From: AAAI Technical Report FS-01-03. Compilation copyright © 2001, AAAI (www.aaai.org). All rights reserved.
Satisficing
Negotiation
for Resource Allocation
with Disputed Resources
WynnC. Stirling
Todd K. Moon*
Electrical and ComputerEngineering Dept.
Brigham YoungUniversity, Provo, UT84602
voice: 801-378-7669 fax:810-378-6586
email: wynn@ee,byu. edu
Electrical and ComputerEngineering Dept.
Utah State University, Logan, UT84322-4120
voice: 435-797-2970fax: 435-797-3054
emaih Todd.Moon@ece.usu. edu
Abstract
Satisficing decisionmakingprovidesa moremalleableframeworkfor negotiation than conventionaltechniquesbased on
optimizationof a utility function. In this paperwesummarize satisficing decisiontheory, whichprovidesa mechanism
for determiningdecisionoptionswhichare "goodenough"as
as tradeoffbetween
a selectabilityfunctionanda rejectability
function,with an indexof cautionas a decisioncontrol parameter.Singleagent satisficing is extendedto multi-agent
satisficing, by whichgrouprationality can be represented;
optionvectorsfor the entire groupare obtainedas a result
of this decisionprocess.Multi-agentsatisficing providesthe
stage uponwhichnegotiationtakes place. Negotiation.in this
context,is the processby whichall agentsdeterminea set
of optionswhichare bothindividuallysatisficing andjointly
satisficing; throughthe courseof negotiationagentscan accommodate
the needsof the group-- compromise
-- through
loweringtheir indexof caution. Anexampleis presentedof
resourceallocationwithdisputedresources.
Introduction
Decisionmakingagents acting together should be influenced
not only by their ownaspirations and budgets but by these
aspects of other agents in the system. To represent this interaction amongagents, a notion of group rationality must
be embodiedin the decision systems of interacting agents.
Grouprationality is’not necessarily a logical consequenceof
rationality based on individual self interest. Undera model
of rationality in whichmaximizationof utility is the operative notion, group behavior obtained by amalgamationof
the individual behaviors is not usually optimizedby optimizing each individual behavior, as is typically done in a gametheoretic setting. Thosewhoput their final confidencein the
limited perspective of exclusive self-interest mayultimately
function disjunctively, and perhapsillogically, whenparticipating in collective activities. Rather than reorient game
theory to accommodatesituations where coordination is a
morenatural operational descriptor of the gamethan is selfinterested conflict, we proposeto describe notions that are
neutral with respect to questions of conflict and coordination.
*Partial support provided by DARPA
under F30602-00-2-0532
106
Beyondsimply taking into account the presence of other
acting agents in a system, there is frequently someform of
sociality that is conduciveto at least a weakform of congruity or mutualagreement.In cooperative scenarios, agents
agree to worktogether; in competitivescenarios, agents tacitly agree to oppose each other. The procedures used to
arrive at these agreements are not determined simply as a
function of the preference structure of the decision makers,
whetherposed in a frameworkof self interest or community
interest. Agreementamongagents is typically obtained via a
process of negotiation, in whichmultiple agents evaluate and
share information whenthey have incentive to strike a mutually acceptable compromise.In the negotiating process, it
is not sufficient for a decision makermerelyto identify an
acceptable joint solution (for the community)according
its own lights. The entire communityshould "buy into" a
joint solution that is mutuallyacceptable.
In negotiation it is rare that all parties involvedwill tip
their handsto reveal all of the factors influencingtheir decisions. In a competitive setting, a policy of secrecy may
keep competitors from exploiting a weakness, or it maybe
used to persuade competitors to a more advantageous position. In a cooperative setting, completedisclosure of information might be precluded due to restrictions in communication bandwidth and/or time. Because of a lack of
disclosure, negotiation mayinvoke principles of inference,
whereinagents attempt to estimate positions or attitudes of
other agents based on the options they bring to the bargaining table.
In light of these observations, someprinciples of negotiation are suggested:
N-1 Negotiators must typically be concerned with meeting
minimumrequirements more than achieving maximum
performance.
N-2 Negotiations should lead to decisions that are both
goodenough for the group as a whole (as established
by a group rationality) as well as goodenoughfor each
individual (as established by local preferences).
N-3 Negotiationis typically an iterative process. Starting
froma set of initial joint options, it is natural to iterate towardsolutions whichare individually acceptable,
rather than attemptingto movedirectly to joint options
whichare a best compromise.
Definition 1 Thesatisficing set Eqis the set of options defined by
N-4 Negotiation mayfrequently incorporate elementsof inference.
A rich model for negotiation should be able to capture
other aspects of the negotiation process, such as recalcitrance (resistance to accede to group preferences), accommodation (openness to group preferences), or annoyance
over extended or unchangingnegotiation positions.
In this paper, wewill briefly reviewthe conceptof praxeic utility decision theory as a meansof implementingsatisficing control, then extendthis to multiple agent decision
makingto modelgroup rationality. Concepts of negotiation
consistent with the principles outlined aboveare established
using this multi-agent satisficing framework.As case study
of a problemfor whicha negotiated solution is reasonable,
a problemof resource allocation with disputed resources is
modeled.
Eq = {u E U: ps(u) >_ qpR(u)
(1)
[]
Single agent satisficing
Thesatisficing set consists of those options for whichthe
benefit exceedsthe cost: the set of alternative whichare arguably "good enough." There may be more than one option
in Eq. Movingawayfrom strict adherence to optimality increases the flexibility, while by not retaining only the best.
Ultimate selection of a single option for action is accomplished by meansof a tie breaking rule, such as most selectable, least rejectable, or maximally
discriminating.
The parameter q in (1) is the index of caution. As q is
increased, fewer options are accepted into the the satisficing
set. As such, the agent exhibits greater caution, accepting
only options of higher merit in comparisonto their cost. We
say that ~q is the satisficing set at level q. Becauseof its
similarity to likelihood ratio tests in conventionaldecision
theory, the test in (1) is referred to as the praxeiclikelihood
ratio test (PLRT).
Satisficing, a term coined by Simon(Simon 1955), refers
to a decision makingstrategy in whichoptions are selected
which are "good enough," differing thereby from conventional approaches whichseek only the best. Fromthe sarisricing viewpoint, being "good enough"is sufficient; insisting on the best and only the best via an optimizingalgorithm
may be an overly restrictive luxury. Froman operational
point of view, however,while establishing that an option is
(at least locally) optimalis at least expressibleas a optimization problem, establishing what is "good enough"appears to
be more elusive. The question of establishing good enough
choices is addressed from a philosophical point of view with
regard to truth systems by Levi (Levi 1984; 1980; 1967).
this epistemological framework,knownas epistemic utility,
options are sought for whichthe amountof information associated with themexceeds the potential for error. All options are deemedacceptable -- goodenough -- that pass a
likelihood ratio test comparinga troth valuation (a probability) and an informationalvalue of rejection (also constructed
as a probability). Applicationof epistemicutility to control
problemsyields praxeic utility theory (Stirling, Goodrich,
&Packard 2001; Stirling, Goodrich, &Frost 1996c; 1996b;
Stirling 1994; Goodrich, Stirling, &Boer 2000; Goodrich,
Stirling, &Frost 1998; Stirling, Goodrich, &Frost 1996a;
Stirling & Goodrich 1999). (For a discussion of praxeic
utility theory in the context of negotiation, and for a more
complete development of these concepts, see (Stirling
Moon
2001).) In this theory, a selectability functionPs (u) is
formedwhich, for each option u available from a universe of
options U available to a decision makingagent X, measures
the degree to which u works toward success in achieving
the agent’s goals. Also, a rejectability function pa(u) is established whichmeasurescosts associated with each option.
This pair of measures,called collectively the satisfiability
functions, are endowedwith the mathematical structure of
probabilities (e.g., they are nonnegativeand sumto 1 on the
U).
Multiple agent satisficing
Satisficing decision theory extends very naturally to multiple agent systems. Satisficing admits degrees of fulfillment, whereas optimization is an absolute concept. While
the statement "Whatis best for meand what is best for you
is also jointly best for us together" maybe nonsense, the
statement "What is good enough for me and what is good
enoughfor you is also jointly good enoughfor us together"
maybe perfectly sensible, especially whenwe do not have
inflexible notions of what it meansto be "good enough."
Satisficing grants roomfor compromise,leaving open the
opportunity for one or moreagents involved to relax standards of individual performancein the interest of the good
of the community.A theory of multi-agent satisficing thus
provides the stage on whichthe act of negotiation can reasonably be presented.
Since they possess the mathematical structure of probabilities, selectability and rejectability can be naturally extended to the multivariate case by defning joint selectability and rejectability measures, whichmaybe used to determinea jointly satisficing set. In addition, individual decision makersmayestablish individual notions of satisficing
by computingmarginal selectability and rejectability functions fromthe joint expressions.
Let X1,... ,XN be N interacting agents, where each
agent has its owndecision space Ui. The joint action space
is the space U = U1 x 0"2 x ... x UN.A joint decision is
an element u = (ul,u2,...
,uN) E U. Wedenote the ith
element of u as u(i).
An act by any single memberof a multi-agent system
has potential ramifications throughoutthe entire community.
And, although a participant mayperform an act either in
its owninterest or for the benefit of others, the act is usually not implementedfree of cost: resources are expended
or risk is taken, perhaps by the single agent, but also perhaps by the entire community.Althoughthese consequences
Satisficing
decision
multiple
making: single
agents
and
107
Theorem1 (The negoth~tion theorem) If u is individually
satisficing for Xi, then it must be the ith element of some
jointly satisficing vector u, i.e., u = u(i) for someu ¯ ~q.
may be defned independently from the benefits, the measures associated with benefits and cost cannot necessarily be
specified independentlyof each other. In light of this, the
object representing the relationships betweenagents in their
systemsregarding their individual and joint selectability and
rejectability is an interdependencemeasurewhichcombines
both rejectability andselectability for all agentsas a multivariate probability function of the form
Proof Without loss of generality, assumei = 1. Let u ¯ E~
(i.e., Psi (u) > qPRI(u)). Toestablish proof by contradiction, assumethat u # u(i) for all u ¯ Eq. It follows that
for all v ¯ U2 x ... x UN, Ps(U,V) < qpR(u,v). Then
PSI,$2,...
,SN,RI,R2,...
,RN(Ul~ ~2 ~ ¯ ¯ ¯ ~ UN~Vl~ V2~¯ ¯ ¯ ~ VN).
This is expressedmorebriefly as PS,R(U,v). Valuesfor the
interdependence measureare typically obtained by meansof
factorization into constituent conditional and marginalprobabilities. In these factorizations, agents mayrepresent how
their selectabilities or rejectabilities are affected by the selectability and rejectability of other agents. Fromthe general
interdependencefunction, the joint selectability function is
obtained by a marginalization
v
[]
which contradicts u ¯ Eq.
Onthe basis of the negotiation theorem, it maybe argued
that each agent has a seat at the negotiation table. Noone is
necessarily frozen out of a deal.
It is important to emphasizewhat the negotiation theorem does not provide. If ul is individually satisficing for
X1, and u2 is individually satisficing for X2, then by the
theorem ul = u(1) and u2 = 0(2) for some u, fi ¯
However,it is not necessarily the case that u = fi: the options that are both individually and jointly satisficing maybe
different for different agents. Thus, the negotiation theorem
does not establish a "solution" to the problem. However,
it provides the basis upon whicha solution maybe sought
through an iterative negotiation scheme. To obtain buy-in
fromall agents, the options that are individually satisficing
for each agent must be elementsof the samejointly satisficing options. Suchoptions are the result of negotiation.
The proof of the negotiation theorem makesuse of the
fact that the same index of caution is used to computeE~
as is used for Eq, but an agent could use a higher q to determine Eq than it does for its ownsatisficing set. Thenegotiation theoremis not necessarily true in this case." Or it
mayhappen that each agent has its own perception of the
interdependencefunction. In this case, we denote Xi’s interdependencefunction as P~,rt, and the correspondingjoint
selectability and rejectability as p~ and ph. Again, the negotiation theorem applies to each agent separately, but not
collectively. Another possibility is that an agent mayuse
joint functions P/s and p~ for establishing Eq, but use individual Ps~ and PR~not computedas marginals ofps and
PR, thereby presenting a different "public face" and "private face." Again, the negotiation theorem does not apply.
(This raises the the question for future investigation: to what
degree can these private satisfiability functions differ from
marginalsatisfiability functions and still havereasonablenegotiations.)
Thenegotiation theorem, and these observations about its
provisional application, motivate the developmentof algorithms for negotiation based on satisficing.
ps(u)= Ps,R(u,
v6U
and similarly
pR(v) = E ps,rt(u,v)
u6U
Definition 2 Themultipartite satisficing decision rule defines the set the multipartite satisficing set by
Eq = {u 6 U: ps(u) > qprt(u)}.
(2)
[:3
Joint options in Eqare those for whichthe benefits exceed
the costs, as viewedfromthe perspective of the group and as
representedby the joint selectability andjoint rejectability.
Thetest in (2) is referred to as the joint praxeic likelihood
ratio test (JPLRT).
Givenjoint selectability andjoint rejectability, an individual agent can computemarginals by
=
ps(u)
{ueU:u(i)=~,}
pR,(v,) =
v
ps(v).
{veU: v(i)=vd
Theresulting individually satisficing set for Xi is then
Alternatively, an agent could employa p& and PR~not obtained as a marginal,presenting a different face to the public
than whatit holds for itself, and use these function to compute E~.
Wewill use the notation u = u(i) to indicate that the
option u 6 Ui is the ith elementof a joint option vector u.
Anoption u that is jointly satisficing for Xi, is not necessarily individually satisficing for Xi. That is, a joint option
u ¯ Eq does not necessarily have u(i) ¯ Eq. The converse,
however,is true: ifu ¯ E~, then u = u(i) for someu ¯ Eq.
This is established by the following.
Satisficing
Negotiation
Multi-agent satisficing is suited to the principles of negotiation outlined in the introduction. By admitting degrees
of fulfillment, satisficing agents can explore options which
are both mutually and individually good enough. In a negotiation process, however,it is not sufficient for a decision
108
makerto identify a solution, even one whichit views as being jointly acceptable. As mentionedabove, other agents in
the system mayhave their ownmodelsof the joint interdependencefunction, and their ownindividual satisficing functions, or maydetermineindividually and jointly satisficing
options not coincident with those of other agents. A negotiated solution should ideally be one in whichall members
of the communitycan individually concur. The multipar7
tite satisficing set Xlqand the individually satisficing set E~
provide each Xi with a basis for negotiation: an assessment
from each agent’s point of view of all options that are good
enoughfor the group, and of the assessmentof all individual
options that are goodenoughfor itself.
Compromiseamonga group of agents involves a lowering of standards by admitting possible actions that an agent,
acting only unilaterally, wouldnot necessarily prefer, but
whichit is willing to admit in the interest of acting as part
of a group. The lowering of standards motivates the use of
a satisficing outlook in the decision theory. Anapproach
based on optimization, particularly one formulated on the
basis of exclusive self-interest, does not admit grades or degrees. A choice is either optimal, or it is not. Compromise
does not necessarily entail completeabolition of any agent’s
standards. An agent feeling that too muchcompromiseis
imposedit maywalk awayfrom the negotiating table.
In the formalismof multi-agent satisficing, an agent’s index of caution qi acts as a parameterrepresenting the degree
of compromisean agent is willing to adopt. By lowering
the degreeof caution, an agent is willing to consider placing
moreoptions in its satisficing set. As qi ~ 0, every option
available to Xi is satisficing for Xi. If all agentsare willing
to sufficiently reducetheir standards, a jointly acceptablesolution can be obtained.
Wewill let q = (ql,... ,qiv) denote the caution vector of the players. The least cautious index is qL =
min{ql,... , qiv). Froman individual perspective, the negotiation theoremapplies if an agent uses its ownindex of
caution, qi to determinethe individually satisficing set, but
uses qL to determineits jointly satisficing set. It is assumed
hereafter that each agent uses q = qL to determineI3q. This
reflects the conservative observationthat the standards of a
group can be no higher than the standards of any memberof
the group.
Therelationship betweenindividually andjointly satisficing sets for an agent is formalizedby the following:
Definition 3 The set of all jointly satisficing vectors in ~
that are also individually satisficing for Xi is the compromise set Ci, defined by
{uj:
uj E
~ Ci.
i=1
Anyjoint option u E NqLis a joint accordoption.
[]
The desired outcomeof a negotiation algorithm is a joint
accord set. FromNqL,a single element joint accord option
maybe selected according to sometie breaking rule, such as
the rule maximizing
joint benefit to cost ratio,
ps(u)
,,eNqLpR(u)
(3)
u* = arg max
This option is called the rational compromise.
If NqL= 0, then there are no decisions whichare jointly
and individually acceptable to all agents in the system.
Definition 5 In the context of multi-agent satisficing theory, negotiation is the process of workingtowarda solution
whichis individually and jointly acceptable to each agent in
the system.
12
¯ Workingtoward a negotiated solution requires at least
one of the agents to lower its standards, then recompute
their compromisesets. If no agent is willing to compromise further (by lowering its ownstandards), then an impasse is reachedin the negotiation process. Until that point
is reached, however,negotiations mayproceedin goodfaith.
The loweringof standards is represented in this context by a
lowering of an agents index of caution. Algorithm1 outlines
a negotiation algorithm based on this observation, termed
the EnlightenedLiberals algorithm ("liberal" in the sense of
being tolerant of views other than one’s own; "enlightened"
in the sharing of information).
Algorithm1 TheEnlightenedLiberals Negotiation Algorithm
Stop 1: Initialize: qi = oqi for i = 1,... , N. qL= min(oqi).
Stop 2: Xi forms lC~Land E~.
i .
Stop 3: X~forms Ci = {u E Zqc’
ul E ~b~i }.
Stop 4: Communicate
C~andq~ to other agents.
Stop 5: Each agent forms Nqc = nN=lci.
Stop 7: If NqL # 0, form the rational
(ul, u2,. ¯ ¯ , uN)according
to (3).
Since qL < qi the negotiation theoremindicates that Ci is
not empty. By the negotiation theorem, if u E ~q~, then
there is someu E Ci such that u = u(i).
Wedefine
---
N
Nql:. ----
Stop 6: If NqL= (~, each agent determineshowmuchto lower
qi, then communicates
q~ with the other agents. Then
repeatfromstep 2.
[]
Ci(j)
Definition 4 Thejoint accordset Nqz is the set of all vectors that are jointly (at caution level qL) andindividually (at
caution level qi) satisficing for all agents. Thatis,
compromise u =
In this algorithm, all agents communicatetheir choices in
the samestep. Thereis no wayto use the partial information
provided by another agent’s compromiseset to modify an
agent’s decisions.
u for someu E Ci}
as the set of all options for Xj in Ci.
109
Inference in negotiation
It maybe noted that Enlightened Liberals is in accord with
the first three principles of negotiation outlined in the introduction. However,no inference is employedin the algorithm as stated, since all agents essentially pass information
simultaneously. The inference problem faced by agent Xi
is to estimate ~, PJR~, PSi, and PR~-- the praxeic system employedby Xj -- given the offered solutions brought
to the negotiating table in the form of Cj. As an estimation problemall the tools of statistical estimation theory can
be brought to bear, such as Bayesian estimates, maximum
likelihood, minimumvariance, maximum
entropy, etc. (The
methodselected is problemdependent.) In the examplepresented below, a heuristic is illustrated whichis similar to
maximum
likelihood.
Incorporation of the inference aspect of the negotiation
is outlined in algorithm 2, whichdiffers from Enlightened
Liberals mostly in the sequence nature of the exchangeof
information.
Algorithm2 TheInferring Liberals NegotiationAlgorithm
Step1: Initialize: qi = oqi fori = 1,... ,N. qL = min(0qi).
Step2: Xiinfers updatesforPsj,i PR~,iPsi andpk, basedon the
compromise
sets for {Cj, j = 1, 2,... , N, j # i}.
initial allocation of resource, each agent must also sustain
a cost of acquisition for each additional resource that is required. Whileexpressed as an abstract "resource allocation"
problem, it maybe helpful to envision the geographical division of a country amongnon-aligned factions. The apportionmentof the land of Israel amongIsraeli and Palestinian
claimants is a recent motivatingexample.Interestingly, data
that might be adaptedfor a problemon a larger scale for Europe in the mid-twentieth century have been the subject of
research in studies in cooperationand complexity(see, e.g.,
(Axelrod 1996)).
Wewill consider specifically only the two agent case; extensions to moreagents is straightforward in principle. We
will denote the allocation decision vector of agent Xi by a
vector u~ E {0, 1}Mwhere u~ = 1 if resource j is selected
by agent i. (A final option denoted as i =0 might al so be
used to indicate that Xi is terminating the negotiation process and walkingawayfrom the negotiating table.) The decision vector ~ is used to indicate the Booleancomplement
of the decision vector ui. For decision vectors u1 and uz, we
will denote by d = ut n u2 the disputed resources claimed
by both agentst Wedenote by ~zi\d the resources claimed
exclusively by Xi. As agents begin the process, they have
someinitial allocation ou1 and ou2, with disputed resources
2od
. =0’0,1I"] 0’0,
Step3: Xi forms :E~Land E~,
Step4: Xi forms Ci.
Step5: Xi communicates
Ci andqi to all other agents.
Step6: Afterall agentshavetransmittedtheir information,each
N Ci.
agent forms NqL= ni=l
Step7: If NqL= @, each agent determines howmuchto lower
qi, then communicates
qi with the other agents. Then
repeatfromstep 2.
Step8: If NqL # 0, form the rational compromise u =
(ul, u2,... , UN)accordingto (3).
Formulation of selectability
and rejectability
The joint interdependence function is Ps,rt(u,v)
pSt,S2,RI,R2(Zq,U2,Vl,V2).
When we want to represent explicitly that this is the interdependence function as perceived by agent Xi, this will be denoted as
P~lsS2,RI,R2(U1,’/£2, Vl, V2). If this is changingwith negotmtion iteration numberr/, we will indicate this with
P~,s~,R,,R~(u~,u2,vl, v2;~7).
Toformulatespecific results, it is expedientto factor the
joint interdependencefunction into conditional and marginal
probability measures.Conditional probabilities, as observed
by Pearl (Pearl 1988)permit local or specific responsesto
characterized. Conditional behavior is behavior at the local
level, with all dependenciesspecified. Suchfactorizations
permit characterization of global behavior in terms of local
relationships, whichare frequently easier to specify. A variety of factorizations are possible, evenfor the simple case of
two agents, andit is not necessaryfor all agentsto invokethe
same factorizations. However,in this example,both agents
will factor the joint interdependencefunction the sameway.
Wewill express the factorization from the point of view
of Xt, then express the inference process from the point of
view of X2(using information from X1). As a shorthand
will represent the factorization of the probability functions
in terms only of their variables, using, for example,$2 ]R2
as a representationfor Ps~IR~(u2 Iv2). A reasonable(but
unique) factorization, expressed from the point of view of
Example:Disputed Resource Allocation
Complexity
is no argumentagainst a theoretical approachif
the complexity
arisesnot out of the theorybut out of the material whichanytheoryoughtto handle.
-- Frank Palmer
Grammar(1971)
Weillustrate the negotiation frameworkoutlined above-including interdependencefactorization, establishing satisfiability functions and inference -- by meansof a resource
allocation problem. Consider a situation in which N agents
are to allocate Mresources amongthemselves in such a way
that all resourcesare allocated to at least oneagent, but more
than one agent mayclaim a resource. A resource with more
than one claimant is a disputed resource. Disputedresources
¯ are of lower value than undisputed resources, both because
utilization of a resource is attenuated by virtue of sharing,
and because of an intrinsic societal valuation that would
avoid dispute. The goal of each agent is to obtain as much
of the resources as possible (or the resource allocation with
the maximum
valuation), while working toward having the
fewest disputed resources as possible. Starting from some
Thenotationul flu2 mightbe moreexactlyrepresentedas ul A
uz, wherea "bitwise" AND
operation is impliedin each element.
However,
the lq notation seemsto be moresuggestive.
110
Costs
XI is
Several elementsof the problemcontribute to an agent’s perception of the cost of the choice.
($1,$2,R1,R2)=
(s~IS~,R~,R~)(R~
IS~,R~)(S~
IR2)(R~).
Reduce disputed resource Each agent evidences the difficulty of sharing the resource by seeking to eliminate the
disputed resources. This not only serves his purposes -since disputed resources maynot be enjoyed at full value
($1, $2, R1, R2) = ($1 IS2, R2) (R1 IS2, R2)($2) (4)
-- but also makesa concessionto the society of the agents,
Thereis an attractive symmetry
in the first twofactors, being
whichwouldprefer undisputed allocations.
the selectability andrejectability (respectively), conditioned
In general, there is a cost function associated with dison those quantities for the other agent. Thefirst two termsof
puted resources, whichfor agent Xi is denoted as 6i (u1 n
this factorization represent X1’s selectability andrejectabilu2). This could dependon a variety of societal or historical
ity, respectively, whenX2places all of its selectability mass
factors. In somedisputed resources, there maybe no cost asand rejectability mass as the conditioning arguments. The
sociated with morethan one claim on the resources, whereas
factorization in (4) is expressedmoreexplicitly
for others there is considerablecost.
A simple modelfor the cost is simply to makethe dispup~,,s~,~,,R~
(ul, u2,vl, v2)=
tation cost function proportional to the value of the disputed
1
x
v
p~,ls~,R2(ullu2,
1 IS2,R=( I~=,v=)Ps=(~’=)PR=(v=)"
v2)PR,
resource to each agent,
Underthe assumptionthat rejectability and selectability are
independentfor a given agent, we obtain
(5)
In the sections below, we describe the parametersthat affect
each of the terms in this factorization.
2
6~(u1 nu2) = 6~(d) oc ~1 +ei.
led
This cost can be placed in the context of the conditional
probability p~, IS=,R2(ul lu2, v2) as follows.
. Whenu2 = ~2, then X2places all of its selectability and
noneof its rejectability on the vector u2, so it is fully com-.
mitted to the option u2. Then it maybe presumedthat
there will be a disputation of d = vl N u2, and the cost
becomes61 (v2 n u2).
. Whenu2 = v2, then X2places all of its selectability as
well as all of its rejectability on u2, and hence is conflicted. In this conflicted state, X1assumeshalf the cost
of disputed territory, ½~1(v2 N u2). (Other options are,
course, possible to deal with this conflicted state.)
. For those territories that X2has indicated that it doesn’t
want(no selectability and high rejectability), there is
cost for a disputedterritory.
Anoverall rejectability function based on disputation can be
formulatedby normalization. Wewill call this rejectability
functionp~ [s2,R~;~ (ul lu2, v2).
Goals
Eachagent wants to maximizethe value of the resources it
claims. There is a functional 9i(u) adopted by Xi evaluating
2its intendedallocation. This mightbe quite simple, as in
gi(u) = ~’ (J)’
jEu
whereei (j) measuresthe intrinsic value of resource j. This
mayinclude the size of the resource as well as other attributes. (In the case of land as a resource, it mightmeasure
attributes such as a harbor or an airport, mineral or agricultural assets, or historical or religious value.) In the case that
the resources are distributed in space, the value maybe determined using less localized measures. For example,there
maybe more value in having the resources as near to each
other as possible, or in a contiguous block. Or there maybe
less value to a resource which is surrounded by resources
claimed by the other agent. (In the case of land apportionment, an agent might prefer large pieces contiguously
joined, with no islands of other agents’ land in the middle.)
All of these variations can be incorporated into gi(u).
Giventhat the agents’ goals are prescribed by the desire to
obtain more resources, we simply normalize the allocation
value to form a probability massfunction:
p~,ls=,R=
(u,I~’=,v=)-- p~,(u,)(xg’(u,).
That is, the the selectability is conditionally independentof
X2’s options. In the joint selectability each agent thus acts
independently:
PS, ,S2 (Ul, U2) = PS, (Ul)PS=(u2).
In (5), the term p~ (u2) is Xl’s model (or perception)
X2’s selectability. ~’his is determinedsimply Xl’s estimate
of g2 (u) (estimated according to X1’s knowledgeof )(2).
2Theset {j E u} appearingin this summation
is a shorthand
for {j: uj = 1}
111
Cost of Acquisition There is a cost associated with acquiring the resources beyondthe initial allocation. (For example, in the case of land resources, simply makingthe decision to acquire the land does not makeit so. It maybe
necessary to deploy troops to enforce the decision, or to
movein colonists, etc.) The cost of acquisition will also
dependon the interest that the competingagent has in the
newterritory. In the context of the conditional probability p~xIS2,R~(ul lu2, v2), the followingobservationscan
made.
. In the case that u2 -- v2 (that is, X2puts all of its selectability andnoneof its rejectability on the vector u2),
then it maybe presumedthat there will be a dispute over
d = vl r’l u2. Thecost of the disputed acquisition is denoted by
)~1 (d; 1, ou2),
whilethe cost of the undisputedacquisition is
reliably suggests the need to estimate this quantity, !f possible, during the negotiating process. Inference of p~j (v)
discussed below.
As the negotiating process begins, someinitial condition
is needed.Oneinitial conditionreflecting this uncertainty is
to assumethat PR2(v2) apportions equal rejectability to all
options. Anotherapproachis to allow PR2(v2) -- as an unconditional measure-- to reflect those aspects of the problem that are most independent of actions or goals of other
agents. In this light, allowingPR2(v2) to be proportional
the cost of acquisition is reasonable,
1, 0u
:~1 (v1 \d; 2)
oU
Thetotal cost is then the sumof these:
X1 (V1 ; OU1 , 0U2) :
f~l (vlkd;oul,
2), 0 ’//,
OU2).~. )~1 (d; ul
¯ Whenu2 = v2; that is, X2is conflicted, placing all of its
selectability as well as all its rejectability on u2, then X1
mightassumethat X2will not be in disputation; then
X1 (v1 ; 0ul , o’/Z2) = 21 (v1 ; u2)
0ul , 0
2 (~2(V2,0ul
~’~(,~)~x2 (v~;0~,)+
¯ For those options on which X2places none of its selectability and all of its rejectability on, wewill assume
that sameresult as whenX2 is conflicted. (Moregenerally, wecould have a reducedcost for acquisition.)
¯ In the more general case, X2 maybe conflicted in some
areas but not in others. In this case, X1only counts as
disputedthose territories whichintersect with its interests
and for whichX2is unconflicted.
Combiningthese costs to~ether and suitably normalizing,
the rejectability function PR,IS2 l~2"x(ul Iu2 v2) is obtained.
N OU2).
It is straightforwardto verify that the joint rejectability
can be computedas
Inference
Weconsider nowthe question of inference of the parameters
of other agents during the course of negotiation, presenting
a methodwhichis reasonable in the context of the present
problem.After its initial decision-makingstep, X1presents
C1 and qx to X2. Basedon this compromiseset and caution
index, what can be inferred about Xl’s satisfiability functions? BecauseX2will be doing its computations based on
the factorization (5), it maybe assumedthat that X2has
model ofps~ (u), since this is based primarily on economic
questions whichare observable by all agents. As mentioned
above,however,p~, (u) is difficult to obtain without further
information. This parameterinfluences the joint rejectability p~ R2, and hence the marginal P~2, so its estimation has
an extendedinfluence in the decision makingprocess. (This
section, for the sake of definiteness, is presented as if X2
were makinginference based on information from X1.)
Given p~ (u) and p~ (u), consider the joint options
C1. Ifp}~ (u) > qlp2~ (u) and u = u(i) for some u E
then the compromiseset provides no information: it reflects
decisions that would be madeby X2 using its estimates.
Also, iffs~(U ) < qlp~(u) and u ¢. Ci(i) then no additional informationis provided: X2did not expect the choice,
and X1did not select it.
However, ifp~(u) >_ qlp~(u) and u ~ Ci(i) then
has rejected option, both individually and jointly, whichaccording to Xu’s model it should have accepted. Furthermore, ifp~(u) ql p2n~(u) but u E Ci(i), th en X1has
accepted options both individually and jointly which, according to X2’s modelit should have rejected. Both of these
circumstances evince that X2’s modelp~. (u) is inaccurate
at u and stands updating. Our inference rule is to changethe
rejectability 2PR~(u) in such a waythat these inconsistencies
are resolved, and in such a waythat the changeat each point
is minimizedwhile ensuring that the probability constraint
is satisfied.
Cost of negotiation An agent may attribute cost to the
process of negotiation. If the negotiation must proceed
through several iterations, an agent maybecomesufficiently
annoyed at the process that its response is to walk away
from the negotiating table. Several factors maybe incorporated into the cost of negotiation, including the numberof
iterations (whichwe denote by r/), or the apparent lack
progress (if the compromisesets comingfrom other agents
appears to be unchanging). A cost based on the number
of iterations can also represent determination of an agent
to with respect to certain options: while the overall boldness is decreasing, the rejectability of someoptions can be
correspondinglyincreased to partially offset the reduction.
Thecost of negotiation is represented by ai(ul, u2, v2;r/);
suitably normalized it becomesthe rejectability function
P~,IS2,n2;,*(uxlu2,V2 ; 9~)
Overall conditional rejectability
The conditional rejectability function in (5) is expressed as a convexsum
the rejectability functions described above:
P~,t s2,R2
(ullug,v~)
l~3P~,lS~,n~;,~(ullu2,v2;rl) (6)
where ~i 31 = 1.
The marginal p]% (v2) and joint rejectability The quantity P~2(v2) in (5) is Xl’s modelof X2’s marginal (unconditional) rejectability. This is viewed(in this formulation)
as separate parameter, not a derived quantity. A variety of
factors influence the joint rejectability. Evenif the factors
could be computedexactly, the weightingfactors in the combination maybe unknown.The difficulty of estimating this
112
Numerical demonstration
Consider a country with four regions as shownin figure 1.
Values are apportioned in such a waythat adjacent regions
have a value greater than the sumof the constituent areas,
and that value increases with moreregions, as shownin table 1. The cost of disputed regions is also shownin table
1 (unnormalized). Cost of acquisition is on a region-byregion basis, as shownin table 2. Theinitial allocation is
0 ul = [1110] (countries 2, 3, and 4) and 2 = [1101].
Figure 2 illustrates the sequenceof estimated probabilities
pXn2(u) and p~t (u) for six iterations. (The abscissa represents the choice u as a decimal representation of the binary decision vector. The lines spaced within an integer
[u, u + 1) represent the probability estimates for different
iterations of the negotiation algorithm.) After several iterations of negotiation, the compromisesets shownin table 3
are obtained, and the joint accord set N join in table 4 is
obtained, where the rational compromiseis indicated with
*. (The integers represent the decimal form of the correspondingbinary option vectors). The individually satisficing sets are Eql = {14} and Eq2 = {5,9,12,13}. The
final boldness reached is q = (1.3, 1.3), after starting
qo = (1.9, 1.9) and decrementing each time by Aq = 0.05.
It is interesting to note that even after negotiation, for the
given value/cost data, the two agents end up with disputed
regions, andthat the initial conditionsstill remainin the joint
accord set. However,having gone through the negotiating
process, while regions must be shared, the agents mayfeel
that they have "boughtin" to this circumstance, since these
options are individually satisficing.
Definethe sets
p2s’(U)
< qlp21(U)or
andu
~Ci(i)
U= {uE Ui:
Pi(~)qlv~l(u) an
ds, e C~
(i)
= {u E Ul:p21(u) < qlPal(U) andu E C/(i)}
IT = {u E Ui: p2st(U) > qiPR,(u) andu ¢. Ci(i)}
Elements in U have rejectability consistent with Ci. Elements in U have rejecta~lity too high to be consistent with
Ci, while elements in U have rejectability too low. For
u E U or u E 0", form updated rejectability functions by
where
q~p-n, tu) U E (7
for somesmall positive e. This introduces a net change in
rejectability
A = E pR,(u)(1-
a(u))
ueOuO
whichis to be distributed amongthe rejectabilities of the
elements in Ui with the smallest change possible without
affecting the decision boundaries. Let
--
2
U>= {u E U: p2sl (u) > qlPR~ (u)}
and let
U< = {u E U:p~,(u) qi p2~(u)}.
Thenotation IUI denotes the numberof elements in the set
U. Thenthe redistribution is as follows:
If A> 0 (i.e., rejectability is added):
¯ Distribute A amongU U D" as equally as_.possible, such
that in U< U 0, PR,
2 ..... (u ) < 1 and in U>, p~, (u)
2
(u).
qlPR.....
Figure 1: Fourresources to be distributed
If A < 0 (i.e., rejectability is removed):
Discussion
Withinthis frameworkfor negotiation there are several observations that maybe made. In some humannegotiations,
parties often repeat a position repetitively, without an apparent changeof state, until at somepoint there is an abrupt
change in feasible options. The procedure represented here
provides a modelfor such behavior: even whenfrom one iteration to the next there might be no change in compromise
sets, each agent is modifyingits modelsof the other agent,
loweringits caution, andpotentially changingits rejectability as a function of the numberof iterations.
¯ Distribute A amongU U U as equally as possible, such
r2
that inU>
> 0 and in U<, p2s~(u) <
_ U6 p.na,new(U)
2
’qlPat,new(U)
This simple-mindedinference does not fully exploit the
information available. For example, if by Ci Xi appears
uninterested in someresource, X2could parameterize an increase its interest in that area either by loweringa rejectability with respect to its acquisition, or by increasing its selectability. However,the inference described above suffices
to demonstratethe concept.
113
C1
U
2
61(u)6~(~)
p~s~(u)
ps~(~)
U
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
(14,1)
(14,3)
(14,4)
(14,5)
(14,6)
(14,8)
(14,9)
(14,12)
(14,13)
(14,15)
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
0.0210
0.0252
0.0546
0.0420
0.0714
0.0756
0.1092
0.0252
0.0504
0.0546
0.0588
0.0714
0.1008
0.1050
0.1345
0
0.0227
0.0182
0.0500
0.0318
0.0636
0.0500
0.0818
0.0455
0.0727
0.0636
0.0727
0.0818
0.1045
0.0955
0.1455
0
3
3
7
6
8
8
15
2
4
4
9
9
13
13
20
0
3
3
7
5
9
9
12
6
9
9
5
12
14
14
20
Other behaviors such as recalcitrance or opennesscan be
modeleddepending on how the boldness is changed.
A concern that maybe raised regarding this procedure is
its computational complexity. A large measureof the complexity arises due to the computationof marginals in the
formulation of PRI,n2. The complexity can be mitigated
somewhatby efficient organization of the computations,using, for example, the factor graph approach described in
(Kschischang, Frey, & Loeliger 2001). Another approach
is to absorb the normalization used determining in Psi,s2
andPR1,R2
into the index of caution q. Oncean initial q can
be determined which provides for meaningfulindividual and
joint solutions, the index of caution is adjusted until a group
accordis established.
In conclusion, the multi-agent satisficing theory provides
a meansof describing solutions whichare individually and
jointly satisficing fromthe perspectiveof an individual agent
in the communityof agents. Wehave provided a definition
of negotiation, whichis the process of workingto achieve
accord amongthe different agents with regard to the solutions they they find acceptable, and provided somealgorithms to implement that process. To demonstrate how the
theory maybe applied to a multi-agent problem, a resource
allocation problem was presented in which agents vie for
disputed resources.
(undisp.) (disp.) (undisp.)
3
6
3
2
2
4
2
6
3
10
5
4
o.1~
0.05
c - ,1111
~oo,[
o.1~
°°
o ,,,
o.1!
2
4
6
8
u
lO
12
(2,13)
(4,13)
(6,13)
(7,13)
(8,13)
(10,13)
(12,13)
(13,13)
(14,14)
(15,13)
Table 4: Joint accordset
Table 2: Cost of acquisition per country
4
(15,9)
(2,12)
(4,12)
(6,12)
(7,12)
(8,12)
(10,12)
(12,12)
(14,12)
(15,12)
N ¯
(14,5)
(14,9)
(14,12)
(14,13)
U
(disp.)
6
4
4
8
(4,9)
(6,9)
(7,9)
(8,9)
(9,9)
(10,9)
(11,9)
(12,9)
(13,9)
(14,9)
Table 3: Compromise
sets after negotiation
Table 1: Valuationsfor resource allocation
0001
0010
0100
1000
C2
(2,5)
(4,5)
(5,5)
(6,5)
(7,5)
(8,5)
(12,5)
(14,5)
(2,9)
(3,9)
14
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