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## Auction Theory

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**Auction Theory**Class 2 – Revenue equivalence**This class**• Revenue in auctions • 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: • nbidders. • 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): 0 x 1**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**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**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: • 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 • 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,…,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.**The revelation principle**• Proof (trivial): The original mechanism: Auction protocol v1s1(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**The revelation principle**Bidders reports their true types, The auction simulates their equilibrium strategies. • Proof (trivial): A direct-revelation mechanism: Auction protocol v1 v1s1(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.**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 • 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.**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:**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.**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 • 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: herbid. • 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?

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