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An argumentation-based approach to multiple

criteria decision

Leila AmgoudJean-Francois BonnefonHenri Prade

Institut de Recherche en Informatique de Toulouse (IRIT)

118, route de Narbonne,

31062 Toulouse Cedex 4 France

{amgoud, bonnefon, prade}@irit.fr

Abstract. The paper presents a first tentative work that investigates

the interest and the questions raised by the introduction of argumenta-

tion capabilities in multiple criteria decision-making. Emphasizing the

positive and the negative aspects of possible choices, by means of ar-

guments in favor or against them is valuable to the user of a decision-

support system. In agreement with the symbolic character of arguments,

the proposed approach remains qualitative in nature and uses a bipolar

scale for the assessment of criteria. The paper formalises a multicriteria

decision problem within a logical argumentation system. An illustrative

example is provided. Various decision principles are considered, whose

psychological validity is assessed by an experimental study.

Keywords: Argumentation; multiple-criteria decision, qualitative scales.

1Introduction

Humans use arguments for supporting claims e.g. [5] or decisions. Indeed, they

explain past choices or evaluate potential choices by means of arguments. Each

potential choice has usually pros and cons of various strengths. Adopting such

an approach in a decision support system would have some obvious benefits. On

one hand, not only would the user be provided with a “good” choice, but also

with the reasons underlying this recommendation, in a format that is easy to

grasp. On the other hand, argumentation-based decision making is more akin

with the way humans deliberate and finally make a choice. Indeed, the idea of

basing decisions on arguments pro and cons is very old and was already some-

what formally stated by Benjamin Franklin [10] more than two hundreds years

ago.

Until recently, there has been almost no attempt at formalizing this idea if

we except works by Fox and Parsons [9], Fox and Das [8], Bonnet and Geffner

[3] and by Amgoud and Prade [2] in decision under uncertainty. This paper

focuses on multiple criteria decision making. In what follows, for each criterion,

one assumes that we have a bipolar univariate ordered scale which enables us

to distinguish between positive values (giving birth to arguments pro a choice

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x) and negative values (giving birth to arguments cons a choice x). Such a scale

has a neutral point, or more generally a neutral area that separates positive and

negative values. The lower bound of the scale stands for total dissatisfaction and

the upper bound for total satisfaction; the closer to the upper bound the value

of criterion cifor choice x is, the stronger the value of ciis an argument in favor

of x; the closer to the lower bound the value of criterion cifor choice x is, the

stronger the value of ciis an argument against x.

In this paper, we propose an argumentation-based framework in which ar-

guments provide the pros and cons of decisions are built from knowledge bases,

which may be pervaded with uncertainty. Moreover, the arguments may not have

equal forces and this make it possible to compare pairs of arguments. The force

of an argument is evaluated in terms of three components: its certainty degree,

the importance of the criterion to which it refers, and the (dis)satisfaction level

of this criterion. Finally, decisions can be compared, using different principles,

on the basis of the strength of their relevant arguments (pros or cons).

The paper is organized as follows. Section 2 states a general framework for

argumentation-based decision, and various decision principles. This framework

is then instantiated in section 3. Lastly, section 4 reports on the psychological

validity of these decision principles.

2A general framework for multiple criteria decision

Solving a decision problem amounts to defining a pre-ordering, usually a com-

plete one, on a set X of possible choices (or decisions), on the basis of the different

consequences of each decision. Argumentation can be used for defining such a

pre-ordering. The basic idea is to construct arguments in favor of and against

each decision, to evaluate such arguments, and finally to apply some principle

for comparing the decisions on the basis of the arguments and their quality or

strengths. Thus, an argumentation-based decision process can be decomposed

into the following steps:

1. Constructing arguments in favor of /against each decision in X.

2. Evaluating the strength of each argument.

3. Comparing decisions on the basis of their arguments.

4. Defining a pre-ordering on X.

2.1Basic definitions

Formally, an argumentation-based decision framework is defined as follows:

Definition 1 (Argumentation-based decision framework). An argumentation-

based decision framework is a tuple <X, A, ?, ?Princ> where:

– X is a set of all possible decisions.

– A is a set of arguments.

– ? is a (partial or complete) pre-ordering on A.

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– ?Princ(for principle for comparing decisions), defines a (partial or complete)

pre-ordering on X, defined on the basis of arguments.

The output of the framework is a (complete or partial) pre-ordering ?Princ, on

X. x1?Princx2means that the decision x1is at least as preferred as the decision

x2w.r.t. the principle Princ.

Notation: Let A, B be two arguments of A. If ? is a pre-order, then A ? B

means that A is at least as ‘strong’ as B.

? and ≈ will denote respectively the strict ordering and the relation of equiva-

lence associated with the preference between arguments. Hence, A ? B means

that A is strictly preferred to B. A ≈ B means that A is preferred to B and B

is preferred to A.

Different definitions of ? or different definitions of ?Princ may lead to differ-

ent decision frameworks which may not return the same results.

Each decision may have arguments in its favor, and arguments against it. An ar-

gument in favor of a decision represents the good consequences of that decision.

In a multiple criteria context, this will represent the criteria which are positively

satisfied. On the contrary, an argument against a decision may highlight the

criteria which are insufficiently satisfied. Thus, in what follows, we define two

functions which return for a given set of arguments and a given decision, all the

arguments in favor of that decision and all the arguments against it.

Definition 2 (Arguments pros/cons). Let x ∈ X.

– ArgP(x) = the set of arguments in A which are in favor of x.

– ArgC(x) = the set of arguments in A which are against x.

2.2Some principles for comparing decisions

At the core of our framework is the use of a principle that allows for an argument-

based comparison of decisions. Below we present some intuitive principles Princ,

whose psychological validity is discussed in section 4. A simple principle consists

in counting the arguments in favor of each decision. The idea is to prefer the

decision which has more supporting arguments.

Definition 3 (Counting arguments pros: CAP). Let <X, A, ?, ?CAP>

be an argumentation based decision framework, and Let x1, x2∈ X.

x1 ?CAP x2 w.r.t CAP iff |ArgP(x1)| > |ArgP(x2)|, where |B| denotes the

cardinality of a given set B.

Likewise, one can also compare the decisions on the basis of the number of

arguments against them. A decision which has less arguments against it will be

preferred.

Definition 4 (Counting arguments cons: CAC). Let <X, A, ?, ?CAC>

be an argumentation based decision framework, and Let x1, x2∈ X.

x1?CAC x2w.r.t CAC iff |ArgC(x1)| < |ArgC(x2)|.

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Definitions 3 and 4 do not take into account the strengths of the arguments. In

what follows, we propose two principles based on the preference relation between

the arguments. The first one, that we call the promotion focus principle (Prom),

takes into account only the supporting arguments (i.e. the arguments PRO a

decision), and prefers a decision which has at least one supporting argument

which is preferred to (or stronger than) any supporting argument of the other

decision. Formally:

Definition 5 (Promotion focus). Let <X, A, ?, ?Prom> be an argumentation-

based decision framework, and Let x1, x2∈ X.

x1?Promx2w.r.t Prom iff ∃ A ∈ ArgP(x1) such that ∀ B ∈ ArgP(x2), A ?

B.

Note that the above relation may be found too restrictive, since when the

strongest arguments in favor of x1 and x2 have equivalent strengths (in the

sense of ≈), x1 and x2 cannot be compared. Clearly, this could be refined in

various ways by counting arguments of equal strength.

The second principle, that we call the prevention focus principle (Prev), consid-

ers only the arguments against decisions when comparing two decisions. With

such a principle, a decision will be preferred when all its cons are weaker than

at least one argument against the other decision. Formally:

Definition 6 (Prevention focus). Let <X, A, ?, ?Prev> be an argumenta-

tion based decision framework, and Let x1, x2∈ X.

x1?Prevx2w.r.t Prev iff ∃ B ∈ ArgC(x2) such that ∀ A ∈ ArgC(x1), B ? A.

Obviously, this is but a sample of the many principles that we may con-

sider. Human deciders may actually use more complicated principles, such as

for instance the following one. First, divide the set of all (positive or negative)

arguments into strong and weak ones. Then consider only the strong ones if any,

and apply the Prevention focus principle. In absence of any strong argument,

apply the Promotion focus principle. This combines risk-aversion in the realm

of extreme consequences, with risk-tolerance in the realm of mild consequences.

3 A specification of the general framework

In this section, we give some definitions of what might be an argument in favor

of a decision, an argument against a decision, of the strengths of arguments,

and of the preference relations between arguments. We will show also that our

framework capture different multiple criteria decision rules.

3.1Basic concepts

In what follows, L denotes a propositional language, ? stands for classical infer-

ence, and ≡ stands for logical equivalence. The decision maker is supposed to be

equipped with three bases built from L:

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1. a knowledge base K gathering the available information about the world.

2. a base C containing the different criteria.

3. a base G of preferences (expressed in terms of goals to be reached).

Beliefs in K may be more or less certain. In the multiple criteria context, this

opens the possibility of having uncertainty on the (dis)satisfaction of the criteria.

Such a base is supposed to be equipped with a total preordering ≥.

a ≥ b iff a is at least as certain as b.

For encoding it, we use the set of integers {0,1,...,n} as a linearly ordered

scale, where n stands for the highest level of certainty and ‘0’ corresponds to

the complete lack of information. This means that the base K is partitioned and

stratified into K1, ..., Kn(K = K1∪ ... ∪ Kn) such that formulas in Kihave

the same certainty level and are more certain than formulas in Kjwhere j < i.

Moreover, K0is not considered since it gathers formulas which are completely

not certain.

Similarly, criteria in C may not have equal importance. The base C is then also

partitioned and stratified into C1, ..., Cn(C = C1∪ ... ∪ Cn) such that all criteria

in Cihave the same importance level and are more important than criteria in Cj

where j < i. Moreover, C0is not considered since it gathers formulas which are

completely not important, and which are not at all criteria.

Each criterion can be translated into a set of consequences, which may not

be equally satisfactory. Thus, the consequences are associated with the satisfac-

tion level of the corresponding criterion. The criteria may be satisfied either in

a positive way (if the satisfaction degree is higher than the neutral point of the

considered scale) or in a negative way (if the satisfaction degree is lower than

the neutral point of the considered scale). For instance, consider the criterion

“closeness to the sea” for a house to let for vacations. If the distance is less than

1 km, the user may be fully satisfied, moderately satisfied if it’s between 1 and 2

km, slightly dissatisfied if it is between 2 and 3 km, and completely dissatisfied

if it is more than 3km from the sea. Thus, the set of consequences will be parti-

tioned into two subsets: a set of positive “goals” G+and a set of negative ones G−.

Since the goals may not be equally satisfactory, the base G+(resp. G−) is also

supposed to be stratified into G+= G+

where goals in G+

and are more important than goals in G+

Gi’s may be empty if there is no goal corresponding to this level of importance.

For the sake of simplicity, in all our examples, we only specify the strata which

are not empty. In the above example, taking n = 2, we have G+

G+

A goal gj

gj

1∪ ... ∪ G+

n(resp. G−= G−

1∪ ... ∪ G−

n)

i(resp. G−

i) correspond to the same level of (dis)satisfaction

j(resp. G−

j) where j < i. Note that some

2= {dist < 1km},

2= {3 < dist}.

1= {1 ≤ dist < 2km}, G−

1= {2 ≤ dist ≤ 3km} and G−

iis associated to a criterion ciby a propositional formula of the form

i→ cimeaning just that the goal gj

irefers to the evaluation of criterion ci. Such