introduction to combinatorial auctions by guy kortsarz
TRANSCRIPT
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.
• Second-price discount: But he is given a discount which is only the value of the second highest bidder.
• His gain v1-b2
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.
Truthfulness
• Let {bi} be the number the bidders bid.
• For simplicity say that b1 and b2 contain the first and second highest bid.
• Let vi be 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 bid bi<vi
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.
First case
• v1≥b2
• Because we are on the case of overbid, namely b1>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.
Remaining cases
• The case that b2>b1
• In this case agent 1 looses 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.
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 than v1.
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.
Many items
• The items are U={1,2,….,m}• For now assume that there is a unique 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 Si
and Si is a disjoint partition of U.• Max I vi(Si) A.K.A Social welfare.
Are values monotone?
• Its is natural to assume that if ST 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 set S
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.
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.
An independent set
• A collection of pairwise non neighbors
An independent set
• A collection of pairwise non neighbors
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- approximation for some .
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 can not be approximated by n
• Amazing: done in 1988, 4 years before the PCP theorem was proved!
• Tool: Randomized graph products.
Implication for CA
• Even if agents are truthful:• Consider one item per edge.• Let V denote 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}=n NA.
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.
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.
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.
An interesting recent paper
• Krysta and Vocking. On-line algorithm with b≥1 copies per item.
• They present an O(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).
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 is NPC (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.
Summary
• A truthful deterministic sqrt{m} approximation is not known
• But is known if we allow randomization.
• In randomization the distribution (in my opinion) should be over dominating strategies.
• Albeit, there are weaker notions (that I don’t like!).
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.
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 possible partitions, 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.
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 player j minus the social welfare the others got (which depends on agent j).
• The second term depends on j.• But the first terms does not depend on j.
Intuitive explanation
• Intuitively the price is the damage of the agent to inflicts on other agents.
• The price pj= the maximum social welfare without player j minus the social welfare the others got (which depends on agent j).
• 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.
Intuitive explanation
• Consider the maximum social welfare without player j.
• 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.
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).
The single item mechanism is VCG
• Player 1 should pay the optimal social welfare, if it does not participate minus the 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 indeed a VCG mechanism.
• Net gain v1-b2
This mechanism is truthful (even if hard to calculate)
• pj = the maximum social welfare without player j 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.
Why is the term not related to i is inserted?
• Player i should pay the optimal social welfare, if it does not participate minus the 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.
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.
Approximation: the missing link
• A quote by Woody Allen:
• When I was kidnap my parents took immediate steps.
Approximation: the missing link
• A quote by Woody Allen:
• When I was kidnap my parents took immediate steps.
• They rented my room!
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!
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.
I wonder
• Is it true that algorithmic mechanism design set minus what 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.
Approximation kills truthfulness!
• 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).
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.
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 gets revenue 4, clearly, best possible.
• The non truthful is charged: 3.1-1.9=2.1.
• Net revenue 0.9. Non truthful mechanism.
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 mi units to agent i so that mi=m and maximize vi(mi)
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.
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!