Presenter: Jen Hua ChiAdviser: Yeong Sung Lin

Network Games with Many Attackers and Defenders

Agenda

Introduction Network Games with Many

Defenders New Strategic Model

3

Introduction

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 graph2. 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

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’

Introduction

Definition: a mixed strategy for player i ∈ N is a

probability distribution over its pure strategy set Si

edge model : defender protects a single link of the network

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.

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 P1, P2, ... and Q1, Q2, ..., Qn, ... are sequences of points in the product space where Qn Q,

Pn P and Qn counters Pn then Q counters P

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.

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

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

Network Games with Many Defenders

Undirected graph G = (V, E) vertex cover CV ⊆ V edge cover CE ⊆ E matching M: m (size) = |M| ≥ |M’| independent set IV ⊆ V if v is incident to an edge e: v ∈ e number of vertices: nV

number of edges: nE

Network Games with Many Defenders

Characterization of (pure) Nash equilibrium game (G) = 〈 , 〉 = A∪ D

G = ∪ strategy set S of the game is Vv x μ

strategy profile s is an element of S s = 〈 1,…, v , S1,… Sμ 〉∈ S

Network Games with Many Defenders

Profit ( ) individual profit of attacker , 1 ≤ i ≤v ( )=

individual profit of defender = |{ : ∃i, 1≤i ≤v , i }|

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.

Network Games with Many Defenders

Theorem 2.3 If the number of attackers v is strictly less

than nV

then an edge model (G) has a Nash equilibrium

if and only if there exist D and A which satisfy

the following conditions:

Network Games with Many Defenders

Definition2.4 For a graph G, we have the following notations

max(G) denotes the game (G) where v = nV -m

and μ= nE

min(G) denotes the game (G) where v = m

and μ= nv-m

where m is the size of a maximum matching in G.

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 CE, there exists a map f : V {0,1} such that

for any multiple-edge star graph of CE with a center , = 0 Theorem 2.7: a game min(G) has a Nash equilibrium if and only

if G satisfies the property Prop (*) .

New Strategic Model

The new model is defined by interchanging the players’ roles.

Attackers Defenders

Original model

Attack a nodeof the network to damage

Protect the network by catching attackersin some part of the network

New model

Damage thenetwork by attacking an edge

Protect the network by choosing a vertex

New Strategic Model

A new strategic game α, δ(G) = 〈 , S 〉

on G is defined as follows S = Eα x Vδ is a strategy set of α, δ(G) s is an element of S s = 〈 e1,…, e α, v1,…,v δ 〉∈ S

Original model

New model

Game (G) = 〈 , 〉

α, δ(G) = 〈 ,

S 〉

S Vv x μ Eα x Vδ

New Strategic Model

Profit ( ) individual profit of attacker , 1≤ i ≤α

individual profit of defender , 1 ≤ j ≤ δ

New Strategic Model

Definition:

Theorem 3.2: |D| ≤ δ and |A| ≤ α where

New Strategic Model

Theorem 3.3 If α is the size of a maximum matching in G and δ=2α, then the game α,

δ(G) has a Nash

equilibrium.

and μ= nv-m

New Strategic Model

Definition 3.4 The graph G is bipartite if V=V0∪V1 for some disjoint vertex sets V0, V1 ⊆ V so that for each edge (u,v) ∈ E, u ∈ V0 and v ∈ V1

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

The End

Thanks for your attention.