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

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.

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

Agents in Network Design 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 s

Goal 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 s

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

Single-Source Connection Game [A, Dasgupta, Tardos, Wexler 2003] Given: G = (V,E), k terminal nodes, costs c e 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

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

Group Network Formation Games Terminal Backup: Each terminal wants to connect to k other terminals.

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

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

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 k other terminals.  “Group Steiner Tree”: Cheapest where every component is a Group Steiner Tree. Corresponds to constrained forest problems, has 2-approx.

Connection Games Given: G = (V,E), k players, costs c e 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) s Goal: minimize payments, while fulfilling connectivity requirements NE

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]

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

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)

Price of Stability with Fairness Price of Anarchy is large Price of Stability is at most log(# players) Proof: This is a Potential Game, so  Nash equilibrium exists  Best Response converges  Can use this to show existence of good equilibrium s t 1 …t k 1k

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)

Example: Self-Interested Behavior t  Demands: 1-t, 2-t, 3-t

Example: Self-Interested Behavior t  Minimum Cost Solution (of cost 1+  )

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

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

Example: Self-Interested Behavior t  so player 3 would deviate

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

Example: Self-Interested Behavior t  so player 2 deviates also

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

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]

Example: Unrestricted Sharing Fair Sharing: differs from the optimal solution by a factor of H k = Θ(log k) Unrestricted Sharing: OPT is a stable solution t 

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

Contrast of Sharing Schemes Unrestricted Sharing Fair Sharing NE don’t always exist NE 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 (P.o.S. = Price of Stability)

Contrast of Sharing Schemes Unrestricted Sharing Fair Sharing NE don’t always exist NE 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)

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

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

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

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

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

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

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

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

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

Contrast of Sharing Schemes Unrestricted Sharing Fair Sharing NE don’t always exist NE 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)

Contrast of Sharing Schemes Unrestricted Sharing Fair Sharing NE don’t always exist NE 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)

Contrast of Sharing Schemes Unrestricted Sharing Fair Sharing NE don’t always exist NE 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 If we really care about efficiency: Allow the players more freedom!

Example: Unrestricted Sharing Fair Sharing: differs from the optimal solution by a factor of H k  log k Unrestricted Sharing: OPT is a stable solution Every player gives what they can afford t 

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

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

Thank you.