An Inflation / Deflation Model for Price Stabilization in Networks
Jun Kiniwa
1
, Kensaku Kikuta
2
and Hiroaki Sandoh
3
1
Department of Applied Economics, University of Hyogo, 8-2-1 Gakuen-nishi, Nishi, Kobe, 651-2197, Japan
2
Department of Strategic Management, University of Hyogo, 8-2-1 Gakuen-nishi, Nishi, Kobe, 651-2197, Japan
3
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan
Keywords:
Multiagent Model, Price Determination, Self-stabilization, Inflation, Deflation.
Abstract:
We consider a simple network model for economic agents where each can buy goods in the neighborhood.
Their prices may be initially distinct in any node. However, by assuming some rules on new prices, we show
that the distinct prices will reach an equilibrium price by iterating buy and sell operations. First, we present a
protocol model in which each agent always bids at some rate in the difference between his own price and the
lowest price in the neighborhood. Next, we show that the equilibrium price can be derived from the total funds
and the total goods for any network. This confirms that the inflation / deflation occurs due to the increment /
decrement of funds as long as the quantity of goods is constant. Finally, we consider how injected funds spread
in a path network because sufficient funds of each agent drive him to buy goods. This is a monetary policy
for deflation. A set of recurrences lead to the price of goods at each node at any time. Then, we compare
two injections with half funds and single injection. It turns out the former is better than the latter from a
fund-spreading point of view, and thus it has an application to a monetary policy and a strategic management
based on the information of each agent.
1 INTRODUCTION
Motivation. Conventionally, the topics of price de-
termination have been discussed in the context of mi-
croeconomics approach(J. E. Stiglitz, 2006). In sup-
ply and demand curves, if the price is higher (resp.
lower) than an equilibrium, there is excess supply
(resp. excess demand) and thus the price moves to the
equilibrium. At the equilibrium price, the quantity of
goods sought by consumers is equal to the quantity
of goods supplied by producers. Neither consumers
nor producers have an incentive to alter the price or
quantity at the equilibrium. Since such a conven-
tional approach cannot capture each person’s behav-
ior, it is difficult to reflect actual economic phenom-
ena. So we considered a multiagent network model(J.
Kiniwa and K. Kikuta, a; J. Kiniwa and K. Kikuta, b),
in which each agent makes auctions and the price of
goods is eventually determined. Our network model
consists of nodes and edges as cities and their links to
neighbors, respectively. Each node contains an agent
which represents people in the city. Agents who want
to buy goods make bids to the lowest-priced neigh-
boring node, if any. Then, agents who want to sell
the goods accept the highest bid. The process of
price stabilization can be shown by using the idea
of self-stabilization in distributed systems(S. Dolev,
2000). From any initial state, self-stabilizing algo-
rithms eventually lead to a legitimate state without
any aid of external actions. We notice that the prop-
erties of self-stabilization resemble those of price de-
termination in convergence to a equilibrium without
external operations.
Problem. The problem in our previous studies (J.
Kiniwa and K. Kikuta, a; J. Kiniwa and K. Kikuta,
b) is an ambiguous relation between the price and the
amount of funds / goods. The most unsuccessful rea-
son is that no other variables than “price” were used.
There was no way to determine the next stage of the
price other than using the prices of buyers and sellers.
So we failed to explain why such an equilibrium price
is determined. To estimate the equilibrium price, we
need auxiliary variables which explain the next stage
of the price under stabilization. In addition, our model
failed to reflect the change of price due to various fac-
tors, called inflation or deflation. To explain the in-
flation / deflation, we need auxiliary variables which
show the flow of money and goods under the process
of such phenomena.
Solution. In this paper, we develop a new model con-
taining a relation between the price and the amount
125
Kiniwa J., Kikuta K. and Sandoh H..
An Inflation / Deflation Model for Price Stabilization in Networks.
DOI: 10.5220/0005186101250132
In Proceedings of the International Conference on Agents and Artificial Intelligence (ICAART-2015), pages 125-132
ISBN: 978-989-758-073-4
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

of funds / goods. We assume that the price is propor-
tional to the amount of funds and inversely propor-
tional to the amount of goods at each node. Further-
more, the volume of trade is assumed to depend on
the price difference between cities. As a result, the
flow of money and goods is determined by the market
principles, and thus the equilibrium price can be ex-
plained reasonably. Furthermore, it confirms that the
inflation / deflation depends on the amount of funds
as long as the amount of goods is constant.
Related Work. The classical theory of price deter-
mination in microeconomics is introduced, e.g., in (J.
E. Stiglitz, 2006; N. G. Mankiw, 2012), and a sur-
vey is in (T. A .Weber, 2012). We review the theory
from multiagent points of view. Though several eco-
nomic network models have been already known(L.
E. Blume, 2009; E. Even-Dar and S. Suri, 2007; S.
M. Kakade and S. Suri, 2004), such models contain
a bipartite structure(E. Even-Dar and S. Suri, 2007;
S. M. Kakade and S. Suri, 2004) or traders who play
intermediary roles (L. E. Blume, 2009). Agent-based
stabilization has been discussed in (J. Beauquier and
E. Schiller, 2001; S. Dolev and J. L. Welch, 2006;
S. Ghosh, 2000; T. Herman and T. Masuzawa, 2001).
Unlike our agents, their ideas are to use mobile agents
for the purpose of stabilization. It is useful in design-
ing protocols by what price we should make a bid.
Several kinds of game theoretic flavors have appeared
in self-stabilization, e.g., time complexity analysis(S.
Dolev and S. Moran, 1995), strategies with optimal
complexity(S. Dolev and P. Tsigas, 2008), relation-
ships between Nash equilibria and stabilization(A.
Dasgupta and S. Tixeuil, 2006; M. G. Gouda and H.
B. Acharya, 2009). Our protocol in Section 3 can
be considered as a kind of consensus algorithm. The
consensus algorithm in decentralized systems is de-
scribed in (N. A. Lynch, 1996), and its self-stabilizing
version is described in (S. Dolev, 2000; S. Dolev and
E. M. Schiller, 2010).
Contributions. We consider an inflation / deflation
network model, where the price is proportional to the
amount of funds, and is inversely proportional to the
amount of goods at each node. First, we present a pro-
tocol in which each agent always offers a fixed price
without considering other bidders’ strategies. Then,
we show that an equilibrium price is determined by
the total amount of funds and goods, and confirm that
inflation / deflation is determined by the amount of
funds. Next we focus on path networks and reveal the
price of each node and the amount of funds of each
node at each time. Finally, we show that the injection
of funds from two points is more effective than that
from a single point.
The rest of this paper is organized as follows.
Section 2 states our model. Section 3 shows that
our protocol can stabilize distinct goods prices. Sec-
tion 4 analyzes the behavior of our protocol in de-
tail. Section 4.1 investigates an equilibrium price in
an arbitrary network. Then, Section 4.2 estimates the
amount of funds at any node at any time for path net-
works. Furthermore, it suggests an effective fund-
injection method for a central bank. Finally, Section 5
concludes the paper.
2 MODEL
Our system can be represented by a connected net-
work G = (V,E), consisting of a set of nodes V and
edges E. In the network G, an arbitrary pair of
nodes i ∈ V and j ∈ V represent cities and an edge
(i, j) ∈ E between them, called neighbors, represents
direct transportation. Let N
i
be a set of neighboring
nodes of i ∈V, and let N
+
i
= N
i
∪{i}. We assume that
each node i ∈V has goods and their initial price may
be distinct. Let p
i
(t), or denoted by p
i
, be the price of
goods at node i for the time step t ∈ T = (0,1, 2,...).
Each node i ∈V has exactly one representative agent
a
i
who always stays at i and can buy goods in the
neighborhood N
i
. Each agent a
i
has funds f
i
, that is,
the total amount of money at i, and the quantity q
i
of
goods at i. The price p
i
is determined by the rela-
tionship between the quantity of goods and the pur-
chasing power, or called supply-demand balance. So
we simply assume that the price is proportional to the
amount of funds for constant goods (Figure 1(a)), and
is inversely proportional to the amount of goods for
constant funds (Figure 1(b)) at each node, that is,
p
i
=
f
i
q
i
. (1)
The buy operation is executed as follows. Each
agent a
i
assigns a value v
j
i
(t), or denoted by v
j
i
, to
the goods of any neighboring node j ∈ N
i
, where the
value means the maximum amount an agent is willing
Figure 1: Price determined by funds and goods at each
node.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
126

to pay. Agent a
i
compares its own goods price p
i
with
the neighboring price p
j
. If the cheapest price in N
i
is
p
j
and is less than p
i
, the agent a
i
wants to buy it and
submits a bid b
j
i
(t), or denoted by b
j
i
, to node j. We
consider v
j
i
(t) = p
i
(t) for any j ∈ N
i
because he can
buy it at price p
i
(t) in his node.
The sell operation is executed as follows. Af-
ter accepting bids from N
j
, agent a
j
contracts with
a
i
∈ N
j
, an arbitrary one of agents who submitted the
highest bid b
j
i
. Then, a
j
passes the goods to (receives
money from) the contracted agent a
i
until the price
p
j
(t +1) becomes b
j
i
derivedfrom the supply-demand
balance. We do not take the carrying cost of goods
into consideration but focus on the change of prices.
In this way, at every time, any price is updated if nec-
essary. The state Σ
i
of each node i ∈V is represented
by the price, the quantity of goods and the amount of
funds (p
i
(t),q
i
(t), f
i
(t)).
We assume a synchronous model, that is, every
agent periodically exchanges messages and knows the
states of neighboring agents. The global state of all
nodes is called a configuration. The set of all config-
urations is denoted by Γ = Σ
1
×Σ
2
×···×Σ
|V|
. An
atomic step consists of reading the states of neigh-
boring agents, a buy / sell operation, and updating
its own state. Then, a configuration is changed from
c
j
∈Γ into c
j+1
∈Γ (or c
j+1
is reached from c
j
) by the
atomic step. An execution E is a sequence of config-
urations E = c
0
,c
1
,... ,c
j
,c
j+1
,... such that c
j+1
∈Γ
is reached from c
j
∈Γ.
3 PROTOCOL DESIGN
In this section, we consider a protocol model, called
FundBidding, in which each agent a
i
always makes
a bid b
j
i
(p
j
(t) ≤b
j
i
≤ p
i
(t)) to an agent a
j
∈N
+
i
with
the lowest price in the neighborhood. For simplicity,
let k be a constant rate so that b
j
i
lies between p
j
(t)
and p
i
(t), where the price may not be an integer.
FundBidding
• Each agent a
i
makes a bid
b
j
i
(t) = p
j
(t) +

p
i
(t) − p
j
(t)
k

, (2)
where k ≥1, to node j ∈N
+
i
which has the lowest-
priced goods in N
+
i
.
• The agent a
j
contracts with the neighboring a
i
who has submitted the highest bid max
i∈N
j
b
j
i
(t).
If a
j
has submitted his bid to neighboring node at
the same time, it is postponed until the next time
1
2
3
4
1
2
3
4
Figure 2: An illustration of protocol FundBidding.
step. The goods of a
j
and the money of a
i
are ex-
changed, that is, the goods are moved from q
j
to
q
i
and the money is moved from f
i
to f
j
as long
as p
i
> p
j
. The prices p
i
(t + 1) and p
j
(t + 1) are
determined by the funds and the amount of goods.
• If several agents make bids to node j with the
same highest price, agent a
j
makes deals with one
of them at random.
Example 1. Figure 2 shows an example of our net-
work system consisting of 4 nodes V = {1,2,3,4}.
For the bidding price (2), let k = 2. At time t,
the prices of goods are (p
1
(t), p
2
(t), p
3
(t), p
4
(t)) =
(50,110,70, 10) as shown in Figure 2(a). Each agent
a
i
wants to buy the lowest-priced goods at node j ∈N
i
if its price is lower than p
i
, that is, p
i
> min
j∈N
i
p
j
.
Thus, agent a
1
makes a bid to node 4 with price
b
4
1
= 30. Likewise, agents a
2
and a
3
make bids to
node 1, respectively. Then, only a
2
’s bid is successful,
and a
2
makes a contract with a
1
.
At time t + 1, the prices become (p
1
(t + 1), p
2
(t +
1), p
3
(t + 1), p
4
(t + 1)) = (80,80, 70,10) as shown in
Figure 2(b). Since price p
1
has been changed, agent
a
1
’s bid b
4
1
is resubmitted as (80+ 10)/2 = 45. Since
the bids b
3
2
and b
4
1
are independent, they are executed
in parallel at time t + 1. ⊓⊔
We are concerned with whether or not the prices
of goods eventually reach an equilibrium price even if
they are initially distinct. So we define the legitimacy
of a configuration as follows.
Definition 1 (legitimate configuration). A configura-
tion is legitimate if the goods in every node have the
same price. ⊓⊔
Let C
t
⊆V be the set of nodes that have updated
their prices from time t to t + 1. The following lemma
proves that the protocol FundBidding is free from
deadlocks.
Lemma 1. The protocol FundBidding is deadlock-
free. That is, there exist some nodes in C
t
as long as
the configuration is illegitimate.
Proof. First notice that no cycle is generated by the
chain of bidding requests, as depicted in Figure 2,
AnInflation/DeflationModelforPriceStabilizationinNetworks
127

because every bidding request occurs from a higher
priced node to a lower priced node.
Next suppose that the configuration is illegitimate
at time t. Then, there is a pair of neighboring nodes
i, j ∈ V such that p
i
(t) = max
h∈N
j
p
h
(t) and p
j
(t) =
min
h∈N
i
p
h
(t), where p
i
(t) − p
j
(t) is the maximum
price difference in the neighborhood. In this case,
agent a
i
makes a bid to node j and agent a
j
accepts
the price. Since p
j
(t) is increased at time t + 1, j ∈C
t
holds. ⊓⊔
In (J. Kiniwa and K. Kikuta, b), we investigated a
condition such that any protocol satisfying the frame-
work of FundBidding achieves price stabilization.
Suppose that agents a
i
and a
j
make bids to node h. We
say that bids have the same order as values if v
h
i
≤ v
h
j
implies b
h
i
≤ b
h
j
for the goods of node h. Next lemma
shows that the bids having the same order as values is
necessary for price stabilization.
Lemma 2. (J. Kiniwa and K. Kikuta, b) If bids do not
always have the same order as values, price stabiliza-
tion is not guaranteed. ⊓⊔
The following theorem further shows that an addi-
tional condition leads to the price stabilization.
Theorem 1. (J. Kiniwa and K. Kikuta, b) Suppose
that bids have the same order as values. If any con-
tract price lies between buyer’s price and seller’s
price, price stabilization occurs. ⊓⊔
Since we assume that v
j
i
(t) = p
i
(t) for any neigh-
boring node j ∈ N
i
and a
i
makes a bid by (2), Fund-
Bidding satisfies the condition above.
4 ANALYSIS
In this section, we investigate several aspects of our
FundBidding for arbitrary networks and path net-
works.
4.1 Arbitrary Network
The following theorem claims that the equilibrium
price is determined by the total amounts of funds and
the goods regardless of the network topology.
Theorem 2. Let F be the total amount of funds, and
Q the total amount of goods. The equilibrium price,
denoted by P
e
, is
P
e
=
F
Q
regardless of the network topology.
Proof. By definition, the price of goods at node i
is p
i
= f
i
/q
i
. Suppose that the equilibrium prices
are different for each stabilization process. Then,
p
i
(t) 6= p
i
(t
′
) for time t and t
′
(t 6= t
′
) holds. Since
f
i
= p
i
(t)q
i
and f
′
i
= p
i
(t
′
)q
′
i
hold for any node i,
where F =
∑
i
f
i
=
∑
i
f
′
i
, we have
p
i
(t) ·
∑
i
q
i
= p
i
(t
′
) ·
∑
i
q
′
i
.
Since the total amount of goods Q is identical, we
have
Q =
∑
i
q
i
=
∑
i
q
′
i
.
Thus we obtain p
i
(t) = p
i
(t
′
), a contradiction. There-
fore, the equilibrium price P
e
is identical for each sta-
bilization process.
Next, since f
i
= P
e
·q
i
holds for every node i, the
total funds sum up to
F = P
e
·Q.
Thus we obtain P
e
= F/Q. ⊓⊔
The theorem above is known as the Fisher’s quan-
tity equation (N. G. Mankiw, 2012) FV = P
e
Q if the
velocity of money V equals to 1. This means the cor-
rectness of our assumption (1) at each node. Thus,
in our inflation / deflation model, the inflation (resp.
deflation) occurs if the total amount of funds increase
(resp. decrease) as long as the total amount of goods
is constant.
4.2 Path Network
In what follows, we restrict our concern to path net-
works. The path networks probably represent the dis-
tance feature in arbitrary networks. Then, we con-
sider how injected funds spread in the path network
because sufficient funds of each agent drives him to
buy goods. This is a monetary policy for deflation.
Section 4.2.1 considers the situation that incremental
funds are injected from a single point. Section 4.2.2
considers the situation that the half of incremental
funds are injected from two points.
4.2.1 Single Injection
We investigate the amount of funds at each node of a
path P = (1,2,...,n) at any time. For simplicity, let
k = 2 and let b
j
i
(t) = (p
i
(t)+ p
j
(t))/2 in (2). Suppose
that we inject funds m into node 1, called an injection
point. Let p
c
i
(t) be the temporary, intermediate price
of node i reached by trading exhaustively for a con-
tract between t and t + 1.
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
128

Lemma 3. Let q
i
be the quantity of goods, and f
i
the
funds of agent a
i
before the trade at node i. Then, the
price after the trade will be
p
c
i
(t) =
f
i−1
+ f
i
q
i−1
+ q
i
.
Proof. Suppose that (q
i−1
, f
i−1
) and (q
i
, f
i
) change
into (q
′
i−1
, f
′
i−1
) and (q
′
i
, f
′
i
) after the trade, respec-
tively. Let F
i−1,i
and Q
i−1,i
be a sum of funds and
a sum of quantities of goods at nodes i−1 and i, re-
spectively. Since no other funds and goods do not
come into these values, we have
f
i−1
+ f
i
= f
′
i−1
+ f
′
i
= F
i−1,i
q
i−1
+ q
i
= q
′
i−1
+ q
′
i
= Q
i−1,i
.
At an equilibrium, since
f
′
i−1
q
′
i−1
=
f
′
i
q
′
i
= P
e
,
q
′
i−1
= f
′
i−1
/P
e
and q
′
i
= f
′
i
/P
e
. Then,
q
′
i−1
+ q
′
i
= ( f
′
i−1
+ f
′
i
)/P
e
,
that is, Q
i−1,i
= F
i−1,i
/P
e
holds. Thus, we have
P
e
=
F
i−1,i
Q
i−1,i
=
f
i−1
+ f
i
q
i−1
+ q
i
.
This means we can find the equilibrium price before
the trade. ⊓⊔
The following Figure and Example present a be-
havior of price diffusion in a path.
Figure 3: Price diffusion in a path.
Example 2. Figure 3 illustrates price diffusion in a
path (1,2,3), where the price at node 1 is initially
higher than others because funds have been injected.
The intermediate state of p
i
between t = 0 and t = 1
is denoted by p
c
i
(0) for convenience. Let k = 2 for the
expression (2). First,
p
1
(1) =
p
1
(0) + p
2
(0)
2
= p
c
2
(0),
and then
p
2
(1) =
p
c
2
(0) + p
3
(0)
2
=
p
1
(1) + p
3
(0)
2
= p
3
(1)
holds. ⊓⊔
Thus, in general, the price p
j
(t) at node j ∈P (2 ≤
j ≤ n−1) can be represented as follows.
p
1
(t) =
1
2
· p
1
(t −1) +
1
2
· p
2
(t −1) (3)
p
j
(t) =
1
2
· p
j−1
(t) +
1
2
· p
j+1
(t −1) (4)
p
n
(t) =
1
2
· p
n−2
(t) +
1
2
· p
n
(t −1) (5)
From (4), we have
∑
t≥1
p
j
(t)x
t
=
1
2
·
∑
t≥1
p
j−1
(t)x
t
+
1
2
·
∑
t≥1
p
j+1
(t −1)x
t
.
Using R
j
(x) =
∑
t≥0
p
j
(t)x
t
, we obtain
R
j
(x) − p
j
(0) =
1
2
(R
j−1
(x) − p
j−1
(0))+
x
2
R
j+1
(x)
2R
j
(x) = R
j−1
(x) + xR
j+1
(x) + (2p
j
(0) − p
j−1
(0))
For simplicity, we assume 2p
j
(0) − p
j−1
(0) = 0
and replace j by j −1. Then,
xR
j
−2R
j−1
+ R
j−2
= 0.
So we have
R
j
= A
1

1+
√
1−x
x

j
+ A
2

1−
√
1−x
x

j
.
Using our initial conditions R
0
(x) =
∑
t≥0
p
0
(t)x
t
= 0
and R
1
(x) =
∑
t≥0
p
1
(t)x
t
≈ p
1
(0),

A
1
+ A
2
= 0
A
1
(
1+
√
1−x
x
) + A
2
(
1−
√
1−x
x
) = p
1
(0)
lead to
R
j
= p
1
(0) ·
x
2
√
1−x
(

1+
√
1−x
x

j
−

1−
√
1−x
x

j
)
.
Using
x
2
√
1−x
= z,
R
j
=
p
1
(0)
2z
{(
1
x
+ z)
j
−(
1
x
−z)
j
}
=
p
1
(0)
2z
{2

j
1

1
x

j−1
z+ 2

j
3

1
x

j−3
z
3
+ ···}
= p
1
(0)
∑
r≥1

j
2r−1

1
x

j−(2r−1)
z
2r−2
= p
1
(0)
∑
r≥1

j
2r−1

1
x

j−(2r−1)

1−x
x
2

r−1
.
Thus,
R
j
= p
1
(0)
∑
r≥0

j
2r+ 1

x
r
·
1
(1−x)
j−1
= p
1
(0)
∑
r≥0

j
2r+ 1

x
r
∑
s≥0

j + s−2
s

x
s
= p
1
(0)
∑
r≥0
∑
0≤s≤r

j
2(r−s) + 1

j + s−2
s

!
x
r
.
AnInflation/DeflationModelforPriceStabilizationinNetworks
129

Therefore, we have
p
j
(t) = p
1
(0)
∑
0≤s≤⌊( j−1)/2⌋

j
2(r −s) + 1

j+ s −2
s

.
Then, p
1
(t) and p
n
(t) can be described as follows:
p
1
(t) =
p
1
(0)
2
t
+
∑
0≤k≤t−1
p
2
(k)
2
t−k
,
and
p
n
(t) =
p
n
(0)
2
t
+
∑
1≤k≤t
p
n−2
(k)
2
t−k+1
.
Let b
j
j−1
(t) = (p
j−1
(t) + p
j
(t))/2 be the bidding
price of node j −1 to node j. Let p
c
j
(t) (or simply
p
c
j
) denote the temporary, intermediate price at node
j between time t andt +1. Then, the amount of goods
q
j
(t + 1) can be determined as follows.
Lemma 4. The amount of goods at time t + 1 is
q
j
(t + 1) =
(b
j
j−1
+ p
j
)(b
j+1
j
+ p
c
j
)
(p
c
j
+ b
j
j−1
)(p
j
+ b
j+1
j
)
·q
j
(t)
Proof. Let x (resp. y) be the amount of goods moved
from node j to node j −1 (resp. node j + 1 to node
j). First, we consider the trade between node j −1
and node j. Notice that the funds of agent j reach
f
j
(t) + x·b
j
j−1
(t) and the amount of goods at node j
becomes q
j
(t)−x. By Lemma 3, when the price p
c
j
(t)
reaches p
c
j
(t) = ( f
j−1
+ f
j
)/(q
j−1
+ q
j
),
f
j
+ x·b
j
j−1
q
j
−x
= p
c
j
x =
p
c
j
q
j
− f
j
p
c
j
+ b
j
j−1
.
Since f
j
= p
j
q
j
, we have
q
c
j
= q
j
−x =
q
j
(b
j
j−1
+ p
j
)
p
c
j
+ b
j
j−1
.
Likewise, for the trade between node j and node
j + 1,
p
c
j
q
c
j
−y·b
j+1
j
q
c
j
+ y
= p
j
y =
p
c
j
q
c
j
−p
j
q
c
j
p
j
+ b
j+1
j
.
Thus,
q
j
(t +1) = q
c
j
+ y
=
q
j
(b
j
j−1
+ p
j
)
p
c
j
+ b
j
j−1
+
p
c
j
q
c
j
−p
j
q
c
j
p
j
+ b
j+1
j
=
(b
j
j−1
+ p
j
)(b
j+1
j
+ p
c
j
)
(p
c
j
+ b
j
j−1
)(p
j
+ b
j+1
j
)
·q
j
(t).
⊓⊔
Theorem 3. The amount of agent a
j
’s funds at time t
is
f
j
(t) = p
j
(t)
t−1
∏
i=1
(b
j
j−1
+ p
j
)(b
j+1
j
+ p
c
j
)
(p
c
j
+ b
j
j−1
)(p
j
+ b
j+1
j
)
·q
j
(0).
⊓⊔
4.2.2 Double Injections of Half Funds
This section considers the half of incremental funds
are injected from two points. Figure 4 illustrates
(a) single injection and (b) double injections of half
funds.
Figure 4: Injection of funds.
First, we focus on the asymptotic behavior of the
terminal agent a
n
. Notice that agent a
n
’s funds only
increases and the amount of goods only decreases
under stabilization. Next, we show that the method
of double injections of half funds is better than that
of single injection from the fund-spreading point of
view. The investigation is motivated by exploring a
good monetary policy.
Lemma 5. Let p
c
n−1
(0) be the price at node n −1
immediately before bidding for node n. Then,
p
c
n−1
(0) =
p
1
(0)
2
n−1
+ p
n
(0)

1−
1
2
n−1

if we assume p
2
(0) = ··· = p
n
(0).
Proof. First, agent a
1
makes a bid to node 2 with
b
2
1
(0) = (p
1
(0) + p
2
(0))/2. Then, agent a
2
makes a
bid to node 3 with b
3
2
(0) = (b
2
1
(0) + p
3
(0))/2, and
so on. The bidding reaches node n with b
n
n−1
(0) =
(b
n−1
n−2
(0) + p
n
(0))/2. Thus, we have
p
c
n−1
(0) =
p
1
(0) + p
2
(0)
2
n−1
+
p
3
(0)
2
n−2
+ ···+
p
n
(0)
2
.
If we assume p
2
(0) = ··· = p
n
(0),
p
c
n−1
(0) =
p
1
(0)
2
n−1
+ p
n
(0)

1
2
+ ···+
1
2
n−1

=
p
1
(0)
2
n−1
+ p
n
(0)

1−
1
2
n−1

.
⊓⊔
ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence
130

After the trade at time t, suppose that agent a
n
’s
funds become f
n
+ x·b
n
n−1
and the amount of goods
becomes q
n
−x. Since the price reaches b
n
n−1
,
f
n
+ x·b
n
n−1
q
n
−x
= b
n
n−1
x =
1
2
q
n
−
f
n
2b
n
n−1
holds. Thus,
q
n
(t + 1) = q
n
(t) −x =
1
2
q
n
+
f
n
2b
n
n−1
.
Let us denote q
t
=
1
2
q
t−1
+
f
t−1
2b
t−1
for simplicity.
Then,
q
n
(t) =
1
2
1
2
q
n
(t −2) +
f
n
(t −2)
2b
n
n−1
(t −2)
!
+
f
n
(t −1)
2b
n
n−1
(t −1)
=
1
2
2
q
n
(t −2) +
1
2
f
n
(t −2)
2b
n
n−1
(t −2)
+
f
n
(t −1)
2b
n
n−1
(t −1)
.
.
.
=
1
2
t
q
0
+
∑
0≤k≤t−1
1
2
k
f
n
(t −1−k)
2b
n
n−1
(t −1−k)
!
.
Since
1
2
t
q
0
→ 0 for large t and
f
0
2b
0
> ··· >
f
t−1
2b
t−1
,
q
t
≤
1
2
t
q
0
+
∑
0≤k≤t−1
1
2
k

f
n
(0)
2b
n
n−1
(0)

Since b
n
n−1
(0) = (p
c
n−1
(0) + p
n
(0))/2, we obtain
1
2
t
q
0
+
2f
n
(0)
p
1
(0)/2
n−1
+ P
e
(2−1/2
n−1
)
< q
t
and
q
t
<
1
2
t
q
0
+
2f
n
(0)
p
1
(0)/2
n−1
+ p
n
(0)(2−1/2
n−1
)
(6)
by Lemma 5.
Consider which is better for money spreading,
from one injection point or from two injection points
with half funds. We compare two cases, (a) one injec-
tion point is node 1, and (b) two injection points are
node 1 and node n. Clearly, the node with the mini-
mum funds at equilibrium, called min-funds node, in
case (a) is node n, and that in case (b) is node ⌈n/2⌉
(simply denoted by n/2). Thus, we have only to com-
pare f
n
in case (a) and f
n/2
in case (b). The following
lemma shows that the quantity of goods at the min-
funds node in case (b) is less than that in case (a).
Notice that q
[m→h]
i
(t) means the quantity of goods
at node i at time t on condition that incremental funds
m are initially injected into node h.
Lemma 6. For any t > 0,
q
[m/2→1,m/2→n]
n/2
(t) < q
[m→1]
n
(t)
holds.
Proof. From equation (6), we have only to compare
q
upper
(n/2), the upper bound of q
n/2
(t) −
1
2
t
q
0
, and
q
lower
(n), the lower bound of q
n
(t) −
1
2
t
q
0
. That is,
they can be described as
q
upper
(n/2) =
2f
n/2
(0)
p
1
(0)/2
n/2
+ p
n/2
(0)(2−1/2
n/2−1
)
and
q
lower
(n) =
2f
n
(0)
p
1
(0)/2
n−1
+ P
e
(2−1/2
n−1
)
.
Since f
n
(0) = f
n/2
(0) and p
n
(0) = p
n/2
(0) at timet =
0, we have
q
lower
(n)
q
upper
(n/2)
=
p
1
(0)/2
n/2
+ p
n
(0)(2−1/2
n/2−1
)
p
1
(0)/2
n−1
+ P
e
(2−1/2
n−1
)
=
p
n
(0)
P
e
·
p
1
(0)/p
n
(0)2
n/2
+ (2−1/2
n/2−1
)
p
1
(0)/P
e
2
n−1
+ (2−1/2
n−1
)
≈ 2
n/2−1
> 1.
Thus, q
upper
(n/2) < q
lower
(n) holds. So the lemma
follows. ⊓⊔
From Lemma 6, we claim the following theorem
because the equilibrium price is equal for each case.
Notice that f
[m→h]
i
means the amount of funds at node
i on condition that incremental funds m are initially
injected into node h.
Theorem 4. At an equilibrium, we have
f
[m→1]
n
< f
[m/2→1,m/2→n]
n/2
.
⊓⊔
The theorem above suggests that the multiple in-
jection points is better than the single injection point
for effective spreading of funds.
5 CONCLUSION
In this paper we considered a new network model for
the price stabilization. First, we presented a system
model in which the price of goods is proportional to
the amount of funds and is inversely proportional to
the amount of goods at each node. Then we provided
a protocol which stabilizes price and moves money /
goods. Next, we showed that the equilibrium price is
determined by the total amount of funds and the to-
tal amount of goods. Then, we concentrated on path
networks to reveal the behavior of the protocol more
precisely. We considered the price under stabilization
at each node. Finally, we investigated which injec-
tion method is better from the fund-spreading point of
view, motivated by an application to monetary policy.
In summary, our network model reveals the fol-
lowing facts.
AnInflation/DeflationModelforPriceStabilizationinNetworks
131

• The equilibrium price of goods can be estimated
if the price is proportional to the amount of funds
and is inversely proportional to the amount of
goods at each node.
• The price under stabilization at each node in a
path is investigated.
• The two injections with half funds is better than
the single injection from fund-spreading point of
view.
Our future work includes investigating an asyn-
chronous system and developing other protocols.
ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant
Number ((B)25285131).
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