Auction Theory Class 2 – Revenue equivalence 1

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

English vs. Vickrey 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. The English Auction: Price starts at 0 Price increases until only one bidder is left. Vickrey (2 nd price) auction: Bidders send bids. Highest bid wins, pays 2 nd highest bid. 3

Dutch vs. 1 st -price Dutch auctions and 1 st 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?” The Dutch Auction: Price starts at max-price. Price drops until a bidder agrees to buy. 1 st- price auction: Bidders send bids. Highest bid wins, pays his bid. 4

Bayes-Nash eq. in 1 st -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. – 1 st price is efficient! 5

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). 6

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) 7

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

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 2 nd -price and 1 st price auctions?

Revenue in 2 nd -price auctions In 2 nd -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

Revenue in 2 nd -price auctions Proof: – assume that v 1 =x. Then, the expected revenue is: We can now compute the expected revenue (expectation over all possible x): 01x

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

Expected order statistics /2 1/32/3 2/41/43/4 One sample Two samples Three samples In general, for the uniform distribution with n samples: k’th order statistic of n variables is (n+1-k)/n+1) 1 st -order statistic: n/n+1

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

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

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

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 – 2 nd -price auctions maximize efficiency in a Bayes-Nash equilibrium Even stronger solution: truthfulness (in dominant strategies). – 1 st price auctions also achieve this goal. Not truthful, no dominant strategies. – Many other auctions are efficient (e.g., all-pay auctions).

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?

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.

Theorem (“The Revelation Principle”): Consider an auction where the profile of strategies s 1,…,s n is a Bayes-Nash equilibrium. Then, there exists a truthful mechanism with exactly the same allocation and payments (“payoff equivalent”). The revelation principle Recall: truthful = direct revelation + truthful Bayes-Nash equilibrium. Basic idea: we can simulate any mechanism via a truthful mechanism which is payoff equivalent.

The revelation principle Proof (trivial): The original mechanism: v1v1 v 1  s 1 (v 1 ) Auction protocol Allocation (winners) payments v2v2 v 2  s 2 (v 2 ) v3v3 v 3  s 3 (v 3 ) v4v4 v 4  s 4 (v 4 ) s 1 (v 1 ) s 2 (v 2 ) s 3 (v 3 ) s 4 (v 4 ) Bidders Auction mechanism

The revelation principle Proof (trivial): A direct-revelation mechanism: v1v1 v 1  s 1 (v 1 ) Auction protocol Allocation (winners) payments v2v2 v 2  s 2 (v 2 ) v3v3 v 3  s 3 (v 3 ) v4v4 v 4  s 4 (v 4 ) v1v1 v2v2 v3v3 v4v4 Bidders reports their true types, The auction simulates their equilibrium strategies. Equilibrium is straightforward: if a bidder had a profitable deviation here, he would have one in the original mechanism.

The revelation principle Example: – In 1 st -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.

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

Revenue equivalence We saw examples where the revenue in 2 nd -price and 1 st -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. 25

Revenue Equivalence Theorem Assumptions: – v i ‘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: 1.(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. 2.(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.

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. 27

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

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

Proof 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). 30 integrating Assume u i (a)=0

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

Revenue equivalence theorem No coincidence! – Somewhat unintuitively, revenue depends only on the way the winner is chosen, not on payments. – Since 2 nd -price auctions and 1 st -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. 33

Remark: Individual rationality The following mechanism gains lots of revenue: – Charge all players $ 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. 34

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

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. 36

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

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 (b 1,b 2 ) in fighting. 38

What did we see so far 2 nd -price, 1 st -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, – … 39

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