1 / 20

Welfare Maximization in Congestion Games

Welfare Maximization in Congestion Games. Liad Blumrosen and Shahar Dobzinski The Hebrew University. Example: File Sharing Systems (Positive Externalities). Example : Scheduling (Negative Externalities). Related Work. Both in game theory and computer science. Examples:

beate
Download Presentation

Welfare Maximization in Congestion Games

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Welfare Maximization in Congestion Games Liad Blumrosen and Shahar DobzinskiThe Hebrew University

  2. Example: File Sharing Systems(Positive Externalities)

  3. Example: Scheduling(Negative Externalities)

  4. Related Work • Both in game theory and computer science. • Examples: • Pure Nash: Rosenthal 73’, Monderer-Shapley 96’ • Price of Anarchy: Roughgarden & Tardos 02’,… • …

  5. Unrestricted externalities negative externalities Positive externalities The Model • nplayers, mresources • vi(j,k) - the (non-negative) benefit for player i from using resource j with exactly k-1 other players • An anonymous model(player-specific games, Milchtaich 96) Value Congestion Goal: welfare maximization (centralized point of view) Find an assignment (S1,…,Sn) of players to resources to maximizes the welfare SjSiSjvi(j,|Sj|) • Each player can be assigned to at most one resource.

  6. Related Work (cont.) • Closest in spirit: paper byChakrabarty, Mehta, Nagarajan and Vazirani 05’ • Cost minimization, negative externalities

  7. Main Results • Relation to combinatorial auctions. • Approximationalgorithms: • Main result: an O(1)-approximation • Even with valuations unresetricted externalities. • Constant approximation lower bounds • Mechanism design: • An O(m½log n) truthful mechanism • with monotone valuations

  8. Congestion Games Players Resource 1 Resource 2 Resource 3

  9. Congestion Games Resource 1 Resource 2 Resource 3

  10. Items Bidder 1 Bidder 2 Bidder 3 Combinatorial Auctions Players • m items for sale. • n bidders, each bidder i has a value vi(S) for every bundle of items S • Goal: find a partition S1,…,Sn of the items such that the total welfare Svi(Si) is maximized. Resource 1 Resource 2 Resource 3

  11. Bidder 1 Bidder 2 Bidder 3 Combinatorial Auctions Every congestion game has a corresponding combinatorial auction with the same optimal welfare Value: 2 Value: 4 Value: 6 Resource 1 Resource 2 Resource 3

  12. Combinatorial Auctions & Congestion Games • Similarities -- “Hierarchy” Results: • Negative externalities  substitutabilities • Specifically, a subclass of XOS • Positive externalities  complementarities • Differences: • Incentives are treated differentlylike a combinatorial auction with strategic items • Different types of reasonable queries.

  13. Main Algorithmic Result An O(1)-approximation algorithm for congestion games with unrestricted externalities

  14. The First Step – Solving the LP • Solve the linear relaxation of the problem: Maximize: Sj,Sxj,Svi(j,|S|) Subject to: • For each player i: Sj,S|iSxi,S ≤ 1 • For each resource j: SSxj,S ≤ 1 • For each j,S: xj,S ≥ 0 • Solvable in polynomial time even though the number of variables is exponential.

  15. The First Lottery Not feasible because players are assigned to more than one resource. • Reminder: Given the fractional solution: Xj,S = the fraction of the set of players S assigned to resource j • Assign to each resource j: • the players in S with probability xj,S/10 • with probability 1-SSxj,S/10 • The expected welfare is OPT/10, but the solution is not necessarily feasible.

  16. After the First Lottery Unassigned Players • Claim: with constant probability the solution produced by the first lottery has the following properties: • The welfare is at least OPT/50 • The total number of assignments after the first lottery is at most n/2 • Each player is assigned to at most one resource in the LP, so the expected number of assignments is n/10.

  17. Same number of users are using the resource before and after second lottery Second Lottery The solution is now feasible! We have enough unassigned players. Unassigned Players • Assign each player i to exactly one resource: • Choose a resource uniformly at random from the set of resources i was assigned to. • Use the unassigned players to keep the congestion on each resource the same.

  18. Analysis of the Approximation Ratio Value: 3 Value: 2 Value: 2 Value: 1 Value: 0 Value: 2 Value: 0 Value: 4 Value: 2 Value: 4 • Lemma: The expected number of resources a player is assigned to after the first lottery is O(1). • Corollary: We lose only a constant fraction of the welfare in the second lottery.

  19. The Algorithm – The Big Picture • Solve the linear relaxation. • First Lottery: use randomized rounding and obtain an (unfeasible) assignment such that: • The welfare is at least OPT/50 • The total number of assignments is at most n/2 • Second Lottery: select uniformly at random one assignment of each player, and arbitrarily replace the rest of the assignments of that player. • We lose only a constant fraction of the welfare because the expected number of assignments per player is constant.

  20. Summary and Open Questions • We study congestion games from a centralized of view. • With different types of externalities. • Open Question: closing the gaps in the approximation ratios. • Open Question: designing polynomial-time truthful mechanisms with better performance. • More generally, design mechanisms for multi-parameter settings with externalities.

More Related