Strategic Network Formation and Group Formation

1. Strategic Network Formation and Group Formation Elliot Anshelevich Rensselaer Polytechnic Institute (RPI)

2. Centralized Control A majority of network research has made the centralized control assumption: Everything acts according to a centrally defined and specified algorithm This assumption does not make sense in many cases.

3. Self-Interested Agents • Internet is not centrally controlled • Many other settings have self-interested agents • To understand these, cannot assume centralized control • Algorithmic Game Theory studies such networks

4. Agents in Network Design s • Traditional network design problems are centrally controlled • What if network is instead built by many self-interested agents? • Properties of resulting network may be very different from the globally optimum one

5. Goal s • Compare networks created by self-interested agents with the optimal network • optimal = cheapest • networks created by self-interested agents = Nash equilibria • Can realize any Nash equilibrium by finding it, and suggesting it to players • Requires central coordination • Does not require central control OPT NE

6. The Price of Stability cost(worst NE) cost(OPT) Price of Anarchy = s [Koutsoupias, Papadimitriou] 1 k cost(best NE) cost(OPT) Price of Stability = t1…tk Can think of latter as a network designer proposing a solution.

7. Single-Source Connection Game[A, Dasgupta, Tardos, Wexler 2003] Given: G= (V,E), k terminal nodes, costs ce for all e  E Each player wants to build a network in which his node is connected to s. Each player selects a path, pays for some portion of edges in path (depends on cost sharing scheme) s Goal: minimize payments, while fulfilling connectivity requirements

8. Other Connectivity Requirements • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation [A, Caskurlu 2009] [A, Dasgupta, Tardos, Wexler 2003]

9. Group Network Formation Games Terminal Backup: Each terminal wants to connect to kother terminals.

10. Group Network Formation Games Terminal Backup: Each terminal wants to connect to kother terminals. “Group Steiner Tree”: Each terminal wants to connect to at least one terminal from each color.

11. Other Connectivity Requirements [A, Caskurlu 2009] [A, Dasgupta, Tardos, Wexler 2003] [A, Caskurlu 2009] • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation: every agent wants to connect to a group that provides enough resources satisfactory group specified by a monotone set function

12. Centralized Optimum • Single-source Connection Game:Steiner Tree. • Sets of nodes: Steiner Forest. • Survivable: Generalized Steiner Forest. • Terminal Backup: Cheapest network where each terminal connected to at least kother terminals. • “Group Steiner Tree”:Cheapest where every component is a Group Steiner Tree. Corresponds to constrained forest problems, has 2-approx.

13. Connection Games s Given: G= (V,E), k players, costs ce for all e  E Each player wants to build a network where his connectivity requirements are satisfied. Each player selects subgraph, pays for some portion of edges in it (depends on cost sharing scheme) NE Goal: minimize payments, while fulfilling connectivity requirements

14. Sharing Edge Costs How should multiple players on a single edge split costs? • One approach: no restrictions... ...any division of cost agreed upon by players is OK. [ADTW 2003, HK 2005, EFM 2007, H 2009, AC 2009] • Another approach: try to ensure some sort of fairness. [ADKTWR 2004, CCLNO 2006, HR 2006, FKLOS 2006]

15. Connection Games with Fair Sharing s Given: G= (V,E), k players, costs ce for all e  E Each player selects subnetwork where his connectivity requirements are satisfied. Players using e pay for it evenly: ci(P) = Σ ce/ke ( ke = # players using e ) e є P Goal: minimize payments, while fulfilling connectivity requirements

16. Fair Sharing Advantages: • Fair way of sharing the cost • Nash equilibrium exists • Price of Stability is at most log(# players) Fair sharing: The cost of each edge e is shared equally by the users of e

17. Price of Stability with Fairness Price of Anarchy is large Price of Stability is at mostlog(# players) Proof: This is a Potential Game, so • Nash equilibrium exists • Best Response converges • Can use this to show existence of good equilibrium s 1 k t1…tk

18. Fair Sharing Fair sharing: The cost of each edge e is shared equally by the users of e Advantages: • Fair way of sharing the cost • Nash equilibrium exists • Price of Stability is at most log(# players) Disadvantages: • Player payments are constrained, need to enforce fairness • Price of stability can be at least log(# players)

19. Example: Self-Interested Behavior t Demands: 1-t, 2-t, 3-t 1 1 1 2 3 1+ 1 2 3 0 0 0

20. Example: Self-Interested Behavior t Minimum Cost Solution (of cost 1+) 1 1 1 2 3 1+ 1 2 3 0 0 0

21. Example: Self-Interested Behavior Each player chooses a path P. Cost to player i is: t cost(P) 1 1 1 cost(i) = 2 3 # using P 1+ 1 2 3 (Everyone shares cost equally) 0 0 0

22. Example: Self-Interested Behavior t 1 1 1 Player 3 pays (1+ε)/3, could pay 1/3 2 3 1+ 1 2 3 0 0 0

23. Example: Self-Interested Behavior t 1 1 1 so player 3 would deviate 2 3 1+ 1 2 3 0 0 0

24. Example: Self-Interested Behavior t now player 2 pays (1+ε)/2, could pay 1/2 1 1 1 2 3 1+ 1 2 3 0 0 0

25. Example: Self-Interested Behavior t so player 2 deviates also 1 1 1 2 3 1+ 1 2 3 0 0 0

26. Example: Self-Interested Behavior Player 1 deviates as well, giving a solution with cost 1.833. This solution is stable/ this solution is a Nash Equilibrium. It differs from the optimal solution by a factor of 1+ + Hk = Θ(log k)! t 1 1 1 2 3 1+ 1 2 3 0 0 0 1 1 2 3

27. Sharing Edge Costs How should multiple players on a single edge split costs? One approach: no restrictions... ...any division of cost agreed upon by players is OK. [ADTW 2003, HK 2005, EFM 2007, H 2009, AC 2009] Another approach: try to ensure some sort of fairness. [ADKTWR 2004, CCLNO 2006, HR 2006, FKLOS 2006]

28. Example: Unrestricted Sharing Fair Sharing: differs from the optimal solution by a factor of Hk = Θ(log k) Unrestricted Sharing: OPT is a stable solution t 1 1 1 2 3 1+ 1 2 3 0 0 0

29. Contrast of Sharing Schemes Unrestricted SharingFair Sharing NE don’t always existNE always exist P.o.S. = O(k)P.o.S. = O(log(k)) (P.o.S. = Price of Stability)

30. Contrast of Sharing Schemes Unrestricted SharingFair Sharing NE don’t always exist NE always exist P.o.S. = O(k) P.o.S. = O(log(k)) P.o.S. = 1 forP.o.S. = (log(k)) for many gamesalmost all games (P.o.S. = Price of Stability)

31. Contrast of Sharing Schemes Unrestricted SharingFair Sharing NE don’t always exist NE always exist P.o.S. = O(k) P.o.S. = O(log(k)) P.o.S. = 1 forP.o.S. = (log(k)) for many gamesalmost all games OPT is an approx. NEOPT may be far from NE (P.o.S. = Price of Stability)

32. Unrestricted Sharing Model • Player i picks payments for each edge e. (strategy = vector of payments) • Edge e is bought if total payments for it ≥ce. • Any player can use bought edges. What is a NE in this model?

33. Unrestricted Sharing Model • Player i picks payments for each edge e. (strategy = vector of payments) • Edge e is bought if total payments for it ≥ce. • Any player can use bought edges. What is a NE in this model? Payments so that no players want to change them

34. Unrestricted Sharing Model • Player i picks payments for each edge e. (strategy = vector of payments) • Edge e is bought if total payments for it ≥ce. • Any player can use bought edges. What is a NE in this model? Payments so that no players want to change them

35. Connection Games with Unrestricted Sharing s Given: G= (V,E), k players, costs ce for all e  E Strategy: a vector of payments Players choose how much to pay, buy edges together  if v does not satisfy connectivity requirements Payments of v otherwise Cost(v) = Goal: minimize payments, while fulfilling connectivity requirements

36. Connectivity Requirements • Single-source: connect to s • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation: every agent wants to connect to a group that provides enough resources satisfactory group specified by a monotone set function

37. Some Results If k=n If k=n OPT is a Nash Equilibrium (Price of Stability=1) • Single-source: connect to s • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation: every agent wants to connect to a group that provides enough resources satisfactory group specified by a monotone set function

38. Some Results =1 =2 =3 =2 OPT is a -approximate Nash Equilibrium (no one can gain more than  factor by switching) • Single-source: connect to s • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation: every agent wants to connect to a group that provides enough resources satisfactory group specified by a monotone set function

39. Some Results =1 =2 =3 =2 If we pay for 1-1/ fraction of OPT, then the players will pay for the rest • Single-source: connect to s • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation: every agent wants to connect to a group that provides enough resources satisfactory group specified by a monotone set function

40. Some Results Can compute cheap approximate equilibria in poly-time • Single-source: connect to s • Survivable: connect to s with two disjoint paths • Sets of nodes: agent i wants to connect set Ti • Group formation: every agent wants to connect to a group that provides enough resources satisfactory group specified by a monotone set function

41. Contrast of Sharing Schemes Unrestricted SharingFair Sharing NE don’t always existNE always exist P.o.S. = O(k)P.o.S. = O(log(k)) P.o.S. = 1 for P.o.S. = (log(k)) for many games almost all games OPT is an approx. NE OPT may be far from NE (P.o.S. = Price of Stability)

42. Contrast of Sharing Schemes Unrestricted SharingFair Sharing NE don’t always exist NE always exist P.o.S. = O(k) P.o.S. = O(log(k)) P.o.S. = 1 forP.o.S. = (log(k)) for many gamesalmost all games OPT is an approx. NEOPT may be far from NE (P.o.S. = Price of Stability)

43. Contrast of Sharing Schemes Unrestricted SharingFair Sharing NE don’t always existNE always exist P.o.S. = O(k)P.o.S. = O(log(k)) P.o.S. = 1 forP.o.S. = (log(k)) for many gamesalmost all games OPT is an approx. NEOPT may be far from NE If we really care about efficiency: Allow the players more freedom!

44. Example: Unrestricted Sharing Fair Sharing: differs from the optimal solution by a factor of Hk  log k Unrestricted Sharing: OPT is a stable solution Every player gives what they can afford t 1 1 1 2 3 1+ 1 2 3 0 0 0

45. General Techniques To prove that OPT is an exact/approximate equilibrium: • Construct a payment scheme • Pay in order: laminar system of witness sets • If cannot pay, form deviations to create cheaper solution

46. Network Destruction Games • Each player wants to protect itself from untrusted nodes • Have cut requirements: must be disconnected from set Ti • Cuttingedges costs money • Can show similar results for: Multiway Cut, Multicut, etc.

47. Thank you.