Network Formation Games Ofir Geri

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: