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## Combinatorial Auctions

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**Reloaded**Combinatorial Auctions**Review**• Optimal allocation • Special case: Single minded bidders • Incentive compatibility**Review**• Greedy algorithm (single minded) • LPR & DLPR • Walrasian Equilibrium • Every bidder receives his demand • “demand” = maximum utility bundle • Poly-time optimal allocation**Review**Social welfare Linear program relaxation (LPR) Item cannot be more than once Bidder has at most one bundle Social welfare**Review**Dual linear program relaxation (DLPR)**Lecture outline**• “The curse of dimensionality” • Bidding languages • Iterative auctions • Queries • Communication complexity • Ascending auctions**The curse of dimensionality**• High-dimensional input data • for all and • LPR • High dimensional solution vector • for all and • DLPR • Enormous amount of constraints Bidding languages Iterative auction**Goals**• Expressiveness • What kind of valuations can we express? • Compactness • Use less bits for “interesting” kinds of valuations**OR bids**• Any subset can be fulfilled • Example • ({a,b}, 12) OR ({c,d}, 8}) • Valuations • v({a}) = 0 • v({a,b}) = 12 • v({a,b,c}) = 12 • v({a,b,c,d}) = 12 + 8 = 20**OR bids**• Formal definition • are called atomic bids. • Intuitively • Take all “valid” collections of items • Choose the one that has maximum value**XOR bids**• Only one subset can be fulfilled • Choose maximum-value subset • Example • ({a,b}, 12)XOR({c,d}, 8) • Valuations • v({a,b}) = 12 • v{c,d}) = 8 • v({a,b,c,d}) = 12**Expression power**• XOR bids can represent any valuation • Just XOR all possible values for all subsets • Can be very inefficient • OR bids – super additive valuations • implies**Expression power**• Additive valuation • Naturally represented by OR bid • Unit-demand valuation • Naturally represented by XOR bid**OR/XOR combinations**• Defined inductively • Let be valuations • Have more representation power**OR/XOR expression power**• Symmetric valuation • depends only on • Downward sloping • Can be represented as with • Downward sloping with cannot be represented by OR bids and needs exponential size XOR bids.**OR/XOR expression power**• Theorem • OR of XOR bids can express any downward-sloping symmetric valuation of items in size • Proof – next slide**OR/XOR expression power**• For each define a clause that offers for any single item • Define the final expression as • Since are non-increasing – first item taken from the second from and so on**OR/XOR expression power**• Example • We have 3 items and we have , and • The expression is**Dummy items**• Goal – reuse single-minded allocation algorithm for OR/XOR bids • Method – represent everything as OR bids • Key observation • OR bids look like a collection of single-minded bids from different players.**Dummy items**• How? Use dummy items! • Example • We have • Add a dummy item d • Rewrite as • Apply recursively for OR/XOR bids • We call this OR* bids**Dummy items**• Theorem: Any valuation that can be represented by OR/XOR formula of size can be represented by OR* formula of size using at most dummy items • Remarks • A valuation in terms of the original formula is translated to in terms of OR* formula, where is the set of all dummy items. • The “size” of a formula is the amount of atomic bids it contains.**Dummy items**• Two stage proof • (1) Prove that we can construct an OR* formula of size s • (2) Prove that we need at-most dummy items in the OR* formula.**Dummy items**• Proof of (1) by induction • Definition • to be the OR* translation of v. • Base • A single pair is also an OR* bid • Step for • Let and • Define to be the union of atomic bids in and . • We got a formula of the same size as .**Dummy items**• Example for XOR • Define dummy items • Translates to**Dummy items**• Step for • Define and • Create dummy items for each pair of atomic bids in and in • Create • Transform each in to become • Similarly, transform each in • We got to be of the same size as.**Dummy items**• Proof of (2) • Dummy item’s purpose is to disallow two atomic bids to be taken concurrently. • Thus we need dummy items – one for each pair of atomic bids that cannot be taken concurrently.**Dummy items**• Conclusion: Every algorithm that can handle single-minded bids in polynomial time can handle any OR/XOR combinations in polynomial time.**Motivation**• Reduce the amount of information transfer • Query mechanism that transmits less bits than OR/XOR? • Expressive power • Preserve some privacy • Bidder limitations • Bidders don’t know their valuation • Need effort to determine valuations • Guide bidders to the data relevant to the mechanism**Goals**• Computational efficiency • How much information is transferred? • How long does it take to determine an allocation? • Incentive compatibility • Why should the bidders answer the queries truthfully?**Queries**• The method of “asking for preferences” • Value query • What is the value of a bundle S? • Demand query • What would you like to buy for those prices? • Formally: Given a set of prices , what is the bundle S that maximizes**Expressive power**• Demand queries are more powerful than value queries • Lemma: • A value query may be simulated using demand queries, where is the number of bits in the representation of bundle’s value. • Exponential number of value queries may be needed to simulate a single demand query. • Demand queries allow solving the LPR problem efficiently**Solve the linear program**• Solve the DLPR • Use a method that doesn’t need all constraints at once • exponential amount of constraints! • Ellipsoid method to the rescue! • Use the solution of DLPR to solve LPR**Ellipsoid method**• Solved LP problems by shrinking an ellipsoid • High level overview • Start with an ellipsoid that contains the solution. • Iteratively create a sequence such that: • contains the solution • results from constraint violation test**Solve the linear program**• Ellipsoid method requirements • Given report weather is feasible or find a single constraint that violates. • DLPR constraints • Treat as utility and as prices • Constraint violation checking – using demand queries.**Solve the linear program**• Demand-query all bidders using the prices . The results are . • Calculate • Using demand queries • Using a value query • If for all have a feasible point • Otherwise – we have a violated constraint.**Solving the primal**• LP • Maximize • Subject to • Dual • Minimize • Subject to**Solving the primal**c b A**Solving the primal**c b A**Solving the primal**c b A**Solving the primal**• Use violated constraints from the Ellipsoid algorithm • Remove all other constraints still the same solution • Ellipsoid algorithm will produce the same final ellipsoid**Solving the primal**c b A**Solving the primal**c b A**Solving the primal**• Solve the reduced-primal with polynomial number of variables! • Assign 0 to the unused ones.**Conclusions**• Facts • WalrasianEquillibrium LPR has integer solution LPR solved optimal allocation • LPR can be solved in polynomial time • LPR solution requires polynomial number of demand queries. • Conclusion • WalrasianEquillibrium Can use demand queries to find optimal allocation in poly-time using polynomial amount of queries.