Auctions and Market Design

Transcription

1 Auctions and Market Design Peter Cramton Professor of Economics University of Maryland, European University Institute, University of Cologne 24 September

2 Introduction

3 Market design Establishes rules of market interaction Economic engineering Economics Computer science Engineering, operations research 3

4 Market design accomplishments Improve allocations Improve price information Reduce risk Enhance competition Mitigate market failures 4

5 Applications Electricity markets Communication markets Financial securities Transportation markets Climate policy Natural resource auctions (timber, oil, diamonds, pollution emissions) Procurement 5

6 Innovation in a few words Electricity Open access Pay for performance Communications Auction spectrum Open access Transportation Price congestion Climate policy Price carbon Financial securities Make time discrete Resistance to market reforms is the norm: Peter to regulator, I can save you $10 billion. Wall Street to regulator, Hey, that s my $10 billion. Distribution trumps efficiency 6

7 Milton Friedman There is enormous inertia a tyranny of the status quo in private and especially governmental arrangements. Only a crisis actual or perceived produces real change. When that crisis occurs, the actions that are taken depend on the ideas that are lying around. That, I believe, is our basic function [as economists]: to develop alternatives to existing policies, to keep them alive and available until the politically impossible becomes politically inevitable. 7

8 Objectives Efficiency Simplicity Transparency Fairness 8

9 Outline Equilibrium of games with incomplete information Mechanism design Auctioning many similar items Revenue equivalence and optimal auctions Applications Electricity Spectrum Mobile communications Transportation Financial securities

10 Readings Demand Reduction and Inefficiency in Multi-Unit Auctions, (with Lawrence M. Ausubel, Marek Pycia, Marzena Rostek, and Marek Weretka) Review of Economic Studies, 81:4, , The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response (with Eric Budish and John Shim), Quarterly Journal of Economics, forthcoming, September An Open Access Wireless Market (with Linda Doyle), Working Paper, University of Maryland, 30 August Price Carbon I will if you will (with David MacKay, Axel Ockenfels and Steven Stoft), Nature, 9 October 2015.

11 First some math Calculating equilibrium behavior in games with incomplete information Warning: A little knowledge is a dangerous thing!

12 Equilibrium in first-price auction Each of n bidder s private value v drawn from distribution F Bidder's expected profit: π(v,b(v)) = (v - b(v))pr(win b(v)). By the envelope theorem, dπ π b π π = dv b v + v = v But then dπ/dv = Pr(Win b(v)) = Pr(highest bid) = Pr(highest value) = F(v) n-1 12

13 Equilibrium in first-price auction By the Fundamental Theorem of Calculus, X v v n 1 = + Z Y X = Z Y 0 0 n 1 π( v) π( 0) F(u) du F(u) du, Substituting into π(v,b(v)) = (v - b(v))pr(win b(v)) yields b( v) X π( v) (n 1) n 1 = v = v F(v) F(u) du. Pr(Win) Z Y v 0 13

14 Example v U on [0,1] Then F(v) = v, so b(v) = v - v/n = v(n-1)/n. The optimal bid converges to the value as n, so in the limit the seller is able to extract the full surplus. In equilibrium, the bidder bids the expected value of the second highest value given that the bidder has the highest value. 14

15 Bargaining: Simultaneous offers (Chatterjee & Samuelson, Operations Research 1983) A seller and a buyer are engaged in the trade of a single object worth s to the seller and b to the buyer. Valuations are known privately, as summarized below Traders Value Distributed Payoff Private Info Common Knowledge Strategy (Offer) Seller s s F on [s, s] u =P s s F, G p(s) Buyer b b G on [ b, b] v =b P b F, G q(b) 15

16 Simultaneous Offers Independent private value model: s and b are independent random variables. Ex post efficiency: trade if and only if s < b. Game: Each player simultaneously names a price; if p q then trade occurs at the price P = (p + q)/2; if p > q then no trade (each player gets zero). 16

17 Simultaneous Offers Payoffs: Seller u(p,q,s,b) = P - s if p q 0 if p > q Buyer R S T R S T v(p,q,s,b) = b - P if p q 0 if p > q where the trading price is P = (p + q)/2 17

18 Example Let F and G be independent uniform distributions on [0,1]. Equilibrium conditions: (1) s [s, s],p(s) argmax E {u(p,q,s,b) s,q( )} p (2) b [b,b],q(b) argmax E {v(p,q,s,b) b,p( )} q b s 18

19 Seller s Problem Assume p and q are strictly increasing. Let x( ) = p -1 ( ) and y( ) = q -1 ( ). Optimization in (1) can be stated as max [(p + q(b)) / 2 s]db p X ZY 1 y(p) First-order condition -y'(p)[p - s] + [1 - y(p)]/2 = 0, since q(y(p)) = p 19

20 Optimization in (2) can be stated as First-order condition Buyer s Problem max [b - (p(s) + q) / 2]ds q X ZY0 x(q) since p(x(q)) = q. x'(q)[b - q] - x(q)/2 = 0, 20

21 Equilibrium Equilibrium condition: s=x(p) and b=y(q) Equilibrium first-order conditions: (1') -2y'(p)[p - x(p)] + [1 - y(p)] = 0, (2') 2x'(q)[y(q) - q] - x(q) = 0. 21

22 Solution Solving (2') for y(q) and replacing q with p yields (2 ) y(p) = p x(p) x (p), so y (p)= x(p)x (p) [x (p)] 2 Substituting into (1') then yields L NM (1 ) [x(p) - p] 3 - x(p)x (p) [x (p)] 2 O L1 p 1 O QP + NM x(p) QP 2 x (p) = 0 22

23 Analytical Solution Linear Solution: x(p) = αp + β. with α = 3/2 and β = -3/8. Using (2") yields y(q) = 3/2 q - 1/8. Inverting these functions results in p(s) = 2/3 s + 1/4 and q(b) = 2/3 b + 1/12 23

24 Figure Offers p, q p(s) q(b) Valuations s, b 24

25 Trade occurs if and only if Outcome p(s) q(b), or b - s 1/4 The gains from trade must be at least 1/4 or no trade takes place. The outcome is inefficient 25

26 Mechanism design General understanding of incentives in decision settings (bargaining, auctions, exchange, public goods, )

27 A General Model (Myerson and Satterthwaite, JET 1983) Direct Revelation Game: Bilateral Exchange with independent private value. s F with positive pdf f on [s, s] b G with positive pdf g on [b,b] F and G are common knowledge In the DRG, the traders report their valuations and then an outcome is selected. Given the reports (s,b), an outcome specifies a probability of trade (p) and the terms of trade (x). 27

28 Definition Direct Mechanism A direct mechanism is a pair of outcome functions p,x, where: p(s,b) is the probability of trade given the reports (s,b), and x(s,b) is the expected payment from the buyer to the seller. 28

29 Payoffs Ex post utilities: Seller's ex post utility: u(s,b) = x(s,b) - sp(s,b) Buyer's ex post utility: v(s,b) = bp(s,b) - x(s,b) Both traders are risk neutral and there are no income effects 29

30 Payoffs Define: b = X Z Y = X Z Y X(s) x(s, b)g(b)db Y(b) x(s, b)f(s)ds b b P(s) p(s, b)g(b)db Q(b) p(s, b)f(s)ds. = X Z Y = X Z Y b X(s) is the seller's expected revenue given s Y(b) is the buyer's expected payment given b P(s) is the seller's probability of trade Q(b) is the buyer's probability of trade s s s s 30

31 Payoffs Interim Utilities: U(s) = X(s) - sp(s) V(b) = bq(b) - Y(b) The mechanism p,x is incentive compatible if for all s, b, s, and b : (IC) U(s) X(s') - sp(s') V(b) bq(b') - Y(b') The mechanism p,x is individually rational if for all and (IR) U(s) 0 V(b) 0. s [s, s] b [b, b] 31

32 Lemma 1 (Mirrlees, Myerson) The mechanism p,x is IC if and only if P( ) is decreasing, Q( ) is increasing, and (IC ) U(s) U(s) P(t)dt = + X Z Y V(b) V(b) Q(t)dt s = + X Z Y s b b 32

33 Lemma 1: Proof Only if: By definition, U(s) = X(s) - sp(s) and U(s') = X(s') - s'p(s'). This and (IC) imply U(s) X(s') - sp(s') = U(s') + (s' - s)p(s'), and U(s') X(s) - s'p(s) = U(s) + (s - s')p(s) Putting these inequalities together yields (s' - s)p(s) U(s) - U(s') (s' - s)p(s') 33

34 Lemma 1: Proof Taking s' > s implies that P( ) is decreasing Dividing by (s' - s) and letting s' s, then yields du(s)/ds = -P(s) Integrating produces (IC') The same is true for the buyer 34

35 Lemma 1: Proof If: To prove (IC) for the seller, note that it suffices to show that s[p(s) - P(s')] + [X(s') - X(s)] 0 for all s, s [s, s] Substituting for X(s') and X(s) using (IC') and the definition of U(s) yields X(s) sp(s) U(s) P(t)dt = + + X Z Y. s s 35

36 Lemma 1: Proof If: Then it suffices to show for every s,s [s, s] that 0 s[p(s) P(s )] + s P(s ) + P(t)dt sp(s) P(t)dt X Z Y X Z Y s = X s s + Z Y X = (s s)p(s ) P(t)dt [P(t) P(s )]dt, s s which holds because P( ) is decreasing. The proof for the buyer is similar. s ZY s s 36

37 Lemma 2 An incentive compatible mechanism p,x is individually rational if and only if U(s) 0 (IR') and V( ) 0. b_ Proof Clearly, (IR') is necessary for p,x to be IR By Lemma 1, U( ) is decreasing; hence, (IR') is sufficient as well 37

38 Theorem: Characterization of IC & IR mechanisms An incentive-compatible, individually rational mechanism p,x satisfies (*) z U(s) + V(b) = X sl b ZY NM b 1 G(b) s g(b) b s F(s) f(s) O QP p(s, b)f(s)g(b)dsdb 0. 38

39 Theorem: Characterization of IC & IR mechanisms Proof Using (IC') and the definition of U(s) yields +zs X(s) = sp(s) + U(s) P(t)dt. s 39

40 Theorem: Characterization of IC & IR mechanisms Taking the expectation with respect to s (and substituting in the definitions of X(s) and P(s)) shows that X b s x(s, b)f(s)g(b)dsdb ZY X Z Y = b s b s U(s) + X Z Y X Z Y b b + X Z Y X Z Y b s s s sp(s, b)f(s)g(b)dsdb p(s, b)f(s)g(b)dsdb. 40

41 Theorem: Characterization of IC & IR mechanisms The third term in the right hand side follows, since X s s ZY X Z Y s s s t s s p(t, b)f(s)dtds X = Z Y X Z Y =X p(t, b)f(s)dsdt Z Y s s p(s, b)f(s)ds. 41

42 Theorem: Characterization of IC & IR mechanisms Preceding analogously for the buyer yields X b s ZY X Z Y b s X b s + Z Y X Z Y b s b s ZY X Z Y b s x(s, b)f(s)g(b)dsdb = -V(b) bp(s,b)f(s)g(b)dsdb X - p(s, b)f(s)[1- G(b)]dsdb. Equating the right-hand sides of the last two equations and applying (IR') completes the proof 42

43 Corollary: Impossibility of efficient trade If it is not common knowledge that gains exist (the supports of the traders' valuations have non-empty intersection), then no incentive-compatible, individually rational trading mechanism can be expost efficient. 43

44 Proof A mechanism is ex-post efficient if and only if trade occurs whenever s b: p(s, b) = R S T 1 if s b 0 if s > b. 44

45 Proof To prove that ex-post efficiency cannot be attained, it suffices to show that the inequality (*) in the Corollary fails when evaluated at this p(s,b). Hence, z b b X ZY min{b,s} s L NM b 1 G(b) s g(b) F(s) f(s) O QP f(s)g(b)dsdb 45

46 Proof b b b b b b b min{b,s} s b s b b b b min{b,s} = [bg(b)+g(b) 1]f(s)dsdb [sf(s)+f(s)]dsg(b)db = [bg(b)+g(b) 1]F(b)db min{bf(b),s}g(b)db (by parts) = [1 G(b)]F(b)db + (b s)g(b)db b = [1 G(b)]F(b)db + [1 G(b)]db (by parts) s = [1 G(t)]F(t)dt < 0, since b < s. b b s s 46

47 Proof The second term in the second line follows, since by integrating by parts X x ZY [sf(s) + F(s)]ds = xf(x). s Since ex-post efficiency is unattainable, we need a weaker efficiency criterion with which to measure a mechanism's performance 47

48 What IC & IR mechanism maximizes expected gains from trade? Ex ante efficient mechanism maximizes the expected gains from trade: X ZY s s U(s)f(s)ds + X Z Y b b V(b)g(b)db subject to IC & IR 48

49 Optimal trading game Myerson and Satterthwaite show that the ex ante efficient decision rule (probability of trade) is: where R S α if c(s, α) d(b, α) p ( s,b) = 1 T0 if c(s, α) > d(b, α) c(,s) = s + F(s) 1 G( b) α α d( α,b) = b α f(s) g( b) and α is chosen so that U( s) = V( ) = 0. b 49

50 Remarks If α = 0, then p α is ex post efficient (all the weight on the objective function). If α = 1, p α maximizes the expression in (*); a constrained maximization. The ex ante efficient trading rule has the property that, given the reports, trade either occurs with probability one or not at all. 50

51 Example Valuations are uniformly distributed on [0,1] Ex ante efficient mechanism: linear equilibrium in which trade occurs if and only if the gains from trade are at least 1/4 (Chatterjee & Samuelson) If the traders cannot commit to walking away from gains from trade, then they would be unable to implement this mechanism So long as it is not common knowledge that gains exist, the traders will, with positive probability, make incompatible demands in situations where gains from trade exist 51

52 Accomplishments Characterization of the set of all BE of all bargaining games in which the players' strategies map their private valuations into a probability of trade and a payment from buyer to seller Proof that ex post efficiency is unattainable if it is uncertain that gains from trade exist Determination of the set of ex ante efficient mechanisms Proof that ex ante efficiency is incompatible with sequential rationality 52

53 Dissolving a Partnership (Cramton, Gibbons, and Klemperer, 1987) n traders. Each trader i {1,...,n} owns a share r i 0 of the asset, where r r n = 1 As in MS, player i's valuation for the entire good is v i The utility from owning a share r i is r i v i Private values, v i s are iid F( ) on [v, v] A partnership (r,f) is fully described by the vector of ownership rights r = {r 1,...,r n } and the traders' beliefs F about valuations 53

54 Dissolving a Partnership MS Case: n = 2 and r = {1,0} There does not exist a BE σ of the trading game such that: (1) σ is (interim) individually rational and (2) σ is ex post efficient CGK Case: If the ownership shares are not too unequally distributed, then it is possible to satisfy both (1) and (2), (satisfying IC, IR, EE and BB) 54

55 Dissolving a Partnership A partnership (r,f) can be dissolved efficiently if there exists a Bayesian Equilibrium σ of a Bayesian trading game such that σ is interim individually rational and ex post efficient 55

56 Theorem The partnership (r,f) can be dissolved efficiently if and only if (*) n i= 1 L NM zv v * i zv v i [1 F(u)]udG(u) F(u)udG(u) * O QP 0 where v i* solves F(v i ) n-1 = r i and G(u) = F(u) n-1 56

57 Example n=3, F(v i ) = v i. Then (*) becomes 3 i= i r

58 Proposition For any distribution F, the one-owner partnership r = {1,0,0,...,0} cannot be dissolved efficiently. The one-owner partnership can be interpreted as an auction Ex post efficiency is unattainable because the seller's value v 1 is private information: the seller finds it in her best interest to set a reserve price above her value v 1 An optimal auction maximizes the seller's expected revenue over the set of feasible (ex post inefficient) mechanisms 58

59 Theorem If a partnership (r,f) can be dissolved efficiently, then the unique symmetric equilibrium of the following bidding game is interim individually rational and achieves ex-post efficiency: given an arbitrary minimum bid b, each player receives a side-payment, independent of the bidding, * * X vi n v i 1 n = Z Y X n Z Y v j= 1 v 1 c (r,,r ) udg(u) udg(u). the players choose bids b i [b, ) the good goes to the highest bidder each bidder i pays n 1 p (b,,b ) = b b n 1 j i i 1 n i j j 59

60 Auctioning many similar items Lawrence M. Ausubel, Peter Cramton, Marek Pycia, Marzena Rostek, and Marek Weretka (2014) Demand Reduction and Inefficiency in Multi-Unit Auctions, Review of Economic Studies, 81:4,

61 Examples of auctioning similar items Treasury bills Stock repurchases and IPOs Telecommunications spectrum Electric power Emission allowances 61

62 Ways to auction many similar items Sealed-bid: bidders submit demand schedules Pay-as-bid auction (traditional Treasury practice) Uniform-price auction (Milton Friedman 1959) Vickrey auction (William Vickrey 1961) P Bidder 1 P Bidder 2 P Aggregate Demand + = Q 1 Q 2 Q 62

63 Pay-as-bid Auction: All bids above P 0 win and pay bid Price Supply P 0 Clearing price Demand (Bids) Q S Quantity 63

64 Uniform-Price Auction: All bids above P 0 win and pay P 0 Price Supply P 0 Clearing price Demand (Bids) Q S Quantity 64

65 Vickrey Auction: All bids above P 0 win and pay opportunity cost Price Residual Supply Q S j i Q j (p) p 0 Demand Q i (p) Q i (p 0 ) Quantity 65

66 Vickrey Auction: m Discrete Items Allocate m items efficiently: m highest marginal values Winning bidder pays k th highest losing bid of others on k th item won Payment = social opportunity cost of items won 3 bidders, 3 items marginal values A B C st nd rd

67 Payment rule affects behavior Price Pay-as-bid Residual Supply Q S j i Q j (p) p 0 Uniform-Price Vickrey Demand Q i (p) Q i (p 0 ) Quantity 67

68 More ways to auction many similar items Ascending-bid: Clock indicates price; bidders submit quantity demanded at each price until no excess demand Clock auction (single price) Clock auction with Vickrey prices (Ausubel 1997) 68

69 Ascending clock: All bids at P 0 win and pay P 0 Price Supply P 0 Clock Excess Demand Demand Q S Quantity 69

70 Ascending clock with Vickrey prices All bids at P 0 win and pay price at which clinched Price Residual Supply Q S j i Q j (p) p 0 Clock Excess Demand Q i (p 0 ) Demand Q i (p) Quantity 70

71 More ways to auction many similar items Ascending-bid Simultaneous ascending auction (FCC spectrum) Sequential Sequence of English auctions (auction house) Sequence of Dutch auctions (fish, flowers) Optimal auction Maskin & Riley

72 Research Program How do standard auctions compare? Efficiency FCC: those with highest values win Revenue maximization Treasury: sell debt at least cost 72

73 Efficiency (not pure common value; capacities differ) Uniform-price and standard ascending-bid Inefficient due to demand reduction Pay-as-bid Inefficient due to different shading Vickrey Efficient in private value setting Strategically simple: dominant strategy to bid true demand Inefficient with affiliated information Dynamic Vickrey (Ausubel 1997) Same as Vickrey with private values Efficient with affiliated information 73

74 Inefficiency Theorem In any equilibrium of uniform-price auction, with positive probability objects are won by bidders other than those with highest values. Winning bidder influences price with positive probability Creates incentive to shade bid Incentive to shade increases with additional units Differential shading implies inefficiency 74

75 Inefficiency from differential shading Large Bidder Small Bidder mv 1 P 0 mv 2 D 1 D 2 b 1 b 2 Q 1 Q 2 Large bidder makes room for smaller rival 75

76 Vickrey inefficient with affiliation Winner s Curse in single-item auctions Winning is bad news about value Winner s Curse in multi-unit auctions Winning more is worse news about value Must bid less for larger quantity Differential shading creates inefficiency in Vickrey 76

77 What about seller revenues? Price Pay-as-bid Residual Supply Q S j i Q j (p) p 0 Uniform-Price Vickrey Demand Q i (p) Q i (p 0 ) Quantity 77

78 Exercise 2 bidders (L and R), 2 identical items L has a value of $100 for 1 and $200 for both R has a value of $90 for 1 and $180 for both Uniform-price auction Submit bid for each item Highest 2 bids get items 3 rd highest bid determines price paid Ascending clock auction Price starts at 0 and increases in small increments Bidders express how many they want at current price Bidders can only lower quantity as price rises Auction ends when no excess demand (i.e. just two demanded); winners pay clock price

79 Uniform price may perform poorly Independent private values uniform on [0,1] 2 bidders, 2 units; L wants 2; S wants 1 Uniform-price: unique equilibrium S bids value L bids value for first and 0 for second Zero revenue; poor efficiency Vickrey price = v (2) on one unit, zero on other 79

80 Standard ascending-bid may be worse 2 bidders, 2 units; L wants 2; S wants 2 Uniform-price: two equilibria Poor equilibrium: both L and S bid value for 1 Zero revenue; poor efficiency Good equilibrium: both L and S bid value for 2 Get v (2) for each (max revenue) and efficient Standard ascending-bid: unique equilibrium Both L and S bid value for 1 S s demand reduction forces L to reduce demand Zero revenue; poor efficiency 80

81 Efficient auctions tend to yield high revenues Theorem. With flat demands drawn independently from the same regular distribution, seller s revenue is maximized by awarding good to those with highest values. Generalizes to non-private-value model with independent signals: v i = u(s i,s -i ) Award good to those with highest signals if downward sloping MR and symmetry. 81

82 Downward-sloping demand: p i (q i ) = v i g i (q i ) Theorem. If intercept drawn independently from the same distribution, seller s revenue is maximized by awarding good to those with highest values if constant hazard rate shifting quantity toward high demanders if increasing hazard rate Note: uniform-price shifts quantity toward low demanders 82

83 But uniform price has advantages Participation Encourages participation by small bidders (since quantity is shifted toward them and less strategy) May stimulate competition Post-bid competition More diverse set of winners may stimulate competition in post-auction market 83

84 Bidding behavior in electricity markets Marginal cost bidding is a useful benchmark, but not a norm of behavior Profit maximization is an appropriate norm of behavior in markets Profit maximization should be expected and encouraged Market rules should be based on this norm 84

85 Uniform-price auction: All bids below p 0 win and get paid p 0 Price Demand Supply (as bid) p 0 (clearing price) q 0 Quantity 85

86 Residual demand removes supply of other bidders p p 0 Demand q 0 Supply p Supply of others = q q i q q i p Supply firm i Residual demand q 86

87 Residual demand curve Price As-bid supply S i (p) p 0 Residual demand D i (p) = D(p) j i S j (p) q i Quantity 87

88 Bidding strategy with perfect competition Price As-bid supply S i = MC i p 0 Loss Residual demand D i q i Quantity 88

89 Incentive to bid above marginal cost: tradeoff higher price with reduced quantity p As-bid supply S i MC i p 0 Gain Loss Residual demand D i q i q 89

90 Optimal bid balances marginal gain and loss p As-bid supply S i p 0 Gain MCi Loss Residual demand D i q i q 90

91 Still bid above marginal cost when others bid marginal cost p Other bidders p Firm i S i S -i =MC -i MC i p -i p 0 p 0 D i D 0 q -i 10 GW 0 q i 1 GW 91

92 Residual demand response reduces incentive to inflate bids p As-bid supply S i MC i p 0 Gain Loss Residual demand D i q i q 92

93 Residual demand is steeper for large bidders p Large bidder p Small bidder S l S s p 0 p 0 D s D l 0 q l 10 GW 0 q s 1 GW 93

94 Large bidder makes room for its smaller rivals p Large bidder p Small bidder S l S s MC s p 0 p 0 D s MC l D l 0 q l 10 GW 0 q s 1 GW 94

95 Economic vs. Physical Withholding p p S i S i MC i MC i p 0 p 0 D i D i q i q e q q i q e q 95

96 Forward contracts mitigate incentive to bid above marginal cost p S i no forward S i with forward p MC i p 0 Forward sale q S Residual demand D i q i q F q i q 96

97 Revenue equivalence and optimal auction in general model

98 Identical items model Seller has quantity 1 of divisible good (value = 0) n bidders; i can consume q i [0,λ i ] q = (q 1,,q n ) Q = {q q i [0,λ i ] & Σ i q i 1} t i is i s type; t = (t 1,,t n ); t i ~ F i w/ pos. density f i Types are independent Marginal value v i (t,q i ) i s payoff if gets q i and pays x i : z0 q i v ( t, y) dy i x i 98

99 Identical items model (cont.) Marginal value v i (t,q i ) satisfies: Value monotonicity non-negative increasing in t i weakly increasing in t j weakly decreasing in q i Value regularity: for all i, j, q i, q j, t i, t i > t i, v i (t i,t i,q i ) > v j (t i,t i,q j ) v i (t i,t i,q i ) > v j (t i,t i,q j ) 99

100 Identical items model (cont.) Bidder i s marginal revenue: marginal revenue seller gets from awarding additional quantity to bidder i 1 F ( t ) v ( t, q ) MR ( t, q ) = v ( t, q ) i i i i i i i i f ( t ) t i i i 100

101 Revenue Equivalence Theorem 1. In any equilibrium of any auction game in which the lowest-type bidders receive an expected payoff of zero, the seller s expected revenue equals E t L NM n q 1z0 i= i ( t ) MR ( t, y) dy i O QP 101

102 Optimal Auction MR monotonicity increasing in t i weakly increasing in t j weakly decreasing in q i MR regularity: for all i, j, q i, q j, t i, t i > t i, MR i (t,q i ) > MR j (t,q j ) MR i (t i,t i,q i ) > MR j (t i,t i,q j ) Theorem 2. Suppose MR is monotone and regular. Seller s revenue is maximized by awarding the good to those with the highest marginal revenues, until the good is exhausted or marginal revenue becomes negative. 102

103 Optimal Auction is Inefficient Assign goods to wrong parties High MR does not mean high value Assign too little of the good MR turns negative before values do 103

104 Three Seller Programs 1. Unconstrained optimal auction (standard auction literature) Select assignment rule and pricing rule to max E[Seller Revenue] s.t. Incentive Compatibility Individual Rationality 104

105 Three Seller Programs 2. Resale-constrained optimal auction (Coase Theorem critique) Select assignment rule and pricing rule to max E[Seller Revenue] s.t. Incentive Compatibility Individual Rationality Efficient resale among bidders 105

106 Three Seller Programs 3. Efficiency-constrained optimal auction (Coase Conjecture critique) Select assignment rule and pricing rule to max E[Seller Revenue] s.t. Incentive Compatibility Individual Rationality Efficient resale among bidders and seller 106

107 1. Unconstrained optimal auction Select assignment rule L n zqi ( t ) q( t ) Q t 0 i i= 1 NM q( t) max E MR ( t, y) dy to Q = { All feasible assignment rules.} O QP 107

108 1. Unconstrained optimal auction (two bidders) price S p 2 MR D 0 q 1 quantity MR 2 d 1 MR 1 d 2 108

109 3. Efficiency-constrained optimal auction Select assignment rule q ( t) max E MR ( t, y) dy q( t ) Q Q R R t L NM n zq 0 i= 1 i ( t ) i to = { Ex post efficient assignment rules.} R O Q P 109

110 3. Efficiency-constrained optimal auction (two bidders) price p 1 S D 0 q=1 quantity MR 2 d 1 MR 1 d 2 110

111 2. Resale-constrained optimal auction Select assignment rule q ( t) max E MR ( t, y) dy q( t ) Q Q R R t L NM n zq 0 i= 1 i ( t ) i = { Resale - efficient assignment rules.} R to O QP 111

112 2. Resale-constrained optimal auction (two bidders) price p 1 S MR R D 0 q R 1 quantity MR 2 MR 1 d 1 d 2 112

113 Theorem. In the two-stage game (auction followed by perfect resale), the seller can do no better than the resale-constrained optimal auction. Proof. Let a(t) denote the probability measure on allocations at end of resale round, given reports t. Observe that, viewed as a static mechanism, a(t) must satisfy IC & IR. In addition, a(t) must be resale-efficient. 113

114 Can we obtain the upper bound on revenue? resale process is coalitionally-rational against individual bidders if bidder i obtains no more surplus s i than i brings to the table: s i v(n q,t) v(n ~ i q,t). That is, each bidder receives no more than 100% of the gains from trade it brings to the table. 114

115 Vickrey auction with reserve pricing Seller sets monotonic aggregate quantity that will be assigned to the bidders, an efficient assignment q * (t) of this aggregate quantity, and the payments x * (t) to be made to the seller as a function of the reports t where q * () * i t ˆ i = i i i i 0 x ( t) v ( t ( t, y), t, y) dy, where { * } tˆ ( t, y) = inf t q ( t, t ) y. i i i i i i t i Bidders simultaneously and independently report their types t to the seller. 115

116 Can we attain the upper bound on revenue? Theorem 5 (Ausubel and Cramton 1999). Consider the two-stage game consisting of the Vickrey auction with reserve pricing followed by a resale process that is coalitionally-rational against individual bidders. Given any monotonic aggregate assignment rule, sincere bidding followed by no resale is an ex post equilibrium of the two-stage game. 116

117 Applications Electricity market design Spectrum auction design Future of mobile communications Future of transportation Future of financial markets

118 Electricity 118

119 Goals of electricity markets Short-run efficiency Least-cost operation of existing resources Long-run efficiency Right quantity and mix of resources Key idea: open access 119

120 Challenges of electricity markets Must balance supply and demand at every instant at every location Physical constraints of network Absence of demand response Climate policy 120

121 Three Markets Short term (5 to 60 minutes) Spot energy market Medium term (1 month to 3 years) Bilateral contracts Forward energy market Long term (4 to 20 years) Capacity market (thermal system) Firm energy market (hydro system) Address risk, market power, and investment 121

122 Long-term market: Buy enough in advance 122

123 Product What is load buying? Energy during scarcity period (capacity) Enhance substitution Technology neutral where possible Separate zones only as needed in response to binding constraints Long-term commitment for new resources to reduce risk 123

124 Pay for Performance Strong performance incentives Obligation to supply during scarcity events Deviations settled at price > $5000/MWh Penalties for underperformance Rewards for overperformance Tend to be too weak in practice, leading to Contract defaults Unreliable resources But not in best markets: ISO New England, PJM 124

125 Spectrum 125

126 Spectrum auctions Many items, heterogeneous but similar Competing technologies and business plans Complex structure of substitutes and complements Government objective: Efficiency Make best use of scarce spectrum Address competition issues in downstream market 126

127 Key design issues Establish term to promote investment Enhance substitution Product design Auction design Encourage price discovery Dynamic price process to focus valuation efforts Encourage truthful bidding Pricing rule Activity rule 127

128 Simultaneous ascending auction

129 Prepare 129

130 Italy 4G Auction, September rounds, 22 days, 3.95B Auction conducted on-site with pen and paper Auction procedures failed in first day No activity rule 130

131 Thailand 3G Auction, October incumbents bid 3 nearly identical licenses Auction ends at reserve price + 2.8% 131

132 Combinatorial Clock Auction 132

133 Combinatorial clock auction Auctioneer names prices; bidder names package Price increased if excess demand Process repeated until no excess demand Supplementary bids Improve clock bids Bid on other relevant packages Optimization to determine assignment/prices No exposure problem (package auction) Second pricing to encourage truthful bidding Activity rule to promote price discovery 133

134 CCA Sucks Look at Switzerland! No: Rules were poor Bidding likely poor Setting was difficult 134

135 Pricing rule 135

136 Bidder-optimal core pricing Minimize payments subject to core constraints Core = assignment and payments such that Efficient: Value maximizing assignment Unblocked: No subset of bidders offerred seller a better deal 136

137 Optimization Core point that minimizes payments readily calculated Solve Winner Determination Problem Find Vickrey prices Constraint generation method (Day and Raghavan 2007) Find most violated core constraint and add it Continue until no violation Tie-breaking rule for prices is important Minimize distance from Vickrey prices 137

138 5 bidder example with bids on {A,B} b 1 {A} = 28 b 2 {B} = 20 b 3 {AB} = 32 b 4 {A} = 14 b 5 {B} = 12 Winners Vickrey prices: p 1 = 14 p 2 =

139 The Core Bidder 2 Payment b 3 {AB} = 32 b 4 {A} = 14 b 1 {A} = 28 Efficient outcome 20 b 2 {B} = 20 The Core 12 b 5 {B} = Bidder 1 Payment 139

140 Vickrey prices: How much can each winner s bid be reduced holding others fixed? Bidder 2 Payment b 3 {AB} = 32 b 4 {A} = 14 b 1 {A} = b 2 {B} = 20 The Core 12 Vickrey prices Problem: Bidder 3 can offer seller more (32 > 26)! b 5 {B} = Bidder 1 Payment 140

141 Bidder-optimal core prices: Jointly reduce winning bids as much as possible Bidder 2 Payment b 3 {AB} = 32 b 4 {A} = 14 b 1 {A} = b 2 {B} = 20 The Core 12 Vickrey prices Problem: bidderoptimal core prices are not unique! b 5 {B} = Bidder 1 Payment 141

142 Core point closest to Vickrey prices Bidder 2 Payment b 3 {AB} = 32 b 4 {A} = 14 b 1 {A} = b 2 {B} = Unique core prices 12 Vickrey prices Each pays equal share above Vickrey b 5 {B} = Bidder 1 Payment 142

143 Activity rule 143

144 Clock stage performs well Proposition: With revealed preference w.r.t. final round If clock stage ends with no excess supply, final assignment = clock assignment Supplementary bids can t change assignment; but can change prices May destroy incentive for truthful bidding in supplementary round Supplementary round still needed to determine competitive prices Possible solutions Do not reveal demand at end of clock stage; possibility of excess supply motivates more truthful bidding (Canada 700 MHz) Do not impose final price cap (UK 4G) 144

145 Summary: CCA is an important tool Eliminates exposure Reduces gaming Enhances substitution Allows auction to determine band plan, technology Readily customized to a variety of settings Many other applications 145

146 Broadcast Incentive Auction 29 March

147 Motivation Value per MHz Value of mobile broadband Value of over-the-air broadcast TV Gains from trade Year 147

148 Voluntary approach TV broadcaster freely decides to Share with another Spectrum freed 0 MHz 3 MHz 6 MHz For simplicity, I assume that channel sharing is only 2:1; other possibilities could also be considered, including negotiated shares with particular partners announced at qualification 148

149 Why voluntary? More likely to quickly clear spectrum Broadcasters benefit from cooperating Lower economic cost of clearing Spectrum given up only by broadcasters who put smallest value on over-the-air signal Market pricing for clearing Avoids costly administrative process Efficient clearing Clear only when value to mobile operator > value to TV broadcaster 149

150 Two approaches Combinatorial exchange Too complex due to repacking Reverse auction to determine supply Optimization gives mandatory repacking options Forward auction to determine demand Market clearing and settlement 150

151 TV broadcaster freely decides to Share with another Reverse auction to determine supply Descending clock adopted with scoring rule Price to clear Price to share Spectrum freed 0 MHz 3 MHz 6 MHz 151

152 Washington DC 0 MHz 3 MHz 6 MHz P = $30 7 Price = $30/MHzPop Reverse auction to determine supply S =

153 Washington DC 0 MHz 3 MHz 6 MHz P = $20 7 Price = $20/MHzPop Reverse auction to determine supply S =

154 Washington DC 0 MHz 3 MHz 6 MHz P = $10 7 Price = $10/MHzPop Reverse auction to determine supply S =

155 P = $20 Supply = S = MHz Mandatory repacking

156 Forward auction to determine demand Mobile operators want large blocks of contiguous paired spectrum for LTE (4G) One to four 2 5 MHz lots Complementaries strong both within and across regions Ascending clock auction adopted Within-region complementarities guaranteed with generic lots Across-region complementarities dealt with in a limited way 156

157 Price P 6 Supply P 5 Forward auction to determine demand P 4 P 3 P 2 P 1 0 Demand Quantity 157

158 Price Supply Forward auction to determine demand P* Demand Q* Quantity 158

159 Price Supply Forward auction to determine demand Broadcasters cannot negotiate ex post with operators, since it is the FCC s repacking that creates value; ex post trades would not benefit from repacking P D P S To Treasury To TV broadcasters Q 0 Q* Demand Quantity 159

160 Example Learnings - Sub-Regions within the US The country can be decomposed into East and West regions The East: 1,341 UHF stations (Blue) & the West: 307 UHF stations Rocky Mountains provide sufficient separation between these two large markets Finer decomposition is not possible due to extensive network interaction among constraints

161 Auction Tools Environment Simulating a single auction involves solving about 800,000 NP Hard computational problems!

162 Platform Overview - Relationships Data Management Platform Development / Analysis Platform GitHub Dropbox Windows Servers Simulation Management Platform Amazon EC2 Linux Servers OpenGrid Server StarCluster

163 Important lessons No auction design is perfect Design must be customized for setting Simultaneous ascending clock Simple settings (upcoming UK) Combinatorial clock Packaging is essential (UK 4G, Canada 700 MHz) Two-sided clock Incentive auction in US Never ignore essentials Encourage participation Demand performance Avoid collusion and corruption 163

164 The Future of Mobile Communications 164

165 An Open Access Wireless Market Supporting Public Safety, Universal Service, and Competition Peter Cramton Linda Doyle 165

166 From monopoly to vibrant competition Monopoly Original Ma Bell telecommunications Oligopoly Spectrum auctions Mobile communications Competition Open access wireless market Internet ecosystem of innovation 166

167 Mobile networks Wholesale market Open access network (ISO) Service providers Proprietary network 1 (MNO 1 ) Mobile virtual network operator (MVNO) Users/devices Retail market MNO 1 Proprietary network 2 (MNO 2 ) MNO 2 A B C D E F G

168 Price ($/GB) Winning buyers Supply Clearing price P* Winning sellers Q* Quantity traded Demand Quantity (GB/h)

169 On peak Off peak Hour 7 Hour 15 Hour 3 Hourly area A Hourly area B Hourly area J Monthly area M Demand Low Medium High

170 Yearly area Monthly area red Monthly area green Hourly area A Hourly area B Hourly area Z Monthly area blue

171 Yearly auction = buy 165 GB per hour, for every peak hour in the year, in the yearly area 5 GB 5 GB 5 GB 5 GB 5 GB 5 GB 5 GB 10 GB 15 GB 15 GB 5 GB 5 GB 5 GB 10 GB 10 GB 10 GB 10 GB 5 GB 5 GB 5GB 5 GB 5 GB 5 GB 5 GB

172 Yearly auction = buy 165 GB per hour, for every peak hour in the year, in the yearly area Monthly auction = sell 8 GB per hour, for every peak hour in the month, in the red area Monthly auction = buy 15 GB per hour, for every peak hour in the month, in the green area 4 GB 4 GB 4GB 7 GB 8 GB 8 GB 4 GB 9 GB 17 GB 17 GB 8 GB 8 GB 4 GB 9 GB 12 GB 12 GB 12 GB 8 GB 4 GB 7GB 7 GB 7GB 7 GB 8 GB Monthly auction = buy 20 GB per hour, for every peak hour in the month, for the blue area

173 Hourly area H Peak hour product Hour Change Net 07 Buy 3 GB 11 GB 08 Sell 2 GB 6 GB 17 Buy 1 GB 9 GB 8 GB

174 Yearly auction Buy 8 GB in each hour Sequence of auctions Monthly auctions April: Sell 1 GB 7 GB net Example Daily auctions of hourly products April 5, hour 15: Buy 3 GB 10 GB net Hourly realization April 5, hour 15: -1 GB deviation 9 GB demand

175 Yearly product Y Monthly product for M M M M M M M M M M M M Day of month in April Hourly product for 5 April H H H H H H H H H H H H H H H H H H H H H H H H Y M H = Auction conducted in December for yearly products = Auction conducted last week of each month for monthly products = Auction conducted every hour for hourly products

176 Sample demand for bidder Manhattan (peak) monthly

177

178 Bidder s screen of supply and demand Manhattan (peak) monthly

179 Auctioneer s screen of supply and demand Manhattan (peak) monthly

180 The Future of Transportation 180

181 The Future of Financial Markets Presentation 181