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Qualitative Heuristics for Balancing the Pros and Cons

Jean-François Bonnefon (bonnefon@univ-tlse2.fr)

Université de Toulouse (CLLE; CNRS, UTM, EPHE)

Didier Dubois and Hélène Fargier

IRIT-CNRS, Toulouse

Sylvie Leblois

Université de Toulouse (CLLE; CNRS, UTM, EPHE)

Abstract.

appealing decision procedure, but one that has received scarce scientific attention

so far, either formally or empirically. We describe a formal framework for pros

and cons decisions, where the arguments under consideration can be of varying

importance, but whose importance cannot be precisely quantified. We then define 8

heuristics for balancing these pros and cons, and compare the predictions of these

to the choices made by 62 human participants on a selection of 33 situations. The

Levelwise Tallying heuristic clearly emerges as a winner in this competition. Further

refinements of this heuristic are considered in the discussion, as well as its relation

to Take the Best and Cumulative Prospect Theory.

Balancing the pros and cons of two options is undoubtedly a very

Keywords: decision heuristics; bipolar information; qualitative information; behav-

ioral data; Take the Best; Cumulative Prospect Theory

1. Introduction

Balancing the pros and the cons is certainly among the most intuitive

approaches one might take to decision making. It was already at the

core of Benjamin Franklin’s “moral algebra” (explained in his famous

1772 letter to Joseph Priestley), and it has certainly not fallen from

grace since then, witness the 93,000,000 web pages featuring both the

words “pro” and “con” as of November 2006. One likely feature of this

kind of decision is that the decision maker will be unable to precisely

quantify how important a given pro or con is, although she may be able

to give a qualitative assessment of this importance.

We can assume that many decision makers attempt to reach a de-

cision by balancing pros and cons, after roughly sorting them out in

different levels of importance. But however appealing this qualitative

balancing act might sound, it has inspired only few mathematical explo-

rations to date, and even fewer psychological investigations. In section 2,

we introduce a formal framework for pros and cons decisions. In sec-

tion 3, we describe eight heuristics for balancing pros and cons. The

predictions of these heuristics are then compared to choices made by

c ? 2007 Kluwer Academic Publishers. Printed in the Netherlands.

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human participants in 33 situations, chosen to emphasize the differences

between the heuristics.

2. Pros & Cons Decisions

Tonight, Emma is going to the cinema, and considers watching one of

two movies. She has listed the pros and cons of each choice. Movie 1 is

one by her favorite director (a strong pro); it will be dubbed, which she

hates, and the movie has attracted terrible critics (two strong cons).

Movie 2 is only given in a remote theater, and she considers it a strong

con that she would need a taxi to get there. On the other hand, movie 2

is a comedy (a genre she likes), it features an actress she likes, and it

is inspired by a book she enjoyed reading. These are three pros, but

Emma does not see them as very decisive: they do matter, but not as

much as the other arguments she listed.

Note that Emma can only give a rough evaluation of how strong a

pro or a con is. She can only say that (a) her liking the director, her

hating dubbed movies, the terrible critics, and movie 2 being given in

a remote theater are four arguments of comparable importance; and

that (b) movie 2’s genre, leading actress, and source of inspiration are

three arguments of comparable importance, but not as important as

the previous ones.

Before we try to predict what Emma’s decision might be, let us for-

malize her problem. Each option (movie) is assessed by a finite subset of

arguments taken from X, the set of all possible arguments. Comparing

two options then amounts to comparing two subsets U, V of 2X. X can

be divided in three disjoint subsets: X+the set of pros, X−the set of

cons, and X0the set of irrelevant arguments (which do not count as a

pro or a con). Any U ⊆ X can likewise be partitioned: let U+= U∩X+,

U−= U ∩ X−, U0= U ∩ X0be respectively the pros, the cons, and

the irrelevant arguments relatively to U.

As in our movie example, all arguments are not equally important,

although it is generally impossible to precisely quantify the importance

of a given argument. Thus, in a purely qualitative, ordinal approach,

the importance of arguments is described on a totally ordered scale of

magnitude L = [0L,1L], by a function π : X ?→ L = [0L,1L]. π(x) =

0L means that the decision maker is indifferent to argument x: this

argument will not affect the decision process. The order of magnitude

1L(the highest level of importance) is attached to the most compelling

arguments that the decision maker can consider. Intermediate values are

attached to arguments of intermediate importance. For any level α ∈ L,

let Uα= {x ∈ U,π(x) = αL}, U+

α= Uα∩ X+, and U−

α= Uα∩ X−.

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Finally, it will be useful to define the order of magnitude M(U) of a set

U as the highest of the order of magnitude of its elements:

∀U ⊆ X,M(U) = max

x∈Uπ(x).

Note that M(U) is a possibility measure on X—for a recent review

on qualitative possibility theory, see Dubois and Prade (2004). We can

now reformulate the Emma case as the comparison between two options

U (movie 1) and V (movie 2).1Option U has one argument in U+

two arguments in U−

arguments in V−

the different heuristics that can be defined for balancing pros and cons

have quite diverging views on the Emma case. Some will prefer movie 1,

some will prefer movie 2, some will regard the two movies as equally

attractive, and some will find it impossible to compare the merits of the

two movies.

αand

α; and option V has one argument in V−

β, where α > β. As we will see in the next section,

αand three

3. Pros & Cons Heuristics

Qualitative heuristics for balancing the pros and the cons (or, tech-

nically, ordinal ranking procedures from bipolar information) have re-

ceived scarce attention so far. Most work on such procedures has come

from the field of Artificial Intelligence, following the renewed inter-

est in argumentative models of choice and inference (Amgoud et al.,

2005; Benferhat et al., 2006).

The heuristics we describe in this section have been formally exam-

ined and axiomatized (Dubois and Fargier, 2005; Dubois and Fargier,

2006). Since the present article takes an empirical rather than ana-

lytical approach to pros and cons heuristics, we will not restate all the

formal properties of the heuristics nor give their axiomatization. We will

nevertheless comment on important properties such as completeness or

transitivity.

3.1. Focus heuristics F1, F2, and F3

The heuristics in the “Focus” family concentrate on the most important

arguments available for the decision, and disregard arguments of lesser

importance.

1In the rest of this article, we will denote an option by the subset of arguments

which is used to assess this option. E.g., movie 1 is assessed by the subset U of

arguments, and will thus be denoted “Option U.”

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3.1.1. The straw and the beam (?F1)

With this heuristic, U is at least as good as V if and only if, at level

M(U ∪ V ) (i.e., the highest level of importance in the current compar-

ison), the presence of arguments for V is cancelled by the existence of

arguments for U, and the existence of arguments against U is cancelled

by the existence of arguments against V . Formally, U ?F1V if and only

if:

M(U ∪ V ) = M(V+) ⇒ M(U ∪ V ) = M(U+)

M(U ∪ V ) = M(U−) ⇒ M(U ∪ V ) = M(V−)

What will Emma do? It turns out that M(U ∪V ) = M(U−); that is,

the strongest con is attached to U. However, it is also true that M(U ∪

V ) = M(V−). The second condition is thus satisfied: the existence of

a strong argument against V offsets the existence of a strong argument

against U. The first condition is satisfied because M(U ∪ V ) is not

M(V+). Consequently, it holds that U ?F1V . Now, it does not hold

that V ?F1U, because while M(U ∪ V ) = M(U+), it is not the case

that M(U ∪V ) = M(V+). There is no strong argument for V to offset

the existence of a strong argument for U. Emma will go and see movie 1.

The relation ?F1is transitive but incomplete. As soon as an option

has both a pro and a con at the highest importance level, it becomes

incomparable to any other option whose description does not feature

pros or cons at this highest importance level.

and

3.1.2. My enemy’s enemies (?F2)

This heuristic treats all arguments against V as arguments for U, and

all arguments for V as arguments against U (and reciprocally). It then

selects the option that is supported by the arguments at the highest

level. Formally, U ?F2V if and only if:

max(M(U+),M(V−)) ≥ max(M(U+),M(V−)).

In the Emma case, max(M(U+),M(V−)) = max(M(U+),M(V−)) =

α. Emma is indifferent, she can toss a coin to decide on a movie. The

relation ?F2is simpler than the relation ?F1, and has the advantage

of being complete, but it is only quasi-transitive: ?F2itself is tran-

sitive, but the corresponding indifference relation is not (e.g., when

M(V+) = M(V−), it is possible to have both U ∼F2V and V ∼F2W,

while U ∼F2W may not hold).

Furthermore, while ?F2is complete, it is not as decisive as ?F1:

it yields indifference more often than ?F1does, as illustrated by the

Emma case. Indeed, ?F1is a refinement of ?F2: U ?F2V ⇒ U ?F1V .

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3.1.3. Pareto dominance (?F3)

This heuristic looks for the option that wins on both the positive and

negative sides. This rule compares the two sets of arguments as a

problem of bi-criteria decision. The first criterion compares negative

arguments according to Wald’s rule (Wald, 1950/1971): U is better

than V on the negative side if and only if M(U−) ≤ M(V−). The sec-

ond criterion compares positive arguments according to the optimistic

counterpart of Wald’s rule. Formally, U ?F3V if and only if:

M(U+) ≥ M(V+)

and

M(U−) ≤ M(V−)

To Emma, there is a strong argument for U, but only weak argu-

ments for V : M(U+) = α > M(V+) = β. In parallel, there are strong

arguments both against U and against V : M(U−) = α = M(V−).

Emma will go and see movie 1.

The relation ?F3is transitive but not complete. For example, as

soon as an option has both pros and cons, whatever their importance,

it becomes incomparable to the null option (no pro, no con).

3.2. Inclusion heuristics I1 and I2

While the heuristics we have considered so far have some intuitive ap-

peal, they all suffer from a notable shortcoming— that is, they do not

satisfy the principle of preferential independence. This principle states

that if U is preferred to V , then this preference should not change

when the descriptions of U and V are enriched by the exact same

set of arguments. Formally: ∀U,V,W such that (U ∪ V ) ∩ W = ∅,

U ? V ⇐⇒ U ∪ W ? V ∪ W.

Consider for example the case of Emma’s cousin, Francine. Francine

must decide whether she will go and see movie 3, about which she knows

nothing, or movie 4, which features an actress she likes (a weak pro). All

three Focus heuristics would (reasonably) predict that Francine will go

and see movie 4. But let us now add the information that both movies

are by Francine’s favorite director (a strong pro in each case). Now, all

three Focus heuristics predict that Francine will be indifferent between

the two movies—a disputable prediction, and a violation of the principle

of preferential independence.

The heuristics in the “Inclusion” family are variants of the Focus

heuristics, which satisfy the principle of preferential independence. They

do so by first cancelling the arguments that appear in the descriptions

of both options, before applying one of the Focus heuristics. Formally:

U ?I1V

U ?I2V

⇐⇒ U \ V ?F1V \ U

⇐⇒ U \ V ?F2V \ U

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