1 / 77

Understanding Price of Anarchy in Cost-Sharing Connection Games

This seminar, supervised by Prof. Michal Feldman at Tel-Aviv University, explores the theoretical framework of connection games in network formation. Agents aim to construct networks to connect various terminals while sharing costs. It delves into general and fair cost-sharing, as well as specific cases like capacities and symmetric scenarios. Key concepts discussed include Nash equilibria, price of stability, and algorithms for efficient payment strategies among players. The insights revealed offer significant implications for network design with selfish agents.

zoltan
Télécharger la présentation

Understanding Price of Anarchy in Cost-Sharing Connection 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. Network Formation GamesOfir Geri Price of Anarchy Seminar Supervised by Prof. Michal Feldman Tel-Aviv University 2/4/2014

  2. Introduction • We discuss games in which agents wish to construct a network • Each agent has different terminals they wish to connect • The cost of the network is shared between the agents • Each agent may contribute to any link

  3. Outline • General cost-sharing connection games • Fair cost-sharing connection games • Capacitated symmetric cost-sharing connection games

  4. General Cost-SharingConnection Games E. Anshelevich, A. Dasgupta, É. Tardos, and T. Wexler, “Near-Optimal Network Design with Selfish Agents”

  5. The Connection Game • There are players • is an undirected graph • Each player must connect a set of terminals in • An edge has a cost • Each player offers payments • The graph of bought edges is where

  6. Basic Properties of Nash Equilibria • The bought edges form a graph that is a forest • A player only contributes to the edges they use • Each edge is either fully paid for or not paid at all

  7. A Game Without Nash Equilibrium

  8. A Nash Equilibrium May RequireCost-Sharing • NE: Player 2 pays 5 foredge , player 1 pays forthe rest • Any NE must buy edge • Player 2 only contributesto • A non-fractional NEdoesn’t exist

  9. The Price of Anarchy • Lower bound:In this example, • The same holds when theedges are directed and thecosts are shared in a fairmanner

  10. The Price of Anarchy • Theorem: In every connection game with players, • Proof: Let be the worst Nash equilibrium • The cost of each player is at most • If a player pays more than , they can buy instead

  11. Single Source Games • All players share a common source , and player has only one other terminal, • In this class of games, the price of stability is 1

  12. Single Source Games: Price of Stability • Denote by the minimum cost Steiner tree that connects all terminal nodes • Consider as the root • Let be the sub-tree thatis disconnected from ifedge is removed

  13. Single Source Games: Price of Stability • We show a Nash equilibrium that buys • We only need to define the payments

  14. Algorithm 1 • For all players and edges , set • For all edges in in reverse BFS order: • For players so that (until is paid for): • If is a cut in , set • Define: • Define to be the cost of the lowest cost path from to in under • Define • Define • Set

  15. Single Source Games: Price of Stability Claim: The algorithm yields a Nash equilibrium Proof: • At any stage, emulates the cost of the lowest cost path that does use an edge • We set the payment for to be at most the cost of the alternative path to • A player cannot reduce their cost by deviating

  16. Single Source Games: Price of Stability • We only need to prove that the algorithm fully pays for • For each edge , the players with terminals in must pay for • Otherwise, each player has a path that explains why they can’t contribute more to • We show that if is not fully paid for, these paths allow us to find a solution that is better than

  17. Single Source Games: Price of Stability • At some stage, denote by the alternate path that costs for player • Choose the one that includes as many ancestors of in as possible • Lemma: is composed of three sub-paths • The first contains only edges from • The second contains only edges from • The third contains only edges from

  18. Single Source Games: Price of Stability Proof: • Once reaches a node in , it will use nodes from since their cost is 0 • Suppose starts with a path that contains edges of , and continues with a path that contains edges of , leading to node in • Let be the common ancestor of and

  19. Single Source Games: Price of Stability (Figure taken from Anshelevich et al., “Near-Optimal Network Design with Selfish Agents”)

  20. Single Source Games: Price of Stability • We prove that • is strictly below • Otherwise, is contained in • We get that • Since , • is not the best deviation

  21. Single Source Games: Price of Stability • Consider an iteration during which player contributed to an edge in • The total payment of is bounded by the cost of any path, including • After reaches , the rest of the path to costs

  22. Single Source Games: Price of Stability • We proved that • If we replace with in • The cost can only decrease • contains a higher ancestor of than () • Hence, a contradiction!

  23. Single Source Games: Price of Stability • We are ready to prove that the algorithm pays fully for • Suppose that for an edge , • Recall that is the alternative path for • The highest ancestor of in that is also in will be denoted (’s deviation point) • Let be the set of the highest deviation points, such that every has an ancestor in

  24. Single Source Games: Price of Stability • Let be the sub-tree rooted at • Assume all players with deviated to • Payments are notincreased • All edges in every are still paid for (Figure taken from Anshelevich et al.,“Near-Optimal Network Design with Selfish Agents”)

  25. Single Source Games: Price of Stability • Each path connects to • pays fully for the edges that are not in • All terminals are connected to after the deviation • The total cost of all players is less than , but is optimal • Hence, a contradiction!

  26. Approximate Nash Equilibria • Definition:A strategy profile is a -approximate Nash equilibrium if no player can decrease their cost by factor of more than by deviating • Intuitively, we want players not to profit much from deviating

  27. Single Source Games • Finding OPT is NP-Hard • Let be an -approximate minimum cost tree • We present a poly-time algorithm for finding a (1+ε)-approximate Nash equilibrium , so that

  28. Single Source Games Algorithm 2 • Define • Use Algorithm 1 to pay for all edges in , with their cost decreased by • is not optimal, so the algorithm may fail to buy an edge • In that case, construct a tree so that and run Algorithm 1 iteratively

  29. Single Source Games • For each player and for each edge , the final payment is

  30. Single Source Games • Algorithm 1 can be run in polynomial time • Algorithm 1 may run at most times • The run-time of Algorithm 2 is polynomial in and the network size

  31. Single Source Games • All edges are fully paid for • Suppose contains edges • Compared to the Nash equilibrium returned by Algorithm 1, the payments increased in

  32. General Connection Games • The price of stability can be • Every NE must buy a path that costs (Figure taken from Anshelevich et al., “Near-Optimal Network Design with Selfish Agents”)

  33. General Connection Games • We show that there is a 3-approximate Nash equilibrium that pays for • Given a set of edges , a stable payment is a payment such that doesn’t have a profitable deviation, assuming the rest of is bought by the rest of the players • A Nash equilibrium consists of stable payments for all players

  34. Stable Payments • Consider a payment scheme • Theorem: If every payment can be divided into at most payments, such that each of them is a stable payment, then is an -approximate Nash equilibrium

  35. Stable Payments Proof: • Let be the best response of player to • Let be the sub-payments of • is still a possible deviation for • We get , hence

  36. Connection Set: Definition • From now on, a player either pays fully or pays nothing for each edge • A set of edges is a connection set of if for every connected component in we have that either • Any player that has terminals in has all of its terminals in , or • Player has a terminal in

  37. Connection Sets • Lemma: A connection set of player is a stable payment of with respect to • Proof: Let be the best deviation of • is a set of edges so that connects all of ’s terminals • If two terminals of another player are in different components of , they are connected in

  38. Connection Sets • Thus, is a possible solution • is optimal, • Therefore,

  39. 3-Approximate Nash Equilibrium • Observe that the set of edges used only by is a connected set, • We want each player to pay for 3 connection sets • will be the first connection set, and from on, assume that all edges are used by at least two players

  40. 3-Approximate Nash Equilibrium • Assume is a path • is the set of terminals at • For each terminal ,define a path (Figure taken from Anshelevich et al., “Near-Optimal Network Design with Selfish Agents”)

  41. 3-Approximate Nash Equilibrium • A payment that contains one edge from for every terminal of except the last terminal is a connection set (Figure taken from Anshelevich et al., “Near-Optimal Network Design with Selfish Agents”)

  42. 3-Approximate Nash Equilibrium • A max-coupled-set is a set of edges such that every is contained in exactly the same paths , for (Figure taken from Anshelevich et al., “Near-Optimal Network Design with Selfish Agents”)

  43. 3-Approximate Nash Equilibrium • Let be a max-coupled-setFor all components of except the two end components, any player that has a terminal in has all its terminals in • Suppose has a terminal in • If has a terminal before or after , the edges in adjacent to can’t be in the same coupled-set

  44. 3-Approximate Nash Equilibrium • A payment that contains at most one max-coupled-set from for every terminal of except the last terminal is a connection set • If a component does not contain a terminal of , it is bordered by edges of the same max-coupled-set

  45. 3-Approximate Nash Equilibrium • Finally, we wish to match at most two connection sets to each player • We form a bipartite matching problem • is the set of all max-coupled-sets • is • There’s an edge between and if there is such that • For players that don’t have a terminal in , form an edge between the last terminal and if

  46. 3-Approximate Nash Equilibrium • We use Hall’s Matching Theorem to assign a node to each max-coupled-set • For , denote by the set of nodes that can be connected to • We need to prove

  47. 3-Approximate Nash Equilibrium • Sort the edges that are part of • If two edges belong to different max-coupled-sets, there must be a path that contains only one of the edges • The player corresponding to must have a terminal between the two edges • There must be a terminal from before the first edge in

  48. 3-Approximate Nash Equilibrium • We have shown that for all • Using the matching, each player can be assigned (at most) two connecting sets • We get a 3-approximate Nash equilibrium • This is expanded by induction to the whole tree

  49. Fair Cost-SharingConnection Games E. Anshelevich, A. Dasgupta, J. Kleinberg, É. Tardos,T. Wexler, and T. Roughgarden, “The Price of Stability for Network Designwith Fair Cost Allocation”

  50. The Fair Connection Game • We consider directed graphs • Each player chooses only which edges to use • We use Shapley (fair) cost-sharing:

More Related