From: AAAI Technical Report WS-02-06. Compilation copyright © 2002, AAAI (www.aaai.org). All rights reserved.
Beyond optimization:
overcomingthe limitations of individual rationality
Wynn C, Stirling
Electrical and ComputerEngineering Department
BrighamYoungUniversity
Provo, Utah 84602, USA
email: wyrm@ee,byu. edu
Generalgame
theoryseems
to be in part a sociological
theorywhichdoesnot includeanysociologicalassumptions.., it maybe too muchto askthat anysociology
bededved
fromthe single assumption
of individual rationality.
R. D. LuceandH. Raiffa
GamesandDecisions(1957)
Abstract
VonNeumann-Morgenstern
gametheory is the multi-agent
instantiationof individualrationality,andis the standardfor
decision-making
in groupsettings. Individualrationality,
however,requires eachplayer to optimizeits ownperformance,regardlessof the effectso doinghas onthe otherplayers. Thisfeaturelimits the ability of game
theoryas a design
paradigm
for groupbehaviorwherecoordinationis required,
since it cannotsimultaneouslyaccommodate
both groupand
individual preferences.Byreplacingthe demand
for doing
the best thing possiblefor the individualwitha mathematically precisenotionof being"goodenough,"
satisficing game
theoryallowsbothgroupandindividualinterests to be simultaneously accommodated.
Introduction
It is a platitude that a decision-makershould makethe best
choicepossible. Typically, this injunction is taken to mean
that a decision-makershould optimize, that is, maximize
expected utility. Althoughthe exigencies of decision-making
under time and computationalconstraints mayrequire the
decision-makerto compromise,resulting in various notions
of boundedoptimization, the fundamentalcommitmentto
optimality usually remains intact. It is almost mandatory
that a decision methodologyincorporate someinstance of
optimization, even if only approximately. Otherwise the
decision-makingprocedureis likely to be dismissed as ad
hoc.
Optimizationis foundedon the principle that individual interests are fundamentaland that social welfare is a
function of individual welfare (Bergson, 1938;Samuelson,
1948). This hypothesis leads to the doctrine of rational
choice, whichis that "eachof the individual decision-makers
Copyright(~) 2002,
American
Association
for Artificial Intelligence(www.aaal.org).
All rights reserved.
99
behavesas if he or she weresolving a constrainedmaximization problem"(Hogarthand Reder, 1986). This paradigm
the basis of muchof the conventionaldecisiontheory that is
used in economics,the social and behavioral sciences, engineering, and computerscience. It relies upontwo fundamental premises:
P-1 Total ordering: the decision-makeris in possessionof
a total preferenceordering(that is, an orderingthat is
reflexive, antisymmetric,
transitive, andlinear) for all of
its possible choicesunderall conditions(in multi-agent
settings, this includes knowledge
of the total orderings
of all other participants).
P-2 The principle of individual rationality: a decisionmakershouldmakethe best possibledecision for itself,
that is, it shouldoptimizewithrespect to its owntotal
preferenceordering(in multi-agentsettings, this ordering maybe influenced by the choices available to the
other participants).
Serf-interested humanbehavior is often consideredto be
an appropriate metaphorin the design of protocols for artificial decision-making
systems.Withsuch protocols, it is
often taken for granted that each member
of a community
of
decision-makersshould try
... to maximizeits owngoodwithout concern for the
globalgood.Suchself-interest naturallyprevailsin negotiations among
independent
businessesor individuals... Therefore, the protocolsmustbe designedusing a noncooperative,
strategic perspective:the mainquestionis whatsocial outcomesfollow givena protocol whichguaranteesthat each
agent’sdesiredlocalstrategyis bestfor that agent----and
thus
the agentwill use it. (Sandholm,
1999,p. 201,202;emphasis
in original).
Whenartificial decision-makersare designedto function
in a non-adversativeenvironment
it is not obviousthat it is
either natural or necessaryto restrict attention to noncooperative protocols, decision-makerswhoare focusedon their
ownself-interest will be driven to competewith decisionmakers whoseinterests might possibly compromisetheir
own.Certainly, conflict cannot be avoided in general, but
conflict can just as easily lead to collaborationas to competition.
Oneof the justifications for adoptingself-interest as a
paradigmfor artificial decision-making
systemsis that it is a
simple and convenientprinciple uponwhichto build a mathemafically basedtheory. Self-interest is the Occam’srazor
of interpersonal interaction and relies only uponthe minimalassumptionthat an individual will put its owninterests
aboveeverything and everyoneelse. This simple principle
allows the decision-makerto abstract the problemfrom its
context and express it in unambiguousmathematicallanguage.Withthis language,utilities can be definedand calculus canbe employed
to facilitate the searchfor the optimal
choice. Thequintessential manifestationof this approachto
decision-making is von Neumann-Morgenstem
gametheory
(yon Neumannand Morgenstem,1944).
Gametheory is built on one basic principle: selfinterest----each player mustmaximize
its ownexpectedutility underthe constraint that other playerswill do likewise.
Suchplayers will seek an equilibrium;that is, a state such
that no individual player can improveits level of satisfaction by makinga unilateral changein its strategy. For twoperson constant-sumgames, this is perhaps the only reasonable, non-vacuousprinciple--what one player wins, the
other loses. Gametheory insists, however,that this same
principle applies to the general case. Thus, even in situations wherethere is the opportunity for group, as well as
individualinterest, onlyindividuallyrational actions are viable. If a joint (that is, for the group)solutionis not individually rational for somedecision-maker,that self-interested
decision-maker
wouldnot be a party to such a joint action.
Coordinatedbehavioris perhaps the most important (and
mostdifficult) social attribute to synthesize in an artificial decision-makinggroup. Achievesuch a design objecfive, however,will be greatly facilitated if decision-making
is basedon rationality principles that permit the decisionmakersto expandtheir spheres of interest beyondthemselves and give deference to others. As Arrowobserved,
whenthe assumptionof perfect competitiondoes not apply,
"the very conceptof [individual] rationality becomes
threatened, because perceptions of others and, in particular, of
their rationality becomepart of one’s ownrationality" (Arrow,1986). Arrowhas put his finger on a critical limitation
of individual rationality. Asis well known,however,a notion of "grouprationality" that requires the groupto do the
best for itself is not compatiblewithindividual rationality
(Luceand Raiffa, 1957). Nevertheless,gametheory is often
used to characterize situations wherecoordinatedbehavior,
wherethe membersof a group coordinate their actions to
accomplishtasks that pursue the goals of both the groupand
its members,is of fundamentalimportance.
In this paperwe first reviewthe various notions of group
preference that have arisen in the context of gametheory,
wethen present an alternative conceptof utility theory and
showdevelopa newclass of games,called satisficing games.
Wethen show howto formulate these games through the
use of conditionalpreferencesand describe howto reconcile
individual and grouppreferences.
Group Preferences
Several attempts have been madeto express group preferences in a game-theoretic context. Shubikoffers two interpretations of this notion, neither of whichgametheorists
view as entirely satisfactory: "Grouppreferences maybe
regarded either as derived from individual preferences by
someprocess of aggregationor as a direct attribute of the
group itself" (Shubik, 1982, p. 109). One way to aggregate a group preference from individual preferences is to
define a "social-welfare"function that providesa total ordering of the group’s strategy profiles. Thefundamentalissue is whetheror not, given arbitrary preference orderings
for each individual in a group,there alwaysexists a wayof
combiningthese individual preferenceorderings to generate
a consistent preference ordering for the group. In an landmarkresult, Arrow(Arrow, 1951; Sen, 1979) showedthat
no social-welfarefunctionexists that satisfies a set of reasonableand desirable properties, each of whichis consistent
withthe notionof self-interested rationality and the retention
of individual autonomy.
ThePareto principle providesa conceptof social welfare
as a direct attribute of the group.Astrategy profile is Pareto
optimal if no single decision-maker,by changingits decision, can increaseits level of satisfaction withoutlowering
the satisfaction level of at least one other decision-maker.
However,if a Pareto-optimalsolution does not provide each
eachplayerat least its security level (i.e., the minimum
payoff it can be guaranteed,evenif all other players conspire
againstit), it couldnot be a party to that decisionandstill be
faithful to individualrationality.
Toimposea strategy profile, suchas a Pareto-optimalsolution, on a groupwouldrequire the existence of a superplayer, or, as Raiffa puts it, the "organization incarnate"
(Raiffa, 1968), whofunctions as a higher-level decisionmaker.Shubikrefers to the practice of ascribing preferences
to a group as a subtle "anthropomorphictrap" of making
a shaky analogy betweenindividual and group psychology.
He arguesthat, "It maybe meaningful.., to say that a group
’chooses’or ’decides’something.
It is rather less likely to be
meaningfulto say that the group’wants’ or ’prefers’ something" (Shubik,1982,p. 124).Raiffa, also, rejects the notion
of a superplayer,but confessesthat he still feels "a bit uncomfortable.., somehow
the group entity is morethan the
totality of its members"
(Raiffa, 1968, p. 237). Arrowexpresses a similar discomfort: "All the writers from Bergson
on agree on avoidingthe notion of a social goodnot defined
in terms of the values of individuals. But where Bergson
seeks to locate social values in welfare judgmentsby individuals, I prefer to locate themin the actions taken by society throughits rules for makingsocial decisions" (Arrow,
1951, p. 106). Evidently, althougha satisfactory accountof
group preferencesmaybe difficult or, perhaps, impossible,
to obtain underindividual rationality, the desire to accommodatethe notion remains.
Perhapsthe source of discomfortis that individual rationality by itself doesnot providethe ecologicalbalancethat a
groupmustachieveif it is to accommodate
the variety of relationships that can exist betweendecision-makersand their
environment.But achieving such a balance should not require the aggregationof individual interests or the fabrication of a superplayer. While such approachesmaybe recommended
as waysto account for group interests, they may
also manifestthe limitations of individualrationality.
i00
Of course, one maysubstitute the interests of others for
one’s ownself-interest, as Sen(1990, p. 19) observed:’~lt
is possible to define a person’sinterests in such a waythat
no matter whathe does he can be seen to be furthering his
owninterests in every isolated act of choice.., no matter
whetheryou are a single-mindedegoist or a raving altruist
or a class-consciousmilitant, youwill appearto be maximizing your ownutility in this enchantedworldof definitions."
Althoughit is certainly possible to suppress one’s preferences in deferenceto others by redefiningone’s ownutility,
doingso is little morethan a devicefor co-optingindividual
rationality into a formthat can be interpreted as unselfish.
Sucha device only simulatesattributes of cooperation,unselfishness, and altruism while maintaininga regimethat is
competitive,exploitive, and avaricious.
Nevertheless, gametheory has been a great success story
for economies,political science, and psychology.Withthese
disciplines, however,gametheory is used primarily as an
analysis tool to explain and predict behavior, and there is
no causal relationship betweenthe performanceof the entities being studied and the modelused to characterize them.
In the engineeringcontext of synthesis, however,the goal
is to designandbuild artificial decision-making
entities and
the modelsused to characterize behaviorare indeed causal.
Although yon Neumann-Morgenstem
game theory has been
successfully applied in manydisciplines, this success does
not implythat self-interest is the onlyprinciplethat will lead
to credible modelsof behavior, it does not implythe impossibility of accommodating
both groupand individual interests in somemeaningfulway,and it does not implythat individual rationality is an appropriate principle uponwhich
to base a theory for the design and synthesis of artificial
decision-making
entities.
Thereis an old saying: "If all I haveis a hammer,
everything looks like a nail." Wemayparaphrasethat sentiment
as followS:"If all I knowhowto do is optimize, everygroup
decision problem looks like avon Neumann-Morgenstern
game."If, however,as Luce and Raiffa conjecture, it is
indeed too muchto ask that a sociology be derived from
the single assumptionof individual rationality, wemaygain
someadvantagein social situations by considering the use
of decision-making
tools that are not foundedon that single
assumptionand hence maybe better suited for the expression of a sociology. Consider, for example,the following
groupdecision scenario.
Example 1 The Pot-Luck Dinner Larry, Curly, and Moe
are goingto havea pot-luck dinner. Larrywill bring either
soup or salad, Curly will provide the main course, either
beef, chicken, or pork, andMoewill furnish the dessert, either lemon custard pie or bananacreampie. The choices
are to be madesimultaneouslyandindividually following a
discussion of their preferences,whichdiscussion yields the
followingresults:
1. In terms of mealenjoyment,if Larrywereto prefer soup,
then Curlywouldprefer beef to chickenby a factor of two,
and wouldalso prefer chicken to pork by the sameratio.
However,if Larry wereto prefer salad, then Curlywould
be indifferent regardingthe maincourse.
101
2. If Curlywereto reject pork as being too expensive, then
Moewouldstrongly prefer (in terms of meal enjoyment)
lemoncustard pie and Larrywouldbe indifferent regarding soupor salad. If, however,Curlywereto to reject beef
as too expensive, then Larry wouldstrongly prefer soup
and Moewouldbe indifferent regardingdessert. Finally,
if Curlywereto reject chickenas too expensive,then both
Larry andMoewouldbe indifferent with respect to their
enjoymentpreferences.
Larry, Curly, andMoeall wish to conservecost but consider
both cost and enjoymentto be equally important. Table 1
indicatesthe total cost (in stoogedollars) of eachof the
possible meal combinations(using obviousabbreviations).
beef (soup/said)
ch~ (soup/said)
pork (soup/said)
lest
23/25
22/24
20/22
bcrm
27/29
26/28
24/26
Table1: Mealcost structure for the Pot-LuckDinner.
Thedecision problemfacing the three participants is for
each to decide independentlywhatto bring to the meal. Obviously, each participant wantshis ownpreferenceshonored,
but no explicit notion of group preference is provided in
the scenario. Adistinctive feature of the preference specification for this exampleis that individual preferencesare
not evenspecified by the participants. Rather, the participants expresstheir preferencesas functionsof other participants’ preferences. Thus,they are not confiningtheir interests solely to their owndesires, but are taking into consideration the consequences
that their possible actions haveon
others. Suchpreferences are conditional. These intercom
nections betweenparticipants mayimply somesort of group
preference, but it is not clear whatthat mightbe. In fact,
if the preferences,either conditionalor unconditional(i.e.,
individual) turn out to be inconsistent, then there maybe no
harmoniousgroup preference, and the group maybe dysfunctional in the sense that meaningfulcooperationis not
possible. But if they are consistent, then someformof harmoniousgroup preference mayemergefrom the conditional
preferences (and any unconditionalpreferences, should they
be provided).If this is the case, then an importantquestion
is howwe might elicit a group decision that accommodates
an emergentgroup preference.
To formulate a von Neumann-Morgenstern gametheoretic solution to this decision problem,each participant
must identify and quantify payoffs for every conceivable
meal configurationthat conformto their ownpreferencesas
well as give due deference to others. Notice that the unconditional preferencesof each of the participants are not
specified nor are all of the possible conditionalpreferences
specified. Unfortunately,individual rationality makesit difficult to obviate such requirements,and the lack of a total
ordering in the problemstatement therefore presents a serious problemfor conventional gametheory. Withoutthis
constraint it is impossibleto apply standard solution concepts such as defining equilibria. The desire to apply game
theory maymotivatedecision-makersto manufactureorderings that that are not warranted.Traditional gametheory is
not an appropriate frameworkfor this problem.
Tosolve this problemin a waythat fully respects the problem statement, we needa solution conceptthat does not depend upontotal orderings. It must, however,accommodate
the fact that, eventhoughagents maybe primarily concerned
with conditional local issues, these concernscan havewidespreadeffects.
Utilities
Thechain of logic that supports gametheory is as follows:
individual rationality leads to optimization, whichrequires
a total orderingof preferences, whichin turn motivatesthe
definingof utility functionsto characterizethese preference
orderings. However,as Raiffa observed: "Onecan argue
very persuasivelythat the weakestlink in any chain of argumentshould not comeat the beginning" (Raiffa, 1968,
p. 130). Thus, if we are to overcomethe limitations imposed by individual rationality, wemustforge a newchain.
To do so, we must: (1) define a newnotion of rationality
that accommodates
a wider sphere of interest; (2) replace
optimization with a criterion that is compatiblewith nonlocalizedrationality; (3) definepreferenceorderingsthat accommodate
both individual and groupinterests, and (4) define utility functionsthat are compatiblewiththese preference orderings.
This paper presents such a chain. To forge it, however,
it is moreconvenientto start at the end and workback to
the beginning. Thus, we start by examiningthe structure of
utility functions,whichleads to an alternative preferenceorderingthat, in turn, leads to a criterion for decision-making
which,finally, definesa newconceptof rationality.
Extrinsic Utilities
Weconcentrateexclusively on finite strategic gamesof complete information. Suchgamesare defined by a payoff array, the entries of whichare the N-tuplesof payoffsto the
players. Eachplayer defines its payoff as a function of the
strategies of all players; that is, the payoffto player i is
7ri(sl,...,s~)
with sj E Uj, j = 1,...,N, where Uj
is player j’s strategy space. In morecompactnotation, we
write 7ri(s)where s = {sx,... , SN } E U = U1×’" × UN
is a strategy profile. Individuallyrational players use such
utilities to makecomparisonsbetweenstrategy profiles and
thereby form solution concepts such as Nashequilibria to
define acceptable strategies. Suchcomparisonsare interstrategyin that they require the comparisons
of the attributes
of eachstrategy to the attributes of all other strategies. Utilities that are usedfor this type of comparisons
are extrinsic.
Theimportantthing to note about the waythese utilities
are usedis that it is not until the payoffsare juxtaposedinto
an arrayso that the payoffs for all players can be compared
that the actual "game"aspects of the situation emerges.It
is the juxtapositionthat reveals possibilities for conflict or
coordination.Thesepossibilities are not explicitly reflected
in the individual payoffs by themselves.In other words,althoughthe individual’spayoffis a functionof other players’
102
strategies, it is not a functionof otherplayers’preferences.
This structure is completelyconsistent with exclusiveselfinterest, whereall a playercares aboutis its personalbenefit
as a functionof its ownandother players’ strategies, without
any regardfor the benefit to the others. Underthis paradigm,
the primarywaythe preferencesof others factor into an individual’s decision-making
is to constrain behaviorso as to
limit the amountof damagethey can do to oneself.
If the gameis not one of pure competition, there maybe
somebenefit to coordinatedbehavior, wherebyplayers take
into considerationthe effect of their actions on the welfareof
others. Onewayto accountfor the interests of others within
the yon Neumann-Morgenstem
frameworkis to transform
the gameby introducing new payoffs of the following form
(Taylor, 1987): 7r~ = EN=ICtij’/r j. Bychoosingthe weights
aq, an altruistic player i maygive deferenceto the preferences of others. This approach,however,requires the imposition of twovery strong assumptions:(a) each player must
precisely knowthe other players’ numericalpayoffs (ordinal rankingsare not sufficient), and 0a) interpersonal comparisons of utility are implied (the addendsmust be commensurable).Evenif these assumptionsapply, choosingthe
weightsaij requires each player to categorically ascribe a
portion of is utility to other players and therefore to defer to someextent to those players in all circumstances.It
does not permit a player to chooseselectively whichof the
other player’s preferencesit will favor or disfavor. Todo so
wouldrequire aij to be functions of other players’ strategies, and the formulation of the gamemayquickly become
intractable.
Perhapsthe critical questionis not whetherit is theoretically possible somehow
to accountfor the interests of others
via extrinsic utilities. Rather, the moreimportantquestion
might be: do they offer an adequate platform by which to
makerationality of others part of one’s ownrationality? The
lack of a definitive notionof grouppreferencethat is consistent with individual rationality wouldseemto cast doubton
an affirmativeanswerto the latter question.
Intrinsic Utilities
In societies that value cooperation,it is unlikely that the
preferences of a given individual will be formedindependently of the preferences of others. Knowledge
about one
agent’s preferences mayalter another agent’s preferences.
Suchpreferences are conditionedon the preferencesof others. In contrast to conditioningonly on the actions of other
participants, conditioningon the preferencesof others permits a decision-makerto adjust its preferencesto accommodate the preferencesof others. It can bestoweither deference
or disfavor to others accordingto their preferencesas they
relate to its ownpreferences.Sincetraditional utility theory
is a functionof participantstrategies, rather thanparticipant
preferences, it cannotbe used to expresssuch relationships.
To address this problem,let us closely examinethe way
preferences are formed. Whendefining preferences one often encountersvaluations that are in opposition. Anygiven
strategy profile will possessattributes that are beneficialand
attributes that are detrimental to each player; such differ-
ences in valuation create dichotomies.By separating the favorableand unfavorableattributes of each strategy profile,
we mayexposethe fundamentalpreference structure. Comparison of the attributes of each profile providesa determination of the benefit that obtains by adoptingit relative to
the cost. Suchdichotomiesare ubiquitous. Peopleroutinely
comparethe upside against the downside,the pros versus
the cons, the pluses versus the minuses,and they can do this
profile by profile withoutdirectly comparing
of one profile
to another. In other words,they performintra-profile comparisons. Such comparisonsare fundamental, and must be
made,evenif implicitly, in order to define the utility function whichcan then be used for inter-profile comparisons
(i.e., total preferenceorderings).
Perhaps,if westart at the headwatersof preferenceformulation, rather than somewheredownstream,we maybe
able to characterize these dichotomousrelationships more
comprehensivelyand systematically. Wethus consider the
formation of two utility functions that accommodate
dichotomies.A first considerationis that, since the twoutility functions are to be compared,they must be expressedin
the sameunits. To avoid arbitrary scalings as well as for
reasons that will soon becomeapparent, it is convenientto
define these utilities as massfunctions. A function p is a
massfunction ifp(s) > 0 and ~)-~s~o(S)= 1. Adoptingthis
conventionmeansthat the player has a unit of massto apportion among
the profiles to weighttheir desirableattributes as
well as a unit of massto apportionto weighttheir undesirable attributes. Theseweightingfunctions thus possess the
mathematicalproperties of probability massfunctions, but
they do notpossessthe samesemanticsand do not admit interpretations of belief, propensity,frequency,or anyother of
the usual probabilistic interpretations. Toemphasize
the distinction betweenthe mathematicalstructure and the interpretation of these functions, wewill refer to the massfunction
that characterizesthe desirableattributes of the strategy profiles as selectability, and we will denotethe massfunction
that characterizesthe undesirableattributes as rejectability.
Whendefining the dichotomousutility functions, operational definitions of whatis selectable and rejectable about
the strategy profiles mustbe provided. Typically, the attributes of a profile that contribute to a fundamentalgoal
wouldbe associated with selectability and those attributes
that inhibit or limit activity wouldbe associated with rejectability. There generally will not be a unique wayto
frame a given decision problem,but regardless of the way
the framingis done,it is essential that the selectability and
rejectability attributes not be restatementsof the samething.
In general, at least for single-agentdecision problems,the
selectability of a strategy shouldbe specifiable withouttaking into considerationits rejectability and vice versa. For
multiple-player problems,however,this independenceneed
not applybetweenplayers; that is, oneplayer’s selectability
or rejectability mayinfluenceanotherplayer’s selectability
or rejectability.
Weare nowin a position to completeour chain that links
utilities to preferenceorderingsto decision rules to rational behavior. Our procedureinvolves makingintra-profile,
rather than inter-profile, comparisons.
Utilities usedfor this
103
purposeare intrinsic, meaningthat they are usedto evaluate
a profiles withrespectto attributes that it possesseswithinitself, independently
howthat profile relates to other profiles.
Withextrinsic comparisons,
the logical notion of rational behavior is to rank-orderthe strategy profiles and chooseone
that is optimal(i.e., to equilibrate). Withintrinsic comparisons, the logical notion of rational behavioris to choosea
profile that is goodenough,in the sense that the gains obtained by choosingit outweighthe losses. This defines a
newnotion of rationality, whichweterm satisficing rationality. This notion is considerably weakerthan individual
rationality, whichasserts that a decision-makermust make
the best choice possible. Put in the vernacular, the essence
of individual rationality is "only the best will do" whilethe
essenceof satisficing rationality is "at least get yourmoney’s
worth."
Whywoulda decision-makerchoose to do anything other
than optimize? For a decision-makerfunctioning in isolation, there is noincentive, ceterisparibus,to eschewan optimal solution. Furthermore,with gamesof pure competition,
there is no incentive to adopt any solution conceptthat does
not maximizethe advantageto the player. But with gamesof
mixedmotive,the notion of being individually optimal loses
muchof its force (as Arrowobserved).Yet, underthe strict
paradigmof individual rationality, a player mustnot modify
its choice to is owndisadvantage,no matter howslight (unless it also redefinesits utility), evenif doingso wouldoffer
a great advantage to others. However,once such a player
starts downthe slippery slope of compromise
by abandoning
individual rationality, there is seeminglyno wayto control
the slide.
Satisficing providesa wayto add somefriction to the slippery slope. While it does indeed abandonindividual rationality, it is not heuristic. Intrinsic utilities are basedon
exactly the sameprinciples of value that are used to define
extrinsic utilities, the valuationsare merelyappliedin a different way.Thus, the satisficing approachis applicable in
situations whererelationships other than pure competition
are relevant.
Thejustificationfor usingthe term"satisficing"is that it is
consistent withSimon’soriginal usageof the terra--to identify strategies that are goodenoughby comparingattributes
of the strategies to a standard.Thisusagediffers onlyin the
standard used for comparison.Simon’sstandard is extrinsic; strategies are compared
to the "aspiration level" of how
gooda solution might reasonablybe achieved(Simon,1955;
Simon,1956; Simon,1990). Satisficing as defined herein,
on the other hand, is intrinsic; the comparisonis donewith
respectto the meritsof the strategy.
Satisficing Decision-Making
To generate a useful theory of decision-making we must
be able to define, in precise mathematicalterms, what it
meansto be goodenough, and we must develop a theory of
decision-makingthat is compatiblewith this notion. Analternative to von Neumann-Morgenstern
N-player gametheory is a newapproach to multiple-agent decision-making
called satisficing games(Stifling and Goodrich,1999b;Stirling and Goodrich,1999a; Goodrichet al., 2000;Stifling,
2002). Twokey features of our developmentare (a) the separationof positive andnegativeattributes of strategy profiles
into selectability andrejectability utility functionsand (b)
the structure of these utility functionsas massfunctions.To
construct these massfunctions, however,wemust first define an even morefundamentalquantity, whichwe term the
interdependencemassfunction, whichaccountsfor linkages
that exist between
selectability andrejectability.
Anact by any individual player has possible ramifications
for the entire group. Someplayers maybe benefited by the
act, somemaybe damaged,and somemaybe indifferent.
Furthermore,althoughan individual mayperformthe act in
its ownbenefit or for the benefit of others, the act is usually not implemented
free of cost. Resourcesare expended,
or risk is taken, or someother penalty or unpleasant consequenceis incurred, perhaps by the individual whoseact
it is, perhaps by other players, and perhapsby the entire
group. Althoughthese undesirable consequences maybe
defined independentlyfromthe benefits (recall the example
of choosing an automobile), the measuresassociated with
benefits and costs cannot be specified independentlyof each
other, dueto the possibility of interaction(e.g., cost preferences for one player maydependuponstyle preferences of
another player). Acritical aspect of modelingthe behavior
of a group,therefore, is the meansof representingthe interdependenceof both positive and negative consequencesof
all possiblestrategy profiles.
Let X1,..., XNbe a group of decision-makers, and let
Ui be the set of strategies available to Xi, i = 1,..., N. The
strategy profile set is the productset U= U1,x --- x U/V.
Let us denote elements of this set as s = {sl,... ,s/v},
wheresi E Ui.
Definition 1 An interdependence function for a group
{Xl.... , X/v}, denotedpsa...sNR,..aN: U x U---, [0, 1],
is a massfunction, that is, it is non-negativeand normalized to unity, whichencodesall of the positive and negative interrelationships betweenthe membersof the group.
Wewill denote this as Psl...s~th...R~(s; r), where s
(sl,... ,s/v) E U represents strategy profiles viewed
terms of selectability and r = (rl,... r/v) E U represents
strategy profiles viewedin terms of rejectability.
[]
The interdependence function provides a complete description of all individualand grouprelationshipsin termsof
their positive and negativeconsequences.Let s and r be two
strategy profiles. PSl...SNRI-..RN
(S; r) characterizesthe simultaneous
disposition of the players withrespect to selecting s and rejecting r. Particularly whens = r, it mayappear
contradictory to consider simultaneouslyrejecting and seleering strategies. It is importantto remember,
however,that
considerationsof selection and rejection involvetwodifferent criteria. It is no contradictionto considerselecting, in
the interest of achievinga goal, a strategy that one would
wishto reject for unrelated reasons, nor is it a contradiction to considerrejecting, becauseof someundesirableconsequences, a strategy one wouldotherwise wish to select.
Evaluatingsuch trade-offs is an essential part of decisionmaking,and the interdependencemassfunction provides a
meansof quantifyingall issues relevantto this trade-off.
Since it is a massfunction, the interdependencefunction
is mathematically
similar to a probability massfunction, but
does not characterize uncertainty or randomness.Nevertheless, it possessesthe mathematicalstructure necessaryto
characterize notions such as independenceand conditioning:
Conditioning
The interdependence function can be rather complexbut,
fortunately, its structure as a rna.~s functionpermitsits decompositioninto constituent parts according to the law of
compound
probability, or chain rule (Eisen, 1969). 1 Applying the formalism(but not the usual probabilistic semantics)
of the chain rule, wemayexpress the interdependencefunction as a productof conditionalselectability and rejectability functions.Toillustrate, considera two-agentsatisficing
gameinvolving decision-makers X1 and X2, with strategy
sets U1and U2, respectively. The interdependencefunction
maybe factored in several ways,for example,we maywrite
PS1S2R1R2
(81,82; rl, 7"2) PSIIS2R1R2(81182; rl, r2
We interpret
this expression
as follows.
PS~IS2R~R2
(Sl [sz, rl, r2) is Xl’Sconditionalselectability
of Sl given that X2 selects s2, X1 rejects rl, and X2
reject rz. Similarly,Ps2IR~R2(s2 Jr1, r2) is X2’sconditional
selectability of sz, given that Xxand X2reject rl and r2,
respectively. Continuing,PR11R~
(rl Iv2) is Xl’Sconditional
rejectability ofrl, giventhat X2rejects r2. Finally, PR~(r2)
is X2’sunconditionalrejectability of r2.
Manysuch factorizations are possible, but the appropriate factorization must be determinedby the context of the
problem. These conditional mass functions are mathematical instantiations of production rules. For example, we
mayinterpret pR~lR2(rl[r2)as the rule: If X2rejects r2,
then X1feels PR~IR2(rl Jr2) strong about rejecting rl.
this sense, they express local behavior, and such behavior is often mucheasier to express than global behavior.
Furthermore,this structure permitsirrelevant interrelationships to be eliminated. Typically, there will be someclose
relationships betweensomesubgroupsagents, while other
subgroupsagents will function essentially independentlyof
each other. For example,supposethat Xl’s selectability has
nothing to do with X2’srejectability. Thenwe maysimplify
PSIIS2RIR2
(81 [82; rl, r2) to becomeps,
ls2R, (sl Is2; r2).
Suchconditioning permits the expressionof the interdependencefunction as a natural consequenceof the relevant
interdependenciesthat exist betweenthe participants. Conditioning is the key to the accommodation
of the interests
~In the probabilitycontext,let Xand Ybe random
variables
andlet x andy possiblevaluesfor Xand Y,respectively.Bythe
law of compound
probability, Pxv(x, y) = PXlY(X[y)PY(Y)
expressesthe joint probabilityof the occurrenceof the event (X
x, Y = y) as the conditionalprobabilityof the event X= x occurringconditioned
on the eventY= y occurring,timesthe probability of Y= y occurring.Thisrelationshipmaybe extendedto
the generalmultivariatecasebyrepeatedapplications,resultingin
whatis calledthe chainrule.
104
of others. For example, if X2were very desirous of implementings if X1were not to implementr, X1 could accommodate
X2’spreferenceby setting PR1Is2 (r]s) to a high
value (close to unity). Then,r wouldbe highly rejectable
to X1if s were highly selectable to X2. Note, however,
that if X2shouldturn out not to highly prefer s and so sets
Ps2(s) ,mO,thenthe joint selectability/rejectabilityof (s; r),
namely,PS2Rt(s; r) =PRllS2(rls)ps2 (s) ,~ SOthejoint
event of X1rejecting r and X2selecting s has negligible
interdependencemass. Thus, X1is not penalized for being
willing to accommodateX2 whenX2 does not need or expect that accommodation.
By controlling the conditioning
values, X1is able to achieve a balance betweenits egoistic
interests and its concernfor others.
Satisficing
the joint selectability andjoint rejectability overthe strategies of all other participants,yielding:
PSi (8i)
= E PSI’"SN (Sl,
Pna(si)----
¯ ¯ ¯ , SN)
RRI’"FIN (81,’’"
, 8N)"
(6)
Definition4 Theindividual satisficing solutions to the satisficing game{U, PS~...SN,PRx...RN} are the sets
= {si e ps,(sO>_ qpm00}-
Games
Fromthe interdependencefunction we mayderive two functions, called joint selectability andjoint rejectability functions, denotedPsa...sz and PR1...RN,respectively, according to the formulas
(5)
(7)
Theproductof the individualsatisficing sets is the satisficing
strategy profile rectangle:
~}~q = ~"~ X ’’-X ~qN = {(81,...
’ 8N): 8i E )"~}. (8)
[]
Psv..s~(s) = ~ PSv..SHRr..RN(S; V)
(2)
vGU
p,, ...,, (s) =ps,..s,,.,.. ,, (v;
(3)
vGU
for all s E U. Thesefunctions are also multivariate mass
functions. Thetwo functions are compared
for each possible
joint outcome,and the set of joint outcomesfor whichjoint
selectability is at least as great as joint rejectability forma
jointly satisficingstrategyprofile set.
Definition 2
A satisficing
game is a triple
{U, psr..S~,PRv..R~}. The joint solution to a saristieing gameis the set
~q = {s E O: Psv..sN (s) > qPR,..RN (S)}, (4)
whereq is the index of caution, and parameterizesthe degree to whichthe decision-makeris willing to accommodate
increased costs to achieve success. Nominally,q = 1, which
attributes equal weightto successand resourceconservation
interests. ~q is termedthe jointly satisficing set, and elementsof ~q are satisficing strategy profiles.
[]
Thejointly satisficing set providesa formaldefinition of
what it meansto be goodenoughfor the group; namely,a
strategy profile is goodenoughit the joint selectability is
greater than or equal to the indexof caution times the joint
rejectability.
Definition 3 A decision-making group is jointly satisficingly rational if the members
of the groupchoosea strategy profile for whichjoint selectability is greater than or
equal to the index of caution times joint rejectability. []
The marginal selectability and marginal rejectability
mass functions for each Xi maybe obtained by summing
105
Definition 5 A decision-makeris individually satisficingly
rational if it choosesa strategy for whichthe marginalselectability is greater than or equal to the indexof caution
times the marginal rejectability.
[]
It is not generallytrue that the satisficing rectangle will
haveany close relationship with the jointly satisficing set.
Whatis true, however,is the followingtheorem:
Theorem1 (The negotiation theorem.) lf si is individually
satisficing for Xi, that is, si E ~, then it mustbe the ith
elementof somejoin@satisficing vector s E ~q.
Proof Wewill establish the contrapositive, namely,that
ifsi is not the ith dement of any s E ~q, then si f[
}3~. Without loss of generality, let i = 1. By hypothesis, Psx...sN (sl, v) qPR~...RN(sl, v) forall v E
(]2
X "" X UN, SO pSx(81)
----
Y~vpS1...SN(Sl,V)
<
q ~v PRr..RN(Sl, V) = qPRI (Sl), hence Sl¢ Eql.
Thus,if a strategyis individuallysatisficing, it is part of a
satisficing strategy profile, althoughit neednot be part of all
satisficing profiles. Theconverse,however,is not true: if si
is the ith elementof a jointly satisficing vector,it is not necessarily individually satisficing for Xi. Thecontent of the
negotiationtheoremis that no one is ever completelyfrozen
out of a deal---every decision-makerhas, from its ownperspective, a seat at the negotiatingtable. Thisis perhapsthe
weakestcondition underwhichnegotiations are possible.
A decision-maker whopossessed a modestdegree of altruism wouldbe willing to undergo somedegree of selfsacrifice in the interest of others. Sucha decision-maker
wouldbe willing to lower its standards, at least somewhat
and in a controlled way,if doingso wouldbe of great benefit to others or to the groupin general. Thenatural wayfor
Xi to expressa loweringof its standards is to decrease its
index of caution. Nominally,we mayset q = 1 to reflect
equal weightingof the desire for successand the desire to
conserveresources. By decreasingq, welower the standard
for successrelative to resourceconsumption
and therebyincreasethe size of the satisficingset. Asq ---, 0 the standardis
loweredto nothing,and eventuallyeverystrategy is satisficing for Xi. Consequently,if all decision-makersare willing
to reduce their standards sufficiently, a compromise
can be
achieved.
ReconcilingGroupandIndividualPreferences
The satisficing conceptinduces a simple preference ordering for individuals, namely,wemaydefine the binary relationships ">-i" and ",~i" meaning"is better than" and "is
equivalentto"
respectively, for player i, such that si >-i s~
¯
i and si "~i si’if either si 6 E~
if si 6 E’q and s!i g[ ~q,
t
i
i
i If si 6 E~, then si
and si 6 Eq or si ¢ ~q and 8~ ~ ~q.
is said to be goodenough¯This interpretation of E ~ differs
fromthe interpretation of conventionalindividual rationality
primarilyin that, in additionto the best strategies it admits
all other strategies that also qualify as goodenough.Animportant feature of the satisficing approach,however,is that
the individualpreferenceorderingsare not specifieda priori,
rather, they emergea posteriori after all of the linkagesbetweenthe players are accountedfor in the interdependence
function. In this sense, individual preferencesare emergent.
Satisficing also induces a preference ordering for the
group, namely, ifs >- s’ifs 6 ~q ands’ ¢ ~q, and
s --, s’ifeithers
6 ~q ands’ E ~q ors t/ ~q and
s’ ¢ ~q. Interpreting this preference ordering, however,
is not straightforward; it is not immediatelyclear what it
meansto be goodenoughfor the group. It wouldseemthat
the notion of group preference should conveythe idea of
harmoniousbehavior or at least someweaknotion of social welfare, but satisficing gametheory is completelyneutral withregardto conflictive andcoordinativeaspectsof the
game. Both aspects can be accommodated
by appropriately
structuring the interdependencefunction. Thefact that the
interdependencefunction is able to accountfor conditional
preference dependenciesbetweenplayers provides a coupling that permitsthemto widentheir spheresof interest beyondtheir ownmyopicpreferences. This wideningof preferences does not guaranteethat there is somecoherent notion of harmonyor disharmony.Althoughsuch implications
are certainly not ruled out, selectability and rejectability do
not favor either aspect. Theymaycharacterize benevolent
or malevolentbehavior and they mayrepresent egoistic or
altruistic interests. Theymayresult in harmonious
behavior
or they mayresult in dysfunctional behavior. Withcompetitive games,conflict can be introducedthroughconditional
selectability and rejectability functionsthat accountfor the
differences in goals and values of the players--the ’group
preference’ will then be to opposeone another. Onthe other
hand, constructive coordinated behavior can be introduced
through the sameprocedure, leading to a group preference
of cooperation. Thus, as is the case with trying to define
grouprationality underthe optimizationregime, the notion
of grouprationality also appearsto be somewhat
elusive under the satisficing regime.
106
However,the apparent elusiveness of a simple interpretation of group rationality is not a weaknessof the satistieing approach. Onthe contrary, it is a strength. Rather
than a notion of group preference being defined as an aggregationof individual interests (a bottom-up,or micro-tomacro,approach)or imposedby a superplayer (a top-down,
or macro-to-micro,approach), group preferences are emergent, in the sense that they are determinedby the totality of
the linked preferences (conditional and unconditional) and
displaythemselvesonlyas the link.~ are forged. It is analogousto makinga cake. Thevarious ingredients (flour, sugar,
water, heat, etc.) influence each other in complexways,but
it is not until they are all combined
in properproportionsthat
a harmoniousgroup notion of"cakeness" emerges.
Thus, just as the satisficing utility functionscompareintrinsic, rather than extrinsic, attributes of strategies, the notions of both individual and group preference that emerge
fromtheir application are also intrinsic. Theydevelopwithin
the groupof playersas they evaluatetheir possibilities. Any
notions of either groupor individual rationality that emerge
need not be anticipated or explicitly modeled.Rather than
being imposedvia either a top-downor bottom-upregime,
such preferences maybe characterized as inside-out, or
meso-to-micro/macro.Both individual and group preferences emergeas consequencesof local conditional interests
that propagate throughoutthe community
from the interdependentlocal to the interdependentglobal and fromthe conditional to the unconditional.
To illustrate the emergenceof individual and grouppreferences, let us nowaddress the Pot-LuckDinner problem
that was introduced in Example1. To examinethis problem
fromthe satisficing point of view, wefirst needto specify
operationaldefinitions for selectability and is a functionof
six independentvariables and maybe factored,rejectability.
Althoughthere is not a uniquewayto framethis problem,let
us take rejectability as cost of the mealand take selectability
as enjoymentof the meal. The interdependencefunction is
a function of six independentvariables and maybe factored,
accordingto the chain rule, as
PSLScSMt~LRCRM
(X, y, Z; U, V, W)
PSclSLSMRLRcRM
(ylx, z; u, v, w)
¯ PSLSMnLRCRM
(X, Z; U, V, W), (9)
where the subscripts L, C, and Mcorrespond to Larry,
Curly, and Moe, respectively.
The mass function
PSclSLSMRLRCRM
(Yl x, Z; U, V, W)expresses the selectability that Curlyplaces on y, given that Larry selects x and
rejects u, that Curlyrejects v, and that Moeselects z and
rejects w. Fromthe hypothesis of the problem,we realize
that, conditionedon Larry’sselectability, Curly’sselectability is independentof all other considerations, thus we can
simplifythis conditionalselectability to obtain
PSoISLS.R~P~RM
(Ylx,Z; U,V,W)= PSclSL(ylx).
Next, weapply the chain rule to the secondterm on the right
handside of (9), whichyields
PSLSMRLRcI=£M (X,
Z; U, "0,
W)
PSLS.IR~ROR,.,
(~:, ~1~,V, ~0)"PRLaOR~
(~, V,
But, given Curly’srejectability, the joint selectability for
Larry and Moeis independentof all other considerations,
2, each of whichis goodenoughfor the group, considered
as a whole. Theindividually satisficing items, as obtained
by computingthe selectability and rejectability marginals,
are also providedin Table2 to be soup, beef, andlemoncustaM.Fortunately,this set of choicesis also jointly satisficing
withoutloweringthe index of caution. Thus,all of the preferences are respected at a reasonablecost and, if pies are
thrown,it is onlyfor recreation,not retribution.
so
PSLSuIRLRcRM(X, ZIU, "0, W) = PSLSMIRC(X, ZI’O
).
By makingthe appropriate substitutions, (9) becomes
PSLScSMRLP~TRM
(X, y; Z, U, 73, W) -~ PSclS~. x)
(Yl
¯ ps~s. w*(x, zlv). pn~noRM
(u,., w).
Jointly Satisficing
Wedesire to obtain~q, the satisficing strategy profiles for
the group and E~ Eqc, and EM,the individually satisficing
strategy sets for Larry, Curly, and Moe,respectively. To do
so, we must specify each of the componentsof (10).
computePSclSL,recall that Curlyprefers beef to chickento
pork by respective factors of 2 conditionedon Larry preferring soupandthat Curlyis indifferent, conditionedon Larry
preferring salad. Wemayexpress these relationships by the
conditionalselectability functions:
Meal
{soup, beef,
{soup, chkn,
{soup, beef,
{sald, pork,
PSclSL (chknlsoup) = 2/7PSclSL (chknlsald) = 1/3
To computePSLSMIRc,
we recall, given that Curly views
pork as completelyrejectable, Moeviews lemoncustard pie
as highly selectable and Larry is indifferent. Giventhat
Curly viewsbeef as completelyrejectable, Larry viewssoup
as selectable, and Moeis indifferent; and given that Curly
views chickenas completelyrejectable, both Larry and Moe
are indifferent. Theserelationships maybe expressedas
0.25
0.25
PsLsMtRc(sald, lcstlchkn ) = 0.25
PSLSMIP,c (sald, bcrmlchkn) = 0.25.
Lastly, we needto specify PRLRCRM,
the joint rejectability function. This is done by normalizing the meal cost
values in Table 1 by the total cost of all meals (e.g,
PnLnCnM
(soup, beef, lcst) = 23/296).
Withthe interdependencefunction so defined and letting
q = 1, the jointly satisficing mealsare as displayedin Table
RM
0.078
0.074
0.091
0.074
Withthe Pot-LuckDinner, we see that, althoughtotal orderings for neither individuals nor the groupare specified,
we can use the a priori partial preference orderings from
the problemstatementto generateemergent,or a posteriori,
groupand individual orderings. A posteriori individual orderings also emergefrom this exercise: Larry prefers soup
to salad, Moeprefers lemoncustard pie to banana cream
pie, and Curlyprefers beef to either chickenor pork. Note,
however,that Curlyis not required to imposea total ordering on his preferences(chicken versus pork)--this approach
does not force the generation of unwarrantedpreferencerelationships. Wesee that a group-widepreferenceof avoiding
conflict emerges,since the individuallysatisficing strategies
are also jointly satisficing. This groupdesideratumwas not
specifieda priori.
PSLSMInc(soup, lcstlpork ) = 0.5
0.0
PSLSMIRc(soup, bcrmlpork)
PsLsMInc(sald, lcstlpork ) = 0.5
PsLsMlac(sald, bcrmlPork) = 0.0,
PSLSMIRc(soup,lcstIchkn ) =
PSLSMinc(soup, bcrmlchkn ) =
PRL Rc
Table2: Jointly and individually satisficing choicesfor the
Pot-LuckDinner.
PSclSL(porklsoup)= 1/7 PSolSL(p°rklsald)= 1/3.
and
0.237
0.119
0.149
0.080
IndividuallySatisficing
Participant Choice
PS
PR
Larry
soup
0.676 0.480
Curly
beef
0.494 0.351
Moe
lcst
0.655 0.459
PSclsL(beefls°up)= 4/7 PSolSL(beeflsald) = 1/3
PSLS, lnc(soup, lcstlbeef ) = 0.5
PSLSMInc(Soup,
bcrmlbeef ) = 0.5
0.0
PSLSMInc(sald,lcst[beef)
PSLSMinc(sald, bcrmlbeef ) = 0.0,
PSLScSM
lcst}
lcst}
bcrm}
lcst}
Conclusion
Satisficing gametheory offers a wayfor the interests of the
group and of the individuals to emergethrough the conditional preferencerelationships that are expressedvia the
interdependencefunction due to its mathematicalstructure
as a probability (but not with the usual semanticsdealing
with randomness
or uncertainty). Just as the joint probability function is morethan the totality of the marginals,the
interdependence
function is morethan the totality of the individualselectability andrejectability functions. It is only
in the case of stochastic independence
that a joint distribution can be constructedfromthe marginaldistributions, and
it is only in the case of completelack of social concernsthat
group welfare can be expressed in terms of the welfare of
individuals.
Optimizationis a strongly entrenchedprocedureand dominates conventional decision-makingmethodologies. There
107
is great comfortin followingtraditional paths, especially
whenthose paths are foundedon such a rich and enduring
tradition as rational choice affords. But whensynthesizing
an artificial system, the designer must employa moresocially accommodatingparadigm. The approach described
in this paper seamlesslyaccountsfor group and individual
interests. Ordercan emergethroughthe local interactions
that occur betweenagents whoshare common
interests and
whoare willing to give deferenceto each other. Rather than
dependinguponthe non-cooperativeequilibria defined by
individual rationality, this alternative maylead to the more
socially realistic andvaluableequilibriumof sharedinterests
and acceptable compromises.
Stirring, W.C. and Goodrich, M. A. (1999a). Satisficing
equilibria: A non-classical approachto gamesand decisions. In Parsons, S. and Wooldridge,M. J., editors, Workshopon Decision Theoretic and GameTheoretic Agents, pages56-70, University College, London,
United Kingdom,5 July.
Stirling, W.C. and Goodrich,M. A. (1999b). Satisficing
games. Information Sciences, 114:255-280.
Taylor, M. (1987). The Possibility of Cooperation. Cambridge Univ. Press, Cambridge.
yon Neumarm,
J. and Morgenstern,O. (1944). The Theoryof
Gamesand EconomicBehavior. Princeton Univ. Press,
Princeton, NJ. (2nd ed., 1947).
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