Auction Theory Class 2 Revenue equivalence 1

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

Dutch vs. 1 st-price 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. • 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? ” 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 1 st-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: $100000 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: 0 • We can now compute the expected revenue (expectation over all possible x): x 1

Order statistics Let v 1, …, vn 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 nth-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 One sample 0 1 1/2 Two samples 0 1/3 1 2/3 Three samples 0 1/4 2/4 3/4 1 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 vi/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 Same revenue as in 2 nd-price auctions. 14

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

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.

The revelation principle Theorem (“The Revelation Principle”): Consider an auction where the profile of strategies s 1, …, sn is 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.

The revelation principle • Proof (trivial): The original mechanism: Bidders v 1 s 1(v 1)) s 1(v 1 v 2 s 22(v 22)) s (v v 3 s 3(v 3)) s 3(v 3 Auction mechanism v 4 s 4(v 4)) s 4(v 4 Allocation (winners) Auction protocol payments

The revelation principle • Proof (trivial): A direct-revelation mechanism: Bidders reports their true types, The auction simulates their equilibrium strategies. v 1 s 1(v 1) v 2 s 2(v 2) v 3 s 3(v 3) v 4 Allocation (winners) Auction protocol payments v 4 s 4(v 4) 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: – 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: 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, – 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’)+ ( v – v’) Qi(v’) We get: truthfulness ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’) 28

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: 29

Proof • We saw: Assume ui(a)=0 integrating • 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

Picture 31

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

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 $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 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