1. Presenter: Jen Hua Chi Adviser: Yeong Sung Lin Network Games with Many Attackers and Defenders

2. Agenda  Introduction  Network Games with Many Defenders  New Strategic Model 2

3. Introduction 3 Mavronicholas et al. started a line of research of network games. (2008) Showing that no game with the defender playing a single edge has a pure Nash equilibrium unless it is a trivial graph. 1. Using bipartite graph 2. Improving non-deterministic algorithm into a deterministic polynomial-time algorithm.

4. Introduction  Theorem if G contains more than one edges, then the game has no pure Nash Equilibrium for any graph G, the game contains a matching mixed Nash equilibrium if and only if the vertices of the graph G can be partitioned into two sets A, B, such that A is an independent set of G and B is a vertex cover of the graph 4

5. Introduction  Definition: matching M a set M ⊆ E is a matching of G if no two edges in M share a vertex vertex cover a set V’ ⊆ V such that for every edge (u, v) ∈ E either u ∈ V or v ∈ V’ edge cover a set E’ ⊆ E such that for every vertex v ∈ V, there is an edge (v, u) ∈ E’ 5

6. Introduction  Definition: a mixed strategy for player i ∈ N is a probability distribution over its pure strategy set S i edge model : defender protects a single link of the network 6

7. How to determine a Nash equilibrium  For n players: n players corresponds to each n-tuple of pure strategies, one strategy being taken for each player. any n-tuple of strategies, may be regarded as a point in the product space obtained by multiplying the n strategy spaces. one such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expectation for its player against, the n - 1 strategies of the other players in the countered n-tuple. a self-countering n-tuple is called an equilibrium point. 7

8. How to determine an equilibrium point : About countering  The correspondence of each n-tuple with its set of countering n-tuples gives a one-to-many mapping of the product space into itself.  The set of countering points of a point is convex.  The closeness is equivalent to saying: if P 1, P 2,... and Q 1, Q 2,..., Q n,... are sequences of points in the product space where Q n  Q, P n  P and Q n counters P n then Q counters P 8

9. How to determine an equilibrium point  Inferring from Kakutani's theorem that the mapping has a fixed point (i.e. point contained in its image). Hence there is an equilibrium point.  Kakutani’s theorem: It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. 9

10. Introduction : Motivation  According to Mavronicholas et al. (2006) research the existence problem of pure Nash equilibria is solvable in polynomial time provided a graph-theoretic characterization of mixed Nash equilibria introduced k-matching configurations that generalize matching configurations provide a characterization of graphs admitting k- matching Nash equilibria 10

11. Introduction : Motivation  According to Mavronicholas et al. (2006) research.. develop a polynomial-time algorithm for computing k-matching Nash equilibria on graphs and exhibit the applicability of the algorithm for bipartite graphs establish that the increased power of the defender results in an improved quality of protection of the network obtain that the gain of the defender, which amounts to the expected number of the arrested harmful procedures, is linear to the parameter k 11

12. Network Games with Many Defenders  Undirected graph G = (V, E) vertex cover C V ⊆ V edge cover C E ⊆ E matching M: m (size) = |M| ≥ |M’| independent set I V ⊆ V if v is incident to an edge e: v ∈ e number of vertices: n V number of edges: n E 12

13. Network Games with Many Defenders  Characterization of (pure) Nash equilibrium game (G) = 〈, 〉 = A ∪ D G = ∪ strategy set S of the game is V v x μ strategy profile s is an element of S s = 〈 1,…, v, S 1,… S μ 〉∈ S 13

14. Network Games with Many Defenders  Profit ( ) individual profit of attacker, 1 ≤ i ≤ v ( )= individual profit of defender = |{ : ∃ i, 1≤i ≤ v, i }| 14

15. Network Games with Many Defenders   s is a Nash equilibrium if and only if there exist D ⊂ and A ⊂ V which satisfy the following conditions: 1. 2. 3. 15

16. Network Games with Many Defenders  Theorem 2.3 If the number of attackers v is strictly less than n V then an edge model (G) has a Nash equilibrium if and only if there exist D and A which satisfy the following conditions: 16

17. Network Games with Many Defenders  Definition2.4 For a graph G, we have the following notations max (G) denotes the game (G) where v = n V -m and μ = n E min (G) denotes the game (G) where v = m and μ = n v -m where m is the size of a maximum matching in G. 17

18. Network Games with Many Defenders  Definition 2.6 : a graph G is said to have the property Prop (*) if and only if for a minimum edge cover C E, there exists a map f : V {0,1} such that for any multiple-edge star graph of C E with a center, = 0  Theorem 2.7: a game min (G) has a Nash equilibrium if and only if G satisfies the property Prop (*). 18

19. New Strategic Model  The new model is defined by interchanging the players’ roles. AttackersDefenders Original model Attack a node of the network to damage Protect the network by catching attackers in some part of the network New modelDamage the network by attacking an edge Protect the network by choosing a vertex 19

20. New Strategic Model  A new strategic game α, δ (G) = 〈, S 〉 on G is defined as follows S = E α x V δ is a strategy set of α, δ (G) s is an element of S s = 〈 e 1,…, e α, v 1,…,v δ 〉∈ S Original modelNew model Game (G) = 〈, 〉 α, δ (G) = 〈, S 〉 S V v x μ E α x V δ 20

21. New Strategic Model  Profit ( ) individual profit of attacker, 1≤ i ≤ α individual profit of defender, 1 ≤ j ≤ δ 21

22. New Strategic Model  Definition:  Theorem 3.2: |D| ≤ δ and |A| ≤ α where 22

23. New Strategic Model  Theorem 3.3 If α is the size of a maximum matching in G and δ =2 α, then the game α, δ (G) has a Nash equilibrium. and μ = n v -m 23

24. New Strategic Model  Definition 3.4 The graph G is bipartite if V=V 0 ∪ V 1 for some disjoint vertex sets V 0, V 1 ⊆ V so that for each edge (u,v) ∈ E, u ∈ V 0 and v ∈ V 1 24

25. New Strategic Model  Theorem 3.5 For a bipartite graph G, a game α, δ (G) has a Nash equilibrium if and only if α, δ ≥ m, where m is the size of the maximum matching in G.  Proof: For a bipartite graph G, if M is a maximum matching and is a minimum vertex cover, then such that 25

26. The End Thanks for your attention.