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Seminar In Game Theory Algorithms, TAU, 2010

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

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

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

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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?

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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

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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:

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

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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

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

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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

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

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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

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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

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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? ?

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

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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

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Incentive Compatible Mechanism Efficient Computable Incentive Compatible Approximation By Factor

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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* 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`

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Incentive Compatible Mechanism Lemma Proof: Denote true bid B(S,v), false bid B*(S*,v*) If B* lose or S*~~ 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
~~

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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

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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

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DLPR Relaxation Solve by limiting lower bounds:

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Walrasian Equilibrium Economy Theory, “The Point where the market clears”

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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

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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

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

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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!

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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

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