1 / 44

Introduction to combinatorial auctions

Introduction to combinatorial auctions. By Guy Kortsarz. A 1 -item auction mechanisms. Each bidder submits a bid in an envelope Auctioneer opens the envelopes, highest bid wins. The method used usually, called VCG method. First-price : the one proposing most money.

Download Presentation

Introduction to combinatorial auctions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Introduction to combinatorial auctions By Guy Kortsarz

  2. A 1-item auction mechanisms • Each bidder submits a bid in an envelope • Auctioneer opens the envelopes, highest bid wins. • The method used usually, called VCG method. • First-price : the one proposing most money. • Second-price discount: But he is given a discount which is only the value of the second highest bidder. • His gain v1-b2

  3. On the importance of being truthful • The bidders may cheat • May give prices different than their true values. • A mechanism is Truthful if the bidders say the correct prize. • Interestingly, a truthful mechanism for this simple problem was only designed in 1961 by Vickrey and since then is called: Vickerey Auction.

  4. Truthfulness • Let {bi} be the number the bidders bid. • For simplicity say that b1 and b2 contain the first and second highest bid. • Let vibe the true values. Say that b1≥ b2 for the moment (in general we do not assume that). • The net value the first agent gets is:v1-b2 • The bidders may bid higher to try to win (it does not change the payment) • And of course may bidbi<vi

  5. On being truthful • Overbidding does not help. Moreover can hurt. Since the agents are rational they will bid truthfuly. • There are three cases depending on v1, b1, and b2. • In any case the assumption is that b1 and b2 are the largest and second largest bid.

  6. First case • v1≥b2 • Because we are on the case of overbid,namelyb1>v1 it is clear that agent 1 wins. • If you overbid, it does not make the true value of the item larger than v1. • Never mind what you overbid, you still get v1-b2 money. No point in cheating.

  7. Remaining cases • The case that b2>b1 • In this case agent1looses so no matter. • The last case remaining is that v1< b2< b1. • This is a case in which agent 1 looses because of over bidding. • v1-b2<0 • Truthful biding will make him loose but at least does not loose money.

  8. But what about underbidding? • b1<v1. Try to save money. • Again let b2 and b1 be the two largest bids. • What if b1<v1<b2? In this case agent 1 does not get the item so truthful or not, same revenue of 0. • If b2<b1. Then the net gain is v1-b2. This is even if the bid is much smaller thanv1.

  9. The last case remaining • b1<b2<v1. • Shows that underbidding was a mistake. • 0 net gain versus v1-b2>0. • We call this a dominating strategy. A strategy that is never worse than the value of any other strategy.

  10. Many items • The items are U={1,2,….,m} • For now assume that there is aunique copy per items. • We have agents and each agent has a value vi(S), for every subset S U.For every i its 2m values! • The goal is to split sets so agent i gets Siand Si is a disjoint partition of U. • Max I vi(Si) A.K.ASocial welfare.

  11. Are values monotone? • Its is natural to assume that if ST then vi(S)≤vi(T). • This assumption is NOT made here. • Very interesting case is SINGLE MINDED BIDDERS. • This means that the bidder wants just one setS and is willing to pay some value for it. But for sets larger than S, his pay is 0! • SMB has very nice theory. But I cal not speak on all subjects.

  12. Social welfare: total happiness. • ivi(Si). Note: uses real values. • Now enters the issue of computability of the partition (in economy, did they even care? I don’t know). • This is hard to compute unless we have exponential time. • Discussion of the Independent set problem.

  13. An independent set • A collection of pairwise non neighbors

  14. An independent set • A collection of pairwise non neighbors

  15. Approximating the IS problem • Hastad: no better than n1- approximation (it is much worse actually but the above enough). • This says the following: its as hard to approximate within n1- is as hard as solving exactly! A remarkable result. • This is a result that follows from the famous PCP theorem. • The PCP theorem basically says that SAT has no 1- approximationfor some .

  16. First hardness • Hastad showed that getting about n size independent set when there is a size n 1-  independent set is as hard as solving the independent set problem exactly! • Not well known: Berman et al showed that if the PCP theorem holds, there is an  so that IS cannot be approximatedbyn • Amazing: done in 1988, 4 years before the PCP theorem was proved! • Tool: Randomized graph products.

  17. Implication for CA • Even if agents are truthful: • Consider one item per edge. • LetVdenote the bidders. • A vertex v ONLY wants its set of edges . • Any solution is an IS. Because single copies. • Say every vertex is willing to pay x for its set but nothing to any other set. • This means that SW roughly sqrt{m}=nNA.

  18. Not possible to approximation within roughly sqrt{m}. • Say every v agrees to pay is a x for his (unique) set but nothing for other sets. • The sum of payments is what auction manager wants to maximize. • The auctioneer can only get |I|*x value with |I| the maximum independent set he can compute. • As m can be about n2 this gives roughly sqrt{m} inapproximability.

  19. State of the art: in the time written and as far as I know. • Nisan and Mualem: one item many units. • Ratio ½ for single minded bidders. • Improve to FPAS by Briest, Krysta and Voecking. • Dobzinski and Nisan: Not single minded bidders. A ½ ratio with Maximum in Range (MIR) algorithm. Hence polynomial time.

  20. More state of the art results • Dobzinski Nisan and Schapira. First to give O(sqrt{m}) randomized truthful algorithm for CA. Note: its for single copy. • See also a paper by Dobzinski in APPROX-RANDOM 2007. • As far as I know, no deterministic algorithm with such results is known. • Big open problem.

  21. An interesting recent paper • Krysta and Vocking. On-line algorithm with b≥1 copies per item. • They present anO(m1/(b+1) log(bm)) on-line competitive ratio algorithm. • The algorith is randomized. • But is a distribution over dominating strategies. • Needs exponential power (not a surprise).

  22. State of the art continued. • This algorithm asks queries such as: given prices {pi} what is the best partition of the set S of elements. • Note that better than O(m1/(b+1) approximation isNPC (albeit the on-line algorithm uses exponential time procedures). • All on-line algorithm are multiplicative update algorithms. Values to items are multiplied at every round. • Popular items get higher values.

  23. Summary • A truthful deterministicsqrt{m} approximation is not known • But is known if we allow randomization. • In randomization the distribution (in my opinion) should be over dominatingstrategies. • Albeit, there are weaker notions (that I don’t like!).

  24. Vickery-Clark-Groves Mechanism • We leave pure computer science and enter the sinister subject of economy. • Because we allow the mechanism to set prices for the agents. The problem becomes a non pure computer science problem. • It turns out that without setting prices it is hard to get truthful mechanism.

  25. We ignore efficiently issues from now almost till the end. • The best social welfare is the one that divides the sets into {Si}, gives agent i the set Si and among all possiblepartitions,it maximizes the social welfare,namely iv(Si) which is the“total hapiness”. • Our goal is to get a truthful mechanism. • Because of the influence of economy, many times exponential time mechanism are allowed. • In any case we allow any time needed, exponential or beyond.

  26. Our goals • High money outcome and truthful. • We set a price pi for agent i. • At the end the net value is v(Si)-pi • Intuitively the price is the damage of the agent to inflicts on other agents. • pj = the maximum social welfare without playerj minus the social welfare the others got (which depends on agentj). • The second term depends on j. • But the first terms does not depend onj.

  27. Intuitive explanation • Intuitively the price is the damage of the agent to inflicts on other agents. • The price pj= the maximum social welfare without playerj minus the social welfare the others got (which depends on agentj). • When j participates, it may be that other agent get less value. • Thus we compute the optimum without j for al other agents minus with j.

  28. Intuitive explanation • Consider the maximum social welfare without playerj. • Clearly this is at least as large as social welfare the others got when j does participate. Because j participates this second value depends on j. • Clearly the optimization without j is the maximum social revenue the others can get. • Thus the above term is at least 0.

  29. In the reverse direction • There are approximation algorithms that use the VCG method. • They define this VCG value on a problem. • They show that under some conditions, there is a good approximation for the problem. Thus the net value VCG is used in approximation algorithm (see some papers by Anupam Gupta and others).

  30. The single item mechanism is VCG • Player 1 should pay the optimal social welfare, if it does not participate minusthe social welfare of the others got from the chosen outcome. • Note: if 1 does not participate the social welfare is v2=b2 because of truthfulness. • SW for others when agent 1 participates 0. • b2-0=0. This is indeed the price for player 1. • Thus b2 price indeeda VCG mechanism. • Net gain v1-b2

  31. This mechanism is truthful (even if hard to calculate) • pj = the maximum social welfare without playerj minus the social welfare the others got. • The first term does not depend on agent j. • So, think of this term as 0. In such a case the value minus discount is vi(Si)+the social welfare the other agents got. • Which is what is maximized. Hence cheating will be a mistake.

  32. Why is the term not related to i is inserted? • Player i should pay the optimal social welfare, if it does not participate minusthe social welfare of the others got from chosen outcome. • This number is clearly at least 0, hence no negative pays. • Also the same reasoning shows that vi-pi is always at least 0. No loss.

  33. Approximation: the missing link • If you want to maximize the social welfare or social revenue, it may be NP-hard,to do so. • Study special cases for example. Special welfare functions for example. • It comes natural to CS people to say: approximate the social welfare/revenue. • We shall see later that approximation can kill truthfulness.

  34. Approximation: the missing link • A quote by Woody Allen: • When I was kidnap my parents took immediate steps.

  35. Approximation: the missing link • A quote by Woody Allen: • When I was kidnap my parents took immediate steps. • They rented my room!

  36. In CS conferences • Here and there people from Economy attend our conferences. • They say: we only care on optimum. • They say: worst case is not a good measure. • They say: your approximation ratios are not practical. • Hence we took immediate steps!

  37. In CS conferences • Here and there people from Economy attend our conferences. • They say: we only care on optimum. • They say: worst case is not a good measure. • They say: your approximation ratios are not practical. • Hence we took immediate steps! • We said that we don’t care.

  38. I wonder • Is it true that algorithmic mechanism designset minuswhat they study in economy is simply approximation algorithms? • Approximation algorithm have all what people in economy may not like. • For example worse case. In Economy is with respect to distributions. • Get a less than optimal (approximation) social welfare. In economy I think its either optimal or not. Note interestes in approximation it seems.

  39. Approximation killstruthfulness! • Example: consider two agents and two items. Both evaluate each item separately by 1.9 and both items by 3.1. • Optimum to maximize social welfare : give each, one item. Total welfare 3.8. • The VCG payment: 3.1-1.9=1.2. • Consider truncating values down Appr’. • If they bid truthfully the approximation algorithm will give values (1,3).

  40. No good deed goes unpunished • With values (1,3) its better to give both items to one of them and get social welfare 3. • The VCG payment of the one who got the items: 3.1 (we use real values for the payment. But using rounded ones fails as well!). • Net revenue is 0.

  41. Cheating is better • Say that exactly one of the players cheats. • Says: for any subset (except the empty set), I am wiling to pay 3. • Giving each one a single item getsrevenue 4,clearly, best possible. • The non truthful is charged: 3.1-1.9=2.1. • Net revenue 0.9. Non truthful mechanism.

  42. Approximation lost us VCG, what to do? • Well, try to find mechanism that will give a good approximation and still are truthful. • Example: say that there is one item in m copies. • Every agent has for every k≤m a value vi(k). It is assumed that vi(k)≤ vi(k+1). • A mechanism has to allocate miunits to agent i so that mi=m and maximize vi(mi)

  43. Solving it even with full knowledge is NPC: Knapsack • An example of a modern result. • Due to Dobzinski and Nisan. • There exists a truthful mechanism that gets at least ½ the optimal social welfare. • This is called ratio 2 at times. The hard part is making it truthful. • The algorithm is efficient as it is maximum in (small) range. You search for the best among few solutions. • Best possible of his type.

  44. Summary • Auctions just one important example. • Voting is another example. • Also extensively studied is price of anarchy and price of stability. Maybe to be described in a future lecture. • Also extensively studied: how hard is it to compute a Nash Equilibrium. This has a complete class of problems already. Thus hardness results. • Leaving cynicism aside: the relation between the community of Algorithmic game theory and economy should be made much closer in my opinion. • To begin with many (but many) results rediscovered in CS. Known from the 1950’s in Economy!

More Related