Shaili Jain September 29 2011 Combinatorial Auction Model Set M of m indivisible items that are concurrently auctioned among a set N of n bidders Bidders have preferences on bundles of items ID: 1018747
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1. CPSC 455/555Combinatorial Auctions, Continued…Shaili JainSeptember 29, 2011
2. Combinatorial Auction ModelSet M of m indivisible items that are concurrently auctioned among a set N of n biddersBidders have preferences on bundles of itemsBidder i has valuation vi Monotone: for S µ T, we have v(S) · v(T)v(;) = 0Allocation among the bidders: S1, …, Sn Want to maximize social welfare: i vi(Si)
3. Iterative Auctions: The Query ModelConsider indirect ways of sending information about the valuationAuction protocol repeatedly interacts with different bidders, adaptively elicits enough information about bidder’s preferencesAdaptivity may allow pinpointing; may not require full disclosureCan reduce complexity, preserve privacy, etc.
4. Iterative Auctions: The Query ModelThink of bidders as oracles and auctioneer repeatedly queries the oraclesWant computational efficiency, both in number of queries and in internal computationsEfficiency means polynomial running time in m and n
5. Types of QueriesValue Query: Auctioneer presents a bundle SThe bidder reports his value v(S) for this bundleDemand Query (with item prices): Auctioneer gives a vector of item prices: p1, …, pmThe bidder reports a demand bundle under these prices, i.e. a set S that maximizes v(S) - i2S pi
6. Value vs. Demand QueriesLemma: A value query may be simulated by mt demand queries, where t is the number of bits of precision in the representation of a bundle’s value. Marginal value query: Auctioneer presents bundle S and j 2 M – SBidder gives v(j|S) = v(S [ {j}) – v(S)
7. Value vs. Demand QueriesHow to simulate a marginal value query using a demand query?For all i 2 S, set pi = 0For all i 2 M – S – {j}, set pi = 1Run binary search on pjNeed up to m marginal value queries to simulate a value query
8. Value vs. Demand QueriesLemma: An exponential number of value queries may be required for simulating a single demand query.Part of your homework… Consider two agents Use the fact that there are exponentially many sets of size m/2
9. An IP FormulationLet xi,S = 1 if agent i gets S, xi,S = 0 otherwise
10. LP Relaxation
11. The Dualmin i2N ui + j2M pjs.t. ui + j2S pj ¸ vi(S) 8 i 2 N, S µ Mui ¸ 0, pj ¸ 0 8 i 2 N, j 2 M
12. Using demand queries…Use demand queries to solve the linear programming relaxation efficientlySolve the dual using the Ellipsoid methodDual is polynomial in number of variables, exponential in the number of constraintsEllipsoid algorithm is polynomial provided that a “separation oracle” is givenShow how to implement the separation oracle via a single demand query to each agent
13. Using demand queries…Theorem: LPR can be solved in polynomial time (in n, m, and the number of bits of precision t) using only demand queries with item prices
14. Proof“separation oracle” either confirms possible solution is feasible or returns constraint that is violatedConsider a possible solution to the dual, e.g. set of ui and pjRewrite the constraints as ui ¸ vi(S) - j2S pjA demand query to bidder i with prices pj reveals the set S that maximizes the RHS
15. Proof ContinuedQuery each bidder i for his demand Di under prices pjCheck only n constraints: ui + j2Di pj ¸ vi(Di)
16. Proof ContinuedNow need to show how the primal is solvedIn solving the dual, we encountered a polynomial number of constraintsCan remove all other constraints Now take the dual of the “reduced dual”Has a polynomial number of variables, has the same solution as the original primal
17. Walrasian EquilibriumGiven a set of prices, the demand of each bidder is the bundle that maximizes her utilityMore formally… For given vi and p1, …, pm, a bundle T is called a demand of bidder i if for every other S µ M, we have: vi(S) - j2S pj · vi(T) - j2T pj
18. Walrasian EquilibriumSet of “market-clearing” prices where every bidder receives a bundle in his demand setUnallocated items have price of 0More formally… A set p*1, …, p*m and an allocation S*1, …, S*m is a Walrasian equilibrium if for every i, S*i is a demand of bidder i at prices p*1, …, p*m and for any item j not allocated, we have p*j = 0
19. An Example2 players, Alice and Bob2 items, {a, b}Alice has value 2 for every nonempty set of itemsBob has value 3 for the whole bundle {a,b} and 0 for any of the singletonsWhat is the optimal allocation?
20. An ExampleOptimal allocation: Both items to BobIn a Walrasian equilibrium, Alice must demand the empty setTherefore, the price of each item must be at least 2The price of whole bundle must be at least 4Bob will not demand this bundle
21. Walrasian EquilibriumWalrasian equilibrium, if they exist, are economically efficient“First Welfare Theorem”Welfare in a Walrasian equilibrium is maximal even if the items are divisibleIf a Walrasian equilibrium exists, then the optimal solution to the linear program relaxation will be integral
22. Walrasian EquilibriumThe existence of an integral optimum to the linear programming relaxation is a sufficient condition for the existence of a Walrasian equilibrium“Second Welfare Theorem”
23. ReferencesThis material was from section 11.3 and 11.5 in the AGT bookFor a good reference on LP-duality, look at “Approximation Algorithms” by Vijay VaziraniQuestions? shaili.jain@yale.edu