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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:

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Welfare Maximization in Congestion Games

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  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.

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