Preference Handling for Belief-Based Rational Decisions
Samy Sá∗, João Alcântara∗
1
Universidade Federal do Ceará (UFC)
MDCC, Campus do Pici, Bl 910 – Fortaleza, Brazil
[email protected],[email protected]
Abstract. We present an approach to preferences suitable for agents that base
decisions on their beliefs. In our work, agents’ preferences are perceived as a
consequence of their beliefs, but at the same time are used to feed the knowledge base with beliefs about preferences. As a result, agents can reason with
preferences to hypothesize, explain decisions, and review preferences in face of
new information. Finally, we integrate utility-based to reasoning-based criteria
of decision making.
1. Introduction
Autonomous agents are frequently required to make decisions and expected to do so according to their beliefs, goals, and preferences, however, beliefs are rarely connected to
the preferences of the agent. Dietrich and List argue in [Dietrich and List 2013] that logical reasoning and the economic concept of rationality are almost entirely disconnected
in the literature: while logical accounts of reasoning rarely gets to deal with rational decisions in the economic sense, social choice is never worried about the origin of agents’
preferences. But if preferences are disconnected from beliefs, how can an agent explain
their decisions? How could we model the influence of new information in an agent’s
preferences?
In order to reason about options available in a decision, an agent needs a complimentary theory of what the best options are. Since rationality involves trying to maximize
gains (utility), the successful integration of rationality and beliefs requires quantifying the
utility of some of those beliefs. By doing so, agents’ preferences are a reflex of beliefs
about the possible decisions and the agent is capable of different reasoning tasks involving
their preferences such as (i) building arguments to explain decisions; (ii) considering alternative scenarios with different outcomes for the decision; (iii) automatically updating
preferences in face of new information; and (iv) hypothesizing with preferences.
Further, reasoning with preferences under uncertainty is a feature rarely considered in the literature (as mentioned in [Domshlak et al. 2011]), even though it plays a key
role in rational choice and game theory [Osborne and Rubinstein 1994]. By accounting
for knowledge bases as answer set programs [Gelfond and Lifschitz 1991], our approach
can model multiple plausible scenarios of a decision, therefore accounting for uncertainty
while making it easy to perform abduction and belief revision as means to evaluate or update an agent’s preferences. In this paper, we show how comparing available options by
∗
This paper is an extended and revised version of [Sá and Alcântara 2013]. Our work is supported by
CNPq (Universal 2012 - Proc. 473110/2012-1), CAPES (PROCAD 2009), and CNPq/CAPES (Casadinho/PROCAD 2011).
weighing their relevant features can successfully connect rationality and qualitative preferences in a way to relate rational decision criteria [Osborne and Rubinstein 1994] and an
inherently qualitative criteria for reasoning-based decisions [Dubois et al. 2008].
The paper is organized as follows. In Section 2, we discuss some concepts of
the formalism used in the paper. In Section 3, we present our preferences model. In
Section 4, we show how beliefs about preferences can be integrated into a knowledge
base to perform different reasoning tasks revolving preferences. Section 5 shows how
agents make decisions and that utility-based decisions are related to beliefs in our model.
Related work is in Section 7, then our conclusions and references complete the paper.
2. Knowledge Bases
In this paper, we consider agents with knowledge bases (or belief sets) as answer set
programs [Gelfond and Lifschitz 1991]. Such programs can present several models called
answer sets, to which we sometimes refer as possible or plausible scenarios conceived by
the knowledge base. The possible presence of multiple models is an interesting feature to
account for uncertainty and we will later show that they are suitable to express uncertainty
over preferences and arguing about them. The notions of preferences here introduced can
be generalized to other kinds of logic languages with unary predicates.
An Extended Logic Program (ELP) or Answer Set Program
[Gelfond and Lifschitz 1991] is defined over a Herbrand Base HB, the set of all
ground atoms the program might resort to. An ELP consists of a set of rules of the form
r : L0 ← L1 , . . . , Lm , not Lm+1 , . . . , not Ln
such that each Li is a literal, i.e., it is either an atom (A) or its negation (¬A). The symbol
not denotes negation as failure (NAF) and not L is a NAF-literal if L is a literal. We
may speak of literals to generalize literals and NAF-literals. In a rule r as above, we
refer to L0 as the head of r and write head(r) to denote the set {L0 }. We refer to the
conjunction L1 , . . . , Lm , not Lm+1 , . . . , not Ln as the body of r, and body(r) denotes the
set {L1 , . . . , Lm , not Lm+1 , . . . , not Ln }. The literals of the positive and negative parts
of the body i.e. the sets {L1 , . . . , Lm } and {Lm+1 , . . . , Ln }, are denoted by body + (r) and
body − (r) respectively. We also indicate the set of NAF-Literals {not Lm+1 , . . . , not Ln }
as not body − (r). A rule may be written as head(r) ← body + (r), not body − (r) as well.
A rule is an integrity constraint if head(r) = ∅ and it is a fact if body(r) = ∅.
We say that a program, rule or literal without variables is ground. A rule with
variables is seen as a succinct manner to represent all of it’s ground instances, which
can be computed by applying every possible substitution θ = {x1 /t1 , . . . xn /tn } from
variables to terms in HB where x1 , . . . , xn are all distinct variables and all ti is a distinct
term from xi . We often refer to ground predicated formulas as propositions.
The semantics of an ELP is given by the Answer Sets Semantics
[Gelfond and Lifschitz 1991]. A program may have one, zero or multiple answer sets.
An answer set S for KB is consistent if S does not simultaneously contain A and ¬A, for
no atom in the language. The program itself will be said consistent if it has a consistent
answer set. Otherwise, the program is inconsistent. Throughout the paper we will assume
only consistent programs. A goal is a conjunction of literals and NAF-literals. We say
that KB credulously (resp. skeptically) satisfies a goal G if some (resp. all) of its answer
sets satisfy G, in which case we write KB |=c G (resp. KB |=s G).
3. Preferences as Utility + Beliefs
In this section, we introduce our notions of preference profiles and quality thresholds.
Agent preferences are drawn on top of beliefs by attributing weights to unary predicates
expressing relevant features (alike with what is done in weighted propositional formulas
[Lafage and Lang 2000]), which results in an unary utility function. Quality thresholds
are employed to classify options as good, poor or neutral in the eyes of an agent.
Definition 1 (preference profile) Let P red be the set of all unary predicates P (x) used
to express possible features of options (e.g. if a car is expensive) in the language of an
agent. A preference profile is a triple P r = hU t, U p, Lwi, involving a utility function
U t : P red → R, and upper and lower utility thresholds U p, Lw ∈ R, U p ≥ Lw.
An agent can have as many preference profiles as desired for each kind of decision
the agent may get involved (as proposed in [Dietrich and List 2013]). Intuitively, each
preference profile specifies a view about (i) which features (predicates) are relevant to
quality available options (they have U t(P (x)) 6= 0); and (ii) what it takes for an option to
be qualified as good, poor or indifferent. The profiles (both utilities and thresholds) can
be designed together with the knowledge base or obtained from other methods, such as
modeling user preferences. In each case, a preference profile gives us two perspectives of
an agents preferences: quantitative and qualitative.
Definition 2 (available options) We write O = {o1 , . . . , on }, n ≥ 2 to refer to the set of
options (outcomes) available in a decision.
The available options are constants in the language of the agent.
Given a program KB and a preference profile P r = hU t, U p, Lwi, to rank the
available outcomes is straightforward, as each option oi ∈ O, has utility
P
U tS (oi ) =
U t(P (x))
P (oi )∈S
in the answer set S. The utility function takes answer sets as a parameter because
the multiplicity of answer sets may suggest uncertainty about the features satisfied by each
option. In a particular answer set S, oi is perceived as a good option if U tS (oi ) ≥ U p, a
poor option if U tS (oi ) < Lw, and as neutral (neither good or poor) otherwise. Clearly, if
U p = Lw, there are only good and poor options.
The concept of preference profile in Definition 1 induces, for each answer set S
of the agent theory, a total preorder1 over the set of available options
<S = {(oi , oj ) | U tS (oi ) ≥ U tS (oj )}.
The proposition oi <S oj is usually read as “oi is weakly preferred to oj
[Fishburn 1999] in S”. If the agent is indifferent about two options oi , oj , i.e., if both
oi <S oj and oj <S oi hold, we write oi ∼S oj .
From now on, we will account for agents as a pair Ag = (KB, P ref s) involving
a program KB and a set P ref s of preference profiles.
1
A total preorder is a relation that is transitive, reflexive and in which any two elements are related.
Total preorders are also called weak orders.
Example 1 Suppose an agent whose knowledge base includes beliefs about Australia (a),
Brazil (b), and Canada (c), possible destinations (D(x)) of a future vacation trip. Interesting features of a destination to the agent include the presence of famous beaches (Bc(x)),
the availability of cheap flights (Ch(x)) and whether there is a spoken language in x that
is unknown to the agent (U L(x)). A destination x will present unknown languages if there
is at least one language y that is spoken there (SL(y, x)) that is not amongst those spoken
by the agent (S(y)). The agent speaks two languages, namely English (en) and German
(ge). Finally the agent knows their partner has a favorite destination (P F (x)), but does
not know which one is their favorite.
We use the semi-colon to separate rules and a period to mark the last one.
KB : D(a) ← ; D(b) ← ; D(c) ← ;
Bc(a) ← ; Bc(b) ← ; ¬Bc(c) ← ;
Ch(c) ← ;
U L(x) ← D(x), SL(y, x), not S(y) ;
S(en) ← ; S(ge) ← ;
SL(en, a) ← ; SL(pt, b) ← ; SL(en, c) ← ; SL(f r, c) ← ;
P F (a) ← not P F (b), not P F (c);
P F (b) ← not P F (c), not P F (a);
P F (c) ← not P F (a), not P F (b).
The agent has a few preference profiles for this decision:
• P r1 = hU t1 = {(Bc(x), 3), (U L(x), −1), (Ch(x), 4), (D(x), 0), (S(x), 0),
(P F (x), 0)}, U p1 = 3, Lw1 = 2i;
• P r2 = hU t2 = {(Bc(x), 3), (U L(x), 2), (Ch(x), 4), (D(x), 0), (S(x), 0),
(P F (x), 0)}, U p2 = 5, Lw2 = 4i;
• P r3 = hU t3 = {(Bc(x), 3), (U L(x), −1), (Ch(x), 4), (D(x), 0), (S(x), 0),
(P F (x), 3)}, U p3 = 5, Lw3 = 4i.
The program has three answer sets S1 = S ∪ {P F (a)}, S2 = S ∪ {P F (b)} and S3 =
S ∪ {P F (c)}, where S = {Bc(a), Bc(b), U L(b), Ch(c)}. For preference profiles P r1
and P r2 , the utilities are the same in all answer sets. If the agent selects the preference
profile P r1 , it would find utilities U tS1 1 (a) = 3, U tS1 1 (b) = 2, and U tS1 1 (c) = 4. In that
case, option c (going to Canada) is the favorite, but going to Australia is also a good
option and the agent is neutral about going to Brazil. If the agent selects the preference
profile P r2 , however, an option will only be good if it holds two or more relevant features,
since no single trait has utility higher than U p2 on its own. In that case, going to Brazil
would be the only good option, while going to Canada would be still considered neutral.
The last profile P r3 suggests uncertainty, as in S1 Australia is the only good option while
going to Brazil is a poor option and going Canada is neither a good nor a poor option
(it is neutral), but the different answer sets S2 and S3 suggests differently. This models a
state of mind where the preference of the agent’s partner is definitive of their own: Each
answer set favors a different destination according to the partner’s favorite. The utilities
and status of the three options according to each profile are summarized in Table 1.
3.1. On Multiple Preference Profiles
In [Dietrich and List 2013], the authors propose that the preferences of an agent are conditioned by a set of reasons M and each subset of M induces a different preference ordering. In our approach, different motivational states are represented as alternative preference
Australia
Brazil
Canada
P r1 in Si
3, good
2, neutral
4, good
P r2 in Si
3, poor
5, good
4, neutral
P r3 in S1
6, good
2, poor
4, neutral
P r3 in S2
3, poor
5, good
4, neutral
P r3 in S3
3, poor
2, poor
7, good
Table 1. The options according to each profile in each answer set Si , i ∈ {1, 2, 3}.
profiles about the same decision. Special propositions in the theory of the agent can be
used to determine which profile is to be assumed. Dietrich and List also introduce two
axioms which should govern the relationship between an agent’s set of beliefs and their
preferences across variations of their motivational state:
Axiom 1 ([Dietrich and List 2013]) For any two options oi , oj ∈ O and any M ⊆ P red,
if {P (x) ∈ M | P (oi ) holds} = {P (x) ∈ M | P (oj ) holds}, then x ∼ y.
Axiom 2 ([Dietrich and List 2013]) For any oi , oj ∈ O and any M, M 0 ⊆ P red such
that M 0 ⊇ M , if {P (x) ∈ M 0 \ M | P (oi ) or P (oj ) hold} = ∅, then x <M y ⇔ x <M 0 y.
Roughly speaking, Axiom 1 states that two options with the exact same characteristics should be equally preferred. Axiom 2 treats the case where new attributes become
relevant to the decision. In that case, the preferences over any two options that do not
satisfy the extra criteria should remain unchanged.
Theorem 1 Each preference relation <S (Sec. 3) satisfies Axioms 1 and 2.
Proof 1 Consider any consistent answer set S of the agent’s knowledge base KB. If two
options satisfy the exact same predicates in S, the utility attributed to both options will
be the same, so Axiom 1 is satisfied. Now suppose a preference profile P r is updated in
a way the predicate P (x) becomes relevant, i.e., in the updated profile P r0 , U t0 (P (x)) 6=
0, while originally, U t(P (x)) = 0. If neither options oi , oj satisfy P (x) in S, i..e, if
P (oi ), P (oj ) 6∈ S, then their utility remains unchanged. Therefore, Axiom 2 is satisfied.
As a consequence, our approach to model preferences of an agent will adequately
relate their belief sets to their preferences. Variations of the agent’s motivational state
are modeled as different profiles for the same decisions. We will see in the next section
how the preferences are integrated in the knowledge base and briefly comment on how
the different profiles can be considered as required.
4. Rules for the Evaluation of Options and Beliefs About Preferences
In this section, we will show how to build a general theory of preferences so agents are
able to reason about the quality of available options. We use the utility function and
the qualitative thresholds of a preference profile (Sec. 3) to devise a set of special rules
that, when appended to the original knowledge base (as a module), promotes conclusions
about the quality of available options. Therefore, rules encoding preferences should not
interfere much with the models of the original program: It is tolerated to add conclusions
about the quality of options and nothing else.
For that matter, we introduce predicates G(x), N (x), standing respectively for
good (G(x)) and neutral (N (x)) options, while options known not to be good are poor
(¬G(x)). We assume G(x), N (x) are reserved, so they do not appear in knowledge bases.
Given an agent Ag = (KB, P ref s) and a selected preference profile P r ∈
P ref s, P r = hU t, U p, Lwi, the set KBP r of preference rules is built as follows:
Let R∗ ⊆ P red be the set of predicates relevant to a decision, i.e., any P (x) with
U t(P (x)) 6= 0, compute the utility of each subset of features from R,
P
R
Ut =
U t(P (x)),
P (x)∈R
for each R ⊆ R∗ . Then, KBP r will be the set of rules r of the form
r : head(r) ← body + (r), not body − (r) where:
• body + (r) = {P (x)|P (x) ∈ R};
• and not body − (r) = {not P (x)|P (x) ∈ R∗ \ R};
• head(r) = G(x) if R U t ≥ U p, head(r) = ¬G(x) if R U t < Lw, or head(r) =
N (x) otherwise;
Example 2 (Continuation of Example 1) For the agent in Example 1, we would have a
complimentary knowledge base KBP r1 , based on profile P r1 , as follows:
KBP r1 : G(x) ← Bc(x), U L(x), Ch(x);
N (x) ← Bc(x), U L(x), not Ch(x);
G(x) ← Bc(x), not U L(x), Ch(x);
G(x) ← Bc(x), not U L(x), not Ch(x);
G(x) ← not Bc(x), U L(x), Ch(x);
¬G(x) ← not Bc(x), U L(x), not Ch(x);
G(x) ← not Bc(x), not U L(x), Ch(x);
¬G(x) ← not Bc(x), not U L(x), not Ch(x);
The resulting agent knowledge base will be KB 1 = KB ∪ KBP r1 .
The inclusion of rules about preferences provides a description with a global perspective of what good/poor/neutral options are, enriching the original knowledge base
with conclusions about the quality of available options. While employing rules as above,
we have the integration of preferences and beliefs in two ways. First, the beliefs promote
the utility of each option and, therefore, the preferences are based on them. Second, the
calculated utilities of each option are used to complement the knowledge base with predicated formulas regarding beliefs about them. Observe the composed program has the
same answer sets as the original with only extra conclusions about the quality of each option, so proposed set of rules also allow different answer sets to have different conclusions
on the quality of available outcomes (such as proposed in Table 1 based on Example 1).
This models uncertainty about the utility and quality of available options.
It can be observed that the number of rules is exponential in the number |R∗ | of
relevant predicates to a profile. However, this calculation needs to be done only once and
the number of predicates is not likely to be high. Also, this reflects the complexity of
making decision by comparing options according to a particular set of relevant properties.
Regarding the multiplicity of profiles, a complimentary agent theory on its motivational states could be devised to conclude which profile to access in each situation. This
can be attained by linking the rules from preference profiles with new formulas reserved
for that purpose and an extra set of rules with such reserved formulas for conclusions.
5. Making Decisions
Now we will show that our approach to preferences successfully connects beliefs and
preferences as we prove that employing utility-based criteria to decision making satisfies
criteria from reasoning-based decision making.
In Dubois et al. [Dubois et al. 2008], attributes are classified as negative (−),
neutral (0) or positive (+). The positive (resp. negative) attributes are then perceived
as arguments in favor (resp. against) the available outcomes and the decision is based
on comparing the sets of arguments against and in favor, which are here introduced as
features (unary predicates). As the utility of features in our approach ranges in the real
numbers, our own model of preferences can be seen as bipolar. Thus, our proposal induces a bipolar decision process and should satisfy the principle of bivariate monotony
[Dubois et al. 2008] which we adapted to deal with uncertainty and multiple scenarios.
In the following sections, consider oi and oj are two options, KB denotes a
S
knowledge base, and the preference profile P r = hU t, U p, Lwi is selected. Let o+
i
denote the set of positive features satisfied in S, i.e., those P (x) such that P (oi ) ∈ S and
S
U t(P (x)) > 0, and o−
denote the set of negative features satisfied in S, i.e., those P (x)
i
such that P (oi ) ∈ S and U t(P (x)) < 0.
5.1. Single Agent Decision
Definition 3 (bivariate monotonicity) A decision criteria satisfies bivariate monotonicity
−S
S
S
S
if, for each answer set S of KB, whenever o+
⊇ o+
⊆ o−
i
j and oi
j available, it prefers
oi at least as much as oj .
Bivariate monotonicity states that whenever an option has all advantages of another (possibly more), but no disadvantages the second does not (possibly less), the first
option is possibly better. Of course, if the set of pros and cons is exactly the same, one
option is just as good as the other.
Theorem 2 When there is no uncertainty, i.e., there is a single plausible scenario, maximizing utility satisfies bivariate monotonicity.
Proof 2 Remember that positive features have positive utility while negative features have
+
−
−
negative utility. Then, in any specific scenario S, if o+
i ⊇ oj while oi ⊆ oj we will have
U tS (oi ) ≥ U tS (oj ). We conclude that maximizing utility satisfies bivariate monotonicity.
5.2. Single Agent Under Uncertainty
Our notion of preference profile relates to game theory, since multiple plausible scenarios
induces the construction of a payoff matrix. See, for instance, that in Example 1, utility
varies amongst answer sets when the preference profile P r3 is assumed. Table 2 below
depicts the induced payoff matrix with Australia, Brazil and Canada as available moves
and the possible states of nature are S1 , S2 , S3 , the answer sets of KB.
For this reason, it is natural to associate decision making under uncertainty to decision criteria from game theory [Osborne and Rubinstein 1994]. We apply the maximin
[Osborne and Rubinstein 1994] criteria to compare options, which consists of trying to
maximize the minimal gains: For each option oi ∈ O, we calculate M IN ({U tS (oi ) | S ∈
AS}), the minimal utility attributed to each oi amongst scenarios, then we compare these
results to induce a partial preorder <M AXIM IN over the set of alternatives.
S1
S2
S3
Australia 6, good
3, poor
3, poor
Brazil
2, poor
5, good 2, poor
Canada 4, neutral 4, neutral 7, good
Table 2. A payoff matrix with possible scenarios S1 , S2 , andS3 .
Definition 4 (maximin criteria) In a decision problem with uncertainty, let AS denote
the set of possible scenarios (answer sets) compatible with the beliefs of an agent. An
option oi is at least as preferred as the option oj by maximin, i.e. oi <M AXIM IN oj iff
M IN ({U tS (oi ) | S ∈ AS}) ≥ M IN ({U tS (oj ) | S ∈ AS}).
The best options are the maximal elements of <M AXIM IN , i.e., those oi such that
oi <M AXIM IN oj for all oj , j 6= i.
Example 3 (Uncertainty) Consider an agent with preferences as depicted in Table 2. The
agent can choose from Australia, Brasil or Canada, and there are three possible scenarios
where the decision may take effect. The agent has no control about which scenario will
be the case, so making a rational choice involves considering all three. The maximin
criteria proposes the agent to maximize the minimal gain, i.e., to get the best guaranteed
satisfaction. Since the minimal payoffs for Australia, Brazil and Canada are, respectively,
3, 2, and 4, by the maximin criteria, the agent should choose to go to Canada.
Theorem 3 The maximin criteria for decision making under uncertainty satisfies bivariate monotonicity (Definition 3) in the case of multiple scenarios.
S
S
S
S
Proof 3 Let oi , oj be any two available options, and suppose o+
⊇ o+
and o−
⊆ o−
i
j
i
j
hold in every S ∈ AS (if any different, the options cannot be directly compared). Let
the quality of any options with the least quality be M IN ({U tS (oj ) | S ∈ AS}) = m.
Let M ∈ AS be any answer set such that U tM (oj ) = m. By Theorem 2, U tM (oi ) ≥
U tM (oj ), so the least utility attributed to oi is M IN ({U tS (oi ) | S ∈ AS}) ≥ m and
oi <M AXIM IN oj . We conclude oi <M AXIM IN oj , i.e., oi is at least as preferred as oj .
5.3. Quality Instead of Utility
Alternatively, the agent can just try to maximize the quality of it’s choice. In that setting, if the agent consider several different options all as good, picking any good option would be rational, even if the choice does not maximize utility. This kind of decision is usually the case in many settings, as argued, for instance in Approval Voting
[Brams and Fishburn 1978] where agents attribute votes to each of those options they
consider to be good enough. As the next theorem states, attempts to maximize quality as
the criteria for rationality also satisfy bivariate monotonicity.
In the following, QS (oi ) stands for the quality of option oi in scenario S. We
compare the quality of different options by considering good > neutral > poor.
Theorem 4 Maximizing quality satisfies bivariate monotonicity (Definition 3).
+
−
−
Proof 4 In a specific scenario S, if o+
i ⊇ oj while oi ⊆ oj , we have, from Theorem 2,
that U tS (oi ) ≥ U tS (oj ). As quality is obtained from utility, QS (oi ) ≥ QS (oj ) follows.
This is also the case with multiple scenarios (uncertainty) and the criteria of choosing the option with maximal minimal quality.
Theorem 5 The maximin criteria for decision making under uncertainty, when interpreted as maximizing the minimal quality of a choice, satisfies bivariate monotonicity
(Definition 3) in the case of multiple scenarios.
S
S
S
S
Proof 5 Let oi , oj be any two available options, and suppose o+
⊇ o+
and o−
⊆ o−
i
j
i
j
hold in every S ∈ AS (if any different, the options cannot be directly compared). Let
the quality of any options with the least quality be M IN ({QS (oj ) | S ∈ AS}) = q and
M ∈ AS be an answer set such that QM (oj ) = q. By Theorem 4, QM (oi ) ≥ QM (oj ), so
the least quality attributed to oi is M IN ({QS (oi ) | S ∈ AS}) ≥ q. As a consequence,
oi <M AXIM IN oj , i.e., oi is at least as preferred as oj .
6. On Reasoning About Preferences
Besides connecting beliefs and preferences, our approach is flexible enough to deal with
different aspects of reasoning, even simultaneously. One key aspect of our model is the
ability to have different perspectives of preferences integrated in a single formalism. We
argue this opens many doors towards reasoning about preferences, such as building arguments to explain decisions and having preferences easily updated in case the agent learns
new information. We briefly explain some possible features.
6.1. Arguments About The Quality of Options
As a desirable capacity of agents in a society, and especially for those that take part
in collective decisions, being able to explain their decisions or opinions in a collective
decision is very important. Intuitively, an argument consists of a conclusion and some
justification to it (a proof) [Sá and Alcântara 2012]. As we introduce rules on the quality
of options (Sec. 4), the agent becomes able to build arguments to explain their decisions
or argue about options. The opinion of an agent about a particular option can also be used
to determine which roles (proponent, opponent) the agent will play in a group decision
dialogue. For instance, if the agent believes an option oi cannot be good, i.e., KB 6|=c
G(oi ), the agent could oppose (argue against) that the group chooses oi .
6.2. Revising Preferences by Revising Beliefs
Adding or removing formulas in an agent theory KB can change its models and, therefore, the conclusions of an agent about how good each available option is. Therefore, performing belief revision [Alchourrón et al. 1985, Krümpelmann and Kern-Isberner 2012]
reflects in automatic updates to the utilities attributed to available options, while any utility
functions originally used are not affected. As a consequence, the rules about preferences
(from Sec. 4) are kept the same and no extra computation is required for these rules, but
the agent is always prepared to re-evaluate options in face of new information.
6.3. Hypothesizing About Preferences
An agent capable of abductive reasoning2 can hypothesize over options by conceiving
alternative scenarios where the quality of available options would be different. This
2
For a survey on abduction in logic programming, please refer to [Denecker and Kakas 2002].
is particularly useful in a number of interactions with other agents such as to to formulate questions for inquiry, build special arguments to seek consensus in deliberation dialogues [Sá and Alcântara 2012] or to to elaborate new proposals in negotiation
[Sakama and Inoue 2007].
6.4. Dealing with New Options
In a decision process, the agent could learn about other options previously unknown.
If equipped with abduction and belief revision, the agent is able to conceive alternative
situations (abductively) in which such options would be good (or neutral, or poor), in
order to inquiry and learn facts (belief revision) to attribute them utility. As shown in
[Kakas and Mancarella 1990] and [Sakama and Inoue 2003], abductive reasoning can be
used to guide knowledge updates, since explanations indicate consistent ways to update
a program in order to satisfy goal formulas. Based on inquiry or dialogues, the agent is
therefore able to learn about the new options and have proper opinions on their quality.
7. Related Work
Closely related to our work, Dietrich and List [Dietrich and List 2013] observe that logical reasoning and the economic concept of rationality are almost entirely disconnected in
the literature. In their work, Dietrich and List propose preference orderings based on alternative logical contexts as being different psychological states of the agent. We connect
logics and utility-based decisions in this paper by attaching utilities to predicates, so the
best outcomes satisfy the most interesting combinations of predicates amongst available
candidates. In particular, we connect rational behavior from game theory to bipolar preferences due to Dubois et al. [Dubois et al. 2008]. On the other hand, Dietrich and List are
not concerned with reasoning about preferences in [Dietrich and List 2013], but restrict
their analysis to how beliefs can influence decisions.
Lafage and Lang propose in [Lafage and Lang 2000] an approach to group decisions based on weighted logics in which formulas represent constraints and weights are
attributed to each formula to quantify importance. This is related to the way we build utility functions and preferences, however they allow formulas with connectives, while we
restrict utility evaluation to ground formulas with unary predicates. Brafman proposes in
[Brafman 2011] a relational language of preferences where rules may include utility values in the conclusions. The author argues the approach is flexible, as their value functions
can handle a dynamic universe of objects. We consider our approach is also very flexible, as an agent can quickly evaluate a new option, since all the weight is in the features
relevant to the agent.
Several notions of preferences in logic programs have been studied in the
literature, but mainly applied to rank the different models a program may have
[Delgrande et al. 2004]. As it can be observed in Delgrande et al. [Delgrande et al. 2004],
most approaches involve introducing priorities to rules to induce an ordering over the answer sets. An example is the work of Brewka and Eiter [Brewka and Eiter 1999]. In
a different approach Sakama and Inoue introduced priorities over propositions (atoms,
ground predicated literals) from the program language in [Sakama and Inoue 2000]. Our
proposal, on the other hand, takes advantage of multiple answer sets to reason over uncertainty as decision are utility-based and built over unary predicates. This kind of approach
is surprisingly not common, as also stated by Doyle in [Doyle 2004]. A notable exception
is [Labreuche 2011], where Labreuche deals with the issue of explaining a decision made
in a multi-attribute decision model where attributes are weighted in an attempt to circumvent the difficulties of properly explaining utility-based decisions. Anyway, our work has
an alike motivation: arguments are better related to logic-based formalisms, so we attach
weights to attributes (predicates) to calculate preferences and translate them back into a
logical language. For an account of approaches and applications of preferences in Artificial Intelligence, see the more recent survey from Domshlak et al. [Domshlak et al. 2011].
8. Conclusions and Future Work
We have introduced an approach to integrate beliefs and rationality in a way utility-based
criteria for decisions is shown to satisfy criteria from reasoning-based decision making.
This is achieved by attributing utility to unary predicates used to encode relevant qualities
of a decision, together with the notion of qualitative thresholds, both parts of our preference profiles. By doing so, agent preferences are simultaneously perceived in different
perspectives: a utility-based cardinal order, a regular ordinal order and a classification in
good/poor/neutral options. All three perspectives are integrated as we devise rules that
encode the preferences of the agent and append them to its knowledge base for reasoning purposes. The result is a formalism capable of modeling reasoning about preferences
where beliefs are the base for rational decisions. Next, we intend to look deeper in the relations between logical consequences and utility-based choices induced by our proposal
to preferences. We also plan to explore how this model can be employed in group decisions by deliberation and voting. Finally, motivated by work in bipolar decisions as
formal argumentation by Amgoud and Vesic [Amgoud and Vesic 2012], we will analyze
how formalisms of decision making by argumentation relate to rationality criteria of decision making in groups if our approach to preferences is used.
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