The Conceptual Framework of Fairness in Consensus Reaching Process Under Fuzziness

Janusz Kacprzyk Dominika Gołuńska Systems Research Institute PhD Studies, Systems Research Institute

Polish Academy of Sciences Polish Academy of Sciences ul. Newelska 6, 01-447 Warsaw, Poland ul. Newelska 6, 01-447 Warsaw, Poland

kacprzyk@ibspan.waw.pl

Department of Automatic Control and Information

Technology Cracow University of Technology ul. Warszawska 24, 31-155 Cracow, Poland dominika.golunska@pk.edu.pl

Abstract - The purpose of this paper is to present briefly the

conceptual framework of fairness in the consensus reaching process with novel elements of grasping imprecision in intentions, preferences and adjustments of individuals. All solutions are based on the idea of “soft” degree of consensus under fuzzy preference relations and a fuzzy majority given as a fuzzy linguistic quantifier. Here, we propose a further extension of the human-consistent consensus reaching support system for group decision-making problems. We enhance a new knowledge-based system with socio-psychological aspects of fair behavior which ensures the satisfaction of justified expectations of its participants. We apply the resource allocation model to express fair influence on consensus reaching process among all decision makers. In this paper we present general conditions for fair solution concept to increase the effectiveness of consensus reaching process and a quality of final decision which becomes highly-justified.

I. INTRODUCTION

It is commonly acknowledged that decision theory appears to be a diverse domain which draws upon ideas from multifarious scientific disciplines, such as philosophy, psychology, sociology, economics, etc. Therefore, recent models of decision making process apply not only to the mathematical approach with strictly specified algorithms and procedures but also to an analysis of human behavior, social interactions and other socio-economic aspects.

An important issue is that a decision maker is hardly ever able to make a goal-directed decision alone. Thus, it may be justified to organize a session with a group of experts and run a collective decision making process with a better understanding of the problem. The main goal of such procedures, in the context considered here, is to attain an agreement among the experts as to the chosen solution, i.e. to achieve a consensus.

We propose a new model which makes the consensus reaching process more human-consistent and plausible by applying the notion of “fairness” in its basic contexts. Thus, we attempt to use sophisticated techniques and formalizations to build computational environment of social behavior, within psychological epistemology.

The purpose of this paper is more conceptual in the sense of presenting a basic framework and philosophy behind the new model enriched with the notion of “fairness”. Section II shows the overall structure of consensus reaching support systems. We will introduce the core and required procedures of an intelligent computational model based on the idea developed and successfully implemented by Kacprzyk and Zadrożny [9]. In Section III we explain the derivation of “fairness” in our novel system and present the conceptual structure of proposed computational technique. We extend the notion of resource allocation problem [13],[14],[15] to express a fair solution concept in consensus reaching process.

II. CONSENSUS REACHING SUPPORT SYSTEM

It is a truth commonly acknowledged that attaining consensus by a group of experts involves: time, creative thinking, active listening, considering the ideas and being open-minded. The model of consensus reaching process is feasible only if individuals are able to negotiate and change their preferences. Fortunately, many psychological studies and experiments have revealed that, in real life, decision makers are not so selfish and self-interested and they tend to cooperate in order to solve a common problem. Thus, we assume that individuals are seriously “committed to reaching consensus” – they are expected to update step-by-step their preferences, and as a result to finally attain a satisfactory agreement.

We assume the topological approach based on distances between decision makers. Initially, experts disagree in their testimonies, i.e. they are far away from consensus. The purpose is to minimize this distance, i.e. get the preferences closer to each other and, as a result, lead the group to the acceptable agreement [7]. What matters here, is that these preliminary differences of opinions are the strength of the group and a key to integrate extra information, providing relevant details and variations and identifying potential solutions with benefits for every decision maker. Combining ideas, advocating pros and cons of various suggestions and clarifying preferences of decision makers take place during the discussion which gives the opportunity to exchange

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knowledge and recognize the position of other participants. In fact, our observation of a group of students in the academic environment confirmed that eagerness in learning the differences between points of views and an interactive character of the group relations turned out to be the root of success in reaching consensus required.

The development of modern, computer-based systems forced scientists to seek sophisticated tools which support and simplify this dynamic process and allow to achieve consensus in a more efficient way. The aim of this paper is to present such a model of consensus support system which improves the consensus reaching process via grasping sociological and psychological course of decision making problems.

In this consensus support system the figure of moderator occurs which plays the role of a guide who based on a set of advice rules helps the individuals change their preferences and knows the appropriate directions of that change in order to lead the group closer to consensus. The moderator measures distances between experts on every stage of the process and evaluates the actual level of agreement among the experts’ preferences. If the consensus level is not acceptable the moderator encourages individuals to discuss their views further in an effort to get them closer to consensus [4]. Therefore, his most important task is to stimulate the exchange of information, suggest suitable arguments, encourage appropriate decision makers to change their testimonies and lead the discussion towards the most promising solutions. The moderator plays his role until the group gets sufficiently close to consensus, i.e. until the individual fuzzy preference relations become similar enough, or until some time limit is reached [7].

We designed a consensus reaching support system which is controlled by a certain advice system with the human moderator. For this purpose we developed and enhanced the discussion part to obtain a specific knowledge about the group members and some useful hints, e.g. as to the most problematic experts or the most controversial options. In other words, we want to substitute the work of a moderator by a feedback mechanism which makes the consensus reaching process more effective and facilitated. In this paper we propose a new model which allows to grasp imprecision in intentions, preferences and adjustment of individuals. The consensus reaching process is run in a fuzzy environment, i.e. with fuzzy preference relations of individuals and a fuzzy majority expressed by a fuzzy linguistic quantifier, and the core of a model is enriched with the notion of “fairness”. A. Fuzzy Preference Relations

We will operate in the following basic settings [5]. There is a finite set of 2≥n alternatives (options, issues),

{ }nsssS ,...,, 21= , and a finite set of 2≥m individuals (experts, agents), { }meeeE ,...,, 21= . Each individual Ek ∈ expresses his or her testimony as to the particular pairs of options in S . These testimonies are assumed to be individual fuzzy preference relation kR defined over the set of alternatives S (i.e. in SS × ).

An individual fuzzy preference relation of expert k , kR , is given by its membership function ]1,0[: →× SS

kRμ .

Namely, 5,0),( >jiR ssk

μ indicates the preference degree of

an alternative is over an alternative js . Respectively,

5,0),( <jiR ssk

μ denotes the preference degree of an option

js over an option is . The third possible relation represented

by 5,0),( =jiR ssk

μ is also acceptable and determines the

indifference between two considered alternatives is and js . We assume card S to be small enough to allow us

represent individual fuzzy preference relation kR by a nn×

matrix ][ kijk rR = , such that ),( jiR

kij ssr

kμ= , i,j=1,…,n;

k=1,…,m. kR is also assumed to be reciprocal, i.e.

1=+ kji

kij rr , moreover, 0=k

iir , for all kji ,, [7]. An important issue of our derivation is that we assume a

conventional form of fuzzy preference relations, which associates with each pair of options a number from the unit interval, instead of recently advocated linguistic fuzzy preference relations whose membership functions take on values in an ordered set of linguistic values, e.g. the preference of alternative is over js can be expressed as much better, worse, etc. Obviously, the latter approach is a step towards a higher human-consistent preference modeling but in this paper we focus on more fundamental and conceptual presentation of the system proposed by authors.

B. Fuzzy Majority and Linguistic Quantifier The key issue while dealing with real-world and human-

consistent perception of consensus is to be able to involve a fuzzy majority in the sense of fuzzy linguistic quantifiers, i.e. most, almost all etc., with a formal representation in the framework of fuzzy logic. In this section we will briefly exemplify fuzzy logic-based calculus of linguistically quantified statements due to Zadeh [19].

A linguistically quantified statement is represented by “most individuals are satisfied” which can be written as

Qy’s are F (1)

where Q is a linguistic quantifier (e.g., most), { }yY = is a set of objects (e.g., individuals) and F is a property (e.g., satisfied).

Our task is to find the degree of truth of this linguistically quantified statement (1). First, a fuzzy linguistic quantifier is equated with a fuzzy set in [0,1]. For instance, “most” may be given as

⎪⎩

⎪⎨

⎧

<≤≤

>−=

3.08.03.0

8.0

06.02

1)(""

xx

x

forforxfor

xmostμ (2)

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which is meant as that if less than 30% of the objects considered own some property, then it is sure that not most of them own, on the contrary, if more that 80% of them own the property, then it is sure that most of them own it [7]. What matters here, is that we consider non-decreasing fuzzy linguistic quantifiers defined as

),''()'(''' xxxx QQ μμ ≥⇒> (3)

for each ]1,0['',' ∈xx . It expresses the attitude “the more the better”, ""mostQ = (2) is definitely non-decreasing.

Property F is defined as a fuzzy set in the set of objects Y, and if { }pyyY ,...,1= , then we suppose that truth

( ) piyisFy iFi ,...1),( == μ . The degree of statement (1), that is, truth (Qy’s are F), is now calculated in two steps:

∑=

=p

iiF y

pr

1

)(1 μ (4)

truth(Qy’s are F) )(rQμ= . (5)

C. “Soft” Degree of Consensus Here, we define a “soft” degree of consensus as proposed

in Kacprzyk [4] and then advanced in Kacprzyk and Fedrizzi [6] and Kacprzyk and Zadrożny [10]. A classically meant concept of consensus occurs only when “all the individuals agree as to all the alternatives”. However, this “full and unanimous agreement” is utopian and unrealistic in practice, because individuals usually reveal meaningful differences in their standpoints, flexibility, tendency to change opinions, etc. The new degree of agreement can be equal to 1, which means full consensus, when “most of the individuals agree in their preferences to almost all of the options.” Except of total agreement between experts as to the chosen solution, this approach allows some partial, acceptable consistency in the range [0,1].

The “soft” degree of consensus in the above sense is now obtained in three steps [7]: 1) for each pair of individuals we compute a degree of agreement as to their preferences between all the pairs of alternatives, 2) we aggregate these degrees to derive a degree of agreement of each pair of experts as to their preferences between 1Q (a linguistic quantifier as, e.g., “most”, “almost all”,…) pairs of options, 3) we combine these degrees to obtain a degree of agreement of 2Q (a linguistic quantifier similar to 1Q ) pairs of individuals as to their preferences between 1Q pairs of alternatives and this is meant to be the degree of consensus.

We start with the degree of strict agreement between individuals 1k and 2k as to their preferences between options

is and js

⎪⎩

⎪⎨⎧ ==

otherwiserrifkkv

kij

kij

ij01),(

21

21 (6)

where, here and later on in this section, if not otherwise specified ,1,...,11 −= mk ,,...,112 mkk += ,1,...,1 −= ni

nij ,...,1+= [7]. Next, according to step two, the degree of agreement

between individuals 1k and 2k as to their preferences between all the pairs of alternatives is:

∑∑−

= +=−=

1

1 12121 ),(

)1(2),(

n

i

n

ijij kkv

nnkkv . (7)

To proceed, the degree of agreement between individuals 1k and 2k as to their preferences between 1Q pairs of

alternatives is:

)},({),( 211211kkvkkv QQ μ= . (8)

In turn, the degree of agreement of all the pairs of individuals as to their preferences between 1Q pairs of alternatives is:

∑ ∑−

= +=−=

1

1 121

1 12

11),(

)1(2 m

k

m

kkQQ kkv

mmv . (9)

Finally, according to the third step, the degree of agreement of 2Q pairs of individuals as to their preferences between 1Q pairs of alternatives, called the degree of consensus, is:

)(),(1221 QQ vQQcon μ= . (10)

Calculating the degree of “soft” consensus derives additionally some partial indicators of consensus, like e.g. the personal consensus degree or the option consensus degree. These consensus indicators point out to the most controversial alternatives and/or individuals isolated in their opinions. Thus, they are used to facilitate the work of moderator by providing him with some hints as to the most promising directions of a further discussion.

III. FAIRNESS

At the beginning of the consensus reaching process, individuals’ preferences are far away from each other and the degree of consensus is very low. In this situation, there is a

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requirement for many changes in order to minimize these distances and lead the group to the acceptable agreement. Our concept of searching for set of preferences to be changed and guiding the further discussion is in line with the fair treatment of all participants. The idea of fairness has become a relevant element of the new knowledge-based computational systems which aim at explaining actual human behavior. It can not go unnoticed that the context of coming to an agreement is essential, especially that we consider system in which human perception or valuation becomes crucial and where we can not ignore human characteristics like variability of opinions, imprecise preferences, etc. [3]

The idea of fairness appears to be a multifarious issue which combines many different scientific disciplines such as mathematics, philosophy, economics and other social sciences, especially social psychology. The last research area is crucial because it gives a response to a question: how do people understand fair behavior? The explanations can be given by the definition of the cooperative game theory which basically is a game where players can support fair behavior. Cooperative game theory is about the distribution of benefits that a group of people gain from collaboration [17]. Furthermore, many psychological studies have revealed that individuals are likely to cooperate and give priority to fairness over grasping behavior [3]. The trust game will transparently perform this activity. In the trust game, person A has an initial amount of money, he or she could either keep or transfer to person B. If A transfers it, the sum is tripled. B could keep this amount, or move it (partially or totally) to A. A traditional game theory suggests that A should keep everything, or if A transfers any amount to B, then B should keep all. Experimental studies have confirmed that people tend to transfer about 50% of their money and this fairness and cooperation is related to all cultures, sexes, etc [1].

One of the definition of fairness says that “fairness means the satisfaction of justified expectations of agents that participate in the system, according to rules that apply in a specific context based on reason and precedent“ [10]. In the light of this definition, our consensus reaching support system should provide the feeling of satisfaction among the decision makers during the discussion and after process completion. Psychological research confirmed that the satisfaction of decision makers has a direct influence on higher quality of final decision and several further activities, e.g. practical implementation of the final decision or survival of the group in the long time period.

During group decision making process, there are always outsiders who are isolated in their opinions as to the rest of the group and they are omitted in consensus building process (it is known as “tyranny of the majority [12]”). Finally, outsiders do not get the satisfaction from the discussion what affects the effectiveness of an entire group. Of course, it does not exclude the final decision achievement, but decreases the opportunity of many further activities. According to these assumptions, we want to ensure “fairness” in the sense of satisfaction among decision makers by neglecting the situation where a small

group of outsiders are isolated and overlooked in their opinions.

A. FAIR ALLOCATION OF RESOURCE The distribution of distances between decision makers is

a significant issue, thus we consider one of fairness judgments identified by a social psychology as a distributive fairness [16] and we try to formalize it with one of the resource allocation model [13].

To get the decision makers closer to each other (thus to obtain a satisfactory degree of agreement), there has to occur some changes in the preferences of individuals as to several pairs of options. These changes in the individuals’ preferences, meant as a difference between current value of preference as to some pair of options and the new one proposed by the moderator, is the resource. It may be also understood as a cost.

The generic resource allocation problem might be stated as follows. There is a set of },...,2,1{ mI = of m activities. There is given a set P of location patterns (location decisions). For each activity i , Ii ∈ , a nonlinear function

)(xfi of the location pattern x has been denoted. This function is defined as an individual objective function and measures the outcome )(xfy ii = of the location pattern for each activity i .

Thus, for a given set of experts moderator wants to allocate the changes of particular preferences in goal-directed behavior, in the sense of minimizing or maximizing the outcome. Here, the outcome indicates the distances, but we want to emphasize that in different models the outcome may represent revenues, profits, throughput (functions )(xfi are strictly increasing) or costs, time, delay, the individuals dissatisfaction of location patterns ( )(xfi are strictly decreasing convex functions) [11]. In our research, the distances between every pair of individuals (7) and the total agreement (10) are considered. Thus we assume the outcome as a mm× matrix ][

21kkyY = , such that

|),(),(| 2121 121kkvQQcony Qkk −= (11)

where ,1,...,11 −= mk ,,...,112 mkk += moreover 011

=kky . In typical formulations, a smaller value of the outcome

(distance) means a better effect. Therefore, this allows us to consider the generic resource allocation problem as the multiple criteria minimization

}:min{}:)(min{ AyyPxxf ∈=∈ (12)

where ),...,,( 21 mffff = is a vector function that maps attainable locations Px ∈ into the feasible outcome vectors

}),(:{ PxxfyRyAy m ∈=∈=∈ [14].

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It is acknowledged, that the resource allocation problem attempts to seek for activity levels that optimize some objective function while satisfying resource constraints [11]. Here, we do not assume any special constraints allowing the feasible set to be a general, discrete set. We have only assumed a finite set of experts for the minimization of the individual outcomes.

Model (12) only says that we are involved in the minimization of all outcome functions if for Ii ∈ . In order to make it operational, we need to assume a fair concept of resource location. Since for each outcome the smaller value is better, some outcome vectors are dominated by others. We state that outcome vector 'y dominates ''y ( ''' yy < ) if

'''ii yy ≤ , for all Ii ∈ where at least one strict inequality holds.

We state that the location pattern Px ∈ is a Pareto-optimal solution of the multiple criteria problem (12), if )(xfy = is nondominated.

Moreover, we agree with the statement that fairness is an abstract notion that implies impartiality and equity [13]. The former, meant also as the anonymity of system entities (decision makers) might be expressed in our multiple criteria optimization problem as:

),...,,(),...,,( 21)()2()1( mm yyyyyy ≅πππ (13)

where )(I∏ indicates the set of all permutations of I . However, the concept of equity causes that the preference model should satisfy the Pigou-Dalton principle of transfers. This principle says that a move of any small amount from one outcome to any other relatively worse outcome gives in a result more desired outcome vector, i.e. whenever ''' ii yy < then

yeey ii ≺''' εε +− for )(0 ''' ii yy −<< ε (14)

Our aim is to find such a distribution of changing preferences that is supposed to be fair by all individuals who participate in this consensus reaching process. Our idea of a fair share is taken from the economic example, namely the European Union Cohesion Policy. It provides a framework for financing a wide range of projects and investments in European Union member states. The European Union Cohesion Policy is navigating towards making regions more competitive, improving life quality, supporting economic growth, job creation and avoiding regional disparities. The main principle here is that the greater part of structural fund resources is concentrated on the poorest regions and countries. The measure of effectiveness reduces regional disparities in Europe by helping those regions to catch up with the ones which are better off. This European territorial cooperation provides fair fund allocation and solidarity among regions. Nevertheless, history showed that whenever a less-developed region grows, other regions can profit as well. After all, the stronger an EU member state is, the stronger the entire

European Union becomes. As a rule, community strength which depends on its unification, cohesion policy and its fair resource allocation ensure the goal which is highly justified. In other words, we discard the scenario where the consensus is reaching only between the experts which are similar in their testimonies, while individuals who are isolated in their opinions are excluded from the process. Moderator can not ignore the “outsiders” in relation to other group members, quite on the contrary his task is to convince them to change their previous preferences. This attitude carries out all of our assumptions, namely, active participation of every individual during the entire consensus reaching process, sense of satisfaction among all decision makers and highly-justified final solution. Thus, the prime version of our fair human-consistent model focuses on these pairs of individuals whose preferences are the furthest from the collective agreement, i.e. on the worst outcomes. We select from the matrix of objective functions these values which exceed our acceptable threshold of dissimilarity

]1,0[,21

∈≥ ρρkky . (15)

The threshold of an acceptable dissimilarity, ρ , is determined by moderator in the beginning of the process and may depend on many factors: complexity of the problem, group dynamics, situational background or the quality of final decision.

The notion of fairness meant here as an equity concerns an equal granularity of resource distribution. Guidance system persuades these pairs of experts which fulfill the assumption (15) to change their testimonies. They update their preferences by turns with the granularity 0.1. Single iteration ends when all selected pairs update with 0.1 their preferences, e.g. as to the most controversial options. When the single iteration ends, moderator measures new distances (11) and checks the inequality (15). This process is repeated until the individual fuzzy preference relations become similar enough, i.e. all pairs of group members fulfill the assumption (15).

Unfortunately, despite the fair final solution and a highly justified total agreement, this solution may provide too big complexity of presented decision making problem. Therefore, authors consider applying maximin discrete location problem [15] as a substitution to the equal granularity of preference adjustment of the most dissimilar pairs of decision makers. Now, resource allocation problem allows to determine the proper quantity of change as to the appropriate pair of options among all pairs of experts who fulfill the assumption (15). Subject function can be written as

))min(max( ijij xd (16)

where ijd defines distance between current value of preference and its next location (may be also understood as a

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cost of change), ijx is a binary variable which denotes location decisions.

In these formulations every single decision maker can affect the final decision. This solution leads to the agreement which satisfies all participants of the consensus reaching process and make it highly-justified.

IV. CONCLUSIONS

In this article we proposed a novel concept of supporting group consensus reaching process. We based our derivation on the approach of “soft” degree of consensus proposed and successfully implemented by Kacprzyk, Zadrożny and Raś [8] enriched by the novel fairness component. We also considered the process which runs in the sophisticated environment under fuzziness, i.e. conventional fuzzy preference relations and a fuzzy majority given as fuzzy linguistic quantifiers. We showed that the notion of a fair treatment should be definitely taken into account while creating knowledge-based support systems, because it is strongly connected with psychology, sociology, economics, etc. In fact, it helps us to understand human behavior within a group of individuals and to design more intelligent and human-consistent systems for supporting consensus reaching in the future development.

Our research yielded conclusion that the degree of consensus obtained by including aspect of fairness would be higher than in the previous approach concerned solely soft consensus with the use of fuzzy logic. We based our consideration on the resource allocation problem which lead us to the corresponding fair solution concept. The proposed systems fulfill significant assumptions, i.e. gaining a satisfactory, highly-justified solution and ensuring the sense of satisfaction among decision makers by avoiding the situation when small groups of outsiders are isolated and omitted in their opinions. Hence, we take the liberty of defining a hypothesis that the concept of novel approach affects the effectiveness of consensus reaching process and the quality of the final decision, which becomes highly justified. The ultimate goal of our further research is the implementation of the formalized model in order to confirm or to reject our assumptions.

ACKNOWLEDGMENT

Dominika Gołuńska’s contribution is partially supported by the Foundation for Polish Science under International PhD Projects in Intelligent Computing. Project financed from The European Union within the Innovative Economy Operational Programme 2007-2013 and European Regional Development Fund.

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