1 / 30

Combinatorial Auction

Seminar In Game Theory Algorithms, TAU, 2010. Combinatorial Auction. Agenda. Introduction Computational Complexity Incentive Compatible Mechanism LP Relaxation & Walrasian Equilibrium . The Problem…. Allocating a set of non-divisible items, among multiple bidders contending for them,

cooper
Download Presentation

Combinatorial Auction

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. Seminar In Game Theory Algorithms, TAU, 2010 Combinatorial Auction

  2. Agenda • Introduction • Computational Complexity • Incentive Compatible Mechanism • LP Relaxation & Walrasian Equilibrium

  3. The Problem… Allocating a set of non-divisible items, among multiple bidders contending for them, In such a way that social welfare is maximized

  4. The Goal! Design a computational efficient mechanism, that will find such a socially efficient allocation

  5. Difficulties • Computational Complexity – • Problem is hard to compute, NP-Complete • Space Complexity – • Values for items is exponential size object • Strategies – • Can we analyze them? Design for them?

  6. Applications • Spectrum Actions – • Selling licenses for bands spectrum using auctions • Transportation Service – • Reverse bid, items value depends on route • Communication Network – • Bid for a path between two edges

  7. Formalization • Valuation: v(S) = r • Assignment of values to sub sets of items • Monotone, Normalized, No Externalities • Allocation: • Assignment of sub sets of items to bidders • Social Welfare:

  8. Single-Minded Case • Valuation function v is called single minded if: • Simplify representation of valuation functions • Biddings are represented in the form (S*,vi)

  9. Complexity • Allocation problem for the single minded case is basically the “Weight-packing” problem • Known to be NP-complete • Proof by reduction from INDEPENDENT-SET

  10. Complexity - Proof • IS Problem: Given Graph G=(V,E), K does the graph has an independent set of size K?

  11. Complexity - Proof • Reduction: • The Set Of Items will be E (graph edges) • The Number of players will be |V| (graph vertices) • The bidding (Vi,Si) of player i: • Vi = 1 (Winning Value is always one) • Si = { e in E | i in e } (Subset of edges containing i ) • Result: • S1,.., Sn Allocation iff • Social Welfare is exactly the size of the independent set

  12. Improved Complexity? • Three possibilities: • Approximation – Compute result close to optimal • Special Cases – Optimized for specific type of input • Heuristic – use heuristics to rapid computation

  13. Complexity - Approximation • Allocation S1,..,Sn is called c-approximation if: • Exists efficient algorithm? NO! • Implies from the NP-completeness reduction • Approximating IS within factor is NP-complete • Approximating allocation within factor is in NP

  14. Complexity – Special Cases • Bidders desire bundles of at most 2: • Eq. Weighted Matching Problem, known to be efficiently solvable • Bidders desire continuous segment of items: • Can be solved efficiently using dynamic programming • Integer programming: • Use known heuristics to solve for integer programs

  15. Incentive Compatible Mechanism • True values are private information of bidders, despite this, can we design a mechanism that will allow the allocation algorithm to optimize social welfare and keeps computation efficiency? • ?

  16. Incentive Compatible Mechanism • Incentive compatible mechanism is one that makes it more worthwhile for bidders to report bids truthfully rather than lie

  17. Incentive Compatible Mechanism • Simple Solution: • Allocation would be the socially efficient one • Payments would be based on VCG • Computationally Intractable • Combined Solution: • Allocation approximation (computed efficiently) • Payments would be based on VCG • Wrong, VCG requires optimal social welfare • Dedicated Algorithm? • Biddings are simple composed of the pair of scalar, item set

  18. Incentive Compatible Mechanism • Efficient Computable • Incentive Compatible • Approximation By Factor

  19. Incentive Compatible Mechanism • Lemma 1.9: mechanism for single minded bidders in which losers pay 0 is incentive compatible iff satisfies: • Monotonicity – if a bid (S,v) is a winning bid the bid (S*,v*) where S*<S or v*>v is also winning. • Critical Payment - A bidder who wins pays the minimum needed for winning • The two conditions are met by the greedy algo`

  20. Incentive Compatible Mechanism • Lemma Proof: • Denote true bid B(S,v), false bid B*(S*,v*) • If B* lose or S*<S – make no sense to use it • Denote p payment for bid B, p* for B* • For every x < p bid (S,x) lose - critical payment • (S*,x) also lose – monotonicity => p* > p • Bidding B~(S,v*) instead (S*,v*) is no worse • B is no worse then B~ since if B wins payment is always p • If B lose, v < p therefore it wont be worth to win

  21. LP Relaxation Formulate allocation problem as integer program: 1.3 – Maximize Social Welfare 1.4 – Each item is allocated to at most one bidder 1.5 – Each bidder wins at most one bundle 1.6 – All values are non-negative

  22. LP Relaxation • LP Relaxation achieve polynomial efficiency by relaxing the variable values from {0,1} to [0..1] • Solution corresponds to fractional allocation assuming items were divisible • LP has exponentially many variables (in the number of items) • For single minded case, simple and efficient, only one variable per player

  23. DLPR Relaxation • Solve by limiting lower bounds:

  24. Walrasian Equilibrium Economy Theory, “The Pointwhere the market clears”

  25. Walrasian Equilibrium • Comes from economic field theory • The set of prices in which demand equals the supply • Demand of a bidder is a bundle T that maximize his utility, i.e. for every other bundle S: • Linear pricing – the price of a bundle of items equals the sum of prices • A pricing and an allocation of items is walrasin equilibrium if for every bidder its allocated bundle is its demand

  26. Walrasian Equilibrium • The First Welfare Theorem: Let p1,…,pm and S1,…,Sn be a Walrasian Equilibrium then the allocation S1,..,Sn maximize social welfare • social welfare is maximized over all fractional allocations as well

  27. Walrasian Equilibrium 1. Walarsian Equilibrium 2. 3. Sum equation in 2 over all n player 4. 5.

  28. Walrasian Equilibrium • Example for non existence of equilibrium: • Two players: Alice & Bob, Two items: A, B • Alice has value 2 for every non empty set • Bob has value 3 for the set {A,B} and 0 for others • Optimal allocation allocates both items to Bob • Alice must demand the empty set in every in equlibrium • Items price must be 2 otherwise Alice will demand them • Now bundle containing both is priced 4 • Bob won’t demand it!

  29. Walrasian Equilibrium • Second Welfare Theorem: if an integral optimal solution exists for LPR then a Walrasian equilibrium allocation is the given solution exists • Proof: based on DLPR and complementary slackness • Corollary: A Warlasian equilibrium exists in a combinatorial auction iff the corresponding LPR admits an integral optimal solution

  30. Questions?

More Related