Complexity of Payment Network
.
Hitoshi Hayakawa
.
Graduate School of Economics, The University of Tokyo
October 2012
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
1 / 13
Problem to analyze
Problem : PCP
How much liquidity is required to settle payments (obligations)
- given subjects, payment pairs, payment amounts
- each obligation is settled with liquidity transfer (no offset)
How about more complex cases?
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
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Importance in Interbank Settlement Systems
Interbank settlement systems
- gross settlement system: each obligation is settled with liquidity
transfer
- payment network is complex
(about 500 participants, 50,000 daily transactions in the BOJ-NET)
- liquidity requirement is an important policy concern
← provision of sufficient intraday liquidity is usually role of Central Banks
← transfers of payments are huge
(about 100 trillion yen on daily basis in the BOJ-NET)
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
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Purpose of this paper
Formulation of the problem (PCP), and Characterization of the
problem in terms of network factors
- Rotemberg(2011)
- limited class of payment network
→ propose a network factor termed as “interconnectedness”
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
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Contribution
With a graph-theoretic framework, successfully treats a general class
of payment network
Characterize with several network factors
propose novel network factors : “a-twisted”, “v-twisted”
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
5 / 13
Framework
base elements V , A
< V , A > : directed graph (vertices, arrows)
additional elements f , s , p
f : A → R+
(flow) : for payment amount
s : A → {1, 2, .., |A |} (sequence) : for payment order
p : V → R0+
(potential) : for initial endowment
Networks
f-Network : < V , A , f >
fsp-Network : < V , A , f , s , p >
Example
{
}
V = {va , vb , vc , vd }, A = (va , vb ), (vb , vc ), (vb , vd ), (vc , vd ), (vd , va ) ,
f ((va , vb )) = f ((vd , va )) = 30, f ((vb , vd )) = 20, f ((vb , vc )) = f ((vc , vd )) = 10,
s ((va , vb )) = 1, s ((vb , vc )) = 4, f ((vb , vd )) = 3, f ((vc , vd )) = 2, f ((vd , va )) = 5,
p (va ) = 30, p (vc ) = 10, p (vb ) = p (vd ) = 0
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
6 / 13
Basic Properties for Networks
closed < V , A , f > : aggregate balanced budget
∀v ∈ V , sum of payments indicated by f are balanced
e-covered < V , A , f , s , p > : sequential balanced budget
covered < V , A , f , s , p >: For payments of < V , A , f >, all the
payments are executable at any point under the order indicated by s
e-covered < V , A , f , s , p >: covered, but no more covered for p ′ ≤ p
(∃v ∈ V , p ′ (v ) < p (v ))
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
7 / 13
PCP (Payment Circulation Problem)
Definition (min(max) Payment Circulation Problem (PCP) x min(max ) (N f ))
Given a closed f-Network N f =< V , A , f >,
∑
min(max)s ,p v ∈V p (v )
s.t., < V , A , f , s , p > is e-covered
.
min PCP : most efficient liquidity circulation
.
max PCP : least efficient liquidity circulation
↔ Rotemberg(2011) examined intermediate efficient liquidity circulation
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
8 / 13
Approach: Closed Cycle Decomposition
Theorem (Closed Cycle Decomposition (fulkerson, 1962))
For any closed f-Network N f =< V , A , f >, there always exists a closed cycle
decomposition.
∑
N f = c ∈C < V c , c , f c >
∑
.
= (1 cycle closed f-Network)
Example for closed cycle decomposition
.
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
9 / 13
PCP in Closed Cycle Decomposition Form
Given a closed f-Network N f , with some closed cycle decompositions
∑
∑
N f = c ∈C < V c , c , f c >, N f = c ∈C ′ < V c , c , f c >,
∑
c
x min (N f ) =
+ R minc
c ∈C f
x max (N f ) =
Example for R
∑
+
R max
′ c
(cycle oriented amount)
+
(residual)
min(max )
C ,{f c }c ∈C
H. Hayakawa (Univ. of Tokyo)
C ,{f }c ∈C
f c ∗ (|V c | − 1)
c ∈C ′
C ,{f }c ∈C ′
=0
Complexity of Payment Network
October 2012
10 / 13
Major Network Factors : a-twisted
x min (N f ) =
∑
c ∈C
fc
Given a closed f-Network N f ,
there exists a-twisted cycles ⇔ ∃C
Example for R minc
C ,{f }c ∈C
H. Hayakawa (Univ. of Tokyo)
+
∈ CN f ,
R minc
C ,{f }c ∈C
R minc
C ,{f }c ∈C
,0
,0
Complexity of Payment Network
October 2012
11 / 13
Major Network Factors : v-twisted
x max (N f ) =
∑
c ∈C
f c ∗ (|V c | − 1)
Given a closed f-Network N f ,
there exists v-twisted cycles ⇔ ∃C
Example for R maxc
C ,{f }c ∈C
H. Hayakawa (Univ. of Tokyo)
∈ CN f ,
+
R maxc
C ,{f }c ∈C
R maxc
C ,{f }c ∈C
,0
,0
Complexity of Payment Network
October 2012
12 / 13
Summary and Remarks
Summary
Formulate a liquidity problem in interbank settlement systems
Originality in its combined aspect of
− “numbering” problem (pure graph theoretic) and
− “flow” problem (applicational fields)
Propose a distinctive approach of Closed Cycle Decomposition
Characterize the problem with network factors
Notions of “a-twisted”, “v-twisted” are originally developed
H. Hayakawa (Univ. of Tokyo)
Complexity of Payment Network
October 2012
13 / 13