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What I Really Wanted To Know About Combinatorial Auctions Arne Andersson Trade Extensions Uppsala University

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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Combinatorial Auction Bid ABid BBid CBid DBid E Item 1100102x Item 210399xx Item 3100xx Item 4105106xx Comb Price 200205305 Single Bids Combinatorial Bids

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Combinatorial Auction Maximize 100 A1 + 103 A2 + 100 A3 + 105 A4 + 102 B1 + 99 B2 + 106 B4 + 200 C + 205 D + 305 E subject to A1 + B1 + C ≤ 1 (only one bid can win Commodity 1) A2 + B2 + D + E ≤ 1(only one bid can win Commodity 2) A3 + D + E ≤ 1 (only one bid can win Commodity 3) A4 + B4 + C + E ≤ 1 (only one bid can win Commodity 4) Bid ABid BBid CBid DBid E Item 1100102x Item 210399xx Item 3100xx Item 4105106xx Comb Price 200205305...and here is why Computer Scientists care about these auctions

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What if.... Bid ABid BBid CBid DBid E Item 1??? Item 2???? Item 3??? Item 4????...we do not allow combinatorial bids, but only single bids? (Simultaneous auction) Will the auctioneer earn higher or lower revenue?

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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Research Background 1985-2000: Research in Algorithms and Data Structures 1995-2000: Applied project on Optimization and Resource Allocation in the Energy Sector Resource allocation handled as Markets Electronic Markets Combinatorial Auctions 2000, co-Founded TradeExtensions 2008, Finally left permanent university position, my hart still belongs to research

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Why do Research? In the beginning: It’s fun! (And it might help the career) After a while: you need to ”build your CV” to get a good job Finally: You can afford to be a bit relaxed on counting publications, and go for the important problems instead.

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What is an important problem? ?

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Decide yourself! !

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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Industrial Background Trade Extensions founded 2000 The world’s first on-line combinatorial auction 2001 Today a world-leading provider of on-line bidding and optimization, handling millions of bid values with complex constraints, where combinatorial bids are just one special case Largest application area is Logistics

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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A Real World Problem A number of items for sale A number of bidders An Auctioneer The Auctioneer’s Goal: (Maximize Efficiency) Maximize Revenue Which auction mechanism should the auctioneer use?

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The Ideal Solution ”Let every bidder tell his true preferences and solve the resulting optimization problem” Bidders Auctioneer BidsAllocations

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The Real World ”Bidders will speculate, and the auctioneer has to be cool about it” Bidders Auctioneer BidsAllocations

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One idea (incentive-compatability) ”Use an auction mechanisms where the bidder’s best strategy is to bid truthfully” Bidders Auctioneer BidsAllocations

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Incentive-compatability is great, but.... It is not a goal in itself, just a tool to reach good efficiency or high revenue Incentive-compatible protocols are often less uesful in practice

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Instead, we should emphasise Simple and practical protocols Example: First-price auctions

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Which protocol is ”best”? An incentive-compatible protocol with low revenue? A simple protocol where the Nash equilibrium is known to give high revenue, but the optimal strategies are unknown? The second one is more likely to be used in practice

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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Combinatorial Auction Bid ABid BBid CBid DBid E Item 1100102x Item 210399xx Item 3100xx Item 4105106xx Comb Price 200205305 Single Bids Combinatorial Bids

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Combinatorial Auction Maximize 100 A1 + 103 A2 + 100 A3 + 105 A4 + 102 B1 + 99 B2 + 106 B4 + 200 C + 205 D + 305 E subject to A1 + B1 + C ≤ 1 (only one bid can win Commodity 1) A2 + B2 + D + E ≤ 1(only one bid can win Commodity 2) A3 + D + E ≤ 1 (only one bid can win Commodity 3) A4 + B4 + C + E ≤ 1 (only one bid can win Commodity 4) Bid ABid BBid CBid DBid E Item 1100102x Item 210399xx Item 3100xx Item 4105106xx Comb Price 200205305...and here is why Computer Scientists care about these auctions

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What if.... Bid ABid BBid CBid DBid E Item 1??? Item 2???? Item 3??? Item 4????...we do not allow combinatorial bids, but only single bids? (Simultaneous auction) Will the auctioneer earn higher or lower revenue?

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Intuition Combinatorial Auction Bid High: No risk of winning just a few items, so I can afford to bid above my single-bid valuation Bid Low: If I win, my combination is part of a puzzle with many other winning combinations. If they bid high I can still bid low and our puzzle wil win anyway. (threshold) Simultaneous Auction Bid High: If I bid high enough, I will beat all others and win my entire combination. Bid Low: Potential risk of winning just a few items, dangerous to bid above single-bid valuation (exposure)

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Previous knowledge For 2 items and 3 bidders, the simultaneous auction gives higher revenue (Krishna & Rosentahl)

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A Nobel Price Problem There exists no theoretical evidence for the belief that combinatorial auctions provide higher revenue Could we provide any such evidence?

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The Plan 1.Provide theoretical evidence that combinatorial auctins give higher revenue 2.Humbly accept the Nobel Price

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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The Proof 1.Formal problem 2.Upper bound on simultaneous auctions 3.Lower bounds on combinatorial auctions 4.Comparison

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The Proof 1.Formal problem 2.Upper bound on simultaneous auctions 3.Lower bounds on combinatorial auctions 4.Comparison

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Formal Problem M items N Single Bidders per item N Synnergy Bidders, each interested in k items, getting a synnergy α iff all are won. Bidders have random valuations, the combinations are randomly selected.

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Bidder ABidder BBidder CBidder DBidder EBidder F Item 10.780.98 Item 20.560.98 Item 30.770.98 Item 40.550.56 Item 50.640.77 Item 6 Item 70.560.98 Item 80.56 Item 90.77 Item 100.77 Synnergy 111 Total Value 0.780.550.646.247.087.92

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Our initial view on the problem Traditional game-theoretic approaches can not be used No hope in deriving the actual equilibrium strategies Try to find some bounds on what is possible to achieve with the two auction protocols.

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The Proof 1.Formal problem 2.Upper bound on simultaneous auctions 3.Lower bounds on combinatorial auctions 4.Comparison

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Upper bound on revenue in simultaneous autcions Main idea: Prove that exposure is a real problem

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Lemma: We can assume the bidders to be ordered, highest, 2nd,... Bidder A Bidder B Bidder C Bidder D Bidder E Proof: Adversary argument

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Observation: You realize synnergy iff you do not collide with any higher bidder Bidder A Bidder B Bidder C Bidder D Bidder E

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Combinatorial argument: You only realize synnergy if you do not collide with a higher bidder Given two synnergy bidders, the probability that they do not collide is The probability that the jth highest bidder gets his synnergy is Summing over all bidders, the expected total realizd synnergy is Adding a maximum valuation of 1 per item, an upper bound on total utility is

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Theorem: Upper bound on revenue for simultaneous auctions

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The Proof 1.Formal problem 2.Upper bound on simultaneous auctions 3.Lower bounds on combinatorial auctions 4.Comparison

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Lower Bound on combinatorial auction Main idea: Prove that free riding / threshold problem does not have a major effect Idea: if a bidder with high valuation bids low, there will be someone else that can benefit from this by raising his bid.

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Proving lower bounds on strategies, the general idea: Strategies are monotone Therefore, a bidder X with low valuation has low probability of winning (since there will be many bidders above him) Therefore, X has low expected revenue Let W be the expected value of a winning bid If W is low enough, X can bid above W and get a higher expected revenue than theoretically possible. We have a contradiction. So, W can not be too low.

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An Example 10 items Combination size k=4 (so only two combinatorial bids can win) Millions of bidders A bidder X with valuation 0.95 per item has very low chance of winning, since there will probably be two non- colliding bidders with higher valuation than 0.95. Suppose the expected value of the lowest winning combinatorial bid is 1.9 per item. Then, since the probability that X does not collide with the other winning bid is quite high, X can get a good expected revenue by bidding 1.91. We have a contradiction

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Asympotic Result Lemma: In the combinatorial auction, as the number of bidders approaches infinity, the lowest winning bid approaches the maximum value k(1+α) Proof: By contradiction: If the winning bids are lower, there will be bidders that can get impossibly high revenue by bidding higher Theorem: In the first-price combinatorial auction as the number of bidders approach infininty, the expected revenue approaches

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A Paremeterized Lower Bound on Combinatorial Auctions

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The Proof 1.Formal problem 2.Upper bound on simultaneous auctions 3.Lower bounds on combinatorial auctions 4.Comparison

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A First Comparison Corollary: As the number of bidders approach infininty, the expected revenue of the first-price combinatorial auction is higher than that of the simultaneous auction, give M ≥ 2k and K > 2.

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A second comparison, finding specific examples

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Content 1.A glance at today’s topic 2.Research Background 3.Industrial Background 4.Auction Protocols 5.The Problem: combiatorial vs. simultaneous auctions 6.The Proof 7.Summary

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Summary Finally, after a couple of years, I know the answer: There is a theoretical support for combinatorial auctions. It does not cover all thinkable cases, but it covers by far more than previous theoretical studies. What’s next?

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