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## Auction Theory

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**Auction Theory**Class 7 – Common Values, Winner’s curse and Interdependent Values.**Outline**• Winner’s curse • Common values • in second-price auctions • Interdependent values • The single-crossing condition. • An efficient auction. • Correlated values • Cremer & Mclean mechanism**Common Values**• Last time in class we played 2 games: • Each student had a private knowledge of xi, and the goal was to guess the average. • Students with high signals tended to have higher guesses. • Students were asked to guess the total value of a bag of coins. • We should have gotten: some bidders overestimate. • Today: we will model environments when there is a common value, but bidders have different pieces of information about it.**Winner’s curse**• These phenomena demonstrate the Winner’s Curse: • Winning means that everyone else was more pessimistic than you the winner should update her beliefs after winning. • Winning is “bad news” • Winners typically over-estimate the item’s value. • Note: Winner’s curse does not happen in equilibrium. Bidders account for that in their strategies.**Modeling common values**• First model: Each bidder has an estimate ei=v + xi • v is some common value • ei is an unbiased estimator (E[xi]=0) • Errors xi are independent random variables. • Winner’s curse: consider a symmetric equilibrium strategy in a 1st-price auction. • Winning means: all the other had a lower signal my estimate should decrease. • Failing to foresee this leads to the Winner’s curse.**Winner’s curse: some comments**• The winner’s curse grows with the market size:if my signal is greater than lots of my competitors, over-estimation is probably higher. • The highest-order statistic is not an unbiased estimator. • With common values:English auctions and Vickrey auctions are no longer equivalent. • Bidders update beliefs after other bidders drop out. • Two cases where the two auctions are equivalent: • 2 bidders (why?) • Private values**A useful notation: v(x,y)**• What is my expected value for the item if: • My signal is x. • I know that the highest bid of the other bidders is y?v(x,y) = E[v1 | x1=x and max{y2,…,yn}=y ] • We will assume that v(x,y) is increasing in both coordinates and that v(0,0)=0.**A useful notation: x-i**• We will sometime use x=x1,…,xn • Given a bidder i, let x-i denote the signals of the other bidders: x-i=x1,…,xi-1,xi+1,…,xn • x=(xi,x-i) • (z,x-i) is the vector x1,…,xnwhere the i’th coordinate is replaced with z.**Second-price auctions**• With common values, how should bidder bid? • Naïve approach: bid according to the estimate you have: v+xi • Problem: does not take into account the winner’s curse. • Bidders will thus shade their bids below the estimates they currently have.**Second-price auctions**In the common value setting: • Theorem: bidding according to β(xi)=v(xi,xi) is a Nash equilibrium in a second-price auction. • That is, each bidder bids as if he knew that the highest signal of the others equals his own signal. • Bid shading increases with competition:I bid as if I know that all other bidders have signals below my signal (and the highest equals my signal) • With small competition, no winner’s curse effect.**Second-price auctions**In the common value setting: • Theorem: bidding according to β(xi)=v(xi,xi) is a Nash equilibrium in a second-price auction. • Equilibrium concept:Unlike the case of private values, equilibrium in the 2nd-price auction is Bayes-Nash and not dominant strategies. • Bidder need to take distributions into account.**Second-price auctions**In the common value setting: • Theorem: bidding according to β(xi)=v(xi,xi) is a Nash equilibrium in a second-price auction. • Intuition: (assume 2 bidders) • b() is a symmetric equilibrium strategy. • Consider a small change of ε in my bid: since the other bidder bids with b(), if his bid is far from b(xi) then an ε change will not matter. • A small change in my bid will matter only if the bids are close. • I might win and figure out that the other signal was very close to mine. • I might lose and figure out the same thing. • I should be indifferent between winning and pay b(x), and losing.**Second-price auctions**In the common value setting: • Theorem: bidding according to β(xi)=v(xi,xi) is a Nash equilibrium in a second-price auction. • Proof: • Assume that the other bidders bid according to b(xi)=v(xi,xi). • The expected utility of bidder i with signal x that bids β is • Where y=max{x-i} • g[y|x] is the density of y given x. • Bidder i wins when all other signals are less than b-1(β)**Second-price auctions**Let’s plot v(x,y)-v(y,y) Recall: v(x,y) increasing in x (for all x,y) y x Utility is maximized when bidding b= β(x)= v(x,x)**Second price auctions: example**• Example: v ~ U[0,1] xi ~ U[0,2v] n = 3 • Equilibrium strategy: • See Krishna’s book for the details.**Symmetric valuations**• The exact theorem and proof actually works for a more general model: symmetric valuations. • That is, there is some function u such that for all i: • vi(x1,….,xn)=u(xi,x-i) • Generalizes private values: vi(x1,….,xn)=u(xi) • It also works for joint distributions, as long they are symmetric.**Game of Trivia**Question 1: What is the distance between Paris and Moscow? Question 2: What is the year of birth of David Ben-Gurion?**Information Aggregation**Common-value auctions are mechanisms for aggregating information. • “The wisdom of the crowds” and Galton’s ox. • In our model, the average is a good estimation • E[ei] = E[v+xi] = E[v] + E[xi] = v+E[xi] ≈ v • One can show: if bidders compete in a 1st-price or a 2nd-price auctions, the sale price is a good estimate for the common value. • Some conditions apply. • Intuition: Thinking that the largest value of the others is equal to mine is almost true with many bidders.**Outline**• Winner’s curse • Common values • in second-price auctions • Interdependent values • The single-crossing condition. • An efficient auction. • Correlated values • Cremer & Mclean mechanism**Interdependent values**• We now consider a more general model: interdependent values • the valuations are not necessarily symmetric. • The value of a bidder is a functions of the signals of all bidders: vi(x1,…,xn) • We assume vi is non decreasing in all variables, strictly increasing in xi. • Again, private values are a special case: vi(x1,…,xn)=vi(xi) • There might still be more uncertainty: then, vi(x1,…,xn) is the expected value over the remaining uncertainty. • vi(x1,…,xn)=E[vi | x1,…,xn ]**Interdependent values**• Example:v1(x1, x2,x3) = 5x1 + 3x2 + x3 v2(x1, x2,x3) = 2x1 + 9x2 + (x3)3v2(x1, x2,x3) = 2x1x2 + (x3)2**Efficient auctions**• Can we design an efficient auction for settings with interdependent values? • No. Claim: no efficient mechanism exists forv1(x1, x2) = x1 v2(x1, x2) = (x1)2 Where x1 is drawn from [0,2]**Efficient auctions**Claim: no efficient mechanism exists forv1(x1, x2) = x1v2(x1, x2) = (x1)2 Where x1 is drawn from [0,2] • Proof: • What is the efficient allocation? • give the item to 1 when x1<1, otherwise give it to 2. • Let p be a payment rule of an efficient mechanism. • Let y1<1<z1 be two types of player 1. Together: y1 ≥ z1 contradiction. y1 z1 1 When 1’s true value is y1: y1-p1(y1) ≥ 0-p1(z1) When 1’s true value is z1:0 - p1(z1)≥ z1 – p(y1) (efficiency + truthfulness)**Single-crossing condition**Conclusion: For designing an efficient auction we will need an additional technical condition. Intuitively: for every bidder, the effect of her own signal on her valuations is stronger than the effect of the other signals. • v1(x1, x2) = x1, v2(x1, x2) = (x1)2 • v1(x1, x2) = 2x1+5x2,v2(x1, x2) = 4x1+2x2**Single-crossing condition**Definition: Valuations v1,…,vn satisfy the single-crossing condition if for every pair of bidders i,j we have: for all x, • Actually, a weaker condition is often sufficient • Inequality holds only when vi(x)=vi(y) and both are maximal. • Single crossing: fixing the other signals, i’s valuations grows more rapidly with xi than j’s valuation.**Single crossing: examples**• For example: when we plot v1(x1, x2,x3) and v2(x1, x2,x3) as a function of x1(fixing x2 and x3) v1(x1, x2,x3) v2(x1, x2,x3) x1 For every x, the slope of v1(x1, x2,x3) is greater.**Single crossing: examples**• v1(x1, x2) = x1 , v2(x1, x2) = (x1)2 are not single crossing. • v1(x1, x2,x3) = 5x1 + 3x2 + x3 v2(x1, x2,x3) = 2x1 + 9x2 + x3v3(x1, x2,x3) = 3x1 + 2x2 + 2x3are single crossing y1 z1 1 x1**An Efficient Auction**Consider the following direct-revelation auction: • Bidders report their signals x1,…,xn • The winner: the bidder with the highest value (given the reported signals). • Argmaxvi(x1,…,xn) • Payments:the winner pays M*(i)=vi( yi(x-i) , x-i)whereyi(x-i) = min{ zi | vi(zi,x-i) ≥ maxj≠ivj(zi,x-i) } • In other words, yi(x-i) is the lowest signal for which iwins in the efficient outcome (given the signals x-i of the other bidders) • Losers pay zero.**An Efficient Auction**What is the payment of bidder 1 when he wins with a signal ? v1(x1, x-i) v2(x1, x-i) v3(x1, x-i) M*(i) x1 y1(x-1)**An Efficient Auction**What is the problem with the standard second-price payment (given the reported signals)? • i.e., 1 should pay v2(x1, x-i)? • In the proposed payments, like 2nd-price auctions with private value, price is independent of the winner’s bid. v1(x1, x-i) v2(x1, x-i) v3(x1, x-i) M*(i) x1 y1(x-1)**An Efficient Auction**Theorem: when the valuations satisfy the single-crossing condition, truth-telling is an efficient equilibrium of the above auction. Equilibrium concept: stronger than Nash (but weaker than dominant strategies): ex-post Nash**Ex-post equilibrium**• Given that the other bidders are truthful, truthful bidding is optimal for every profile of signals. • No bidder, nor the seller, need to have any distributional assumptions. • A strong equilibrium concept. • Truthfulness is not a dominant strategy in this auction. • Why? • My “declared value” depends on the declarations of the others.If some crazy bidder reports avery high false signal, I may win and pay more than my value.**An Efficient Auction:proof**Proof: • Suppose i wins for the reports x1,…,xn, that is, vi(xi,x-i) ≥ maxj≠ivj(xi,x-i). • Bidderipaysvi(yi(x-i) ,x-i), where yi(x-i) is its minimal signal for which his value is greater than all others. • vi(yi(x-i) ,x-i) < vi(xi ,x-i) non-negative surplus. Due to single crossing: • For any bid zi>yi(x-i), his value will remain maximal, and he will still win (paying the same amount). • For any bid zi≤yi(x-i), he will lose and pay zero. No profitable deviation for a winner.**An Efficient Auction:proof**Proof (cont.): • Suppose i loses for the reports x1,…,xn,that is, vi(xi,x-i) < maxj≠ivj(xi,x-i). • xi< yi(x-i) • Payoff of zero • To win, I must report zi>yi(x-i). • Still losing when bidding lower (single crossing). • Then payment will be: M*(i) = vi( yi(x-i) , x-i) > vi(xi, x-i)generating a negative payoff.**Weakness**Weakness of the efficient auction: seller needs to know the valuation functions of the bidders • Does not know the signals, of course.**Outline**• Winner’s curse • Common values • in second-price auctions • Interdependent values • The single-crossing condition. • An efficient auction. • Correlated values • Cremer & Mclean mechanism**Revenue**• In the first few classes we saw: with private, independent values, bidders have an “information rent” that leaves them some of the social surplus. • No way to make bidders pay their values in equilibrium. • We will now consider revenue maximization with statistically correlated types.**Discrete values**• We will assume now that signals are discrete • drawn from a distribution on Xi={Δ, 2Δ, 3Δ,….,TiΔ}(For simplicity, let Xi={1, 2, 3,….,Ti} ) • think about Δ as 1 cent • The analysis of the continuous case is harder. • We still require single-crossing valuations, with the discrete analogue: for all i and k, and every xi,vi(xi, Δ+x-i) - vi(xi,x-i)≥ vk(xi, Δ + x-i) - vk(xi,x-i)**Correlated values**For the Generalized-VCG auction to work, signals are not necessarily statistically independent: correlation is allowed. Which one is not a product of independent distributions?: Independent distributions:f1(1)=1/6, f1(2)=1/3, f1(3)=1/2 f2(1)=1/4, f2(2)=1/2, f2(3)=1/4 A joint distribution x2 x2 x1 x1**Revenue**• Example: let’s consider the joint distribution • Let’s consider 2nd-price auctions: • Expected welfare: 14/6 • Expected revenue for the seller: 10/6 • Expected revenue with optimal reserve price (R=2): 11/6 • Can the seller do better? • Intuitively, information rent should be smaller (seller can gain information from other bidders’ values)**Revenue: example**• Consider the following auction: • Efficient allocation (given the bids), ties randomly broken. • Payments: see table for payment for bidder 1 Claim: the auction is truthful • Example: when x1=2, assume bidder 2 is truthful. • u1(b1=2)= 0.25*(2-0) + 0.5*(0.5*2-1) + 0.25*(-2) • u1(b1=1) = 0.25*(0.5*2+1/2) +0.5*(0) + 0.25*(-2) = - 0.125 • Note: although bidder 1 bids 1, the true probabilities are according to x1=2. • u1(b1=3) = 0.25*(2-0) + 0.5*(2-2) + 0.25*( 0.5*2 –3.5 ) = -0.125 =0**Revenue: example**• Consider the following auction: • Efficient allocation (given the bids), ties randomly broken. • Payments: see table for payment for bidder 1 Claim:E[seller’s revenue]=14/6 • Equals the expected social welfare • Easy way to see: the expected surplus of each bidder is 0.**Revenue**• Conclusions from the previous example: • An incentive compatible, efficient mechanism that gains more revenue than the 2nd-price auction • Revenue equivalence theorem doesn’t hold with correlated values. • The expected surplus of each bidder is 0 • Seller takes all surplus. No information rent. • Is this a general phenomenon? • Surprisingly: with correlated types, the seller can get all surplus leaving bidders with 0 surplus. • Even with slight correlation.**Revenue**• The Cremer-Mclean Condition: the conditional correlation matrix has a full rank for every bidder. • That is, some minimal level of correlation exists.**The correlation matrix**Pr(x1,…,xn) Pr(x-i | xi) x-i Correlated Full rank (3) xi Rank 1 independent**Revenue**• The Cremer-Mclean Condition: the conditional correlation matrix has a full rank for every bidder. • That is, some minimal level of correlation exists. • Theorem (Cremer & Mclean, 1988):Under the Cremer-Mclean condition, then there exists an efficient, truthful mechanism that extracts the whole surplus from the bidders. • That is, seller’s profit = the maximal social welfare • The expected surplus of each bidder is zero.**Revenue**• We will now construct the Cremer-Mclean auction. • Idea: modify the truthful auction (“generalized VCG”) that we saw earlier. • Remark: The Cremer-Mclean auction is • not ex-post individually rational • (sometimes bidders pay more than their actual value) • Interim individually rational • Given the bidder value, he will gain zero surplus in expectation (over the values of the others).**Reminder:”Generalized VCG”**• Bidders report their signals x1,…,xn • The winner: the bidder with the highest value (given the reported signals). • Payments:the winner pays Mi*=vi( yi(x-i) , x-i)whereyi(x-i) = min{ zi | vi(zi,x-i) ≥ maxj≠ivj(zi,x-i) } + ci(x-i) • A general observation: adding to the payment of bidder any term which is independent of her bid will not change her behavior. • Mi*=vi( yi(x-i) , x-i) + ci(x-i)**The trick**• The expected surplus of each bidder: As before, Qi(x1,…,xn) is the probability that bidder i wins. • For every i, we would like now to find values ci(x-i) such that and for every xi: That’s the conditional probability for which the Cremer-Mclean condition applies**The trick (cont.)**• If we could find such values ci(x-i), we will add it to the bidders’ payments. • As observed, it will not change the incentives. • The expected surplus of bidder i is now: =Ui* by definition =Ui* due to the choice of ci(x-i)