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
(Interior stability)
2) for any
,there is an imputation
such that
(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
game, then
Largeness
be a totally balanced
Extendibility
Exactness
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-NPcomplete (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:
s
s1
t
t1 s2
D1
D2
Combination in serial:
s
D1
t
D2
Combination in parallel:
D1
s
D2
t
t2
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:
s
D1
t
D2
Combination in Parallel:
D1
s
D2
t
Exactness, Extendable,Largeness
 Proof cont.
 (d)=>(e):
D is a DAG, so consider the topological sort of the
vertex set.
Cut(v1,..,vi; vi+1,..,vn)
>Max Flow= min cut
vi
dummy arc
So, D contains no dummy arc, so D is balanced.
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
THANKS~~