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Lecturer: Moni Naor

Algorithmic Game Theory Uri Feige Robi Krauthgamer Moni Naor Lecture 11: Combinatorial Auctions. Lecturer: Moni Naor. Announcements. Course resumes to 1400:-16:00 . The setting. Set of alternatives A Who wins the auction Which path is chosen Who is matched to whom

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Lecturer: Moni Naor

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  1. Algorithmic Game TheoryUri Feige Robi Krauthgamer Moni NaorLecture 11: Combinatorial Auctions Lecturer:Moni Naor

  2. Announcements • Course resumes to 1400:-16:00

  3. The setting • Set of alternatives A • Who wins the auction • Which path is chosen • Who is matched to whom • Each participant: a value function vi:A  R • Can pay participants: valuation of choice a with payment pi is vi(a)+pi Quasi linear preferences

  4. Mechanism Design • Mechanisms • Recall: We want to implement a social choice function • Need to know agents’ preferences • They may not reveal them to us truthfully • Example: • One item to allocate: • Want to give it to the participant who values it the most • If we just ask participants to tell us their preferences: may lie • Can use payments result is also a payment vector p=(p1,p2, … pn)

  5. The Revelation Principle Theorem: if there exists an arbitrary mechanism implementing a social choice function f in dominant strategies, then there exists an incentive compatible mechanism that implements f The payments of the players in the incentive compatible mechanism are identical to those obtained at equilibrium in the original mechanism Proof: by simulation

  6. Direct Characterization A mechanism is incentive compatible iff the following hold for all i and all vi • The payment pi does not depend on vi but only on the alternative chosenf(vi, v-i) • the payment of alternative a is pa • The mechanism optimizes for each player: f(vi, v-i) 2 argmaxa (vi(a)-pa)

  7. Expected Revenues Theorem [Revenue Equivalence]: under very general conditions, every twoBayesian Nash implementations of the same social choice function if for some player and some type they have the same expected payment then • All types have the same expected payment to the player • If all player have the same expected payment: the expected revenues are the same Not true when there are reservation prices!

  8. Range Voting • Each voter ranks the candidates in a certain range (say 0-99) • The votes for all candidates are summed up and the one with highest total score wins Can be considered as a generalization of approval voting from the range 0-1 No incentive for voter to rate a candidate lower than a candidate they like less.

  9. Vickrey Clarke Grove Mechanism • f(v1, v2, … vn)maximizes i vi(a) over A • Maximizes welfare • There are functions h1, h2,… hn where hi: V1 V2 …  Vn R does not depend on vi we have that: pi(v1, v2, … vn) = hi(v-i) - j  i vj(f(v1, v2,… vn)) Clark Pivot rule: Choosing hi(v-i) = maxb 2 Aj  i vj(b) Payment of i when a=f(v1, v2,…, vn): pi(v1, v2,… vn) = maxb 2 Aj  i vj(b) -j  i vj(a) Depends only on chosen alternative Does not depend on vi Social welfare when he does not participate Social welfare (of others) when he participates

  10. Examples of Auctions • Sponsored Search: • Buying keywords (Google, Yahoo, MS) • Spectrum Auctions • Ebay • Government procurement • Privatization FCC Auction 2008

  11. המשך המכרז: אלג'ם העלו ל-3.8 מיליארד, וסבן העלה ל-4.11 מיליארד, וניצח. הפרטת בזק מקור: "הארץ", 29 למאי 2005: • שתי קבוצות התמודדו במכרז: קבוצת איייפקס-סבן, וקבוצת אלג'ם. • ההצעות ההתחלתיות - אלג'ם: 3.2 מיליארד, סבן: 3.6 מיליארדש"ח • התכנון המקורי של משרד האוצר: לחשוף את ההצעות, ולקבל הצעות חדשות • לערוך מכרז אנגלי עולה. • נוכח הפער הגדול בין שתי ההצעות: באוצר חששו להמשיך לפי התכנון המקורי, שמא אלג'ם תתייאש ותפרוש. • היועץ המקצועי, פרופ' מוטי פרי מירושלים, התעקש להמשיך לפי התכנון: "אין ביטחון מלא, אבל בדרך כלל השקיפות מביאה לתוצאות הטובות ביותר. יש למסור לכל אחד מהמשתתפים את תוצאות הסבב הראשון... ההצעה העיקרית שלי במכרז הייתה ליצור מערכת שקופה לחלוטין, שבה כל אחד מהצדדים יודע בדיוק מה קורה ומקבל את כל המידע. ככל שלמתחרים יש יותר ידע, רמת הסיכון שלהם יורדת, מרכיב ההימור בעסקה יורד, והם מוכנים להסתכן בהצעות גבוהות יותר". From Ron Lavi’s slides

  12. VCG: turning mechanisms truthfulUsing exact optimization Exact optimization algorithms for social welfare f VCG prices Strategic players truthful mechanism

  13. VCG and Computational Complexity Several optimization must be performed in VCG: • f(v1, v2, … vn)maximizes i vi(a) over A • Clark Pivot rule:hi(v-i) = maxb 2 Aj  i vj(b) These may be hard optimization problems. What happens when we replace then with approximations? Incentive compatibility not guaranteed!

  14. Combinatorial Auctions • Set M of m indivisible items • Set N of n bidders • Preferences are on subsets S µ M – bundles – of items • Valuation function vi: 2M R • vi(S) - value bidder i receives on bundle S • Must satisfy: monotonicity: v(s) · v(T) for S µ T • Normalized: v() = 0 Allocation: subsets S1, S2, … Sn where SiÅ Sj =  for i≠j Social welfare of allocation: i=1n vi(Si)

  15. Combinatorial Auctions: Example Interested in any one of red subsets Allocation: subsets S1, S2, … Sn where SiÅ Sj =  for i≠j Social welfare of allocation: i=1n vi(Si) items{1,…, m} players indivisible item

  16. Issues • Computational Complexity of finding the allocation maximizing the social welfare • Representation and communication of the valuation functions vi • May be large since defined for every subset • How to analyze the strategy of the bidder in light of the obstacles

  17. Today • Complexity of Approximation • Tight bounds for the single minded bidder • Connection with communication complexity • Demand queries and the relaxed LP formulation

  18. Single Minded Auctions A valuation v is single minded if there is a bundle of items S* and value v* 2 R such that • v(S) = v* if S* µ S • v(S) = 0 for all other S • Very simple to represent: (Si*, vi*) Allocation problem for single minded bidders: • Given{(Si*, vi*)}i for bidders i=1..n • Find a subset W of winning bids such that Si* Å Sj*=  with maximum social welfarej 2 W vj*

  19. Single Minded Auctions Example Allocation problem for single minded bidders: • Given{(Si*, vi*)}i for bidders i=1..n • Find a subset W of winning bids such that Si* Å Sj*=  with maximum social welfarej 2 W vj* Example: communication links in a tree • bidders want a path between some two nodes in the tree • Since there is only one path between any pair of nodes, the bidders are single-minded Bidder i wants to connect si to ti • Items = edges of tree • Si*= {set of edges connecting (si, ti) • vi* = 1

  20. Complexity of Single Minded Auctions Allocation problem for single minded bidders: • Given{(Si*, vi*)}I for bidders i=1..n • Find a subset W of winning bids such that Si* Å Sj*=  with maximum social welfarej 2 W vj* Claim: Allocation problem for single minded bidders is NP-hard Proof: Reduction from imaximum independent set. For a Graph G=(V,E): • each node  bidder • each edge  item • Si* = {e 2 E|I 2 e} and vi*=1. Winning set W must correspond to an independent set Welfare of W is its size |W| Size of max W is exactly size of max independent set! Size m Size n

  21. Computationally easy cases When each Si is of size 2 • Allocation problem corresponds to maximum weighted matching in the corresponding graph • Item  node • Bidder  edge, vi* is the weight Can be solved in polytime When the items are on a line. Each Si is a contiguous segment Finding max weight independent set in in interval graphs possible in poly time vi* item vi*

  22. Approximation of Single Minded Auctions Allocation problem for single minded bidders: • Given{(Si*, vi*)}I for bidders i=1..n • Find a subset W of winning bids such that Si* Å Sj*=  with maximum social welfarej 2 W vj* Size of max W is exactly size of max independent set What about approximating the best allocation? W is a c-approximation: if for any other allocation W’ j 2 W’ vj* / j 2 W vj* · c Limited by the approximation ratio of independent set: Cannot be better than n1- [Hastad] In terms of the number of items: cannot be better than m1/2-

  23. Incentive Compatible Approximation Mechanism Want to satisfy both incentive compatibility and computational efficiency Lemma: a mechanism for single-minded bidders where losers pay 0 is incentive compatible iff • Monotonicity: a bidder who wins with (vi* , Si*) also wins with (vi’, Si’) for any vi’ ¸ vi* and Si’ µ Si* • Critical payment: a bidder who wins with pays the minimum value needed for winning: infimum of all vi’, where (vi’, Si*) wins.

  24. Incentive Compatible Approximation Mechanism Want to satisfy both incentive compatibility and computational efficiency Given{(Si*, vi*)}I for bidders i=1..n • Rank bidders by vi* / √ |Si*|: • v1* / √ |S1*|¸ v2* / √ |S2*|¸ … • Run a greedy algorithm, starting from large to small • Add to W if not adjacent to any current member of W • Allocation: the set W • Payments: for i 2 W: pi = vj*/ √ (|Sj*| / |Si*|) where j is smallest index that • Si* Å Sj*≠ • j wins without i and 0 if none exists The value that would have made j appear before i

  25. Approximation Ratio of the Mechanism Theorem: Let OPT be allocation maximizing j 2 OPT vj* and W the output of the greedy algorithm. Then j 2 OPT vj*· √mj 2 W vj* Proof: For each i 2 W let OPTi={j 2 OPT, j ¸ i|Si*Å Sj*≠ } The set of elements that did not enter W “because” of i We know that OPTµ[i2WOPTi Will show: j 2 OPTi vj*· √m vi* For all j 2 OPTi we know that vj*· vi*/ √ (|Sj*|/|Si*|) j 2 OPTi vj*· vi*/ √ |Si*| j 2 OPTi√ |Sj*|

  26. Approximation Ratio of the Mechanism Will show: j 2 OPTi vj*· √m vi* For all j 2 OPTi we know that vj*· vi*/ √ (|Sj*|/|Si*|) j 2 OPTi vj*· vi*/ √ |Si*| j 2 OPTi√ |Sj*| By Caushy-Schwarz: j 2 OPTi√ |Sj*| · √ |OPTi | √ j 2 OPTi |Sj*| For j 2 OPTi : Si*Å Sj*≠ Since OPT is an allocation: • these intersections are disjoint and |OPTi | · |Si*| • j 2 OPTi |Sj*| · m j 2 OPTi√ |Sj*| · √|Si*| √m

  27. Today • Complexity of Approximation • Tight bounds for the single minded bidder • Connection with communication complexity • Demand queries and the relaxed LP formulation

  28. Communication Complexity and Bidding • Two (or more) parties • Each party i can compute valuation function vi: 2M R by an oracle call • No succinct description • Want to find optimal allocation by sending as few bits to each other. • Suppose vi: 2M {0,1} Claim: need to exchange an exponential number of bits to find optimal allocation

  29. Alice Bob Communication Complexity x2X y2Y • Let f:X xY Z • Input is split between two participants • Want to compute outcome: z=f(x,y) • while exchanging as few bits as possible

  30. Alice: 0 Bob: 1 Alice: 0 Bob: 1 A protocol is defined by the communication tree z5 z0 z1 z2 z3 z4 z5 z6 z7 ...

  31. A Protocol A protocol P over domain X x Y with range Z is a binary tree where • Each internal node v is labeled with either • av:X {0,1} or • bv:Y {0,1} • Each leaf is labeled with an element z 2 Z • The value of protocolP on input (x,y) is the label of the leaf reached by starting from the root and walking down the tree. • At each internal node labeled avwalk • left if av(x)=0 • right if av(x)=1 • At each internal node labeled bvwalk • left if bv(y)=0 • right if bv(y)=1 • The cost of protocolP on input (x,y) is the length of the path taken on input (x,y) • The cost of protocolPis the maximum path length

  32. Motivation for studying communication complexity • Originally for studying VLSI questions • Connection with Turing Machines • Data structures and the cell probe model • Boolean circuit depth • Combinatorial Auctions • …

  33. Communication Complexity of a function • For a function f:X x Y Z the (deterministic) communication complexity of f (D(f)) is the minimum cost of protocol P over all protocols that compute f Observation: For any function f:X x Y Z D(f) ≤ log |X| + log |Z| Example: let x,y µ {1,…,n} and let f(x,y)=max{x [ y} Then D(f) · 2 log n

  34. Combinatorial Rectangles • A combinatorial rectangle in X x Y is a subset R µ X x Y such that R= A x B for some A µ X and B µ Y Proposition: R µ X x Y is a combinatorial rectangle iff (x1,y1) 2 R and (x2,y2) 2 R implies that (x1,y2) 2 R For Protocol P and nodevlet Rv be the set of inputs(x,y) reaching v Claim: For any protocol P and nodevthe set Rv is a combinatorial rectangle Claim: given the transcript of an exchange between Alice an Bob (but not x and y) possible to determine z=f(x,y)

  35. Fooling Sets • For f:X x Y Z a subset R µ X x Y is f-monochromatic if f is fixed on R • Observation: any protocol P induces a partition of X x Y into f-monochromatic rectangles. • The number of rectangles is the number of leaves in P • A set Sµ X x Y is a fooling set for f if there exists a z 2 Z where • For every(x,y) 2 S, f(x,y)=z • For everydistinct (x1,y1), (x2,y2) 2 S either • f(x1,y2)≠z or • f(x2,y1)≠z Property: no twoelements of a fooling set S can be in the same monochromatic rectangle Lemma: if f has a fooling set of size t, then D(f) ≥ log2 t

  36. Examples Equality: Alice and Bob each hold x,y 2 {0,1}n • want to decide whether x=y or not. • Fooling set for Equality S={(w,w)|w 2 {0,1}n } Conclusion: D(Equality) ¸ n Disjointness: let x,y µ {1,…,n} and let • DISJ(x,y)=1 if |x  y|¸ 1 and • DISJ(x,y)=0 otherwise • Fooling set for Disjointness S={(A, comp(A))|A µ {1,…,n} } Conclusion: D(DISJ) ¸ n

  37. Bidding as a function • In bidding the parties are interested in finding a good allocation • Can be viewed as computing a relation • Can view the

  38. Communication Complexity and Bidding Theorem: every protocol that find the optimal allocation for every pair of {0,1} valuation v1,v2 must use {m choose m/2}¼ 2m/√ (/2 m) bits. Proof: by a fooling set argument • For every valuation v define “dual” v*: v*(S)=1-v(Sc) • dual” v* is monotone Claim: Let v ≠ u: then the sequence of bits transmitted on (v,v*)cannot be the same as that transmitted on (u,u*) Claim  Theorem: there are at least {m choose m/2}different valuations ‘1’ m/2 ‘0’ Number of 1’s

  39. Claim: Let v≠ u: then the sequence of bits transmitted on (v,v*) cannot be the same as that transmitted on (u,u*) Proof: if v≠ u but same transcript on (v,v*) and (u,u*): then same transcript on (v,u*) and (u,v*). Same allocation, {S, Sc}, is produced in all 4 cases For some T we have that v(T) ≠ u(T). WLOG: v(T) =1 andu(T)=0, u*(Tc)=1. Hence: v(T) + u*(Tc)=2. Therefore v(S) + u*(Sc) ¸ 2. However: v(S) + v*(Sc) + u(S) + u*(Sc)=2 We get that u(S) + v*(Sc)· 0 which is suboptimal • Allocation produced • Should be optimal

  40. Approximation? • Approximating the social welfare with a factor strictly smaller than min{n,m1/2-} requires exponential communication

  41. Today • Complexity of Approximation • Tight bounds for the single minded bidder • Connection with communication complexity • Demand queries and the relaxed LP formulation

  42. Demand Queries v(S) never explicitly represented! Demand query: auctioneer presents a vector of item prices p1, p2, …, pn the bidder reports a demand bundle for the prices: a set S maximizing v(S) +j 2 S pi Value query: auctioneer presents a bundle S, the bidder reports his value v(S) for this bundle.

  43. Linear Programming Formulation For single minded bidder: just xi,S* Linear Programming Relaxation: • Variable xi,S for each bidder i 2 N and subset S Max i,S vi(S) xi,S s.t. i,S s.t. j 2 Sxi,S· 1 for all j 2 M S µ Mxi,S· 1 for all i 2 N xi,S¸0 for all S µ M, i 2 N The integer Program: xi,S2{0,1} Exponential number of variables xi,S =1 iff i receives bundle S

  44. Dual Linear Programming Relaxation • Variable ui for each bidder i 2 N and pj for each item j 2 M Minimize i 2 N ui + j 2 M pi s.t. ui + j 2 S pj¸vi(S) for all S µ M, i 2 N ui¸0, pj¸ 0 for all i 2 N, j2 M Interpretation: pj: price of item j Ui: demand of user i Exponential number of constraints

  45. Demand Queries v(S) never explicitly represented! Demand query: auctioneer presents a vector of item prices p1, p2, …, pn the bidder reports a demand bundle for the prices: a set S maximizing v(S) +j 2 S pi Theorem: Linear Programming Relaxation can be solved in poly time (in n,m and number of bits of precision) using only demand queries with item prices.

  46. Back to the RLP • Use the dual problem (n+m variables, exp # constraints) • Ellipsoid Method: • Uses Separation oracle for : when given a candidate solution, either confirms that it is feasible or respond with a violated constraint • Given (u,p) need to check ui ¸ vi(S) -j 2 S pj • query user i with demand query p – response Di • See whether ui + j 2 Di pj ¸ vi(D_i) • ui + j 2 S pj¸vi(S)for S µ M, i 2 N • ui¸0, pj¸ 0for i 2 N, j2 M Value query

  47. From the Dual to the RLP • The Ellipsoid Algorithm makes a polynomial number of calls to the separation oracle • Each time a constraint is returned: poly # altogether • Remove all other constraints and obtain a “reduced” dual LP • Ellipsoid Algorithm still returns the same result • Use the dual of the reduced dual problem to get a solution to the original primal • With a polynomial number of variable • It is also a solution to the original (with 0’s on all “non”-variables) What to do with the relaxed solution? Many approximation algorithms use it: Randomized rounding is the obvious choice Encountered in Separation • ui + j 2 S pj¸vi(S)for S µ M, i 2 N • ui¸0, pj¸ 0for i 2 N, j2 M

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