COMP558 Network Games Martin Gairing University of Liverpool, Computer Science Dept 2nd Semester 2013/14 COMP558 – Network Games 0.1 ·
Topic 4: Network Formation Games Ch.19 Local Connection Game Model Characterizing Solutions and Price of Stability Price of Anarchy Global Connection Game Model Price of Anarchy Price of Stability Facility Location Model Existence of pure NE and PoS Utility games and PoA COMP558 – Network Games Network Formation 4.1 ·
4 Network Formation Games Scenario ◮ Consider users constructing a shared network ◮ Each user has its own interest and is driven by: ◮ Minimising the price he pays for creating/using the network ◮ Receiving a high quality of service ◮ We wish to model the networks generated by such selfish behaviour of the users and compare them to the optimal networks COMP558 – Network Games Network Formation 4.2 ·
Objectives ◮ How to evaluate the overall quality of a network? ◮ social cost = sum of players’ costs ◮ What are stable networks? ◮ we use Nash equilibrium as solution concept ◮ we refer to networks corresponding to Nash equilibrium as being stable ◮ Our main goal: bounding the efficiency loss resulting from selfishness ◮ Price of Anarchy ◮ Price of Stability COMP558 – Network Games Network Formation 4.3 ·
Local vs. Global Connection Game ◮ Local connection game: ◮ users buy edges ◮ bought edge can be used by all users ◮ users wish to minimise distance to all other nodes ◮ while minimizing the number of edges they buy ◮ Resembles formation of P2P networks ◮ Global connection game: ◮ users want to connect two nodes s i , t i in the network ◮ users share the cost of used edges ◮ users minimise their cost ◮ Resembles use of a large scale shared network COMP558 – Network Games Network Formation 4.4 ·
Local Connection Game Topic 4: Network Formation Games Local Connection Game Model Characterizing Solutions and Price of Stability Price of Anarchy Global Connection Game Model Price of Anarchy Price of Stability Facility Location Model Existence of pure NE and PoS Utility games and PoA COMP558 – Network Games Network Formation 4.5 ·
Local Connection Game Model Local Connection Game Model ◮ n players: nodes in a graph G on which the network will be build ◮ Strategy S u of player u ∈ V is a set of undirected edges that u will build (all are incident to u ) ◮ For a strategy vector S , the union of all edges in players’ strategies form a network G ( S ) ◮ α ... cost for building an edge ◮ dist S ( u , v ) ... distance (number of edges on shortest path) between u and v in G ( S ) ◮ Cost of player u ∈ V : � C u ( S ) = dist S ( u , v ) + α · n u , v ∈ V where n u is number of edges bought by player u COMP558 – Network Games Network Formation 4.6 ·
Local Connection Game Model Local Connection Game ◮ A network G = ( V , E ) is stable for a value α , if there is a NE S that forms G . Social Cost of a Network G = ( V , E ) ( = sum of players’ costs) � SC ( G ) = dist ( u , v ) + α · | E | u � = v Observations ◮ Since the graph is undirected an edge ( u , v ) is available to both u and v ◮ At Nash equilibrium at most one of the nodes u , v pays for the edge ( u , v ) ◮ At Nash equilibrium we must have a connected graph, since dist ( u , v ) = ∞ , if u and v are not connected COMP558 – Network Games Network Formation 4.7 ·
Local Connection Game Characterizing Solutions and Price of Stability Characterizing Solutions and Price of Stability Lemma 4.1 (Lem. 19.1) If α ≥ 2 then any star is an optimal solution, and if α ≤ 2 then the complete graph is an optimal solution. Lemma 4.2 (Lem. 19.2) If α ≥ 1 then any star is a Nash equilibrium, and if α ≤ 1 then the complete graph is a Nash equilibrium. Remark: There are also other Nash equilibria. Theorem 4.3 (Thm. 19.3) If α ≥ 2 or α ≤ 1, the price of stability is 1. For 1 < α < 2, the price of stability is at most 4/3. COMP558 – Network Games Network Formation 4.8 ·
Local Connection Game Price of Anarchy Price of Anarchy To prove upper bound on the Price of Anarchy we bound the diameter of a stable (NE) graph 1 use diameter to bound cost 2 Lemma 4.4 (Lem. 19.4) If a graph G at Nash equilibrium has diameter d , then PoA ( G ) = O ( d ) . V e P v u e v v ’ e ’ u ’ P u COMP558 – Network Games Network Formation 4.9 ·
Local Connection Game Price of Anarchy Price of Anarchy Theorem 4.5 (Thm. 19.5) The diameter of a stable graph G is at most 2 √ α , and hence PoA ( G ) = O ( √ α ) . Theorem 4.6 (Thm. 19.6) The price of anarchy is O ( 1 ) whenever α = O ( √ n ) . More generally, α PoA = O ( 1 + √ n ) . A w B u t w u v d ′ COMP558 – Network Games Network Formation 4.10 ·
Global Connection Game Model Topic 4: Network Formation Games Local Connection Game Model Characterizing Solutions and Price of Stability Price of Anarchy Global Connection Game Model Price of Anarchy Price of Stability Facility Location Model Existence of pure NE and PoS Utility games and PoA COMP558 – Network Games Network Formation 4.11 ·
Global Connection Game Model Global Connection Game Model ◮ directed G = ( V , E ) with non-negative edge cost c e ◮ k players , each player i ∈ [ k ] has a source s i and sink node t i ◮ A strategy for a player i is a path P i from s i to t i in G ◮ Given each players strategy we define the constructed network to be ∪ i P i ◮ Players who use edge e divide the cost c e according to some cost sharing mechanism. ◮ We will consider the equal-division mechanism: c e � cost i ( S ) = k e e ∈ P i ◮ S = ( P 1 , . . . , P k ) ◮ k e ... number of players whose path contains e COMP558 – Network Games Network Formation 4.12 ·
Global Connection Game Model Existence of Stable Networks ◮ Does every global connection game have a pure Nash equilibrium? ◮ Yes. ◮ Why? ◮ It is a special congestion game! ◮ Rosenthal’s potential function k e c e � � Φ( S ) = j e ∈ E , k e > 0 j = 1 � = c e · H k e e ∈ E ( H k is k-th harmonic number) COMP558 – Network Games Network Formation 4.13 ·
Global Connection Game Price of Anarchy Price of Anarchy Social Cost (total cost of used edges) SC ( S ) = � i ∈ [ k ] cost i ( S ) Example t 1 , t 2 , ..., t k ◮ optimal network has cost 1 ◮ best NE: all players use the left edge ⇒ PoS = 1 1 k ◮ worst NE: all players use the right edge ⇒ PoA = k s 1 , s 2 , ..., s k Theorem 4.7 For any global connection game with k players, PoA ≤ k . COMP558 – Network Games Network Formation 4.14 ·
Global Connection Game Price of Stability Price of Stability: a lower bound ( ε > 0 arbitrary small) t 1 1 1 1 1 2 3 k -1 k s 1 s 2 s 3 s k -1 s k 1 + ε v ◮ optimal network has a cost of 1 + ε Is it stable? ◮ cost of unique stable network: � k 1 j = H k j = 1 COMP558 – Network Games Network Formation 4.15 ·
Global Connection Game Price of Stability Price of Stability: an upper bound Lemma 4.8 (Lem. 19.8) For any strategy profile S = ( P 1 , . . . , P k ) we have SC ( S ) ≤ Φ( S ) ≤ H k · SC ( S ) Lemma 4.8 and Lemma 3.18 directly imply: Theorem 4.9 (Thm. 19.10) The price of stability in the global connection game with k players is at most H k . Remark: H k = Θ( log k ) COMP558 – Network Games Network Formation 4.16 ·
Facility Location Topic 4: Network Formation Games Local Connection Game Model Characterizing Solutions and Price of Stability Price of Anarchy Global Connection Game Model Price of Anarchy Price of Stability Facility Location Model Existence of pure NE and PoS Utility games and PoA COMP558 – Network Games Network Formation 4.17 ·
Facility Location Model Facility Location Game Model ◮ k service providers and a set of clients [ m ] ◮ each provider i ∈ [ k ] has a set of possible locations A i where he can locate his facility; denote A = ∪ i ∈ [ k ] A i ◮ c js i .. cost of serving customer j ∈ [ m ] from location s i ∈ A i ◮ π j .. value of client j ∈ [ m ] for being served A 1 A 2 Possible facilities j c js Clients s COMP558 – Network Games Network Formation 4.18 ·
Facility Location Model Facility Location Game Model cont. ◮ If provider i serves client j from A 1 A 2 location s i at a price p Possible ◮ π j − p .. benefit for client j facilities j c js ◮ p − c js i .. profit for provider i Clients ◮ ⇒ social value: π j − c js i s (independent of price) ◮ To simplify notation assume π j ≥ c js i for all j , i and s i ∈ A i . ◮ This does not change social value. ◮ Given s = ( s 1 , . . . , s k ) , each client j ◮ is assigned to provider i with lowest cost c js i , ◮ pays price p ij = min i ′ � = i c js i ′ ◮ total social value of s = ( s 1 , . . . , s k ) : � V ( s ) = ( π j − min i ∈ [ k ] c js i ) j ∈ [ m ] COMP558 – Network Games Network Formation 4.19 ·
Facility Location Existence of pure NE and PoS Facility Location Game is Potential Game Theorem 4.10 (Thm. 19.16) V ( s ) is a potential function for the facility location game. Corollary 4.11 Each facility location game admits a pure NE. 1 Every optimum is also a NE and thus PoS = 1. 2 Up next: ◮ bound price of anarchy ◮ we do this for the more general class of utility games COMP558 – Network Games Network Formation 4.20 ·
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