A Study on Negotiation for Revealed Information with Decentralized

Asymmetric Multi-objective Constraint Optimization

Toshihiro Matsui

Nagoya Institute of Technology, Gokiso-cho Showa-ku Nagoya Aichi 466-8555, Japan

Keywords:

Privacy, Decentralized, Multi-objective, Asymmetric Constraint Optimization, Multiagent, Negotiation.

Abstract:

The control of revealed information from agents is an important issue in cooperative problem solving and

negotiation in multi-agent systems. Research in automated negotiation agents and distributed constraint opti-

mization problems address the privacy of agents. While several studies employ the secure computation that

completely inhibits to access the information in solution process, a part of the information is necessary for

selﬁsh agents to understand the situation of an agreement. In this study, we address a decentralized framework

where agents iteratively negotiate and gradually publish the information of their utilities that are employed

to determine a solution of a constraint optimization problem among the agents. A beneﬁt of the approach

based on constraint optimization problems is its ability of formalization for general problems. We represent

both problems to determine newly published information of utility values and to determine a solution based

on published utility values as decentralized asymmetric multi-objective constraint optimization problems. As

the ﬁrst study, we investigate the opportunities to design constraints that deﬁne simple strategies of agents

to control the utility values to be published. For the objectives of individual agents, we also investigate the

inﬂuence of several social welfare functions. We experimentally show the effect and inﬂuence of the heuristics

of different criteria to select the published information of agents.

1 INTRODUCTION

The control of revealed information from agents is an

important issue in cooperative problem solving and

negotiation in multi-agent systems. Research in au-

tomated negotiation agents (Kexing, 2011) and dis-

tributed constraint optimization problems (Yeoh and

Yokoo, 2012; Fioretto et al., 2018) address the pri-

vacy of agents. While several studies employ the se-

cure computation that completely inhibits to access

the information in solution process (L

eaut

e and Falt-

ings, 2013; Grinshpoun and Tassa, 2016; Tassa et al.,

2017; Tassa et al., 2019), a part of the information is

necessary for selﬁsh agents to understand the situation

of an agreement.

A class of asymmetric constraint optimization

problems (Grinshpoun et al., 2013) has been proposed

to represent the situation where agents have different

individual utility functions (Petcu et al., 2008; Mat-

sui et al., 2018a; Matsui et al., 2018b). This prob-

lem can be extended as the basis of a negotiation

framework that publishes/reveals selected partial in-

formation of constraints and solves the problems with

the published information. A beneﬁt of the approach

based on constraint optimization problems is its abil-

ity of formalization for general problems.

In a previous work (Matsui, 2019), a solution

framework consisting of asymmetric constraint opti-

mization problems with the publication of private util-

ity values and a centralized local search method as

a mediator has been presented. With the approach,

agents gradually reveal their information until they

agree on the ﬁrst globally consistent solution. While

the agents locally select utility values to be published

by referring the information from the mediator, pub-

lished utility values of individual agents are globally

aggregated with the traditional summation operator in

the optimization process based on the published infor-

mation.

However, there are opportunities to design de-

centralized framework without the central node. In

addition, for the problems that consider the utilities

and the cost values of published information for indi-

vidual agents, criteria of multiple objectives should

be employed. For this class of negotiation, there

are opportunities to apply the decentralized constraint

optimization with multiple objectives for individual

agents (Matsui et al., 2018a).

Matsui, T.

A Study on Negotiation for Revealed Information with Decentralized Asymmetric Multi-objective Constraint Optimization.

DOI: 10.5220/0010349601490159

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 1, pages 149-159

ISBN: 978-989-758-484-8

149

i,j

j,i

i,j

j,i

0 0 0 0

...

... ... ... ...

0 0 0 0

variable of agent a

asymmetric constraint

local valuation of a

Figure 1: Asymmetric multi-objective DCOP with publica-

tion cost.

For the problem settings similar to the previous study,

we address a decentralized framework where agents

iteratively negotiate and gradually publish the infor-

mation of their utilities that are employed to deter-

mine a solution of a constraint optimization problem

among the agent. In the proposed framework, util-

ity values in constraints for agents are incrementally

published to restrict revealed information, and prob-

lems with only published information are solved until

the agreement or termination condition of the negoti-

ation.

We represent both problems to determine newly

published information of utility values and to deter-

mine a solution based on published utility values as

decentralized asymmetric multi-objective constraint

optimization problems. As the ﬁrst study, we inves-

tigate the opportunities to design constraints that de-

ﬁne simple strategies of agents to control the utility

values to be published. For the objectives of individ-

ual agents, we also investigate the inﬂuence of several

social welfare functions.

Our contribution is the fundamental design of

the negotiation framework that applies a decentral-

ized constraint optimization approach with criteria for

multiple objectives among individual agents, and in-

vestigation of composite constraints of fundamental

strategies to determine newly published utility values.

Our experimental results demonstrate the inﬂuence

and effect of criteria to select published information

of constraint.

The rest of the paper is organized as follows. In

the next section, we present preliminaries of our study

including fundamental problem deﬁnitions, criteria of

optimization, and basis of solution methods by refer-

ring the previous study. Then we present a framework

with complete decentralized constraint optimization

methods that iterates rounds of negotiation process to

gradually publish utility values considering publica-

tion cost in Section 3. We experimentally investigate

the proposed approach in Section 4. We address dis-

cussions in Section 5 and conclude our study in Sec-

tion 6.

aggregation of

objectives

solution

selection of

newly published utility values

optimal solution for

published utility values

negotiation round

Figure 2: A negotiation framework.

2 PRELIMINARY

2.1 Asymmetric Constraint

Optimization Problem with

Publication of Private Information

In a previous study, a fundamental solution frame-

work based on an asymmetric constraint optimization

problem with publication of constraints and a cen-

tral processing that performs a local search has been

proposed (Matsui, 2019). Here, we address a similar

problem with a decentralized solution framework.

In an asymmetric constraint optimization problem

with publication of constraints, agents select the pub-

licity of each utility value in the constraints by con-

sidering their privacy cost values. This problem is

deﬁned by hA,X,D,C,P,X

i. Here, A is a set of

agents, X is a set of variables, D is a set of the do-

mains of variables, and C is a set of constraints that

deﬁne the utility values for the assignments to several

variables in X.

Agent a

∈ A has variable x

∈ X and unary con-

straint u

∈ C for the variable. For several pairs of

variables x

and x

, asymmetric binary constraints

i, j

j,i

∈ C are deﬁned. It is assumed that each agent

knows the domains of peer agents related to its asym-

metric binary constraints. Agent a

has u

i, j

, and a

has u

j,i

. Each agent aggregates its own utility value

for unary and binary constraints that are related to the

agent. The utility values are deﬁned as u

: D

→ N

and u

i, j

: D

× D

→ N

. The functions are also de-

noted by u

(d) and u

i, j

P is a set of privacy cost values for utility val-

ues in constraints. p

∈ P corresponds to u

, and

i, j

∈ P corresponds to u

i, j

. The privacy cost values

are individually evaluated by each agent to select its

related utility value to be published. The cost val-

ues of constraints are deﬁned as p

: D

→ N

and

i, j

: D

× D

→ N

. The functions are also denoted

by p

(d) and p

i, j

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

150

An example of directed constraint graph and re-

lated function tables for above basic parts of a prob-

lem is shown in Figure 1.

and D

are a set of binary variables and a set

of the domains of the variables that represent the pub-

lication of the utility values, respectively. When a

utility value in a constraint is published, its corre-

sponding variable takes 1. Otherwise, the variable

takes 0. Variable x

i,(d)

∈ X

corresponds to utility

value u

(d) in unary constraint u

. Similarly, variable

i, j,(d,d

)

∈ X

corresponds to utility value u

i, j

(d, d

)

of assignments d ∈ D

and d

∈ D

in asymmetric bi-

nary constraint u

i, j

. If at least one of x

i, j,(d,d

)

takes 1,

i,(d)

is set to 1.

For each variable x

∈ X , a value d

∈ D

repre-

sents a special case where an agent does not coop-

erate with any other agents. Value d

is selected by

several agents when any solution cannot be globally

evaluated with utility functions whose utility values

have been partially published. For assignment d

a variable x

, the corresponding utility value u

)

and privacy cost value p

) take zero. Similarly,

i, j

,d), u

i, j

(d, d

), and corresponding privacy cost

values take zero. Value d

is given as a common

value in the domains of all variables, and correspond-

ing zero utility/privacy-cost values are also deﬁned as

common knowledge. Unpublished utility values re-

lated to assignment d

are deﬁned as −∞. The evalua-

tions v

(d) and v

i, j

(d, d

) for a unary constraint and an

asymmetric binary constraint are represented by the

followings.

(d) =



(d) if x

i,(d)

= 1

−∞ otherwise

(1)

i, j

(d, d

) =



i, j

(d, d

) if x

i, j,(d,d

)

= 1

−∞ otherwise

(2)

With an assignment to related variables in X

, the lo-

cal valuation f

) for agent a

and an assignment

to related variables in X is deﬁned as the summa-

tion of the evaluations for its own unary constraint

and asymmetric binary constraints for neighborhood

agents in Nbr

that are related to i by constraints:

) = v

) +

∑

j∈Nbr

i, j

) , (3)

where d

and d

are the value of x

and x

in assign-

ment D

The goal the of problem is the maximization of the

globally aggregated valuations for all f

maximize ⊕

∈A

) (4)

While a typical aggregation operator of the valuations

is the summation of the values, different operators

can be applied in the case of multi-objective problems

among individual utilities of agents as shown in Sec-

tion 2.2.

To avoid the utility value of −∞ shown in Equa-

tions (1) and (2), value d

can be assigned to several

agents’ variables if any other solutions cannot be eval-

uated with published part of utility functions. How-

ever, such a situation will also be avoided, since the

utility value corresponds to value d

is zero.

The agent can partially publish the utility values

of its own utility functions. The total published cost

of agent a

is deﬁned as follows.

∑

d∈D

s.t. x

i,(d)

(d) + (5)

∑

j∈Nbr

,d∈D

∈D

s.t. x

i, j,(d,d

)

i, j

(d, d

)

A problem is to determine the utility values to be

published within a budget. We consider a scheme of

negotiation process where agents iteratively publish

their utility values based on situations in each round

of the process.

2.2 Social Welfare Criteria

When we consider objective of individual agents, sev-

eral operators ⊕ and corresponding criteria to aggre-

gate/compare the objectives as shown in Equation (4).

For multi-objective problems, several types of social

welfare (Sen, 1997) and scalarization methods (Mar-

ler and Arora, 2004) are employed to handle the ob-

jectives. In addition to the traditional summation op-

erator and the comparison on scalar values, we em-

ploy leximin and maximin with a tie-break by a sum-

mation value, that are also employed in a solution

method (Matsui et al., 2018a). Since we employ a

solution method in the previous study to solve our

problem, we also inherit these operators and crite-

ria. While those operators and criteria are designed

for maximization problems of utilities, those can be

easily modiﬁed for minimization problems.

Summation

∑

∈A

only consider the total util-

ities. While maximin maxmin

∈A

improves the

worst case utility value, it does not consider the whole

utilities, and is not Pareto optimal. Therefore, ties

are additionally broken by comparing summation val-

ues. Leximin can be considered as an extension of

maximin where the objective values are represented

as a vector whose values are sorted in ascending or-

der, and the comparison of two vectors is based on a

dictionary order of values in the vectors. The maxi-

mization with leximin is Pareto optimal and relatively

improves fairness among objectives. See the litera-

A Study on Negotiation for Revealed Information with Decentralized Asymmetric Multi-objective Constraint Optimization

151

ture for details (Sen, 1997; Marler and Arora, 2004;

Matsui et al., 2018a).

In addition, to evaluate the fairness among agents

in a resulting solution, we also employ the Theil index

that is deﬁned as an inverted value of entropy. T =

∑

i=1





, where x denotes the average value

for all x

. T takes zero if all x

are equal.

2.3 Decentralized Dynamic

Programming Method for

Asymmetric Multi-objective

Constraint Optimization Problems

A class of asymmetric constraint optimization prob-

lems with individual objectives of agents and decen-

tralized complete solution methods have been pre-

sented (Matsui et al., 2018a). The goal of this study

was the application of a social welfare criteria called

leximin to pseudo-tree-based solution methods to im-

prove the fairness among individual agents. As men-

tioned above section, with the leximin, individual ob-

jectives of agents are aggregated and optimized con-

sidering the worst case and fairness among the ob-

jectives. In addition to the leximin, several social

welfare criteria including traditional summation were

compared.

The solution methods are extended versions of dy-

namic programming and tree-search based on pseudo-

trees of constraint graphs. A pseudo tree is a struc-

ture on a constraint graph that is employed to decom-

pose problems. A pseudo tree is based on a depth-

ﬁrst search tree on a constraint graph, and optimiza-

tion methods are performed along the spanning tree

with back edges that are not included in the edges

of the spanning tree. For asymmetric problems, the

pseudo-tree and solution methods have been modiﬁed

so that both two agents that relate each other by a pair

of asymmetric constraints can evaluate assignments to

opposite agent’s variable. In part of Figure 2, simple

examples of the modiﬁed pseudo trees are illustrated.

See the literature for details (Matsui et al., 2018a).

In this study, we employ a variant of pseudo-tree-

based dynamic programming method DPOP (Petcu

and Faltings, 2005) that has been extended in the pre-

vious study (Matsui et al., 2018a), because it is rel-

atively simple as a basis of our extension. Note that

its scalability is limited for complex problems. When

problems are densely constrained, the tree width of

pseudo trees that corresponds to the number of com-

binations of variables in partial problems for agents

grows and cause a combinational explosion. How-

ever, for the fundamental investigation, we concen-

trate on the problems that can be addressed with this

class of complete solution methods. We develop two

variants of the solution method that are adjusted for

our problem deﬁnitions.

3 A DECENTRALIZED

SOLUTION FRAMEWORK

THAT ALTERNATES

SELECTING PUBLISHED

UTILITY VALUES AND

SOLVING PUBLISHED

PROBLEMS

3.1 Basic Framework

We propose a framework that alternates two opti-

mization methods to select utility values to be pub-

lished and to solve the current problem with published

utilities. For both optimization methods, we em-

ploy a decentralized dynamic programming method

based on pseudo trees as mentioned above. For sim-

plicity, two methods employ a common pseudo tree.

For the ﬁrst investigation, we focus on the process

that incrementally publishes the utility values of con-

straints/functions. We assume that arbitrary condi-

tions and parameters to control and stop the publica-

tion process can be deﬁned.

The framework iterates rounds of negotiation

among agents and utility values are incrementally

published. In each round, the utility values that

are published for the next problem are determined,

and the next problem with published utility values is

solved.

Similar to the previous methods, in the optimiza-

tion for the problem with published utility values, the

total utility for all agents is maximized.

3.2 Selection of Utility Values to be

Published

In the phase of selecting utility values to be published,

agents iteratively publish a part of utility values of

their own constraints in each round of negotiation. As

deﬁned in Section 2.1, the decision variables of agent

in this problem are originally x

i, j

∈ X

for Nbr

In the initial state, all the variables in X

are initial-

ized by 0, and they are set to 1 when their correspond-

ing utility values are published. However, the solution

space for the variables in X

is too huge to explore for

the complete solution method that is employed in this

study. Therefore, we deﬁne another problem for this

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

152

negotiation to restrict solutions in the original solution

space.

The problem is deﬁned by hA,X

,D,C

i. Here,

we introduce new decision variables X

. For agent

, a decision variable x

∈ X

is deﬁned. Variable x

takes a value from D

∈ D that is the same as a part of

decision variables in the original problem. Assigning

d and d

to x

and x

represents that the utility values

(d) and u

i, j

(d, d

) are published in the next round of

negotiation. When x

takes the special variable value

, it means that no utility values relating x

are newly

published. In addition, if a utility value corresponding

to d and d

have been published, the published utility

value does not change the current situation. In the

end of each round, several original variables in X

are set to 1 by translating the solution for X

. If an

assignment to a variable x

∈ X

or x

i, j

∈ X

changed

from 0 to 1, its corresponding utility value is newly

published.

The structure of this problem determining the util-

ity values to be published for the next problem resem-

bles the maximization problem with published utility

values. The variables and their domains are identical

to the maximization problem for utility values. How-

ever, it is deﬁned as a minimization problem with cost

values for published utility values. The constraints

∈ C

are asymmetrically deﬁned for each agent a

Here, cost values related to a constraint can take real

number to represent ratio values. Several cost values

for the constraints can be deﬁned based on different

publication strategies, and the cost values can be inte-

grated as vectors with a structure. We investigate fun-

damental strategies as shown in Section 3.3 and 3.4.

3.3 Criteria to Evaluate Newly

Published Utility Values

The following criteria evaluate the situation of newly

published utility values that are represented by an as-

signment to variables in X

. We assume that, some

information about status of published utility values is

available in the evaluation. The information can be

collected by simple additional protocols.

3.3.1 R) Degree of Privacy Cost for Published

Utility Values

Ratio of revealed privacy cost: we assume that the in-

formation of publish cost values can be employed in

the negotiation of agents by modifying them to ratio

values of locally aggregated cost values so that the

ratio values only represent a normalized degree of un-

satisfactory. For each agent, the ratio of privacy cost

values between the total values for published utilities

and total values for all constraints is deﬁned as a cri-

terion.

) =

i,(d

)

∗ p

) +

∑

j∈Nbr

i, j,(d

)

∗ p

i, j

))/

(

∑

d∈D

(d) +

∑

j∈Nbr

,d∈D

∈D

i, j

(d, d

)) , (6)

where, d

is the value of x

in assignment D

. Values

of variables x

i,(d)

and x

i, j,(d,d

)

are equal to x

i,(d)

and

i, j,(d,d

)

respectively if x

and x

does not represent

the publication of corresponding utility values. Oth-

erwise, x

i,(d)

and x

i, j,(d,d

)

are set to 1.

Since agents might relate different numbers of

constraints and utility values, the above ratio value is

also considered.

3.3.2 A) Degree of Unpublished Utility Values

Agreement opportunity: for each value d of each

agent’s variable x

, the number of unpublished utility

values that are related to the variable’s value is con-

sidered as a criterion. For x

= d

) =

2|{(d

)| j ∈ Nbr

∈ D

= d

∨ d

= d

}| +

|{x

i, j,(d

)

| j ∈ Nbr

∈ D

6= d

∧ d

6= d

i, j,(d

)

= 0}| (7)

With this criterion, we consider that when the number

of unpublished utilities that relates the assignment for

newly published utility values is relatively large, the

opportunity of agreement among agents with the as-

signment is relatively small. For the assignment of d

that represents no publication for a pair of variables’

values, the count of an unpublished utility for the as-

signment is doubled for emphasis.

3.3.3 U) Degree of Progress in Publication

Process

Utility publication progress: since, the cost values for

revealed information restrict the publish of utility val-

ues, a counter part of the cost values is necessary to

continue publication process. Here, we employ a ratio

of published utility values.

) = |{x

i, j,(d

)

| j ∈ Nbr

i, j,(d

)

= 1}/

∑

j∈Nbr

| × |D

| (8)

As addressed below, this cost value is combined with

other criteria so that it has a higher priority than oth-

ers.

A Study on Negotiation for Revealed Information with Decentralized Asymmetric Multi-objective Constraint Optimization

153

3.3.4 T) A Measurement of Trade-off between

Publication Cost and Utility

Trade-off measurement: above criteria do not em-

ploy the information of utility value. To consider the

expected gain of utility by publishing utility values,

the information of utility values should be consid-

ered. However, originally the utility values should be

hidden until their publication. As abstract informa-

tion, we employ the ratio of trade-off values which

are locally aggregated as difference values between

expected utility values and cost values of published

utility values.

Since the optimal utility value is unknown, each

agent employs rough upper and lower bound values

of trade-off.

trd

)

= u

∑

j∈Nbr

i, j

)−rvl

) (9)

trd

)

⊥

= u

) − rvl

) (10)

rvl

) =

i,(d

)

∗ p

) +

∑

j∈Nbr

i, j,(d

)

∗ p

i, j

)

(11)

While arbitrary estimation values between the bounds

can be employed, we investigate simple cases for

the fundamental analysis. The estimation trade-off

value trd

) is set to trd

)

⊥

, (trd

)

trd

)

⊥

)/2 or trd

)

. Then the ratio of the

trade-off value is employed as a criterion.

) = 1−(trd

)−trd

⊥

)/(trd

−trd

⊥

) (12)

trd

= u

) +

∑

j∈Nbr

max

∈D

i, j

) (13)

trd

⊥

= −(

∑

d∈D

(d) +

∑

j∈Nbr

,d∈D

∈D

i, j

(d, d

))

(14)

3.3.5 S) Degree of Newly Published Utility

Values

Switch of trade-off measurement: the above cost val-

ues except U need the energy to continue publication

process. We modiﬁed the previous criteria T to con-

ditionally continue publication process. The modiﬁed

cost c

) is deﬁned as follows. First, we evaluate

the cost value for newly published utility values

rvl

new

) = (x

i,(d

)

− x

i,(d

)

) ∗ p

) +

∑

j∈Nbr

i, j,(d

)

− x

i, j,(d

)

) ∗ p

i, j

)

(15)

If rvl

new

= 0, then c

) = 0. Otherwise, if

trd

) > 0,

) = −(trd

) − trd

⊥

)/(trd

−trd

⊥

) . (16)

In other cases, c

) = c

With this criterion, the publication is preferred by

a negative cost value when trd

) > 0.

3.3.6 H) Hard Constraint for Termination

Hard constraint for termination: we also introduce a

cost value c

) of hard constraint for the assign-

ments that should be avoided. The value of c

) is

1 or 0 that represents violation or satisfaction. This

cost value should have the ﬁrst priority. Here, we

employ c

) for a termination condition based on

above trade-off value. c

) = 1 if rvl

new

) >

0 ∧ trd

) < 0. Otherwise, c

) = 0. This con-

straint inhibits the publication with negative trade-off

value where an estimation utility value is less than the

total publish cost.

3.4 Integrating Criteria for Published

Utility Values

The above criteria can be combined as hierarchically

structured cost vectors, and the optimization method

solves a minimization problem on the vectors to de-

termine the utility values to be newly published. Hard

constraint H is most prioritized. Since criteria R, A,

T with higher priorities will cause earlier convergence

without publication of sufﬁcient utility values, cost U

should be secondary prioritized. S can be used with-

out U. T and S consider a trade-off between an esti-

mation utility and to total publish cost, while R and

A does not evaluate utility values. Considering these

properties, we investigate the following combinations

of criteria.

• S: c

 c

• UAR: c

 c

• URA: c

 c

• UT: c

 c

Here, any aggregated cost values do not exceed a cost

value with a higher priority.

For two hierarchically structured cost vectors v

v =

,··· ,v

] and v

= [v

,··· ,v

], the aggregation of the

vectors is deﬁned as v

v ⊕ v

= [v

⊕ v

,··· ,v

⊕ v

The comparison v

v < v

is deﬁned as ∃t, ∀t

< t,v

∧ v

< v

We also investigate the cases where the ele-

ments of the vectors are aggregated and compared

with several criteria for multiple objectives for in-

dividual agents. Here, the lexicographic augmented

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

154

weighted Tchebycheff function (Marler and Arora,

2004) where ties of maximum values are broken by

summation values, and leximax that is a modiﬁed lex-

imin for minimization problems are applied in addi-

tion to traditional summation.

3.5 Solution with Published Utility

Values

In each round of the negotiation to determine newly

published utility value, agents can solve a problem

with currently published utility values as shown in

Section 2.1. While this result can be employed as a

feedback to agents’ strategies, we only evaluate the

anytime property for rounds of negotiation as the ﬁrst

study.

4 EVALUATION

4.1 Settings

We empirically evaluated the proposed approach.

Due to the limitation of dynamic programming

on a pseudo-tree, we solved relatively small scale

and sparse problems. A problem instance has

n variables/agents/unary-constraints and c pairs of

asymmetric binary constraints. For a pair of vari-

ables/agents, a pair of asymmetric constraints were

deﬁned. The size of variables’ domain including d

was commonly set to four. Utility values of con-

straints were set to random integer values in [10,50]

based on uniform distribution. Privacy cost values

were set as follows.

• equ: A privacy cost value for a utility value u is

an integer values bmax(1,u/10)c so that there is a

positive correlation.

• inv: A privacy cost value for a utility value u is an

integer values bmax(1,(50 − u + 1)/10)c so that

there is a negative correlation.

• rnd: Random integer values in [1,5] based on uni-

form distribution.

As shown in Section 3.4, we compared the inﬂu-

ence of several combinations of criteria for selecting

published utility values that are denoted by S, UAR,

URA and UT. In addition, as shown in Section 3.3.4,

the estimated trade-off values for the criterion T, S,

and H is set to the minimum/average/maximum val-

ues of lower and upper bounds that are denoted by

tmin/tave/tmax.

We also applied different aggregation/comparison

operators of objectives, including summation, min-

imum/minimum value with tie-break by summa-

tion, and leximin/leximax, on utility/privacy-cost val-

ues/vectors.

• sum: Summation for both minimization problems

determining published utility values and maxi-

mization problems of utilities under published

utility values.

• ms: The maximum/minimum value with a

tie-break by summation for the minimiza-

tion/maximization problems for publica-

tion/utility.

• lxm: Leximax/leximin for the minimiza-

tion/maximization problems for publica-

tion/utility.

• summs: Summation value for the minimization

problems for publication and the minimum value

for the maximization problem for utility.

• sumlxm: Summation value for the minimization

problems for publication and leximin for the max-

imization problem for utility.

Here, we assumed a simple termination condition

where an accumulated publish cost exceeds estimated

utility for next publication. In this case, each agent

does not select the corresponding assignment. When

all agents do not select new publication, the negotia-

tion process terminates.

For each setting, results are averaged over ten in-

stances.

4.2 Results

Figures 3-6 show typical anytime-curves of ‘utility’

and ‘trade-off’. Each curve corresponds to an agent.

While the utility values almost converge in earlier

rounds, trade-off values decrease until a termination.

Therefore, an issue is the selection of published util-

ity values within a budget, and we investigate the such

opportunities. In these problem settings, the trade-

off values are better in earlier steps for most agents

due to the scale of accumulated publication cost val-

ues are relatively greater than that of utility values.

This situation basically depends on publication cost

parameters of the problem settings, and by simply set-

ting smaller cost values, the peaks will move to later

rounds. Here, we do not focus on this issue because

the parameter design and more sophisticated strate-

gies of agents to handle such peaks will be included

our future work based on this investigation.

Table 1 shows the results in the ﬁnal round of pub-

lication process in the case of n = 10 variables/agents,

the number of pairs of asymmetric binary constraints

c = 20, equ, and sum. Here, the following results are

A Study on Negotiation for Revealed Information with Decentralized Asymmetric Multi-objective Constraint Optimization

155

Table 1: Inﬂuence of priorities on criteria for utility values to be published (n = 10, c = 20, equ, sum).

alg. term. utiliy trade-off ratio. rvl. num. ratio. rvl. cost

round sum. min. ave. max min. ave. max. min. ave. max. min. ave. max.

sum, S, tmin 6.7 1620.9 97.4 162.1 225.0 76.9 134.1 194.2 0.181 0.291 0.429 0.178 0.292 0.435

sum, S, tave 21.2 1763.2 110.2 176.3 248.8 46.7 88.3 127.9 0.763 0.856 0.934 0.805 0.884 0.961

sum, S, tmax 21.8 1773.1 103.0 177.3 254.3 30.9 77.3 115.4 1 1 1 1 1 1

sum, UAR, tmin 6.1 1560.6 95.6 156.1 223.4 72.8 124.2 187.3 0.245 0.357 0.482 0.211 0.330 0.478

sum, UAR, tave 13.9 1770.1 106.3 177.0 251.9 39.2 84.4 123.0 0.859 0.929 0.985 0.854 0.929 0.988

sum, UAR, tmax 11.9 1773.1 103.0 177.3 254.3 30.9 77.3 115.4 1 1 1 1 1 1

sum, URA, tmin 6.0 1526.6 90.2 152.7 219.0 64.7 121.1 181.9 0.248 0.359 0.488 0.212 0.328 0.472

sum, URA, tave 14.8 1762.1 107.1 176.2 253.1 38.5 83.0 124.3 0.881 0.938 0.991 0.865 0.934 0.987

sum, URA, tmax 13.7 1773.1 103.0 177.3 254.3 30.9 77.3 115.4 1 1 1 1 1 1

sum, UT, tmin 5.5 1598.4 97.6 159.8 225.4 76.2 130.5 190.5 0.193 0.307 0.438 0.191 0.305 0.453

sum, UT, tave 16.3 1764.2 103.2 176.4 251.9 41.3 86.8 129.4 0.754 0.873 0.950 0.800 0.898 0.964

sum, UT, tmax 13.4 1773.1 103.0 177.3 254.3 30.9 77.3 115.4 1 1 1 1 1 1

Table 2: Inﬂuence of priorities on criteria for utility values to be published (n = 10, c = 20, equ, S).

alg. term. utiliy trade-off ratio. rvl. num. ratio. rvl. cost

round sum. min. ave. max theil min. ave. max. theil min. ave. max. theil min. ave. max. theil

sum, S, tmin 6.7 1620.9 97.4 162.1 225 0.032 76.9 134.1 194.2 0.039 0.181 0.291 0.429 0.032 0.178 0.292 0.435 0.035

sum, S, tave 21.2 1763.2 110.2 176.3 248.8 0.031 46.7 88.3 127.9 0.041 0.763 0.856 0.934 0.002 0.805 0.884 0.961 0.002

sum, S, tmax 21.8 1773.1 103 177.3 254.3 0.035 30.9 77.3 115.4 0.061 1 1 1 0 1 1 1 0

ms, S, tmin 6.6 1617.9 110.5 161.8 226.9 0.026 86.9 133.3 194.5 0.031 0.185 0.298 0.434 0.034 0.184 0.299 0.457 0.037

ms, S, tave 20.7 1704.4 126.5 170.4 231.3 0.019 56.9 82.2 118.3 0.025 0.755 0.853 0.940 0.002 0.807 0.885 0.955 0.001

ms, S, tmax 19.7 1710.2 126.6 171.0 232.5 0.020 44.9 71.0 102.8 0.033 1 1 1 0 1 1 1 0

lxm, S, tmin 6.5 1600.3 109.2 160.0 218.9 0.025 84.4 131.5 185.8 0.031 0.183 0.295 0.439 0.036 0.181 0.301 0.471 0.040

lxm, S, tave 20 1657.2 125.4 165.7 224 0.020 51.5 78.1 113.1 0.032 0.741 0.849 0.935 0.002 0.784 0.879 0.957 0.002

lxm, S, tmax 19.8 1653.8 126.6 165.4 219.4 0.015 38.6 65.4 90.3 0.033 1 1 1 0 1 1 1 0

Table 3: Inﬂuence of priorities on criteria for utility values to be published (n = 10, c = 20, equ, S).

alg. term. utiliy trade-off ratio. rvl. num. ratio. rvl. cost

round sum. min. ave. max theil min. ave. max. theil min. ave. max. theil min. ave. max. theil

sum, S, tmin 6.7 1620.9 97.4 162.1 225 0.032 76.9 134.1 194.2 0.039 0.181 0.291 0.429 0.032 0.178 0.292 0.435 0.035

sum, S, tave 21.2 1763.2 110.2 176.3 248.8 0.031 46.7 88.3 127.9 0.041 0.763 0.856 0.934 0.002 0.805 0.884 0.961 0.002

sum, S, tmax 21.8 1773.1 103 177.3 254.3 0.035 30.9 77.3 115.4 0.061 1 1 1 0 1 1 1 0

summs, S, tmin 6.7 1576.5 99.8 157.7 216.4 0.029 79.1 129.6 184.4 0.036 0.181 0.291 0.429 0.032 0.178 0.292 0.435 0.035

summs, S, tave 21.2 1697.5 126.6 169.8 227 0.019 54.7 81.7 114.3 0.026 0.763 0.856 0.934 0.002 0.805 0.884 0.961 0.002

summs, S, tmax 21.8 1710.2 126.6 171.0 232.5 0.020 44.9 71.0 102.8 0.033 1 1 1 0 1 1 1 0

sumlxm, S, tmin 6.7 1573.4 99.8 157.3 214.6 0.028 79.1 129.3 182.6 0.034 0.181 0.291 0.429 0.032 0.178 0.292 0.435 0.035

sumlxm, S, tave 21.2 1673.8 126.6 167.4 216.8 0.016 53.7 79.3 106.9 0.022 0.763 0.856 0.934 0.002 0.805 0.884 0.961 0.002

sumlxm, S, tmax 21.8 1653.8 126.6 165.4 219.4 0.015 38.6 65.4 90.3 0.033 1 1 1 0 1 1 1 0

Table 4: Inﬂuence of priorities on criteria for utility values to be published (n = 10, c = 20, inv, S).

alg. term. utiliy trade-off ratio. rvl. num. ratio. rvl. cost

round sum. min. ave. max theil min. ave. max. theil min. ave. max. theil min. ave. max. theil

sum, S, tmin 8.6 1703.0 107.2 170.3 235.4 0.029 87.1 145.6 205.4 0.034 0.218 0.381 0.540 0.037 0.206 0.356 0.541 0.041

sum, S, tave 22.6 1773.1 103 177.3 254.3 0.035 50.8 105.4 159.6 0.052 0.962 0.990 1 0.0001 0.940 0.986 1 0.0003

sum, S, tmax 21.8 1773.1 103 177.3 254.3 0.035 49.2 104.3 159 0.054 1 1 1 0 1 1 1 0

ms, S, tmin 8.5 1668.5 114.5 166.9 233.2 0.027 94.4 141.7 202.5 0.031 0.217 0.387 0.546 0.036 0.208 0.363 0.548 0.040

ms, S, tave 21.7 1711.7 126.6 171.2 232.5 0.020 69.2 99.2 142.1 0.024 0.963 0.991 1 0.0001 0.942 0.987 1 0.0003

ms, S, tmax 19.6 1710.2 126.6 171.0 232.5 0.020 68.1 98.0 141.3 0.025 1 1 1 0 1 1 1 0

lxm, S, tmin 8.3 1656.8 116.3 165.7 228.7 0.023 96 140.9 199.5 0.027 0.216 0.386 0.556 0.037 0.206 0.357 0.537 0.039

lxm, S, tave 21.4 1653.8 126.6 165.4 219.4 0.015 64 93.4 129.5 0.021 0.965 0.990 1 0.0001 0.945 0.987 1 0.0003

lxm, S, tmax 19.8 1653.8 126.6 165.4 219.4 0.015 63.3 92.4 128.7 0.022 1 1 1 0 1 1 1 0

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

156

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10111213141516171819

utility

round

(Each curve corresponds to an agent.)

Figure 3: Utility of agents (n = 10, c = 20, equ, S, lxm).

100

150

200

250

300

1 2 3 4 5 6 7 8 9 1011 1213141516171819

trade

-off (utlity

-cost)

round

Figure 4: Trade-off of agents (n = 10, c = 20, equ, S, lxm).

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

utility

round

Figure 5: Utility of agents (n = 10, c = 20, equ, URA, lxm)

100

150

200

250

300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

trade

-off (utlity

-cost)

round

Figure 6: Trade-off of agents (n = 10, c = 20, equ, URA,

lxm).

evaluated. The minimum/average/maximum value is

evaluated for agents.

• utility: The optimal utility value of the problem

with published utility values.

• trade-off: The actual trade-off values that is the

difference between ’utility’ and the total privacy

cost value for published utility values.

• ratio. rvl. num.: The ratio of revealed number of

utility values.

• ratio. rvl. cost.: The ratio of total privacy cost

values for revealed utility values.

The result shows that the publication process contin-

ued until all the utility values are published when the

case of estimation trade-off tmax, since this estima-

tion value is too optimistic. In this problem setting,

earlier termination of publication process is relatively

better to save the privacy cost obtaining some utility.

‘Trade-off’ values of criteria S and UT that consider

trade-off between the estimation utility and the total

cost of published utility value are relatively greater

than that of other criteria in the case of tmin that ter-

minates in relatively earlier rounds. In the case of

tmin, ‘utiliy’ values of UAR is relatively better than

URA, since it considers the opportunities of aggrega-

tion. Basically, U enforces publication and dominates

other criteria. As a result, the total publication cost is

relatively greater than S.

Table 2 shows the results in the ﬁnal round of pub-

lication process in the case of n = 10, c = 20, equ,

and S. Here, the Theil index that is a measurement of

in equality among agents is also evaluated. When all

agents have the same value, the Theil index value is

zero. Note that there is an inherent trade-off between

the summation/average value and fairness, while it is

often preferred for selﬁsh agents without other mech-

anisms to trade their proﬁts. The result shows that the

Theil index values of ‘trade-off‘ in ms and lxm are

relatively smaller than that of sum, since those crite-

ria consider the improvement of the worst case. How-

ever, lxm did not overcome ms although lxm consider

the inequality, it reveals that the difﬁculty to design

appropriate estimate trade-off value to be well opti-

mized. Table 3 shows the results of the same problem

settings, while the optimization criteria for publica-

tion process is sum. Since the publication process is

the same, the result shows that lxm is the fairest crite-

rion for ‘utility’ and it affects ‘trade-off’.

Table 4 shows the results in the ﬁnal round of pub-

lication process in the case of n = 10, c = 20, inv, and

S. In this case, the Theil index of lxm for ‘trade-off’

was relatively better.

Table 5 shows the results in the case of n = 10, c =

20, rnd, and S. Due to the problem settings, ‘trade-off’

values were negative in a few instances in the case

of tave and tmax. For such cases, the Theil index of

A Study on Negotiation for Revealed Information with Decentralized Asymmetric Multi-objective Constraint Optimization

157

Table 5: Inﬂuence of priorities on criteria for utility values to be published (n = 10, c = 20, rnd, S).

alg. term. utiliy trade-off ratio. rvl. num. ratio. rvl. cost

round sum. min. ave. max theil min. ave. max. theil min. ave. max. theil min. ave. max. theil

sum, S, tmin 6.4 1649.5 102.8 165.0 237.5 0.03414 80.4 137.4 207.2 0.043 0.172 0.261 0.379 0.032 0.158 0.249 0.364 0.035

sum, S, tave 18.6 1760.7 105 176.1 250.6 0.03417 46.3 88.1 132 0.052 0.660 0.767 0.862 0.003 0.648 0.758 0.862 0.004

sum, S, tmax 22.1 1770.9 101.8 177.1 260.7 0.037 20.6 61.2 100.5 (0.061) 0.962 0.993 1 0.0001 0.954 0.992 1 0.00022

ms, S, tmin 6.6 1619.2 102.5 161.9 236.5 0.032 75 133.3 203.8 0.042 0.184 0.274 0.410 0.034 0.161 0.259 0.388 0.039

ms, S, tave 18.4 1675.8 117.6 167.6 236.9 0.026 46.3 79.5 119.2 0.040 0.644 0.772 0.872 0.004 0.641 0.760 0.865 0.004

ms, S, tmax 20.4 1673.8 120.5 167.4 229 0.0227 14.4 51.5 84.8 (0.048) 0.952 0.992 1 0.0002 0.944 0.991 1 0.0003

lxm, S, tmin 6.5 1624 101.6 162.4 240.6 0.033 74.3 133.4 207.6 0.045 0.183 0.278 0.419 0.037 0.162 0.262 0.404 0.041

lxm, S, tave 17.7 1644.5 117.6 164.5 232.7 0.0231 44.6 77.2 117 0.040 0.660 0.767 0.859 0.003 0.639 0.752 0.845 0.004

lxm, S, tmax 20.8 1642.1 120.5 164.2 230.8 0.020 11.8 48.2 79.9 (0.043) 0.958 0.993 1 0.0002 0.952 0.992 1 0.00025

Table 6: Inﬂuence of priorities on criteria for utility values to be published (n = 20, c = 20, equ, S).

alg. term. utiliy trade-off ratio. rvl. num. ratio. rvl. cost

round sum. min. ave. max theil min. ave. max. theil min. ave. max. theil min. ave. max. theil

sum, S, tmin 7.2 2166.6 54.4 108.3 183.5 0.057 38.3 84.7 150.3 0.066 0.254 0.463 0.683 0.031 0.262 0.472 0.707 0.032

sum, S, tave 15.1 2231.9 57.2 111.60 195.8 0.055 27.8 63.2 104.9 0.053 0.715 0.889 0.992 0.00347 0.771 0.913 0.996 0.0024

sum, S, tmax 17.1 2232.3 57 111.62 198 0.056 24.4 58.3 96.6 0.052 0.940 0.992 1 0.00024 0.965 0.995 1 0.00008

ms, S, tmin 7.2 2150.1 61.8 107.5 182.7 0.049 46 84.1 150.3 0.055 0.251 0.461 0.692 0.034 0.258 0.470 0.710 0.0331

ms, S, tave 15.1 2169.6 69.3 108.5 189.8 0.044 27.4 60.2 96.2 0.044 0.715 0.883 0.995 0.00357 0.771 0.907 0.998 0.00247

ms, S, tmax 16.4 2164.6 69.5 108.2 190.3 0.044 21 55.0 86.6 0.050 0.924 0.990 1 0.000335 0.956 0.994 1 0.000117

lxm, S, tmin 7 2108.9 61.8 105.4 171.4 0.040 46.6 82.2 139.0 0.045 0.254 0.459 0.692 0.033 0.256 0.468 0.710 0.0332

lxm, S, tave 14.2 2098.8 68.9 104.9 173.8 0.033 30.3 57.0 88.5 0.035 0.707 0.873 0.992 0.00354 0.760 0.900 0.997 0.00255

lxm, S, tmax 15.6 2102.9 69.5 105.1 171.8 0.031 24.6 51.9 79.5 0.039 0.925 0.990 1 0.000329 0.957 0.994 1 0.000116

‘trade-off’ is evaluated only for positive values and

denoted by parentheses. In this case, the beneﬁt of

lxm seems to small. These results reveal the inﬂuence

of correlation between utility values and their publish-

cost values. Table 6 shows the results in the case of

n = 20, c = 20, equ, and S. The results resemble in

the case of n = 10 and c = 20.

The average execution time of our experimental

implementation on a computer with g++ (GCC) 4.4.7,

Linux version 2.6, Intel (R) Core (TM) i7-3770K

CPU @ 3.50GHz and 32GB memory was 973 sec-

onds in the case of n = 10, c = 20, inv, lxm, S, and

tave.

5 DISCUSSION

In the previous work (Matsui, 2019), similar prob-

lem was solved using a mediator agent that performs

a centralized local search. The goal of the study is

to ﬁnd the ﬁrst solution where all agents can agree

with published utility values. Therefore, the solu-

tion process only ﬁnds one of combinations of parts

of constraints that involves an assignment to all vari-

ables, and other possible complete solutions are not

explored. In addition, the criteria to aggregate and

evaluate publish cost values and utility values is the

summation.

Our study investigates the negotiation process on sim-

ilar problem with a decentralized complete solution

method employing several criteria that consider pref-

erences of individual agents. We also allow to con-

tinue the search for other solutions with better utility

values, because the complete solution method ﬁnds

the ﬁrst solution after the ﬁrst round. On the other

hand, due to the limitation of dynamic programming

on a pseudo-tree, we solved relatively small scale and

sparse problems in comparison to the problems in the

previous work.

We assumed that it is accepted to reveal some ab-

stract information to determine utility values to be

published, since the aim of this work is a fundamental

investigation of the proposed approach. There are op-

portunities to employ a secure computation in part of

the negotiation process of publication. In such situa-

tions, the information to be ﬁnally published so that

agents can understand the reason of an agreement on

a solution will be an issue.

We employed complete solution methods to solve

problems in each negotiation round so that local

parts of negotiation are based on optimal solutions

as a baseline. However, incomplete solution meth-

ods are necessary for practical and large-scale prob-

lems. There are opportunities to develop such solu-

tion methods for composite criteria considering social

welfare among agents. For initial investigation, we

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

158

concentrated on fundamental benchmark problems.

Applying the proposed approach with more scalable

solution methods to practical resource allocation and

collaboration problems will be a goal of future study.

6 CONCLUSION

In this study, we addressed a negotiation framework

based on asymmetric constraint optimization prob-

lems, where agents iteratively publish utility values

of their constraints and solve the problem with pub-

lished utility values. We studied applying a decentral-

ized complete solution method to solve both phases in

each negotiation round. The proposed approach em-

ploys two solution methods based on pseudo-trees to

select utility values to be published and to solve the

problem with the published utility values. As our ﬁrst

investigation, we evaluated the criterion of the dedi-

cated optimization problems and aggregation opera-

tors, and demonstrated is inﬂuence and effect.

Since we employed a complete solution method

based on pseudo-trees, the scalability for complex

problems is limited. A focus of our future work

will be decentralized solution methods for large scale

problems in practical domains. Improvement of the

proposed criterion and termination condition consid-

ering agreement among agents with dedicated pricing

of privacy and utility will also be included our future

work.

ACKNOWLEDGEMENTS

This work was supported in part by JSPS KAKENHI

Grant Number JP19K12117.

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