From: AAAI Technical Report WS-02-06. Compilation copyright © 2002, AAAI (www.aaai.org). All rights reserved.
Decisions and Gamesfor BD Agents
Mehdi Dastani
Institute of Informationand Computing
Sciences
Utrecht University
The Netherlands
email: mehdi@cs.uu.nl
Abstract
Strategic gamesmodelthe interaction betweensimultaneous
decisionsof a groupof agents.Thestarting pointof strategic games
is a set of players(agents)havingcertainstrategies
(decisions)andpreferencesonthe game’soutcomes.In this
paperwedo not assumethe set of decisionsand preferences
of agentsto be given,butderivethemfromtheir mentalattitudes. In particular,weintroducea rule-basedarchitecture
for agentswithbeliefs anddesiresandexplainhowtheir decisions andpreferencescan be derived. Wespecify groups
of suchagents,definea mapping
fromtheir specificationto
the specificationof the gametheyplay,anduse somefamiliar
notionsfromgametheory,suchas ParetoefficiencyandNash
equilibrium,
to characterizethe interactionbetween
their decisions. Wealso discussa reversemapping
fromthe specificationof games
that a groupof agentplayto the specifications
of thoseagents.Thismapping
can be usedto specifygroups
of agentsthat canplaya ceratingame.
Introduction
One of the main problemsin agent theory is the distinction betweenformal theories and tools developedfor individual autonomous
agents, and theories and tools developed
for multi agent systems.In the social sciences, this distinction is called the micro-macrodichotomy.Theprototypical
exampleis the distinction betweenclassical decision theory
based on the expected utility paradigm(usually identified
with the work of Neumannand Morgenstem(von Neumann
&Morgenstern1944) and Savage(Savage 1954)) and classical gametheory (such as the workof Nashand morerecently
the workof Axelrod). Whereasclassical decision theory is
a kind of optimization problem(maximizingthe agent’s expectedutility), classical gametheory is a kind of equilibria
analysis.
There are several approachesin artificial intelligence,
cognitive science, and practical reasoning(within philosophy) to describe the decision makingof individual agents
and to bring the micro and macrodescriptions together.
In these approaches, the decision makingof individual autonomous
agents is describedin terms of other conceptsthan
maximizing
utility. For example,since the early 40sthere is
a distinction betweenclassical decisiontheory and artificial
Copyright©2002,American
Associationfor Artificial Intelligence(www.aaai.org).
All rights reserved.
37
Leendert van der Torte
Department
of Artificial Intelligence
Vrije Universiteit Amsterdam
The Netherlands
email: torre@cs.vu.nl
intelligence basedon utility aspiration levels and goal based
planning (as pioneered by Simon (Simon 1981)). Moreover, in cognitive science and philosophythe decision making of individual agents is described in terms of concepts
fromfolk psychologylike beliefs, desires and intentions. In
these studies, the decision makingof individual agents is
characterized in terms of a rational balance betweenthese
concepts, and the decision makingof a group of agents is
described in terms of conceptsgeneralized from those used
for individual agents, such as joint goals, joint intentions,
joint commitments,etc. Moreover,also new concepts are
introducedat this social level, suchas norms(a central concept in mostsocial theories).
It is still an open problem howthe micro-macro dichotomyof classical decision and gametheory is related
to the micro-macrodichotomyof these alternative theoties. Has or can the micro and macro level be brought
together by replacing classical theories by alternative theodes? Before this question can be answered, the relation
betweenthe classical and alternative theories has to be clarified. Doyle and Thomason(Doyle & Thomasonsummer
1999) argue that classical decision theory should be reunited with alternative decision theories in so-called qualitative decisiontheory, whichstudies qualitative versionsof
classical decision theory, hybrid combinationsof quantitative and qualitative approachesto decision making,and decision makingin the contextof artificial intelligence applications such as planning, learning and collaboration. Qualitative decision theories have been developedbased on beliefs (probabilities) anddesires (utilities) usingformaltools
such as modallogic (Boutilier 1994)and on utility functions
and knowledge(Lang 1996). Morerecently these beliefsdesires modelshave been extended with intentions or BDI
models (Cohen & Levesque 1990; Rao & Georgeff 1991a;
1991b).
In this paper, we introduce a rule based qualitative decision and gametheory for agents with beliefs and desires. Likeclassical decisiontheorybut in contrast to several
proposals in the BDI approach (Cohen &Levesque 1990;
Rao&Georgeff1991a), the theory does not incorporate decision processes, temporal reasoning, and scheduling. We
also ignore probabilistic decisions and related gameswith
mixedstrategies. In particular, we explain howdecisions
and preferences of individual agents can be derived from
their beliefs and desires. Wespecify groupsof agents and
discuss the interaction betweentheir decisions and preferences. This enables us to define a mappingfrom agent specification to the specificationof the gamethat they play. This
mappingconsiders agent decisions as agent strategies and
the consequencesof decisions as the outcomesof the game.
Finally, we discuss the reverse mappingfrom the specification of a gameto the specification of the agents that play
the game.This mappingprovides the beliefs and desires of
agents that can play a certain game.
A qualitative decision and gametheory
has the input T, is the extensioncalculated basedon _Rand
T. Thisis formalizedin the followingdefinition.
Definition 1 (Extension) Let R C_ LAWx LAWbe a set of
rules and T c LAWbe a set of sentences. The consequents
of the T-applicablerules are:
R(T) = {y ] x ~ y ¯ R,x ¯
andthe R extension ofT is the set of the consequentsof the
iteratively T-applicablerules:
ER(T) ¢’~c_x,R(c,~aw(X))c_x X
The qualitative decision and gametheory introducedin this
section is developedfor agents that have conditional beliefs and desires, i.e. the beliefs and desires of agents
are sets of rules. The architecture and the behavior of
this type of agents are studied in (Broersen et al. 2001b;
Dastani &van der Torre 2002). Here, we analyze this type
of agents froma decision and gametheoretic point of view
by studying possible decisions of individual agents and the
interaction betweenthese decisions. Wedo so by defining
an agent systemspecification that indicates possible decisions of agents. Weshowhowwe can derive agent decisions
and preferencesfroman agent systemspecification. First we
considerthe logic of rules we adopt.
Logic of rules
Thestarting point of any theory of decision is a distinction
between choices madeby the decision makerand choices
imposedon it by its environment. For example, a software upgrade agent (decision maker) mayhave the choice
to upgrade a computersystem at a particular time of the
day. The software company(the environmen0mayin turn
allow/disallow such a upgrade at a particular time. Let
S = {al,...,c~,} be the society or set of agents, then
we therefore assumen disjoint sets of propositional atoms:
A = AI U... U A, = {a, b, c,...} (agents’ decision variables (Lang1996) or controllable propositions (Boutilier
1994)) and W= {p, q, r,...} (the world parametersor
controllable propositions). In the sequel we consider each
decision makeras entities consisting of rules. Sucha decision makergenerates its decisions by applyingsubsets of
rules to its input. Beforewe proceedsomenotations will be
introduced.
¯ LA,, Lwand LAWfor the propositional languages built
up fromthese atomsin the usual way, and variables z, y,
... to stand for any sentencesof these languages.
¯ CnA,, Cnwand GnAWfor the consequence sets, and
~A~, ~Wand ~aw for satisfiability,
in any of these
propositionallogics.
¯ z =~ y for an ordered pair of propositional sentences
calleda rule.
¯ ER(T)for the R extension of T, as defined in Definition
1 below.
In our frameworkthe generation of decisions are formalized basedon the notion of extension. In particular, the decision of an agent, whichis specified by a set of rules R and
38
Wegivesomeproperties
oftheR extension
ofT inDefinition
1. First note that ER(T)is not closed underlogical
consequence. The following proposition showsthat ER(T)
is the smallest superset of T closed underthe rules R interpreted as inferencerules.
Proposition 1 Let
¯ E°(T) =
¯ E~(T) = EiR-I(T) U R(CnAw(EiR-l(T)))fori
We have ER(T) = U~°E~(T).
This definition of extensionsallowsinconsistent decisions
as illustrated by the followingexample.
Example 1 Let R = {T =~ p,a =~ ~p} and T = {a},
where T stands for any tautology like p V -~p. Wehave
ER(O) = {p} and ER(T) = {a,p,-~p}, i.e. the R extension ofT is inconsistent.
Of course, allowing inconsistent decisions maynot be
intuitive. Weare here concernedabout possible decisions
rather than reasonableor feasible decisions. Later wewill
define reasonableor feasible decisions by excludinginconsistent decisions.
Agent system specification
Anagent systemspecification given in Definition 2 contains
a set of agentsandfor eachagenta descriptionof its decision
problem. The agent’s decision problemis defined in terms
of its beliefs and desires, whichare consideredas belief and
desire rules, a priority orderingon the desire rules, as well as
a set of facts and an initial decision(or prior intentions).
assumethat agents are autonomous,in the sense that there
are no priorities betweendesires of distinct agents.
Definition 2 (Agentsystem specification) An agent system specification is a tuple AS = (S, F, B, D, >_, °) t hat
contains a set of agents S, and for each agent ~i a finite
n F,
set of facts Fi C Lw (F = [.Ji=l
i), afinite set of belief
rules Bi C LAW× Lw(B = uin=l Bi), a finite set of desire rules Di C_ LAWx LAW(D = Ui~l Di), a relation
_>~_cD, x D~(>_=[.J~n__
1 _>,) whichis a total ordering(i.e.
reflexive, transitive, andantisymmetricandfor any two elements dl and d2 in Di, either da >_i d2 or d2 >_i dl), and
a finite initial decision 6° C LA (6o = Uinl 6O). For an
agent ai ¯ S we write x ~i y for one of its rules.
In general, a belief rule is an orderedpair x =~4y with
x E LAWand y E Lw. This belief rule should be interpreted as ’the agentai believesy in contextx’. Adesire rule
is an ordered pair x =~i Y with x E LAWand y E LAW.
Thisdesire rule shouldbe interpreted as ’the agent desires y
in contextx’. It impliesthat the agent’sbeliefs are aboutthe
world(x =~4P), and not about the agent’s decisions. These
beliefs can be about the effects of decisions madeby the
agent(a =~i P) as well as beliefs aboutthe effects of parameters set by the world(p =~4q). Moreover,the agent’s desires
canbe aboutthe world(x =~i P, desire-to-be), but also about
the agent’s decisions (x =~i a, desire-to-do). Thesedesires
can be triggered by parametersset by the world (p ~i Y)
well as by decisions madeby the agent (a =~i y). Modelling
mentalattitudes such as beliefs and desires in terms of rules
results in whatmightbe called conditional mentalattitudes
(Broersenet al. 2001b).
Agent Decisions
The belief roles of the form LA ~ Lw can be used to determine the expected consequencesof a decision, where a
decision54 of agent ai is any subsetof LA, that containsthe
initial decision5i°. Theset of expectedconsequences
of this
decision 5i is the belief extensionof Fi U 5i. Moreover,we
considera feasible decisionas a decision that doesnot imply
a contradiction.
Definition 3 (Decisions) Let
AS
=
({al ....
,a,},F,B,D,>,50)
be an agent system
specification. AnAS decision profile 5 for agents al .....
an is 5 = (51,... 5n) where5i is a decision of agent
such that
50 C 5i C LA, fori = 1...n
A feasible decisionfor agentai is 54 suchthat
EB,( Fi U5i ) is consistent
A feasible decisionprofile is a decisionprofile suchthat
EB(Fu 5) is consistent
Thefollowingexampleillustrates the decisionsof a single
agent.
Example2Let A1 = {a, b, c, d, e}, W = {p,q} and
AS = ({al }, F, B, D, E, °) with F1 - -- { ~p}, B1 ={c
q,d ~ q,e =~ ~q}, DI = {T =~ a,T =~ b,b ~ p,T =~
q,d =~ q}, >1= {b =*- p > T =~ b}, and5° = {a}. The
initial decisiono51 reflects that the agenthasalreadydecided
in an earlier stage to reachthe desire T =~a. Notethat the
consequentsof all B1 rules are sentences of Lw, whereas
the antecedentsof the B1rules as well as the antecedents
and consequentsof the D1 rules are sentences of LAW.We
havedue to the definition of ER(S):
EB(FU {a}) = {--,p, a}
EB(Fu {a,b}) = {--,p,a,b}
Es(FU {a,c}) = {-~p,a,c,q}
EB(FU{a,d}) = {--,p,a,d,q}
EB(FU {a, e}) = {--,p, a, e, ~q}
EB(FU{a,d,e})
= {-~p,a,d,e,q,-~q}
39
Therefore{a, d, e} is not a feasible ASdecision profile, becauseits belief extension is inconsistent. Continuedin Example5.
According
to definition3, the feasibility of a decisionprofile, whichindicates the decisions of individual agents, is
formulatedin terms of the consistencyof the extension that
is calculated basedon the decision profile. However,there
are two different waysto calculate such an extension dependingon whetheragents can apply rules of each other or
not. First, the extension can be calculated by demanding
that each agent generatesits decisions by applyingonly its
ownbelief and desire rules. This wayof calculating the extension assumesthat mental attitudes of agents are private
and inaccessible to each other. Wewill call this notion of
feasible decision profile individualistic. Second,the extension can be calculated by allowingagents to apply belief and
desire rules of each other. This notion of feasible decision
profile, whichwill be called collective, assumesthat mental
attitudes of agents are public.
Definition4 Feasibledecision profiles can be defined in the
following two ways:
¯ An individualistic feasible decisionprofile 5 is a decision
profile if EB( F 0 5) = ~i=l EBi(Fi U54) is consistent.
¯ A collective feasible decisionprofile5 is a decisionprofile
if EB(FU S) = EB,u...uB, (F1 U 51 U . . . UUSn
) is
consistent.
Thefollowingexampleillustrates individualistic and collective feasible decisionprofiles.
Example3 Let A1 = {a}, A2 = {b}, W = {p} and
AS = ({al,a2},F,B,D,E,50
) with F1 = F2 = 0,
B1 = {a =~ p},
B2 = {b =~ -~p},
D1 = D2 = 0,
>_ is the identity relation, and 5o = 5° = 0. The decision profile 5 = (51,52) = (0, {b}) is both individualistic and collective since EB~uB2(F1 0 51 0 F2 0 52)
EBb(F1U 51) U EB2(F2U (I2) {b,-~p}. Bu t, th e decision profile 5 = (51,52) = ({a}, {b}) is neither individualistic nor collective since EB~UBz(FI
U 51 U F2 U 52)
EB, (z~’~l U 51) E,2(F2U 52) = {a,p, b, ~p}.
Finally, the followingexampleillustrates that a decision
profile canbe individualisticbut not collectivefeasible decision profile.
Example 4 Let A1 = {a}, A2 = {b}, W= {p, t} and
AS = ({al, a2}, F, B, D, >, 5°) with F1 = F2 = 0,B1 =
{a =~ p}, B2 = {b ~ t,p ~ ~t}, D1 = D2 = 0, >_ is
the identity relation, and60 =60 =0. Thedecision profile
5 = (51,52) = ({a}, {b}) is individualistic feasible since
EBa(F1USI)UEB2(F2U52)= {a,p}U{b,t} = {a,p,b,t}.
However,
5 is not a collective feasible decisionprofile since
EBIuBz(F1U51 UF2 052) = {a,p,b,t,-~t}.
In the following,we use the definition of individualistic
feasible decisionprofiles.
Agent preferences
In this section weintroduce a wayto comparedecisions. We
comparedecisions by comparingsets of desire rules that are
not reachedby the decisions. Since an agent systemspecification providesonlythe orderingon individual desire rules,
andnot orderingonsets of desire rules, wefirst lift the ordering on individual desire rules to an ordering on sets of
desire rules. In this way, we makeit possible to compare
sets of desire rules using the preferenceon individualdesire
rules.
Definition 5 Let AS = (S, F, B, D, >, 6O) be an agent systern specification,Di, ~Di" two subsets of Di, and Di \ i
the set of Di elements whichare not D[ elements. Wehave
D[ _D
>-i ifVd
"" " ED#i i\Di3d
,
i Di\DEi isuch
thatd I >
dII.
Wewrite D[ >- D[’ if D[_>-Di" and Di" ~ Di,’ and we write
D~~- D~’ if D~~_ D" and D~’~ D~.
Thisdefinitionstates that an agent~ prefers a set of desire
rules D~ to anotherset of desire rules D"if for eachdesire
rule in D"there exists a preferable desire rule in D~. Thus,
if D’ contains only one desire rule whichis preferable by c~
to any desire rule from D", then a prefers D~ to D". The
followingpropositionsshowthat the priority relation ___is
reflexive, anti-symmetric,
and transitive.
Proposition2 Let DI (£ D3and D39~ D1. For finite sets,
the relation ~- is reflexive ( VDD >- D), anti-symmetric
VD1, D2 (D1 ~- D2 ^ D2 ~- D1) ~ D1 = D2, and
transitive, i.e. D1~- D2and D2>- Daimplies Dz ~- D3.
The desire rules are used to comparethe decisions. The
comparisonis based on the set of unreacheddesires and not
on the set of violated or reached desires. A desire z =~y
is unreachedby a decision if the expectedconsequencesof
this decision implyx but not F- Thedesire rule is violated
or reached if these consequencesimply x A --,y or x A y,
respectively.
Definition 6 (Comparingdecisions) Let AS
=
(S, F, B, D, >_, °) be a n agent s ystem specification
and 6 be a AS decision. The unreacheddesires of decision
6for agent ai are:
Decision 6 is at least as good as decision 6’ for agent
ai, written as 6 >iv6’, iff
u,(6’)
_>u,(6)
Decision 6 dominatesdecision 6~ for agent ai, written as
6 >6’, iff
6 6’ and 6’ 6
Thus,a decision6 is preferredby agent c~ to decision6~ if
the set of unreached
desire rules by 6 is less preferredby ~ to
the set of unreacheddesire rules by 6~, i.e. the unreached
desire rules by 6 are less importantthan unreacheddesire rules
by 6’. The followingcontinuation of Example2 illustrates
the comparisonof decisions.
Example 5 (Continued) Wehave:
U({a}) = {T ~ b,T ~ q},
U({a, b}) = {b =~ p, T =~ q},
U({a, c}) = {T =>. q},
4O
U({a, d}) = {T::~ b, d =~ q},
U({a, e}) = {T:=~ b, T :=~ q},
U({a,b, c})= {bp}.
Wethus havefor examplethat the decision {a, c} dominates
the initial decision {a}, i.e. {a, c} >u {a}. Thereare two
decisions for whichtheir set of unreachedcontains only one
desire. Dueto the priority relation, wehavethat {a, c} > u
{a, b, c}.
From Agent System Specifications
to Game
Specifications
In this subsection, we consider agents interactions based
on agent system specifications, their correspondingagent
decisions, and the ordering on the decisions as explained
in previous subsections. Gametheory is the usual tool to
modelthe interaction betweenself-interested agents. Agents
select optimal decisions under the assumptionthat other
agents do likewise. This makesthe definition of an optimaldecision circular, and gametheory therefore restricts
its attention to equilibria. For example,a decision is a
Nash equilibrium if no agent can reach a better (local)
decision by changing its own decision. The most used
concepts from gametheory are Pareto efficient decisions,
dominantdecisions and Nash decisions. Wefirst repeat
somestandard notations from gametheory (Binmore1992;
Osborne&Rubenstein 1994).
As mentioned,we use 6i to denote a decision of agent t~
and 6 = {61,..., 6n) to denote a decision profile containing one decision for each agent. 6-i is the decision profile
of all agents except the decision of agent o~i. (6_~, 6[) denotes a decision profile whichis the sameas ~ except that
the decision of agent i from 6 is replacedwith the decision
of agent i from6’. 6/ >iv 6i denotesthat decision~[ is better than 6i accordingto his preferences>iv and 6~>iv6i if
better or equal.A is the set of all decisionprofiles for agents
al,..., c~n, Af C A is the set of feasible decision profiles,
and Ai is the set of possible decisionsfor agent ai.
Definition 7 (GameSpecification) A gamespecification is
a tuple (S, A, (>_i)) where
¯ S is a set (of agents)
¯ A = Az x ... x A,~ where Ai is a set of decisions of
agent oq
¯ (>_i) is a partial orderon A (agent i’s preferences)
Wenowdefine a mappingAS2ACfrom an agent specification to a gamespecification.
Definition 8 (AS2AG)Let AS
=
=
(S
{al,..., t~n}, F, B, D, >_, 6O) be specification of agent
systemin S, Ai be the set of ASfeasible decisions of agent
o~i according to Definition 3, A = A~ x ... x A,, and
> vi be the AS preferencerelation of agent ai defined on A
accordingto definition 6. Then, the gamespecification of
ASis the tuple (S, A, (>iv)).
Wenowconsider different types of decision profiles
which are similar to types of strategy profiles from game
theory.
Definition9 Let AS be the set of feasible decisionprofiles.
A PSdecisionprofile 6 = (61,..., 6n) E Af is:
Paretodecision if there is no 6’ = (5~,..., 6~) E Af for
which6~ >u6ifor all agents ~i.
stronglyParetodecisionif there is no U= (6’x, . . , 6~) E
A f for which6[ >_iv6i for all agentsoq and63 >u gj for
someagents ~j.
dominantdecision if for all 6’ E AI and for every agent
v ’ 6i)
i it holds: (6~_i, 61) >_i
(6_i,
’ i.e. a decisionis dominant if it yields a better payoff than any other decisions
regardlessof whatthe other agentsdecide.
Nashdecision if for all agents i it holds: (6-i, 6i) _>iv
(6_i,6~i)for all i E A}
It is a wellknown
fact that Paretodecisionsexist (for finite
games), whereasdominantdecisions do not have to exist.
Thelatter is illustrated by the followingexample.
Example6 Let ~1 and a2 be two agents, FI = F2 = 9,
and initial decisions 6° = 6° = 9. They have the following
beliefs en desires:
Do, = {T =~p,T =~ q}
>at=
T :> p > T => q > T =~ -w > T => -~
p
B,~2 = {b ::> q,-~b ~ -~q}
FromGameSpecifications to Agent
Specifications
D~,= {T~ -,p, T ~ ~q}
>a,= T=~. ~p> T =>. ~q> T=>q> T=>p
Let AS be feasible decision profiles, EB be the outcomesof the decisions, and U(ti) be the set of unreached
desires for agent~i.
AI
EB
(a, b)
{p, q}
U6,
Example8 Let A1 = {a} (oq cooperates), A2 = {b} (52
cooperates), and AS be an agent system specification with
D1 = {T :=> ~aAb, T =~ b, T => ~(aA~b)}, D2= {T =>
a A -~b, T ~ a, T =¢,--~(--~a A b)}. Theonly Nashdecision
is {-~a, -~b}, whereasboth agentswouldprefer {a, b}.
Starting from an agent systemspecification, we can defive the gamespecification and in this gamespecification we
can use standard techniques to for examplefind the Pareto
decisions. However,the problemwith this approachis that
the translation froman agent systemspecification to a game
specification is computationallyexpensive. For example,a
compactagent representation with only a few belief and desire rules maylead to a huge set of decisions if the number
of decisionvariables is high.
The mainchallenge of qualitative gametheory is therefore whether we can bypass the translation to gamespecification, and define properties directly on the agent system
specification. For example,are there particular properties of
agent systemspecification for whichwe can provethat there
alwaysexists a dominantdecision for its correspondingderived gamespecification? A simple exampleis an agent system specification in whicheach agent has the samebelief
and desire rules.
U~:
{T =4--,p,
T =’;, -,q}
{T =’,, --,p}
{a,-~b) {p,--q}
{T =~, q}
{--p, q}
{T => p}
{T =:> ~q}
(-~a,b)
(--,a,-~b) {-~p,--,q} {T =:,’- p, T =~ q} 0
0
According
to definition 5, for Ctl :
U((a,b>)
>U((a,-~b>)
U(
(-~a,b))
>U(
(-~,-~b))
andfor oc2:
U((-~a,-,b))
> u(( ~a,b)) > U((a,-~b))
> u((a,b)).
Noneof these decision profiles are dominantdecisions, i.e.
the agents specifications has no dominantsolution with
respectto their unreached
desires.
The followingexampleillustrates a typical cooperation
game.
Example7 B1 = {a =¢ p, b => -~p A q}, D1 = {T =>
pAq} B2 = {c ::~ q, d =~ pA-~q} D2 = {T ~ pAq}.
The agents have a commongoal p A q, whichthey can only
reach by cooperation.
Thefollowingexampleillustrates a qualitative version of
the notorious prisoner’s dilemma,wherethe selfish behavior
of individual autonomousagents leads to global bad decisions.
41
In the previoussection a mappingfromagent systemspecifications to standard gamespecifications has beendefined. In
this section the questionis raised whetherthe notion of agent
systemspecification is expressive enoughfor gamespecifications, that is, whetherfor each possiblegamespecification
there is an agent systemspecification that can be mappedon
it. Weprovethis propertyin the followingstandardway:
1. First, we define a mappingfrom gamespecifications to
agent systemspecifications;
2. Second, we showthat the composite mappingfrom game
specifications to agent systemspecifications and back to
gamespecificationsis the identity relation.
The second step showsthat if a gamespecification GSis
mappedin step 1 on agent system specification AS, then
this agent systemspecification AS is mappedon GS. Thus,
it showsthat there is a mappingfromagent systemspecifications to gamespecifications for every gamespecification
GS.
Unlike the second step, the composite mappingfrom
agent systemspecifications to gamespecifications and back
to agent systemspecifications is not the identity relation.
Thisis a direct consequence
of the fact that there are distinct
agent system specifications that are mappedon the same
gamespecification. For example, agent system specifications in whichthe variable namesare uniformlysubstituted
by new names.
The mappingfrom gamespecifications to agent system
specificationsconsists of the followingsteps:
1. Copythe set of agents;
2. Introducea set of decision variables andinitial decisions
that will generatethe set of decisions;
3. Introduce a set of parameters, and for each agent sets of
initial facts, beliefs, and desires, such that they generate
the preferenceorder for each agent.
Westart with the set of decision variables. Weintroduce
for each decision a separate decision variable. Moreover,
we assumeinitial decisionsthat see to it that the agent selects at least onedecision variable, andat mostone decision
variable.
Definition 10 (Decisions) Let A = {~1,..., ~,~} be a set
of decisions (of someagent). Thenwe introduce the set
decision variables A = {dl ..... d~} (for this agent) and
the followinginitial decision(for this agent):
dl V ... V dn
i 5~ j =~-,(di A dj)
Moreover,wedefine the sets of desires.
Definition11 Considera set of decisions with a partial ordering on it (for an agent). For eachdecision 6, for which
weintroducedthe decision variable d, we introducethe followingdesire:
T=~
V d’A A -’,d’
6’>~ 6~’~’ and z,~
The following proposition showsthat the mappingfrom
gamespecifications to agent systemspecifications leads to
the desiredidentity relation for the composite
relation.
Proposition 3 Let GSbe a gamespecification as in Definition Z Moreover,let AS be an agent system specification
suchthat the decisionvariables,the initial decisionsandthe
desires are as definedin Definition10and11, andlet the set
of parameters,the sets of beliefs, the set observationsbe
empty. Themappingfrom this agent system specification to
gamespecification defined in Definition 8 mapsAS to GS.
Proof. Assumeany particular GS. Construct the AS as
above. Nowapply Definition 6. The unreached desires of
decision6 for agentc~i are:
Ui(~) = {x =~ y E D I EB(Fu~) ~ x EB(FUtJ) ~ y}
The subset ordering on these sets of unreacheddesires reflects exactlythe original orderingon decisions.
The followingtheoremfollows directly fromthe proposition.
Concluding remarks
Weare using a very simple approach, probablythe simplest
approachpossible, basedon a twosorted propositional language. This simplicity is not a disadvantagebut an advantage, for the following reasons: it enables us to focus on
essentials (comparehowGelfondand Lifschitz (Gelfond
Lifschitz 1993) simple .,4 languagewhichfocussedplanning
logics on essentials), it is easy to implementor generalize
to classical formalisms, and it showswhyexactly we need
morecomplexstuff such as causal theories and action languages. Our motivation comesfrom the analysis of rulebased agent architectures, whichhave recently been introduced. However,the results of this paper maybe relevant
for a muchwider audience. For example, Dennett argues
that automatedsystemscan be analyzed using conceptsfrom
folk psychologylike beliefs and desires. Our workmaybe
used in the formal foundationsof this ’intentional stance’
(Dennett 1987).
In this paper we have defined a qualitative decision and
gametheory in the spirit of classical decision and gametheory. The theory illustrates the micro-macrodichotomyby
distinguishing the optimization problem from gametheoretic equilibria. Wealso link decision theory to gametheory
by showinghowagent system specification can be mapped
to gamespecification and back. Wethink that the methodof
this paperis moreinteresting than its formalresults. Thedecision and gametheory are based on several ad hoc choices
which need further investigation. For example,the desire
rules are defeasible but the belief rules are not (the obvious extension leads to wishful thinking problemsas studied in (Thomason2000; Broersen et aL 2001a)). Wehope
that further investigationsalongthis line brings the theories
and tools used for individual agents and multi agent systems
closer together.
There are several topics for further research. The most
interesting question is whetherbelief and desire rules are
fundamental,or whetherthey in turn can be represented by
someother construct. Othertopics for further research are
the developmentof an incremental any-time algorithm to
find optimal decisions, the developmentof computationally
attractive fragmentsof the logic, and heuristics of the optimization problem.
References
TheoremI The set of agent system specifications with
emptyset of parameters,beliefs rules and observations is
expressive enoughfor gamespecifications.
Proof. Followsfromconstruction in Proposition3.
Theaboveconstruction raises the question whetherother
sets of agent systemspecifications are completetoo, suchas
for examplethe set of agent systemspecifications in which
the initial decisionis an emptyset. Or the set of agent systemspecifications in whichthe set of desires contains only
desire-to-be desires. Weleave these questions for further
research.
42
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