=Paper= {{Paper |id=Vol-2018/paper-05 |storemode=property |title=Agents’ Interaction Algorithm in a Strongly Coupled System with a Transferable Utility |pdfUrl=https://ceur-ws.org/Vol-2018/paper-05.pdf |volume=Vol-2018 |authors=Michael Geraskin,Olga Kuznetsova }} ==Agents’ Interaction Algorithm in a Strongly Coupled System with a Transferable Utility== https://ceur-ws.org/Vol-2018/paper-05.pdf

Agents’ Interaction Algorithm in a Strongly
Coupled System with a Transferable Utility

Michael I. Geraskin and Olga A. Kuznetsova

Samara National Research University, Samara, Russia,
innovation@ssau.ru

Abstract. The problem of the utility distribution mechanism analyzing
in a strongly coupled system with the utility (profit) transfer possibility
by the aggregated utility criterion is considered. The utility distribution
algorithm in the system with complementary demand functions is devel-
oped. The strategic behavior algorithm under stability and the individual
rationality conditions as well as Pareto efficiency fulfillment is confirmed
by numerical simulation of the mechanism for the “retailer-bank-insurer”
system. Experimental modeling proves that the application of the pro-
posed algorithm with the increasing risk rates claimed by agents leads
to a decrease in the profit of the meta-agent while preserving the prof-
its of the agents. The situations of the environment’s domination and
the meta-agent’s domination with similar initial parameters of the sys-
tem lead to the same dynamics of the agent’s profits with an increase
in the risk rate. Thus, distortion of information about the risk rate to-
wards the increase is an advantage for agents and a disadvantage for the
meta-agent.

Keywords: distribution mechanism, tightly coupled system, anonymous
agent, aggregated utility, complementary demand, transferable utility,
nash equilibrium, retailer, bank, insurer

1 Introduction

The strongly coupled organizational and economic systems are formed under
the subjective factors influence, for example, as a result of some legal entities
with other affiliation by virtue of a controlling stake disposal. Also, these sys-
tems are formed under the objective factors influence of some agents’ prevailing
economic activity in comparison with others. A typical microeconomic problem
that illustrates the agents’ heterogeneity in economic activity levels is the agents’
interests reconciliation in integrated systems with complementary demand. This
situation arises when conditionality of the buyer’s need in one commodity by
the fact of acquiring other goods. Agents, whose goods initiate the demand for
other agents’ goods, are characterized by predominant economic activity. The
complementary demand effect is often observed for specific goods. Under these
conditions, either of the agents or the agents’ association in the case of affiliation
may have the status further defined as “meta-agents”. Other agents may delegate
to the meta-agent the right to redistribute (transfer) the integration effect in the
system. The status “meta-agents” is realized in the form of possessing informa-
tion about the true utility functions or utility values of other agents, as well as
the right to choose the distribution mechanism of the system integration effect.
Therefore, the agents’ utilities in this case can be considered as transferable.
Transferable utility distribution algorithms were developed for systems in
which agents were anonymous, that is, they had equivalent criteria. In this case,
the priorities of the criteria were not taken into account when aggregating. In
particular, Pareto efficiency was justified [3, 10, 21] for an algorithm in which a
minimum between the agent’s optimum and the average undistributed system
utility was determined. If the agents’ criteria [5, 6, 19] have different priorities,
then the distribution algorithms [14, 15, 18] were reduced to a median multicri-
teria choice [4, 7, 16].This solution was not Pareto efficient in the general case,
however, mechanisms for anonymous symmetric coalitions [1,2] were Pareto effi-
cient in particular cases. The distribution mechanisms were constructed for the
utility distribution problem represented as a multicriteria choice problem: stud-
ies with additive aggregation of agents’ utilities were made for nontransferable
utilities [11, 13, 17] and transferable utility [12, 20].
Further we consider the distribution mechanism, optimal by the multiplica-
tive utility criterion, which under certain conditions leads to Nash equilibrium
(compatibility with stimuli) and Pareto efficiency. The problem is to develop
an algorithm for distributing the transferable utility between the agents of the
“retailer-bank-insurer” system [9] based [8] on the results obtained for anonymous
agents.
Thus, this article’s subject is the study of the utility distribution mecha-
nism from the standpoint of resistance to the agents’ strategic behavior and the
individual rationality conditions and Pareto efficiency.

2 Methods and materials

We introduce the following assumptions about the price’s functions and agents’
costs functions.
1. Agents operate in the monopolistic competition’s markets, which causes
the decreasing demand curves, simulated in the form of power functions (“inverse
functions of demand”),
b
pk(n) = ak(n) Qkk(n) , ak(n) > 0, bk(n) < 0, |bk(n) | < 1, k ∈ K, n = 0, 1, 2, (1)

where pk(n) is k-th agent’s goods price, ak(n) , bk(n) are the price function coeffi-
cients for n-th variant of the system organization (n = 0 is lack of integration,
n = 1 is the integrated system for Q∗−l = Q∗l , l ∈ K, n = 2 is the integrated
system for Q∗−l < Q∗l , l ∈ K); K is the agents’ set; |K| is the amount of elements
of set K; the meta agent is indicated by the symbol l ∈ K, the environment
is indicated by the «−l» symbol, the agent’s optimum is indicated by the «∗»
symbol.
2. Sales growth occurs with a constant return expansion, i.e. the agent’s
marginal costs are constant ck = const, k ∈ K, agents have risky costs ρk =
const, k ∈ K аnd integration costs uk = const, k ∈ K. Risk costs characterize
the share of the agent’s average proceeds probable losses from the goods price.
The ones for the retailer mean the banks overdue debt on loans for goods sold.
The ones for the bank mean the overdue debts on loans granted, which are
taken into account by the discount factor. The ones for the insurer mean the
payments on insurance cases, which take into account the probability of their
occurrence. Integration costs uk > 0, k ∈ K are interpreted as price discounts
or commissions in favor of agents with greater economic activity. If integration
costs uk < 0, k ∈ K, then they represent the income of a more active agent as
a transfer in the form of price premiums or commissions from other agents for
participating in an integrated system.
3. The system is characterized by complementary demand. We consider the
sales volumes of all agents expressed in one measure and assume that the volume
of the meta-agent’s sales through the coefficient of its demand function depends
on the volume of the environment’s sales in the following form (in this case, the
relationship between demands for environment goods is neglected):
(
∗ αlk(1) , n = 1,
al = αlk al0 Qk , k ∈ K\l, αlk = (2)
αlk(2) , n = 2,

where αlk is the coefficient of complementarity of the k-th and l-th goods,
αlk(1) > αlk(2) > 0 are constants; the complementarity effect is expressed in
the fact that for a retailer the growth in lending at a low interest rate (n = 1)
leads to faster growth in goods turnover, and at a high interest rate (n = 2) leads
to a slow growth in turnover. In other words, in the piecewise constant model
αlk , the property of growth of the complementarity effect is simplified with a
decrease in the complement’s price.
Based on these assumptions, we present models for the optimal actions choice
in the following form:

Q∗k = arg max πk (Qk ),
Q∈AQk
bk(n) +1
(3)
π (Q ) = p̄
k k k(n) Qk − ck Qk , k ∈ K,

where p̄k = ak − uk − ρk > 0, k ∈ K; πk (Qk ) is the function of the agent’s profit.
The criteria for the agents (3) are obviously strictly concave, twice continuously
differentiable, and the optima of the agents are finite, that is, the solutions of
problems (3) are internal.
We introduce the following utility distribution mechanism:
(
0 πkmax − µπ̄, k ∈ K1 , |K|
πk = max
µ= ≥ 1, (4)
πk , k ∈ M, |K| − |M |

where πk0 is the agents utility after distribution; M is the minority agents’ set for
which the profit unconditional maximum is lower than the average profit loss in
the system; |M | is the amount of elements of set M ; K1 is non-minority agents’
set; these sets are defined in the form

M = {k ∈ K : πkmax < π̄}, K1 = {k ∈ K\M : πkmax ≥ π̄}.

The notation (4) !
1 X X
π̄ = πkmax − πk∗ ≥0
κ
k∈K k∈K

is an average profit loss in the system; πk∗ = πk (Q∗k ) is conditional maximum of
agent’s profit by criterion

Q∗k = arg max πk (Qk ), k ∈ K\l, k
AQk\l = {Qk ∈ R+ , Qk ≤ Q∗l , k ∈ K\l},
Qk ∈AQk\l

which is found taking into account that the meta-agent has already chosen its
optimum; πkmax is the unconditional maximum of the agent’s profit according to
the criterion

πkmax = max πk (Qk ), k
AQk = {Qk ∈ R+ , k ∈ K}.
Qk ∈AQk

The problem is to analyze the influence of agent-type parameters ck , uk , ρk ,
the parameters of the markets under consideration ak , bk and the parameters of
the market’s relationship αlk on the resulting utility distribution (4).

3 Results
The necessary optimality condition written for the k-th agent has the following
form
0 b
πkQ k
= p̄k(n) (bk(n) + 1)Qkk(n) − ck = 0, k ∈ K. (5)
The optimum of the agent taking into account (5) for the agent’s optimum has
the form:
1/bk(n)
∗ ck
Qk = , k ∈ K, n = 0, 1, 2, (6)
p̄k(n) (bk(n) + 1)
which for n = 0 characterizes the individual optimum of the non-integrated
agent Q∗k(0) .
A sufficient maximum condition has the form
00 b −1
πkQ k
= p̄k(n) (bk(n) + 1)bk(n) Qkk(n) < 0, k ∈ K, n = 0, 1, 2. (7)

The analysis of condition (7) shows that, taking into account the restrictions
on the coefficients of the price functions (1), it is satisfied under condition when
p̄k(n) > 0, k ∈ K.
The algorithm for analyzing the influence of agent’s type parameters, market
parameters, and the parameters of the markets relationship to the resulting
utility distribution includes the following steps.
1. Input of market environment parameters ak , bk and cost parameters ck , uk ,
as well as the complementarity parameters αlk , the accuracy parameter of
the iterative process ε.
2. Setting the step number t = 1.
3. Entering agent messages about the type parameters ρkt .
4. The agents’ optimal sales volume calculation of Q∗kt by (6).
max
5. The agents’ profit functions calculation (profits πkt ) according to (3).
6. The determination of meta-agent by the condition of the maximum value of
the agents’ profit
lt = arg max Q∗kt .
k∈K

7. The system organization determination (the meta-agent’s domination n = 1
or the environment domination n = 2) by the rule

1, max Q−lt > Q∗lt ,
−l∈K
nt =
2, max Q−lt ≤ Q∗lt .
−l∈K

8. The demand function coefficient alt determination for the meta-agent prod-
uct according to (2).
9. The meta-agent conditional optimum calculation Q̄∗lt by (2) with the value
of alt found in step 8.
∗
10. The meta-agent’s conditional maximum profit calculation πlt = πl (Q̄∗lt )
by (3).
11. The calculation of profit in the integrated system according to the mecha-
0
nism (4) as a vector πkt , k ∈ K.
12. The termination condition verification: if ∀t > 1
X
0 0
|πkt − πkt−1 | ≤ ε, k ∈ K,
k∈K

then iteration ends.
0
13. The distribution analysis by agents πkt , k ∈ K.
14. The step number setting t = t + 1.
15. The agent’s messages about the type parameters ρkt and proceeding to step
4 entering.

4 The effect distribution simulation according to the
algorithm
The simulation is carried out on the basis of integrated system parameters sets,
indicated k = 1, 2, 3, respectively. The choice of the meta-agent is made ac-
cording to the maximum profit criterion. The prices are presented as inverse
demand functions (1). The situations of the meta-agent’s domination and the
environment’s domination are considered.
The initial data for the first variant of the simulated systems are presented
in Table 1.
Table 1. Parameters of the model and criteria vector

Parameter Element
of the
model 1 2 3
ak 5000.00 0.75 0.35
bk -0.06 -0.19 -0.20
ck 2250.00 0.06 0.03
uk -1000.00 0.03 0.02
Q∗k 214744.64 195682.87 176234.17
αlk 4.5 · 10−6

It is obvious from the table data that the environment domination situation
is realized.
The figures below show the agents’ profit and the total (Σ) profit graphics
depending on the declared risk of the agent.
The declared agents’ risk rate pk vary in steps in the range from 0 to 0.23
with step of 0.01. If the agent’s risk rate does not change, then it is set at the
base level.
In Fig. 1 π10 is retailer’s profit, after profit distribution in the integrated
system, reduced by 10,000 times, π20 is the bank’s distributed profit, π30 is the
insurer’s distributed profit in the integrated system.

Fig. 1. The agents’ distributed profit dynamics in the integrated system with the in-
crease in the declared risk rate of all agents simultaneously

Figure 1 shows that the bank’s and the insurer’s profits are incomparably less
than the retailer’s one. The system’s total profit is approximately equal to the
retailer’s profit. The retailer acts as a meta-agent and transfers part profit among
agents who lost the profit in the integration. Thus, in the integrated system
after the profits’ transferring the bank’s and the insurer’s profits are constant,
regardless of the change in the risk rate. Changes in the declared values of all
agents’ risk rates will only affect the retailer’s profit.
The effect of an increase in the individual declared agents’ risk on the meta-
agent profit is reflected in Fig. 2.

Fig. 2. The meta-agent’s distributed profit dynamics in the integrated system

0
In the Fig. 2 π1(1) is the retailer’s profit with the increase in the risk rate
0
declared by the retailer; π1(2) is the retailer’s profit with the increase in the risk
0
rate declared by the bank; π1(3) is the retailer’s profit with the increase in the
risk rate declared by the insurer; the profit’s amount in the integrated system is
reduced by 10,000 times.
When declared the risk rate is increased by one of the agents, the retailer’s
profit as a meta-agent and, accordingly, the system profit is reduced. The greatest
impact on the system profit is the change in the declared retailer risk rate.
Insignificant impacts on the system profit are the change in the declared bank’s
and insurer’s risk rates.
0
In Fig. 3 πΣ1 is the system total profit with an increase in the risk rate
0
declared by the retailer; πΣ2 is the system total profit with an increase in the
0
risk rate declared by the bank; πΣ3 is the system total profit with an increase in
the risk rate declared by the bank; the profit amount in the integrated system
is reduced by 10,000 times.
The system total profit shows the dynamics similar to the retailer’s profit.
The agents’ profit received at integration before transferring differs from the
declared profit (Fig. 4).
In Fig. 4 π1∗ is the retailer’s profit conditional maximum reduced by 10,000
times; π2∗ is the bank’s profit conditional maximum; π3∗ is the insurer’s profit
∗
conditional maximum; π2(0.03) is the bank’s profit conditional maximum with
∗
real risk rate (0.03); π3(0.02) is the insurer’s profit conditional maximum with
real risk rate (0.02).
In the integrated system the agents receive a part of the profit independently,
and the meta-agent then replenishes the amount to the value of the agents’ profit
Fig. 3. The integrated system profit dynamics when the declared one of the agents’
risk rate changes

Fig. 4. Agents’ profit dynamics in the integrated system

without integration. It is stimulus for the agents to overestimate the values of
the declared risk rates in order to obtain additional profit from the meta-agent.
Let’s consider a model similar to the previous one, but in the situation of the
meta-agent’s domination (Table 2).
A significant amount of retailer’s profit compared to other agents makes
inefficient change of the meta-agent in this model.
The meta-agent’s profit dynamics (Fig. 5) shows that an increase in the risk
rate declared by one of the agents leads to a stable decrease in the meta-agent
profit.
0
In Fig. 5 π1(1) is the retailer’s profit with the increase in the risk rate declared
0
by the retailer; π1(2) is the retailer’s profit with the increase in the risk rate
0
declared by the bank; π1(3) is the retailer’s profit with the increase in the risk
rate declared by the insurer; the profit’s amount in the integrated system is
reduced by 1,000 times.
Table 2. Parameters of the model and criteria vector

Parameter Element
of the
model 1 2 3
ak 5500.00 0.75 0.35
bk -0.06 -0.19 -0.20
ck 2520.00 0.06 0.03
uk -200.00 0.03 0.02
Q∗k 159037.68 195682.87 176234.17
αlk 4 · 10−6

Fig. 5. Meta-agent’s distributed profit dynamics in the integrated system

When the declared risk rate is increased by one agent, the retailer’s profit as
a meta-agent and, accordingly, the total profit is reduced. The greatest impact
on the system’s profit is the change in the declared retailer risk. Insignificant
impact on the total profit is the change in the bank’s and the insurer’s declared
risk rates.
The transferred profit dynamics in the system is the same for both the envi-
ronment’s domination and the meta-agent’s domination. Thus, there is no need
to demonstrate other similar graphics.

5 Conclusion
The algorithm of simulation of the agents’ interaction in a strongly coupled
system with a transferable utility is developed. The algorithm makes it possible
to simulate the states of the system for specified parameters. The optimizing
of the utility function taking into account the priorities leads to a distribution
based on the minimax principle (guaranteed result), therefore, determines the
Pareto efficient equilibrium.
It is shown that an increase in the any agents declared risk rate leads to a
decrease in the meta-agent’s profit in any case and does not change the transfer
profit of the agents themselves. Thus, the overestimation of the risk rates is
beneficial to all agents of the system in addition to the meta-agent.
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