Combinatorial Auction. Conbinatorial auction t 1 =20 t 2 =15 t 3 =6 f(t): the set X  F with the highest total value the mechanism decides the set of.

Slides:



Advertisements
Similar presentations
Combinatorial Auction
Advertisements

Truthful Mechanisms for Combinatorial Auctions with Subadditive Bidders Speaker: Shahar Dobzinski Based on joint works with Noam Nisan & Michael Schapira.
Combinatorial Auctions with Complement-Free Bidders – An Overview Speaker: Michael Schapira Based on joint works with Shahar Dobzinski & Noam Nisan.
6.896: Topics in Algorithmic Game Theory Lecture 21 Yang Cai.
Algorithmic mechanism design Vincent Conitzer
6.896: Topics in Algorithmic Game Theory Lecture 20 Yang Cai.
Blackbox Reductions from Mechanisms to Algorithms.
Online Mechanism Design (Randomized Rounding on the Fly)
On Optimal Single-Item Auctions George Pierrakos UC Berkeley based on joint works with: Constantinos Daskalakis, Ilias Diakonikolas, Christos Papadimitriou,
CPSC 455/555 Combinatorial Auctions, Continued… Shaili Jain September 29, 2011.
Approximating optimal combinatorial auctions for complements using restricted welfare maximization Pingzhong Tang and Tuomas Sandholm Computer Science.
Seminar in Auctions and Mechanism Design Based on J. Hartline’s book: Approximation in Economic Design Presented by: Miki Dimenshtein & Noga Levy.
Combinatorial auctions Vincent Conitzer v( ) = $500 v( ) = $700.
An Approximate Truthful Mechanism for Combinatorial Auctions An Internet Mathematics paper by Aaron Archer, Christos Papadimitriou, Kunal Talwar and Éva.
Multi-item auctions with identical items limited supply: M items (M smaller than number of bidders, n). Three possible bidder types: –Unit-demand bidders.
Preference Elicitation Partial-revelation VCG mechanism for Combinatorial Auctions and Eliciting Non-price Preferences in Combinatorial Auctions.
A Sufficient Condition for Truthfulness with Single Parameter Agents Michael Zuckerman, Hebrew University 2006 Based on paper by Nir Andelman and Yishay.
Seminar In Game Theory Algorithms, TAU, Agenda  Introduction  Computational Complexity  Incentive Compatible Mechanism  LP Relaxation & Walrasian.
Yang Cai Oct 15, Interim Allocation rule aka. “REDUCED FORM” : Variables: Interim Allocation rule aka. “REDUCED FORM” : New Decision Variables j.
Complexity ©D Moshkovitz 1 Approximation Algorithms Is Close Enough Good Enough?
SECOND PART: Algorithmic Mechanism Design. Implementation theory Imagine a “planner” who develops criteria for social welfare, but cannot enforce the.
6.853: Topics in Algorithmic Game Theory Fall 2011 Matt Weinberg Lecture 24.
Bundling Equilibrium in Combinatorial Auctions Written by: Presented by: Ron Holzman Rica Gonen Noa Kfir-Dahav Dov Monderer Moshe Tennenholtz.
Algorithmic Applications of Game Theory Lecture 8 1.
Complexity 16-1 Complexity Andrei Bulatov Non-Approximability.
Ron Lavi Presented by Yoni Moses.  Introduction ◦ Combining computational efficiency with game theoretic needs  Monotonicity Conditions ◦ Cyclic Monotonicity.
Item Pricing for Revenue Maximization in Combinatorial Auctions Maria-Florina Balcan, Carnegie Mellon University Joint with Avrim Blum and Yishay Mansour.
Limitations of VCG-Based Mechanisms Shahar Dobzinski Joint work with Noam Nisan.
An Improved Approximation Algorithm for Combinatorial Auctions with Submodular Bidders Speaker: Shahar Dobzinski Joint work with Michael Schapira.
SECOND PART: Algorithmic Mechanism Design. Mechanism Design MD is a subfield of economic theory It has a engineering perspective Designs economic mechanisms.
The selfish-edges Minimum Spanning Tree (MST) problem.
Frugal Path Mechanisms by Aaron Archer and Eva Tardos Presented by Ron Lavi at the seminar: “Topics on the border of CS, Game theory, and Economics” CS.
Bayesian Combinatorial Auctions Giorgos Christodoulou, Annamaria Kovacs, Michael Schapira האוניברסיטה העברית בירושלים The Hebrew University of Jerusalem.
A Truthful 2-approximation Mechanism for the Steiner Tree Problem.
Near-Optimal Network Design with Selfish Agents By Elliot Anshelevich, Anirban Dasgupta, Eva Tardos, Tom Wexler STOC’03 Presented by Mustafa Suleyman CIFTCI.
Truthful Mechanisms for One-parameter Agents Aaron Archer, Eva Tardos Presented by: Ittai Abraham.
SECOND PART: Algorithmic Mechanism Design. Implementation theory Imagine a “planner” who develops criteria for social welfare, but cannot enforce the.
Computability and Complexity 24-1 Computability and Complexity Andrei Bulatov Approximation.
SECOND PART: Algorithmic Mechanism Design. Mechanism Design MD is a subfield of economic theory It has a engineering perspective Designs economic mechanisms.
VC v. VCG: Inapproximability of Combinatorial Auctions via Generalizations of the VC Dimension Michael Schapira Yale University and UC Berkeley Joint work.
SECOND PART: Algorithmic Mechanism Design. Suggested readings Algorithmic Game Theory, Edited by Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V.
Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders Speaker: Michael Schapira Joint work with Shahar Dobzinski & Noam Nisan.
Competitive Analysis of Incentive Compatible On-Line Auctions Ron Lavi and Noam Nisan SISL/IST, Cal-Tech Hebrew University.
SECOND PART on AGT: Algorithmic Mechanism Design.
SECOND PART: Algorithmic Mechanism Design. Mechanism Design Find correct rules/incentives.
VCG Computational game theory Fall 2010 by Inna Kalp and Yosef Heskia.
Auction Seminar Optimal Mechanism Presentation by: Alon Resler Supervised by: Amos Fiat.
More on Social choice and implementations 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A Using slides by Uri.
By: Amir Ronen, Department of CS Stanford University Presented By: Oren Mizrahi Matan Protter Issues on border of economics & computation, 2002.
Yang Cai Oct 08, An overview of today’s class Basic LP Formulation for Multiple Bidders Succinct LP: Reduced Form of an Auction The Structure of.
TECH Computer Science NP-Complete Problems Problems  Abstract Problems  Decision Problem, Optimal value, Optimal solution  Encodings  //Data Structure.
SECOND PART on AGT: Algorithmic Mechanism Design.
Slide 1 of 16 Noam Nisan The Power and Limitations of Item Price Combinatorial Auctions Noam Nisan Hebrew University, Jerusalem.
Algorithmic Mechanism Design: an Introduction Approximate (one-parameter) mechanisms, with an application to combinatorial auctions Guido Proietti Dipartimento.
The Minimum Spanning Tree (MST) problem in graphs with selfish edges.
The Stackelberg Minimum Spanning Tree Game J. Cardinal, E. Demaine, S. Fiorini, G. Joret, S. Langerman, I. Newman, O. Weimann, The Stackelberg Minimum.
Yang Cai Oct 06, An overview of today’s class Unit-Demand Pricing (cont’d) Multi-bidder Multi-item Setting Basic LP formulation.
Algorithmic Mechanism Design: an Introduction (with an eye to some basic network optimization problems) Guido Proietti Dipartimento di Ingegneria e Scienze.
SECOND PART on AGT: Algorithmic Mechanism Design.
6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 22.
Algorithmic Mechanism Design Shuchi Chawla 11/7/2001.
Combinatorial Auction. A single item auction t 1 =10 t 2 =12 t 3 =7 r 1 =11 r 2 =10 Social-choice function: the winner should be the guy having in mind.
One-parameter mechanisms, with an application to the SPT problem.
Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders Speaker: Shahar Dobzinski Joint work with Noam Nisan & Michael Schapira.
SECOND PART: Algorithmic Mechanism Design.
Comp/Math 553: Algorithmic Game Theory Lecture 13
Combinatorial Auction
Combinatorial Auction
Combinatorial Auction
Combinatorial Auction
Presentation transcript:

Combinatorial Auction

Conbinatorial auction t 1 =20 t 2 =15 t 3 =6 f(t): the set X  F with the highest total value the mechanism decides the set of k winners and the corresponding payments Each player wants a bundle of objects t i : value player i is willing to pay for its bundle if player i gets the bundle at price p his utility is u i =t i -p F={ X  {1,…,N} : winners in X are compatible} v 1 =20 v 2 =16 v 3 =7

Combinatorial Auction (CA) problem – single-minded case Input: n buyers, m indivisible objects each buyer i: Wants a subset S i of the objects has a value t i for S i Solution: X  {1,…,n}, such that for every i,j  X, with i  j, S i  S j =  Measure (to maximize): Total value of X:  i  X t i

CA game each buyer i is selfish Only buyer i knows t i (while S i 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 v i Computes a solution using an output algorithm g( ٠ ) takes payments p i from buyer i using some payment function p

More formally Type of agent buyer i: t i : value of S i Intuition: t i is the maximum value buyer i is willing to pay for S i Buyer i’s valuation of X  F: v i (t i,X)= t i if i  X, 0 otherwise SCF: a good allocation of the objects w.r.t. the true values

How to design a truthful mechanism for the problem? Notice that: the (true) total value of a feasible X is:  i v i (t i,X) the problem is utilitarian! …VCG mechanisms apply

VCG mechanism M= : g(r): arg max x  F  j v j (r j,x) p i (x): for each i: p i =  j≠i v j (r j,g(r - i )) -  j≠i v j (r j,x) g(r) has to compute an optimal solution… …can we do that?

Approximating CA problem within a factor better than m 1/2-  is NP-hard, for any fixed  >0. Theorem proof Reduction from independent set problem

Maximum Independent Set (IS) problem Input: a graph G=(V,E) Solution: U  V, such that no two verteces in U are jointed by an edge Measure: Cardinality of U Approximating IS problem within a factor better than n 1-  is NP-hard, for any fixed  >0. Theorem

the reduction CA instance has a solution of total value  k if and only if there is an IS of size  k G=(V,E) each edge is an object each node i is a buyer with: S i : set of edges incident to i t i =1 …since m  n 2 …

How to design a truthful mechanism for the problem? Notice that: the (true) total value of a feasible X is:  i v i (t i,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!

A problem is binary demand (BD) if 1. a i ‘s type is a single parameter t i  2. a i ‘s valuation is of the form: v i (t i,o)= t i w i (o), w i (o)  {0,1} work load for a i in o when w i (o)=1 we’ll say that a i is selected in o

An algorithm g() for a maximization BD problem is monotone if  agent a i, and for every r -i =(r 1,…,r i-1,r i+1,…,r N ), w i (g(r -i,r i )) is of the form: Definition 1 Ө i (r -i ) riri Ө i (r -i )  {+  }: threshold payment from a i is: p i (r)= Ө i (r -i )

Our goal: to design a mechanism satisfying: 1. g( ٠ ) is monotone 2. Solution returned by g( ٠ ) is a “good” solution, i.e. an approximated solution 3. g( ٠ ) and p( ٠ ) computable in polynomial time

A greedy  m-approximation algorithm 1. reorder (and rename) the bids such that 2. W   ; X   3. for i=1 to n do 1. if S i  X=  then W  W  {i}; X  X  {S i } 4. return W v 1 /  |S 1 |  v 2 /  |S 2 |  …  v n /  |S n |

The algorithm g( ) is monotone Lemma proof It suffices to prove that, for any selected agent i, we have that i is still selected when it raises its bid Increasing v i can only move bidder i up in the greedy order, making it easier to win

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

Computing payment p i v 1 /  |S 1 |  …  v i /  |S i |  …  v n /  |S n | Consider the greedy order without i index j Use the greedy algorithm to find the smallest index j such that: 1. j is selected 2. S j  S i  p i = v j  |S i |/  |S j |

Let OPT be an optimal solution for CA problem, and let W be the solution computed by the algorithm, then Lemma iWiW  i  OPT v i   m  i  W v i proof OPT i ={j  OPT : j  i and S j  S i  }  i  W OPT i =OPT since it suffices to prove:  j  OPT i v j   m v i crucial observation for greedy order we have v i  |S j | iWiW  j  OPT i  |S i | v j 

proof we can bound Cauchy–Schwarz inequality iWiW  j  OPT i v j   j  OPT i vivi  |S i |  |S j |  j  OPT i  |S j |   |OPT i |  j  OPT i |S j | ≤|S i | ≤ m   m v i   |S i |  m

Cauchy–Schwarz inequality y j =  |S j | x j =1 n= |OPT i | for j=1,…,|OPT i | …in our case…