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**Combinatorial Auction**

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**Conbinatorial auction**

f(t): the set XF with the highest total value t1 =20 v1=20 t2=15 v2=16 t3=6 v3=7 the mechanism decides the set of winners and the corresponding payments Each player wants a bundle of objects ti: value player i is willing to pay for its bundle F={ X{1,…,N} : winners in X are compatible} if player i gets the bundle at price p his utility is ui=ti-p

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**Combinatorial Auction (CA) problem – single-minded case**

Input: n buyers, m indivisible objects each buyer i: Wants a subset Si of the objects has a value ti for Si Solution: X{1,…,n}, such that for every i,jX, with ij, SiSj= Measure (to maximize): Total value of X: iX ti

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**CA game each buyer i is selfish**

Only buyer i knows ti (while Si is public) We want to compute a “good” solution w.r.t. the true values We do it by designing a mechanism Our mechanism: Asks each buyer to report its value vi Computes a solution using an output algorithm g(٠) takes payments pi from buyer i using some payment function p

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**More formally Type of agent buyer i: Buyer i’s valuation of XF:**

ti: value of Si Intuition: ti is the maximum value buyer i is willing to pay for Si Buyer i’s valuation of XF: vi(ti,X)= ti if iX, 0 otherwise SCF: a good allocation of the objects w.r.t. the true values

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**How to design a truthful mechanism for the problem?**

Notice that: the (true) total value of a feasible X is: i vi(ti,X) the problem is utilitarian! …VCG mechanisms apply

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**pi =j≠i vj(rj,g(r-i)) -j≠i vj(rj,x)**

VCG mechanism M= <g(r), p(x)>: g(r): arg maxxF j vj(rj,x) pi(x): for each i: pi =j≠i vj(rj,g(r-i)) -j≠i vj(rj,x) g(r) has to compute an optimal solution… …can we do that?

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Theorem Approximating CA problem within a factor better than m1/2- is NP-hard, for any fixed >0. proof Reduction from independent set problem

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**Maximum Independent Set (IS) problem**

Input: a graph G=(V,E) Solution: UV, such that no two verteces in U are jointed by an edge Measure: Cardinality of U Theorem Approximating IS problem within a factor better than n1- is NP-hard, for any fixed >0.

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**the reduction G=(V,E) each edge is an object**

each node i is a buyer with: Si: set of edges incident to i ti=1 G=(V,E) CA instance has a solution of total value k if and only if there is an IS of size k …since m n2…

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**How to design a truthful mechanism for the problem?**

Notice that: the (true) total value of a feasible X is: i vi(ti,X) the problem is utilitarian! …but a VCG mechanism is not computable in polynomial time! what can we do? …fortunately, our problem is one parameter!

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**A problem is binary demand (BD) if**

ai‘s type is a single parameter ti ai‘s valuation is of the form: vi(ti,o)= ti wi(o), wi(o){0,1} work load for ai in o when wi(o)=1 we’ll say that ai is selected in o

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Definition An algorithm g() for a maximization BD problem is monotone if agent ai, and for every r-i=(r1,…,ri-1,ri+1,…,rN), wi(g(r-i,ri)) is of the form: 1 Өi(r-i) ri Өi(r-i){+}: threshold payment from ai is: pi(r)= Өi(r-i)

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**Our goal: to design a mechanism satisfying:**

g(٠) is monotone Solution returned by g(٠) is a “good” solution, i.e. an approximated solution g(٠) and p(٠) computable in polynomial time

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**A greedy m-approximation algorithm**

reorder (and rename) the bids such that W ; X for i=1 to n do if SiX= then W W{i}; X X{Si} return W v1/|S1| v2/|S2| … vn/|Sn|

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**Lemma The algorithm g( ) is monotone proof**

It suffices to prove that, for any selected agent i, we have that i is still selected when it raises its bid Increasing vi can only move bidder i up in the greedy order, making it easier to win

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**Computing the payments**

…we have to compute for each selected bidder i its threshold value How much can bidder i decrease its bid before being non-selected?

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**Computing payment pi pi= vj |Si|/|Sj| pi= 0 if j doen’t exist**

Consider the greedy order without i v1/|S1| … vi/|Si| … vn/|Sn| index j Use the greedy algorithm to find the smallest index j (if any) such that: 1. j is selected 2. SjSi pi= vj |Si|/|Sj| pi= 0 if j doen’t exist

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**for greedy order we have**

Lemma Let OPT be an optimal solution for CA problem, and let W be the solution computed by the algorithm, then iOPT vi m iW vi proof iW OPTi={jOPT : j i and SjSi} since iW OPTi=OPT it suffices to prove: vj m vi iW jOPTi crucial observation for greedy order we have vi |Sj| vj jOPTi |Si|

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** proof iW vi |Sj| vj m vi |Si| |Sj| |OPTi| |Sj|**

Cauchy–Schwarz inequality we can bound |Sj| |OPTi| |Sj| |Si|m jOPTi jOPTi ≤|Si| ≤ m

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**Cauchy–Schwarz inequality**

…in our case… xj=1 n= |OPTi| for j=1,…,|OPTi| yj=|Sj|

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An Approximate Truthful Mechanism for Combinatorial Auctions An Internet Mathematics paper by Aaron Archer, Christos Papadimitriou, Kunal Talwar and Éva.

An Approximate Truthful Mechanism for Combinatorial Auctions An Internet Mathematics paper by Aaron Archer, Christos Papadimitriou, Kunal Talwar and Éva.

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