Class 2 – Revenue equivalence Auction Theory Class 2 – Revenue equivalence

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

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.

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?”

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!

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).

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)

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?

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?

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

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): x 1

Order statistics Let v1,…,vn be 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

Expected order statistics 1 1/2 1/3 2/3 2/4 1/4 3/4 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) 1st-order statistic: n/n+1

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.

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

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

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: 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).

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. Truthful mechanisms: direct-revelation mechanisms where revealing the truth is (a Bayes Nash) equilibrium. Other solution concepts may apply. 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 s1,…,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: Auction protocol v1 v1s1(v1) s1(v1) Allocation (winners) v2 v2 s2(v2) s2(v2) payments v3 v3 s3(v3) s3(v3) v4 s4(v4) s4(v4) v4

The revelation principle Proof (trivial): A direct-revelation mechanism: Bidders reports their true types, The auction simulates their equilibrium strategies. Auction protocol v1 v1 v1s1(v1) Allocation (winners) v2 v2 v2 s2(v2) payments v3 v3 v3 s3(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.

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 mechanism.

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

Revenue equivalence We saw examples where the revenue in 2nd-price and 1st-price auctions is the same. Can we have a general theorem?

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.

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. Due to the revelation principle, we will look at truthful auctions.

=ui(v’)+ ( v – v’) Qi(v’) 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’)

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:

Proof Assume ui(a)=0 We saw: We know: And conclude: Of course: Interpretation: expected revenue, in equilibrium, depends only on the allocation. same allocation same revenue. integrating

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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.

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.

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

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.

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!

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.

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, …

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