Nash equilibria in Voronoi Games on Graphs

Transcription

1 Nash equilibria in Voronoi Games on Graphs Christoph Dürr, Nguyễn Kim Thắng (Ecole Polytechnique) ESA, Eilat October 07

2 Plan Motivation : Study the interaction between selfish agents on Internet k players, each one chooses a vertex in graph G and gains the area of Voronoi cell. The existence of pure Nash equilibrium depends on (k,g) and deciding the existence is NPcomplete. The difference of social cost between pure Nash equilibria is bounded by Ω( (n/k)), O( (kn))

3 Related works Competitive facility location: Voronoi Games on continuous surface [Hee-Kap et al 04, Cheong et al 04]. Service Provider Games [Vetta 02]

4 Social cost discrepancy the worst Nash equilibrium the best Nash equilibrium optimum social cost price of anarchy (PoA) social cost discrepancy price of stability (PoS) Social cost discrepancy: worst Nash / best Nash Ideas: Unfair to compare the cost with OPT in selfish setting. Measure the degree of choice in a game.

5 The Game Given G(V,E), k players. Player s strategy set is V. A vertex (customer) is assigned in equal fraction to the closest players. Payoff = the fractional amount of vertex assigned to the player. A pure Nash equilibrium is a strategy profile in which no one can unilaterally increase her payoff Social cost = sum of distances over all vertices to their closest player = problem minimum k-median

6 Non convergence on the cycle The game in continuous setting a player doesn t increase her payoff if she stays in the interval with the same neighbors, player A who moves to the same location as player B gains 1/2 of the old gain of B. In discrete setting, it is different: etc. old new payoff 1

7 A gadget Lemma: There is no Nash equilibrium with k=2 players. Proof : By sym. player 1 is on u 2 (or u 1 ). Then player 2 may gain 5 (or 6) by moving to u 6. Now player 1 may increase his payoff by moving to u 7 and so on... u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9

8 A gadget Lemma: There is no Nash equilibrium with k=2 players. Proof : By sym. player 1 is on u 2 (or u 1 ). Then player 2 may gain 5 (or 6) by moving to u 6. Now player 1 may increase his payoff by moving to u 7 and so on... u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9

9 A gadget Lemma: There is no Nash equilibrium with k=2 players. Proof : By sym. player 1 is on u 2 (or u 1 ). Then player 2 may gain 5 (or 6) by moving to u 6. Now player 1 may increase his payoff by moving to u 7 and so on... u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9

10 existence of equilibrium? 3-Partition (unary NP-hard) Theorem: input: a 1,,a 3n,B such that i B/4<a i <B/2, a i =nb output: whether there exists a partition into n triplets, each of sum B Given G(V,E) et k, deciding the existence of pure Nash General game (unary NP-hard) (positive weight w on vertices, strategy set is U V) equilibrium is NP-complete. Original game (binary NP-hard)

11 General games <G(V,E),U,w,k>: each vertex v has weight w(v) and the strategy set is restricted to U. w(v)-1 U T

12 ( ) Proof Construction vertices of weight a i c a vertex by triplet w(v 0 ) = 1 w(v 1 ) = a 1 c w(u 1,2,3 ) = 1 k=n+1 players U w(u 1 ) = d the gadget w(u 3 ) = d w(v 2 ) = a 2 c w(u 1,2,4 ) = 1 w(u 2 ) = d w(u 4 ) = d w(v 3 ) = a 3 c w(u 1,2,5 ) = 1 w(u 5 ) = d w(u 6 ) = d w(u 7 ) = d w(u 8 ) = d w(v 3n ) = a 3n c w(u 3n 2,3n 1,3n ) = 1 w(u 9 ) = d where c,d are functions of n and B

13 Lower bound of Cost Discrepancy b a equilibrium : cost Θ(kb+ka 2 ) equilibrium : cost Θ(kab+ka 2 ) worst ratio : Ω( (n/k)) for b=a 2.

14 Upper bound of Cost Discrepancy Let and be two equilibria. Idea: these equilibria are not far from the one to the other. We group all Voronoi cell generated by into regions.

15 Delaunay graph -- Stars Delaunay graph: G(E,V) and an equilibrium, (i,j) in H if they are neighbors. Star: G(V,E), A is a star if A 2 and a vertex in A connecting to all other vertices in A.

16 Delaunay graph -- Stars Delaunay graph: G(E,V) and an equilibrium, (i,j) in H if they are neighbors. Star: G(V,E), A is a star if A 2 and a vertex in A connecting to all other vertices in A. Fact: Any connected graph can be partitioned into stars.

17 Upper bound of Cost Discrepancy For a fixed region, let r be the maximal distance of vertexplayer. 4r r Lemma: there is at least one player of whose distance to a player of the star is at most 4r.

18 Upper bound of Cost Discrepancy For a fixed region, let r be the maximal distance of vertexplayer. 4r r Lemma: there is at least one player of whose distance to a player of the star is at most 4r. Theorem: for any connected graph G(V,E) with k players, the cost discrepancy is O( (kn)).

19 And now Close the gap between (n/k) and (kn) Study the cost discrepancy in the others games.

20 And now Close the gap between (n/k) and (kn) Study the cost discrepancy in the others games. Thank you!