Core Stability of Simple Flow Game

Core Stability of Simple Flow Game Jan 2007 Qizhi Fang, Xiaoxun Sun City University of Hong Kong, Hong Kong Ocean University of China, Qingdao Jian Li Fudan University

Cooperative Game Theory • Profit Game • A set of Player: N={1,….,n} • Coalition: Sµ N • Characteristic Function: v(S) for Sµ N, s.t. (Superadditivity Property)

Cooperative Game Theory • Individual Rationality Condition Def: Imputation: The vector s.t.

Cooperative Game Theory • Collective(Group) Rationality Condition Def: Core: A imputation belongs to core if

Cooperative Game Theory Def: imputation  is said to dominate imputation in coalition S (denoted as ) if Def: imputation  is said to dominate imputation  if there is a coalition S for which , we donate it as

Cooperative Game Theory Def: Stable Set S (NM-solution) 1) implies or 2) for any ,there is an imputation such that (Interior stability) (exterior stability)

Cooperative Game Theory THM: Core is the set of nondominant imputations. If core is nonempty and stable set exists, it contains the core. But, stable set may not exists So, It is a natural question: WHEN STABLE SET=CORE?

Cooperative Game Theory • Known Results for core stability • Determining the existence of a stable set is not known to be computable, and it is still open (Deng and Papadimitriou (1994)) • Core stability of assignment games (Solymosi and Raghavan (2001)) • Core stability of minimum coloring games (Bietenhader and Okamoto(2005)) • For convex game, the core is the unique stable set

Cooperative Game Theory

Cooperative Game Theory • THM: (Sharkey,82; Biswas etc 99) Let be a totally balanced game, then Exactness Largeness Extendibility Core Stability

Flow Game Given a network D(V,A;s,t,w) • V: vertex set • A: Arc set • s,t: source and sink • w: A->R+ capacity function

Flow Game The Flow Game over D: Player: arc set A Characteristic funtion: v(S)={maximum s,t-flow on D(V,S;s,t,w), Sµ A} It is easy to check superadditivity holds. Simple Flow Game: w(e)=1 for all e2 A

Flow Game • Known Results for Flow Game • The core of a flow game is always non-empty, a core member can be found in polynomial time • When D is simple, the core is exactly the convex hull of the characteristic vectors of minimum cuts of D (Kalai and Zemel (1982)) • For general case, the problem of testing whether a given imputation belongs to core is co-NP-complete (Fang et al. (2001))

Flow Game • Our Result: (1) A simple flow game has a stable core if and only if the underlying network is a balanced DAG(directed acyclic graph). (balanced: outdegree(v)=indegree(v) for all v2 V except s and t)

Flow Game • Our Result: (2)The flow game =(A,v) is defined on the directed graph D(V,A;s,t).The following statement are equivalent: (a) The flow game  is exact (b) The flow game  is extendable (c) The core C() is large (d) Every (s,t)-cut contains a minimum (s,t)-cut (e) D is a balanced serial parallel digraph

Core Stability • In D(V,A;s,t), An arc e2 A is a Dummy Arc if max-s,t-flow(D)=max-s,t-flow(D\{e})

Core Stability • Lemma 1: Core C() is stable if and only if for any y2 Imp()\C(), There is an (s,t)-path P s.t. z(P) = 1 and z(e) >y(e) for all e2 P. • Lemma 2: Let e be a dummy arc,  and * be the flow games with respect to network D and D\{e}. Then C() is stable => C(*) is stable.

Core Stability • THM: The flow game has a stable core iff the corresponding network D constains no dummy arc. Proof (sketched): (<=) easy (=>) Induction on number of dummy arcs, and repeatedly apply lemma 2.

Core Stability • THM: A simple network D contains no dummy arc iff D is balanced DAG. Proof (sketched): (<=) easy (=>) easy

Exactness, Extendable,Largeness • THM: The flow game =(A,v) is defined on the directed graph D(V,A;s,t).The following statement are equivalent: (a) The flow game  is exact (b) The flow game  is extendable (c) The core C() is large (d) Every (s,t)-cut contains a minimum (s,t)-cut (e) D is a balanced serial parallel digraph

Exactness, Extendable,Largeness

Exactness, Extendable,Largeness 2-terminal SP graph: • Base case: • Inductive step: Combination in serial: Combination in parallel: s t D1 D2 s1 s2 t2 t1 D1 D2 s t D1 t s D2

Exactness, Extendable,Largeness (d)(e): (d) Every (s,t)-cut contains a minimum (s,t)-cut (e) D is a balanced serial parallel digraph NOTE: (d) is equivalent to the following statement which we call maxmin cut property: maximum minimal cut = minimum cut

Exactness, Extendable,Largeness • Proof cont. • (e)=>(d): Induction on number of edges Combination in serials: Combination in Parallel: D1 D2 s t D1 t s D2

Exactness, Extendable,Largeness • Proof cont. • (d)=>(e): D is a DAG, so consider the topological sort of the vertex set. So, D contains no dummy arc, so D is balanced. Cut(v1,..,vi; vi+1,..,vn) >Max Flow= min cut vi dummy arc

Exactness, Extendable,Largeness • Proof cont. • (d)=>(e): A graph W is homeomorphic to H if it contains four distinct vertices a,b,c,d and 5 pairwise internally vertex disjoint paths Pab,Pbc,Pcd,Pac,Pbd.

Exactness, Extendable,Largeness • THM(Duffin 65): • A 2-terminal DAG D is a 2-terminal DSP graph if and only if D doesn't contain a subgraph homeomorphic to H.

Exactness, Extendable,Largeness The minimal cut C with k+1 edges consists: • 4 edges: the last edges of psb and pab and the first edges of pcd and pct. • k-3 edges: all edges of (s,vi) in D n D3. H D3

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Core Stability of Simple Flow Game