectrum Auctions • A Vickrey (2nd price) auction was run in New Zealand to sale a bunch of auctions. (In 1990) • Winning bid: $100000 Second highest: $6 (!!!!) Essentially zero revenue. • NZ Returned to 1st price method the year after. • After that, went to a more complicated auction (in few weeks). • Was it avoidable?

9. 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 expected revenue gained by 2nd-price and 1st price auctions?

10. Revenue in 2nd-price auctions • In 2nd-price auction, the payment is the minimum of the two values. • E[ revenue] = E[ min{x,y} ] • Claim: when x,y ~ U[0,1] we have E[ min{x,y} ]=1/3

11. Revenue in 2nd-price auctions • Proof: • assume that v1=x. Then, the expected revenue is: • We can now compute the expected revenue (expectation over all possible x): 0 x 1

12. Order statistics Let v1,…,vnbe n random variables. • The highest realization is called the 1st-order statistic. • The second highest is the called 2nd-order statistic. • …. • The smallest is the nth-order statistic. Example: the uniform distribution, 2 samples. • The expected 1st-order statistic: 2/3 • In auctions: expected efficiency • The expected 2nd-order statistic: 1/3 • In auctions: expected revenue

13. Expected order statistics One sample 0 1 1/2 Two samples 0 1 1/3 2/3 Three samples • In general, for the uniform distribution with n samples: • k’th order statistic of n variables is (n+1-k)/n+1) • 1st-order statistic:n/n+1 0 1 1/4 2/4 3/4

14. Revenue in 1st-price auctions • We still assume 2 bidders, uniform distribution • Revenue in 1st price: • bidders bid vi/2. • Revenue is the highest bid. • Expected revenue = E[ max(v1/2,v2/2) ] • = ½ E[ max(v1,v2)] • = ½ × 2/3 • = 1/3 Same revenue as in 2nd-price auctions.

15. 1st vs. 2nd price • Revenue in 2nd price: • Bidders bid truthfully. • Revenue is 2nd highest bid: • Revenue in 1st price: • bidders bid • Expected revenue is What happened? Coincidence?

16. This class • Revenue in auctions • Connection to order statistics • The revelation principle • The revenue equivalence theorem • Example: all-pay auctions.

17. Implementation Our general goal: given an objective (for example, maximize efficiency or revenue), construct an auction that achieves this goal in an equilibrium. • "Implementation” • Equilibrium concept: Bayes-Nash For example: when our goal is maximal efficiency • 2nd-price auctions maximize efficiency in a Bayes-Nash equilibrium • Even stronger solution: truthfulness (in dominant strategies). • 1st price auctions also achieve this goal. • Not truthful, no dominant strategies. • Many other auctions are efficient (e.g., all-pay auctions).

18. Terminology Direct-revelation mechanism: player are asked to report their true value. • Non direct revelation: English and Dutch auction, most iterative auctions, concise menu of actions. • Concepts relates to the message space in the auction. Truthful mechanisms: direct-revelation mechanisms where revealing the truth is (a Bayes Nash) equilibrium. • Other solution concepts may apply. • Alternative term: Incentive Compatibility. • What’s so special about revealing the truth? • Maybe better results can be obtained when people report half their value, or any other strategy?

19. The revelation principle • Problem: the space of possible mechanisms is often too large. • A helpful insight: we can actually focus our attention to truthful (direct revelation) mechanisms. • This will simplify the analysis considerably. • “The revelation principle” • “every outcome can be achieved by truthful mechanism” • One of the simplest, yet trickiest, concepts in auction theory.

20. The revelation principle Theorem (“The Revelation Principle”): Consider an auction where the profile of strategies s1,…,snis a Bayes-Nash equilibrium. Then, there exists a truthful mechanism with exactly the same allocation and payments (“payoff equivalent”). Recall:truthful = direct revelation + truthful Bayes-Nash equilibrium. • Basic idea: we can simulate any mechanism via a truthful mechanism which is payoff equivalent.

21. The revelation principle • Proof (trivial): The original mechanism: Auction mechanism Bidders Auction protocol v1s1(v1) s1(v1) v1 Allocation (winners) v2 s2(v2) s2(v2) v2 payments v3 s3(v3) v3 s3(v3) v4 s4(v4) s4(v4) v4

22. The revelation principle Bidders reports their true types, The auction simulates their equilibrium strategies. • Proof (trivial): A direct-revelation mechanism: Auction protocol v1 v1s1(v1) v1 Allocation (winners) v2 v2 s2(v2) v2 payments v3 v3 s3(v3) v3 v4 s4(v4) v4 v4 Equilibrium is straightforward: if a bidder had a profitable deviation here, he would have one in the original mechanism.

23. The revelation principle • Example: • In 1st-price auctions with the uniform distribution: bidders would bid truthfully and the mechanism will “change” their bids to be • In English auctions (non direct revelation):people will bid truthfully, and the mechanism will raise hands according to their strategy in the auction. • Bottom line:Due to the revelation principle, from now on we will concentrate on truthful mechanisms.

24. This class • Revenue in auctions • Connection to order statistics • The revelation principle • The revenue equivalence theorem • Example: all-pay auctions.

25. Revenue equivalence • We saw examples where the revenue in 2nd-price and 1st-price auctions is the same. • Can we have a general theorem? • Yes. Informally:What matters is the allocation. Auctions with the same allocation have the same revenue.

26. Revenue Equivalence Theorem Assumptions: • vi‘s are drawn independently from some F on [a,b] • F is continuous and strictly increasing • Bidders are risk neutral Theorem(The Revenue Equivalence Theorem): Consider two auction such that: • (same allocation) When player i bids v his probability to win is the same in the two auctions (for all i and v) in equilibrium. • (normalization) If a player bids a (the lowest possible value) he will pay the same amount in both auctions. Then, in equilibrium, the two auctions earn the same revenue.

27. Proof • Idea:we will start from the incentive-compatibility (truthfulness) constraints.We will show that the allocation function of the auction actually determines the payment for each player. • If the same allocation function is achieved in equilibrium, then the expected payment of each player must be the same. • Note: Due to the revelation principle, we will look at truthful auctions.

28. Proof • Consider some auction protocol A, and a bidder i. • Notations: in the auction A, • Qi(v) = the probability that bidder i wins when he bids v. • pi(v) = the expected payment of bidder i when he bids v. • ui(v) = the expected surplus (utility) of player i when he bids v and his true value is v. ui(v) = Qi(v) v - pi(v) • In a truthful equilibrium: i gains higher surplus when bidding his true value v than some value v’. • Qi(v) v - pi(v) ≥ Qi(v’) v - pi(v’) =ui(v) =ui(v’)+ ( v – v’) Qi(v’) We get: truthfulness  ui(v)≥ ui(v’)+ ( v – v’) Qi(v’)

29. Proof • We get: truthfulness  ui(v)≥ ui(v’)+ ( v – v’) Qi(v’) or • Similarly, since a bidder with true value v’ will not prefer bidding v and thus ui(v’)≥ ui(v)+ ( v’ – v) Qi(v) or Let dv = v-v’ Taking dv  0 we get:

30. Assume ui(a)=0 Proof integrating • We saw: • We know: • And conclude: • Of course: • Interpretation: expected revenue, in equilibrium, depends only on the allocation. • same allocation  same revenue (as long as Q() and ui(a) are the same).

31. Picture

32. Example: 2 players, uniform dist. Q1(v)= v pi(1/2)= 1/2*1/2*1/2 v*v*1/2=v2/2 pi(v)= The expected revenue from bidder 1: For 2 bidders: E[revenue]=1/6+1/6=1/3 1/2 1/2

33. Revenue equivalence theorem • No coincidence! • Somewhat unintuitively, revenue depends only on the way the winner is chosen, not on payments. • Since 2nd-price auctions and 1st-price auctions have the same (efficient) allocation, they will earn the same revenue! • One of the most striking results in mechanism design • Applies in other, more general setting. • Lesson:when designing auctions, focus on the allocation, not on tweaking the prices.

34. Remark: Individual rationality • The following mechanism gains lots of revenue: • Charge all players $10000000 • Bidder will clearly not participate. • We thus have individual-rationality (or participation) constraints on mechanisms:bidders gain positive utility in equilibrium . • This is the reason for condition 2 in the theorem.

35. This class • Revenue in auctions • Connection to order statistics • The revelation principle • The revenue equivalence theorem • Example: all-pay auctions.

36. Example: All-pay auction (1/3) • Rules: • Sealed bid • Highest bid wins • Everyone pay their bid • Claim: Equilibrium with the uniform distribution: b(v)= • Does it achieve more or less revenue? • Note: Bidders shade their bids as the competition increases.

37. All-pay auction (2/3) • expected payment per each player: herbid. • Each bidder bids • Expected payment for each bidder: • Revenue: from n bidders • Revenue equivalence!

38. All-pay auction (3/3) • Examples: • crowdsourcing over the internet: • First person to complete a task for me gets a reward. • A group of people invest time in the task. (=payment) • Only the winner gets the reward. • Advertising auction: • Collect suggestion for campaigns, choose a winner. • All advertiser incur cost of preparing the campaign. • Only one wins. • Lobbying • War of attrition • Animals invest (b1,b2) in fighting.

39. What did we see so far • 2nd-price, 1st-price, all pay: all obtain the same seller revenue. • Revenue equivalence theorem:Auctions with the same allocation decisions earn the same expected seller revenue in equilibrium. • Constraint: individual rationality (participation constraint) • Many assumptions: • statistical independence, • risk neutrality, • no externalities, • private values, • …

40. Next topic • Optimal revenue: which auctions achieve the highest revenue?