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Quickly locating efficient, equitable deals in automated negotiations under two-sided information uncertainty

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Quickly locating efficient, equitable deals in automated negotiations under two-sided information uncertainty
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  Quickly locating ef  fi cient, equitable deals in automated negotiations under two-sidedinformation uncertainty Hemalatha Chandrashekhar  a, ⁎ , Bharat Bhasker  b a Indian Institute of Management, Ranchi, India b Indian Institute of Management, Lucknow, India a b s t r a c ta r t i c l e i n f o  Article history: Received 22 January 2009Received in revised form 9 March 2011Accepted 25 June 2011Available online 2 July 2011 Keywords: Automated negotiationBilateralMulti-issueMechanism designAgent designUncertainty This paper develops an automated negotiation procedure inclusive of mechanism design and agent design forbilateral multi-issue negotiations under two-sided information uncertainty. The proposed negotiationmechanism comprises a protocol called MUP (Monotonic Utility-granting Protocol) and a matching strategycalled WYDIWYG (What You Display In fl uences What You Get). The proposed preference elicitationprocedure makes the agents faithful surrogates of the user they represent while the proposed FrontierTracking ProposalConstruction Algorithm (FTPCA)makes them learn theopponent's fl exibility in negotiationandrespondappropriately.Themechanismdesignandtheagentdesigntogetherhelpinlocatingef  fi cientandequitable deals quickly.Theef  fi ciency,stability, simplicity, distribution symmetry andincentive compatibilityof the proposed procedure are demonstrated through negotiation simulation experiments.© 2011 Elsevier B.V. All rights reserved. 1. Introduction The negotiation problem is considered complex mainly due to theindeterminacy of its outcome. For instance, any of a huge number of feasible solutions could be the outcome of a negotiation between abuyer and a seller (who have different preferences as regards to themultiple attributes  —  price, quality, payment terms, delivery termsetc.) of a product/service. Not all feasible solutions would be  ef   fi cient  . “  An economic allocation or decision is ef   fi cient if and only if there is noother feasible allocation that makes some individuals better off without making other individuals worse off  ”  [11]. Not all  ef   fi cient   solutionswould be  equitable . As noted in Ref. [25] negotiation outcomes coulddepend on the strategic interaction between the bargainers i.e., thebetter negotiator may get more utility from the derived solution thanthe other(s). Such solutions may not be fair to all parties involved. Ineconomic allocations, a fair share translates to an equitable share butnot necessarily an equal share.  Equitable  solutions in bargainingsituations correspond to the Nash solution. Nash solution point is theone where the product of the corresponding individual utilities ismaximized [20]. Under total information uncertainty, negotiating parties may not have much idea of the ef  fi ciency of the derivedsolution, but they may have some idea of its equitability based on thefairnessofthederivedsolution.Onlyfair/equitablesolutionswouldbe stable  in the long run. For instance, a negotiated contract that is notfair to all concerned parties would entail the risk of being reneged bythe disgruntled party sooner or later. Deriving ef  fi cient and equitabledeals may be straightforward when all the negotiating parties sharetheir private information. In reality, parties tend to hide orstrategically misrepresent private information for fear of beingexploited by the opponent(s). It is this total information uncertaintythat complicates and delays the location of ef  fi cient and equitabledeals in negotiations. However, by adhering to the negotiationmechanism and agent deliberation model proposed in this paper,agents can still hope to quickly locate ef  fi cient and equitable deals inautomated negotiations under two-sided information uncertainty (ortotal information uncertainty). 1.1. Research objective Game Theoretic literature addresses some mechanism designissues (protocols and strategies) in negotiation. It discusses charac-teristicsofperfectsolutionsbutnotthepreciseprocedurestoarriveatsuch solutions. Bilateral negotiations are treated as two person non-zero sum games and values for such a game under completeinformationarefoundinRef.[20],solutionsunderperfectequilibrium are determined in Ref. [26] and Nash solutions for bargainingsituations under incomplete information are determined in Ref. [9].Game theoretic approaches assume idealized conditions such ascomplete rationality of the bargainers, equal bargaining skills,capability of each player to accurately compare ones own desires forvariousthingsandacompleteknowledgeoftheopponent'stastesandpreferences [20]. Mechanism design for interactions between Decision Support Systems 52 (2011) 157 – 168 ⁎  Corresponding author. Tel.: +91 9798009003. E-mail addresses:  hemalatha@iimranchi.ac.in, hemshekh@rediffmail.com(H. Chandrashekhar), bhasker@iiml.ac.in (B. Bhasker).0167-9236/$  –  see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.dss.2011.06.004 Contents lists available at ScienceDirect Decision Support Systems  journal homepage: www.elsevier.com/locate/dss  multiple agents investigated in Ref. [21] again is con fi ned to thecomplete information situation. Works that do address informationuncertainty, assume knowledge of a subjective probability distribu-tion of the opponent's preferences as in Refs. [4,9] while in realityeven assessing probability distributions is not easy. AutomatedNegotiation literature takes inputs from Game theory and dealsmostly withthe agentdesignaspects. Onestreamof literature focuseson the knowledge elicitation aspect [3,18,19] while another on thedecision-making models [2,5,6,8,13 – 15,23,32] of agents. A classi fi ca-tion scheme for negotiation in electronic commerce is presented inRef. [17] while some prospects and challenges in automatednegotiations are presented in Ref. [12]. A good automated negotiation procedure can emerge only when the mechanism design is suitablefor use by autonomous agents and when the agent design comple-ments the mechanism design to derive solutions with the desirablecharacteristics. However, there is a dearth of Automated Negotiationmodels that deal with both the mechanism design and agent designaspects in a comprehensive manner.Besides,there is a major needforimprovement in mechanism design for negotiations under totalinformation uncertainty. Thus our objective was to develop a goodautomated negotiation procedure for bilateral multi-issue negotia-tions under two-sided information uncertainty, by addressing themechanism design aspect as well as the agent design aspect in anintegrated manner.Contributions of this research include the development of (1) anincentive compatible negotiation mechanism inclusive of the MUP(Monotonic Utility-granting Protocol) and the WYDIWYG (What YouDisplay In fl uences What You Get) strategy (2) an agent deliberationmodel inclusive of a preference elicitation procedure to make agentsfaithful surrogates of the user they represent and the FrontierTracking Proposal Construction Algorithm (FTPCA) to make agentslearn the opponent's  fl exibility over the various negotiation issues inthe ongoing negotiation and respond accordingly.Contributionstopracticearequiteafew.Unsophisticatedbargainers(buyers/sellers of products/services in B2C e-commerce) operating indynamic or time-constrained environments, can adopt the proposedautomated negotiation procedure to quickly locate the best mutuallybene fi cialdeal.Eventherelativelymoresophisticatedparties(asinB2Be-commerce) can adopt the proposed procedure to avoid deadlocksituations(innegotiations)likelytoariseduetocomparablebargainingskills/powers or information uncertainty or both. 2. Background of the work   2.1. Mechanism design Akin to the concepts of   ef   fi ciency ,  equitability  and  stability  thatwere introduced earlier in the context of negotiation outcomes, thereare certain essential attributes that a chosen mechanism forautomated negotiation should exhibit. They are  ef   fi ciency ,  stability , simplicity  and  distribution - symmetry  [24, page 44]. Ef  fi cient mecha-nisms lead to Pareto optimal/ef  fi cient solutions. When one agentfollows some publicly known strategy, if the other agent  fi nds it bestto follow the same publicly known strategy and has no incentive todeviate from the same, then the mechanism is deemed  stable . Thecomputational burden and communication burden of a mechanismdetermine its  simplicity . The  symmetric distribution  criterion checks if boththeagentsareequallyplacedinthejointsearchprocess,i.e.ifthederived deal would be the same, irrespective of who made the  fi rstmoveinthenegotiation.Itisquitefeasibletodevisemechanismswiththe above-mentioned desirable characteristics in complete informa-tion situation as evident from some of the mechanisms presented inRef. [24] but not as much when there is total information uncertainty.Besides, mechanisms under total information uncertainty need tosatisfy one another criterion  —  incentive compatibility . An incentivecompatible mechanism will encourage agents to honestly revealinformation during negotiation. This is an essential characteristicbecause it is directly connected with ef  fi ciency. If in a bilateralnegotiation, both agents strategically misrepresent their value trade-offs, inef  fi cient contracts will result [22, page 144]. The need for therightkindofatmospheretoencouragetruthfulrevelationofinformationand the need for a suitable mechanism possessing the desiredcharacteristics in total information uncertainty have motivated thedevelopment of the protocol and strategy proposed in this paper.  2.2. Distributed vs. integrated search methods In devising procedures that can handle total information uncer-tainty, research in Automated Negotiations progressed along twodistinct paths: one that employs an integrated search of the solutionspace and the other that advocates a distributed search. Theintegrative search methods employ a mediator, who interacts withthe negotiators, learns their individual preferences and then suggestsa solution agreeable to both the parties. The  ‘ Adjusted Winner ’  [30] isone such procedure. When the search happens in multiple stages, themediator takes feedback from the individual agents after each stageand proceeds with the search of the solution space only in thedirection which improves the joint gains of the negotiators, till asolution which can no further be improved is reached. The SingleNegotiation Text (SNT) approach [16] and those in Refs. [7,31] fall in this category. Unlike the integrative search methods, the distributedsearch methods [1,5,6,8,13 – 15,18,29,32] allow the protagonists tohaggle over the negotiation objects to  fi nd a suitable deal. The agentsin this kind of set up are self-centered in the sense that they try tomaximize their own utility. For any agent, striking a deal anywhere inthe agreeable zone is always considered better than breaking off anegotiation; this forces an agent to try to appease the opponentthrough an offer that shall be more agreeable to the opponent whilenot compromising too much on one's own utility. As a result theagents iteratively converge to a mutually agreeable solution point.Under total information uncertainty, though integrative search mayseem idealistic, reality necessitates a distributed search due to lack of trust on a mediator. The need for decentralized methods (distributedsearch methods) has been highlighted in Refs. [10,18]. Quite a few distributed search methods [5,6,8,13 – 15,32] learn the preferences of the opponent and modify the successive proposals such that theybecome progressively more agreeable to the opponent. These workshave in fl uenced the development of the learning agents proposed inthis paper. Speci fi cally the rationale provided in Ref. [8] for makingtradeoffs in multi-issue negotiations to locate social welfare maxi-mizing deals has encouraged us to incorporate the tradeoff strategywithin the context of our proposed WYDIWYG strategy.  2.3. Preference elicitation Though initial attempts at precisely acquiring user preferenceshave been made [3,18,19] the need for good preference elicitationtechniques as pointed out in Ref. [12] still seems to exist. When therearemultipleissues,notonlydoesitbecomecomplexforindividualstoexpress utility values to each of the exponentially large number of feasible solutions but even a preferential ranking of the variousproposals goes out of question. The overall utility of a proposal that iscomposed of a particular combination of values for the various issuesmay not be a simple summation of the individual utility values foreach of the negotiation issues. The user may attach differentimportance weights to different issues and these differences wouldeventually translate to the tradeoff preferences of the user over thevarious negotiation issues. Eliciting a user's tradeoff preferences is noeasy task because it is quite natural for individuals to be very unsurewhen they are directly asked to express their tradeoff preferences orassign importance weights to the various negotiation issues. Thepreference elicitation problem in multi-issue negotiations being 158  H. Chandrashekhar, B. Bhasker / Decision Support Systems 52 (2011) 157  – 168  similar to that in multi criteria decision-making (MCDM), adoptingthe techniques used in MCDM seem reasonable. Multi attribute utilitytheory (MAUT) principles have already been used in the negotiationmodel suggested in Ref. [1]. Since the normalized weight vector (priorities vector) of the various attributes obtained in AnalyticHierarchy Process (AHP) [27] represents the tradeoff preferences of the user over the various attributes, we were motivated to replicateAHP's priorities vector determining procedure within the context of multi-issue negotiations to determine the normalized weight vectorand thus the tradeoff preferences of the user. 3. The proposed automated negotiation procedure Some of the assumptions are as follows: 1. The negotiations areone off negotiations. (There is no negotiation history.) 2. Theindividual negotiating agents are sel fi sh. (They aspire to maximizetheir own utility. Successive proposals of each agent will be in thedecreasing order of its own utility.) 3. The utility values of the variousproposalsareindependentofthetimedimensioni.e.;aproposalmadeataparticularnegotiationroundwouldhavethesameutilityifitweremade at an earlier instance or later. (This could be a reasonableassumption because negotiations under the proposed procedure willconclude(agreement ornoagreement)quickly.)4.Theagentsarenotonly committed to their current proposal but also automaticallybound to all of their previous proposals. 5. There is no interdepen-dency between the various negotiation issues. The utility contribu-tions from each of the multiple issues are plain additive (a weightedsumatthemost).6.Thenegotiationsareofthealternatingofferstype.Thenegotiationendswhenthereisanagreementorwhenatleastoneof the agents prefers to break off.  3.1. Proposed protocol and strategy The suggested procedure advocates a distributed search of thesolution space by sel fi sh negotiating agents in the presence of anintervener. Among the four different types of interveners identi fi ed inRef. [22, pages 22 – 23], the current procedure requires the services of an intervener assuming the roles of a facilitator as well as a rulesmanipulator. The individual negotiating agents do not have to shareanysortofprivateinformationwitheachotherorwiththeintervener.The intervener in the role of a facilitator makes the agents to (1)agree on the negotiation objects (issues under negotiation) (2)announce their initial offer (3) commonly agree upon a scale overwhich they shall relatively (but privately) evaluate the variousnegotiation issues and (4) unanimously  fi x on the least concessionstepsizefor each of theissuesunder negotiation.Since the initialoffersof both the agents are already on the table, the negotiation typicallybegins with one agent (does not matter which of the two) making the fi rst counter offer. It is at this point that the intervener in the rules-manipulator's role mandates that  the successive proposals from eachagent should be non-decreasing in utility to the opponent  . This rulebasically ensures convergence to a mutually agreeable solution andmay be called the Monotonic Utility-granting Protocol (MUP). Whilethe Monotonic ConcessionProtocol (MCP) [24, pages40 – 41] is de fi nedfor task-oriented domains and for negotiations under completeinformation,MUPisanadaptationofMCPtosuitnegotiationsinvolvingtwo-sided information uncertainty. The MUP, just like other multi-stage negotiation protocols requires the negotiation to proceed in asequence of rounds. Negotiation round #1 (NR=1) comprises theinitialoffersofboththeagentsputforwardsimultaneously.Fromround#2 onwards (NR=2), just as is done in the well-known AlternatingOffers Protocol, agents alternate in making a negotiation decision. Thenegotiation decision could be (1) to put forward a counter proposal or(2) to accept the opponent's most recent proposal or one of theopponent's past proposals as the deal. An agent accepts one of theopponent's proposals as the deal only if the utility of that proposal toitself is greater than or equal to the utility of the counter proposal it isabout to put forward in the current round. Otherwise the agenttentativelyrejectstheopponent'sproposalandgoesaheadwithitsowncounter proposal where the agent grants extra utility to the opponent.When the MUP requires an agent to grant utility to the opponentmonotonically through its counter offers, the utility referred to is notthepubliclyknownutility(becausethereisnopubliclyknownutilityinnegotiations under two-sided information uncertainty) but it is thepublicly perceived utility. The perception is based on what theopponent agent has displayed by means of its own successive offers.Anagent'ssuccessiveoffersaresubtlepointerstothevaluefunctionsof theagentprovidedtheagenthasbeenputtingforthitssuccessiveoffersinitsowntrulypreferredorder.MUPthus,justpersuadeseachagenttobereceptiveto suchpointersfrom theopponentand accommodate theopponent's needs as far as possible. Indirectly MUP also coerces eachagent to honestly reveal information through its own series of offers.Since MUP mandates each agent to accommodate the opponent'spreferences only as inferred from the opponent's series of offers, anyagent can hope to derive maximum bene fi t from its opponent only byprojectingitspreferencestruthfullythroughitsownseriesofoffers.Therules manipulator checks each agent's compliance with MUP onlybased on what its counterpart has displayed by way of its ownsuccessive offers. A point of difference between the MUP and the MC'Pis that while MCP permits each agent to either make a concession inevery round or repeat the previous offer, MUP necessitates each agenttograntsomepubliclyperceivedutilitytotheopponentineveryround.Besides, an agent abiding by the MUP announces its breaking off fromthe negotiation when it is unable to grant any more utility to theopponent and simultaneously unable to accept any of the opponent'sproposals. Interestingly, the de fi nition of the MUP has packed withinitself a suggestion for a matching strategy too. That is, the agents haveto plan their current proposal so as to accommodate the opponent'spreferences as publicly displayed by the opponent. Thus the strategythe agents need to adopt to design their current moves may be aptlycalled the WYDIWYG (What You Display In fl uences What You Get)strategy. Agents adopting the WYDIWYG strategy need to observe theopponent's concession pattern in the negotiation. Concessions in allthoseissueswheretheopponentitselfhasconcededrelativelylessmaybe assumed to provide relatively more utility to the opponent andconcessions in all those issues where the opponent itself has concededrelatively more may be assumed to provide relatively lesser utility tothe opponent. If an agent grants utility to the opponent through aconcession scheme over the various negotiation issues based on thisreasoning, it could very well be sure of convincing the rulesmanipulator of its compliance with the MUP. The WYDIWYG strategythus becomes a more or less default strategy when negotiating agentsfollowtheMUP,andnoagentwouldhaveanyincentivetodeviatefromthe strategy thereby rendering the mechanism to be a stable one.Further,sincethismechanism(comprisingtheMUPandtheWYDIWYGstrategy) encourages only truthful revelation of information by thenegotiating agents, it seems to be an incentive compatible one. Owingto the incentive compatibility, this mechanism is bound to derive onlyef  fi cient solutions. Negotiation simulation experiments (Section 4)besides demonstrating the desired characteristics exhibited by thismechanism are evidence to the simplicity and implementationfeasibility of this mechanism.  3.2. Proposed deliberation model for individual negotiating agents Each agent has its own domain of acceptable offers, its ownreservation values for the various issues under negotiation and itsown distribution of preference/importance weights over the variousnegotiation issues. In multi-issue negotiations literature,  Weight   of anissue refers to the importance of an issue. Here we introduce anadditional term called  Flexibility . Flexibility refers to the inclination toprovide a concession in an issue. The more the  weight   of an issue, the 159 H. Chandrashekhar, B. Bhasker / Decision Support Systems 52 (2011) 157  – 168  lesser would be the  fl exibility  on that issue. Each agent keeps itsinformation private and does not know the private information of thenegotiating counterpart. Given the initial offers of both agents in abilateral multi-issue negotiation, the offer comprising the middlevalue in each of the issues is commonly interpreted (in GameTheoretic literature) as the reference point/focal point. An agentfollowing our proposed deliberation model will agree to enter into anegotiation with its counterpart if the reference point/focal point lieswithin its own domain of acceptable offers. If that is not the case, themid point in at least half of the total number of negotiation issuesshould fall within the reservation value of the agent in the respectiveissues. Otherwise the agent may not expect to get a fair deal andtherefore would not enter into a negotiation with its counterpart.Assuming a similar deliberation model of the opponent, each agentthus needs to put forth reasonable initial offers. Too ambitious or toogreedy initial offers carry the risk of being ousted even before thecommencement of a negotiation. The initial offer of any agent wouldrepresent the agent'smostpreferredoffer as it wouldbe comprised of theoptimalvalueforeachissuefromtheagent'sownperspective,andrender the maximum possible utility to the agent. After exchangingthe initial offers, if the agents mutually agree to commence anegotiation, they need to (1) Fix the least concession step size foreach issue under negotiation (Since the agents learn their opponent'srelative fl exibility over the various issues from the concession patternof the opponent, there should be no ambiguity as to what isconsidered a concession in any particular issue). (2) Commonlyagree upon a scale over which they shall relatively (but privately)evaluate the various negotiation issues. (Since the agents try toaccommodate the opponent's needs to the extent possible byexamining the opponent's preferences relative to one's own prefer-ences, it becomes absolutely necessary for the agents to adhere to thesame scale while they individually and privately evaluate the relativeimportance of thevariousnegotiation issues.Besidesthere shallbenoincentive for any agent to deviate from this commonly agreed uponscale when they abide by the MUP). Since Saaty's procedure forpreference elicitation is suggested (Section 3.2.1) for the individualagents, Saaty's scale for evaluating the various negotiation issuesbecomes the natural choice for both the agents. An overview of thedeliberation model of a negotiating agent in a bilateral multi-issuenegotiation under the proposed mechanism is presented in Fig. 1. Theactions described in solid rectangles will be supervised for confor-mance with the mechanism by the intervener while the actionsdescribed in perforated rectangles are the private actions of thenegotiating agent. The bottom most two rectangles describe theactionsthat happenrepeatedlytill a dealis reachedor the negotiationgets aborted.  3.2.1. Eliciting user preferences and determining Self-Flexibility The negotiating agents in our model elicit user preferences byposing pair wise comparison questions to the user as is done in themulti criteria decision-making tool called Analytic Hierarchy Process(AHP) [27]. Saaty's scale [27] is adapted (refer to Fig. 3) to suit the multi-issue negotiation context (the difference is only in theinterpretation of the absolute values on the scale as explained incolumn 3 of  Fig. 3) and the user responses are obtained on the same.Fora negotiation involving  n issues,a n × n reciprocalmatrix M  is builtfrom the responses. If the response to a pair-wise comparisonquestion  ‘ How important is issue  i  as compared to issue  j ? ’  is 7 thenthe matrix entry at  a ij  is 7while that at  a  ji  is 1/7. All diagonal elementsare 1 since they represent the comparison of an issue with itself. Thenormalized principal Eigen vector of such a reciprocal matrix  M  represents the relative priorities/importance of the issues beingcompared [27] and thus the normalized weight vector. Howeverwhen there are multiple issues, an examination of all the responsesputtogethermayincertaininstancesrevealsomeinconsistencyintheuser's evaluation. As a result, the pair wise comparison matrix wouldbecome inconsistent thereby rendering the Eigen vector of theinconsistent matrix to be a meaningless estimate of the weightvector. More details about this inconsistency issue and ways to dealwith the same may be obtained in Ref. [28]. In order to obtain a meaningful estimate of the user's weight vector, our proposednegotiating agent partly induces consistency and partly forces itwithout affecting the overall objective or the end result in any way.The agent  fi rst gets the user to rank the various negotiation issues inthe descending order of importance and then formulates a set of pairwise comparison questions by pairing up every issue only with theuser's fi rstmostimportantissue.Thustheuseris assistedinprovidinghis responses on the Saaty scale more or less consistently. For the restof the pair wise comparisons the agent simply computes theconsistent response and  fi lls up the reciprocal matrix such that thepairwisecomparisonmatrixremainsperfectlyconsistent (witheveryentry in the matrix satisfying the condition  a ik = a ij • a  jk ). Thenormalized weight vector obtained from the consistent reciprocalmatrixservestwopurposes.Firstly,ithelpsindeterminingtheoverallutility of a proposal that could practically comprise any combinationof values for the various issues out of the huge number of feasiblecombinations. Secondly, it leads to the determination of a  fl exibilityvector  Self-Flexibility  that represents the relative  fl exibility the userhasoverthevariousnegotiationissuesinprovidingconcessionswhenconcessions are solicited during the course of the negotiation. Sincethe relative importance of an issue and the relative  fl exibility on thatissue are opposing concepts, the agent obtains another reciprocal Fix negotiation issues in conjunction with counterpartExchange initial offer with counterpartCheck if negotiation may be commenced with counterpartFix the least concession step size for each issue in concurrence with counterpartCommonly agree on a scale (to be used by both agents) to relatively evaluate the importance of the various negotiation Elicit user preferences over the various negotiation issues and determine Self-Flexibility (one’s own relative flexibility over the various negotiation issues)Initially assume opponent is equally flexible in all the negotiation issuesObserve opponent’s most recent proposal and Update Opponent FlexibilityConstruct current counter proposal that provides extra utility (perceived utility) to opponentIf Utility to self from one’s own current counter proposal > Utility to self from the opponent’s most recent proposal then present current counter proposalIf Utility to self from one’s own current counter proposal Utility to self from one of opponent’s past proposals then accept that proposal of opponent as the dealIf it is not possible to provide extra utility to opponent because reservation value is reached in every issue, and if it is not possible to accept any of the opponent’s proposals, communicate to opponent the decision to break off from negotiation.In case of unwillingness to enter into a negotiation with counterpart, communicate the same and exitIn case of willingness to enter into a negotiation with counterpart, proceed thusissues Fig. 1.  The deliberation model of an individual negotiating agent.160  H. Chandrashekhar, B. Bhasker / Decision Support Systems 52 (2011) 157  – 168  matrix  M  ′  by replacing each entry in  M   with its reciprocal value. Thenormalized principal Eigen vector of   M  ′  represents the normalized fl exibility vector. The normalization is removed (by dividing eachcomponent in the vector by the least value component in the vector)to obtain the  fl exibility vector  Self-Flexibility .  3.2.2. Learning and updating Opponent-Flexibility Atthebeginningofthenegotiation,theagentassumestheopponentagent to be equally  fl exible in all the negotiation issues and henceinitializes the vector  Opp-Flexibility  with a value of 1 in each of the  n dimensions/issues. This vector is then updated with each proposalreceived fromtheopponentbycomparingtheoffervalueineachof theissuesinthecurrentofferwiththatcorrespondingtothepreviousoffer.If in a particular issue, one or more concession(s) is/are received in thecurrentround(NR)ascomparedtotheoffervalueinthepreviousround(NR-1) then the  fl exibility value corresponding to that issue in thevector  Opp-Flexibility  is incremented by the number of concessionsreceived. If instead status quo is maintained or one or more concession(s) is/are removed in that issue as compared to the offer value in theprevious round then the  fl exibility value corresponding to that issue isleft unaltered. Thus at any stage, the ratio of the  fl exibility values (asobtained from  Opp-Flexibility ) between any two issues shall approxi-matetheopponent'stradeoffratiobetweenthosetwoissues.Oneotherthing the agent does every time after receiving an offer from theopponentistocompute one'sownutility fortheopponent'sofferusingone'sownnormalizedweightvectorandupdatethevariable U  self  (Opp-offer) representingtheutilitytoselffortheopponent'smostrecentoffer.  3.2.3. Constructing the new counter proposal to be presented in thecurrent round/turn An agent's own initial offer provides the maximum possible utilityto the agent. The agent's own successive offers tend to beprogressively decreasing in utility to the agent owing to theprogressively increasing amount of concessions it provides to theopponent through these offers. Both the agents provide suchconcessions to its counterparts as a result of its compliance with theMUP. However to enable a thorough search of the frontier solutionspace, at each round only a certain minimum amount of additionalconcessionisrequiredtobeprovidedbyeachagenttoitsopponent.Inother words, the number of concession units an agent needs toprovide in any round shall be a function of the negotiation round# (NR). Say when the negotiation begins i.e., at NR=2 only 1 unit(NR-1=1) of concession needs to be provided. (It may be recalledhere that at NR=1 both the agents put forth their initial offerssimultaneously and negotiation practically begins only from NR=2).  A unit of concession  for any agent will be equivalent to one concessionin its own most  fl exible issue. A single unit of concession is analogousto the  ‘ Minimal suf  fi cient concession ’  described in Ref. [24, page 44].Thevalueofaconcessioninanyparticularissueintermsofconcessionunits to an agent will depend on the tradeoff ratio between that issueand the agent's most  fl exible issue. An agent's tradeoff ratio betweenany two issues is the ratio of the  fl exibility values of the two issues asobtained from  Self-Flexibility .In a particular stage of negotiation, when there is no agreementbetween the agents, the agent whose turn is to present the newcounter offer,  fi rst determines  reqd_conc   i.e., the number of concessions (in terms of its own concession units) it has to providein the current round. Then it decides on the appropriate scheme toallocate the concessions over the various issues. The agent's best(self-centered) scheme for allocating the concessions would be toprovide maximum possible concessions in its own most  fl exibleissue followed by concessions in the other issues sorted in thedecreasing order of one's own  fl exibility. However from theopponent's perspective, the best scheme would be to provide themaximum possible concessions in the opponent's least  fl exible issuefollowed by concessions in the other issues sorted in the ascendingorder of the opponent's  fl exibility. In order to closely track thePareto frontier, the agent making the current offer follows a hybridscheme of concession allocation that emerges by accommodatingthe latter (sel fl ess) scheme as much as possible into the former(self-centered) scheme. For instance, if the sel fl ess scheme suggestsa concession in issue A while the self-centered scheme suggests aconcession in issue D,  fi rstly a comparison is made between the costof giving a concession in issue A (tradeoff ratio between the most fl exible issue of self and issue A) and  reqd_conc   at that stage. Only if  reqd_conc   is greater, it is partially justi fi ed to adopt the sel fl essscheme. For complete justi fi cation, a comparison between the D-to-A tradeoff ratio for self and that for the opponent is made. If theopponent's tradeoff ratio is greater than the self-tradeoff ratio, itimplies that more value could be provided to the opponent bymeans of the tradeoff without diminishing the value to self. Thus theagent could adopt the sel fl ess scheme and allocate a concession inissue A; On the other hand, if the opponent's tradeoff ratio is lesserthan the self-tradeoff ratio, it indicates that no bene fi t to theopponent is likely to result by way of the tradeoff. Hence the agentcan go ahead with the self-centered scheme and offer a concessionin issue D. By far it is this hybrid scheme of concession allocationthat helps the agents to almost always pick the frontier points as itsnew offers despite having no knowledge of the solution space andits bounds and this logic is deduced from the following facts: (1) In abilateral negotiation, given a particular utility to one agent, thefrontier point is the one that provides the maximum possible utilityto the other agent. (2) The opponent shall get maximum gain inutility with maximum concessions in the opponent's least  fl exibleissues. Despite following the above-mentioned logic, the agentssometimes misjudge the frontier point and end up giving a proposalthat is slightly off the frontier. However these misjudgments can beentirely attributed to the information uncertainty at that stage of negotiation. For example, consider a hypothetical stage early on in afour-issue (the four issues being A, B, C, D) negotiation where anagent A1 has given some concessions in issues A and C but none inissues B and D. The opponent, agent A2 though clearly understandsthat A1 is less  fl exible in issues B and D, it has no information aboutthe opponent's fl exibility ordering within these two issues. In realityif A1 was least  fl exible in issue D and A2 decides to provideconcessions in issue B before trying to provide concessions in issueD while it might as well have done the other way around withoutcausing any harm to itself, then the proposal chosen by A2 would beaway from the frontier. However, when A1 during its successiveturns begins to relent in issue B while still holding tight on issue D orconcedes more in issue B than in issue D, A2 would correct its judgment about its opponent's  fl exibility ordering within the issuesB and D and then substitute one or more concessions already givenin issue B with one or more in issue D depending on its own tradeoff ratio between issues B and D and thus return to the frontier pointthat would provide the maximum possible utility to the opponent atthat stage of negotiation.An agent while allocating the concessions required to be given in aparticular round may end up giving a bit extra too in some particularrounds.Considerasituation whereanagentinthe processof allocatingthe concessions (requiredto be providedin thatparticularround) overthevariousissuesfollowingthehybridschemeisleftwithonemoreunitof concession to be provided through the current offer. If the currentallotment of concessions is such that the agent has provided themaximum possible concessions in its most  fl exible issue (i.e. thereservation value is reached in the most  fl exible issue) then it has nootherchoicethanallocatingonemoreconcessioninits2ndmost fl exibleissue or 3rd most  fl exible issue or so on in that order depending uponwhere some space for providing concessions is  fi rst found (i.e. wherereservation value is not yet reached). In doing so, the agent overshootsthe required number of concession units in the current round since aconcessioninanyissueotherthanthemost fl exibleissueiscostlierand 161 H. Chandrashekhar, B. Bhasker / Decision Support Systems 52 (2011) 157  – 168
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