An Agenda Based Framework for Multi-Issue Negotiation

Transcription

1 An Agenda Baed Framework for Multi-Iue Negotiation Shaheen S. Fatima, Michael Wooldridge Department of Computer Science, Univerity of Liverpool, Liverpool L69 7ZF, U.K. Nichola R. Jenning Department of Electronic and Computer Science, Univerity of Southampton, Southampton SO17 1BJ, U.K. Atract Thi paper preent a new model for multi-iue negotiation under time contraint in an incomplete information etting. The iue to e argained over can e aociated with a ingle good/ervice or multiple good/ervice. In our agenda aed model, the order in which iue are argained over and agreement are reached i determined endogenouly, a part of the argaining equilirium. In thi context we determine the condition under which agent have imilar preference over the implementation cheme and the condition under which they have conflicting preference. Our analyi how the exitence of equilirium even when oth player have uncertain information aout each other, and each agent information i it private knowledge. We alo tudy the propertie of the equilirium olution and determine condition under which it i unique, ymmetric, and Pareto-optimal. Key word: Multi-iue negotiation, Game theory, Agenda, Intelligent agent. 1 Introduction Negotiation i a mean for agent to communicate and compromie to reach mutually eneficial agreement [18,30,29,11,40,14]. In uch ituation, agent have a common interet to cooperate, ut have conflicting interet over exactly how to Correponding author. addree: S.S.Fatima@cc.liv.ac.uk (Shaheen S. Fatima), M.J.Wooldridge@cc.liv.ac.uk (Michael Wooldridge), nrj@ec.oton.ac.uk (Nichola R. Jenning). Preprint umitted to Artificial Intelligence 28 March 2003

2 cooperate. Put differently, agent can mutually enefit from reaching agreement on an outcome from a et of poile outcome, ut have conflicting interet over the et of outcome. In thi context, the main prolem that confront agent i to decide how to cooperate efore they actually enact the cooperation and otain the aociated enefit. On the one hand, each agent would like to reach ome agreement rather than diagree and not reach any agreement. But, on the other hand, each agent would like to reach an agreement that i a favourale to it a poile. To thi end, a numer of negotiation model that addre thi prolem have een developed and applied to data allocation in information erver, reource allocation and tak ditriution [32,27,31,18,19]. Apart from thee, another application area in which agent-mediated negotiation ha received coniderale attention i in the field of electronic commerce [36,35,24,22,23]. In thi domain, which i the main focu of thi paper, the aim i to uild oftware agent that will optimally negotiate with other agent on ehalf of uer for uying and elling good/ervice. Here we look at one-to-one negotiation etween a uyer and a eller. In order to develop oftware agent for uch ilateral encounter, we firt examine the important feature of reallife argaining ituation that need to e incorporated in the oftware agent. To thi end, the three crucial feature of mot practical argaining procee are a follow [28]: (1) The time contraint of the argainer. (2) The information tate of the argainer. (3) The numer of iue to e argained over. We firt explain the role of time in negotiation. Conider an e-commerce cenario in which a uyer agent and a eller agent negotiate over the price of a good or ervice. The uyer clearly prefer a low price, while the eller prefer a high one (hence the competitive nature of the encounter). In addition to attempting to otain the et price, agent alo uually need to enure that negotiation end efore a certain deadline. However, the end point may not e the only way in which time influence negotiation ehaviour. Conider the cae in which the ervice i provided immediately after negotiation end uccefully (ay at price È and time Ì ). In ome ituation, it i not ufficient merely for an agent to enure that Ì i any time le than it deadline. Thi may e the cae, for intance, ecaue one of the agent, ay the uyer, could e loing utility with time a a reult of not getting the ervice. On the other hand, the eller may perhap gain more utility y providing the ervice a late a poile. Thu, in thi cae, the eller trie to maximize Ì (within the limit of it deadline) and the uyer trie to minimize Ì. In hort, it i clear that agent can have different attitude toward time. Generally peaking, the mot common time effect in argaining ituation are time dicounting and deadline [21,10]. An agent that gain utility with time and ha the incentive to reach a late agreement (within it deadline) i aid to e a trong or patient player. An agent that loe utility with time and trie to reach an early agreement i aid to e a weak or impatient player. A we will how, thi dipoition and the actual deadline itelf 2

3 trongly influence the negotiation outcome. The econd crucial feature of a argaining proce i information. During negotiation, each agent ha to make deciion aout generating offer and counter-offer in uch a way that it own utility from the final agreement i maximized. An eential input to thi deciion making proce i information; here defined a knowledge aout all factor which affect the aility of an individual to make choice in a given ituation. For intance, in argaining etween a uyer and a eller, information include not only information aout an agent own parameter (uch a it reervation price or it preference over poile outcome), ut alo thoe of it opponent. In mot realitic cae agent have only incomplete information aout their opponent. To thi end, game theoretic model have already een propoed for argaining with incomplete information. For intance, Ruintein [34] developed a model in which agent have incomplete information aout time preference. Fudenerg et al [12] analye uyer-eller negotiation in which reervation price are uncertain. Sandholm and Vulkan [37] conider uncertainty over agent deadline. All thee model are uilt on the aumption that information aout the uncertain parameter (in the form of poile value and a proaility ditriution over them) i the agent common knowledge. However, in practice, perhap the main way of acquiring information aout the opponent i through learning from previou encounter. In uch cae, an agent elief aout it opponent will not e known to it opponent. We therefore tudy the trategic ehaviour of agent y treating each agent information a it private knowledge. The third key feature i the numer of iue that have to e negotiated. In many of the application that are conceived in the domain of e-commerce, it i important that the agent hould not only argain over the price of a product, ut alo take into account iue uch a the delivery time, quality, payment method, and other product pecific propertie. In uch multi-iue negotiation, one approach i to undle all the iue and dicu them imultaneouly a a complete package. Thi allow the player to exploit trade-off among different iue, ut require complex computation to e carried out [17,4]. The other approach, which i computationally impler, i to negotiate the iue equentially. A econd and more important reaon why partie may chooe to ettle iue one y one i the trategic implication of the choice of the negotiation procedure (i.e., iue-y-iue v. complete package). When there are two oject to negotiate, the deciion to negotiate them imultaneouly or one y one i y no mean neutral to the outcome [38,2]. Although iue-y-iue negotiation minimize the complexity of the negotiation procedure, an important quetion that arie i the order in which the iue are argained over. Thi ordering i called the negotiation agenda and it ha een hown to e one of the factor that determine the outcome of negotiation [9]. For intance, if there are two iue, and, the two agenda and can lead to two different outcome. The agent need not have identical preference over thee 3

4 outcome and one of them may prefer the agenda to, while the other may prefer to. Given thi fact, exploring the role of the argaining agenda, and how player might manipulate it, i timely, epecially given that many real-life negotiation involve multiple iue. There are two way of incorporating agenda in the negotiation model. One i to fix the agenda exogenouly a part of the negotiation procedure. Conidering the aove example, one of the agenda, ay i impoed exogenouly. Then the argainer have to ettle firt, and will e allowed to negotiate only after i ettled. The other way, which i more flexile, i to allow the argainer to decide which iue they will negotiate next during the proce of negotiation. Thi i called an endogenou agenda [16] and i the approach we explore in thi paper. Exiting game theoretic model for iue-y-iue negotiation [1,9,16], which are aically extenion of [33,34], have two main hortcoming. Firtly, they tudy the trategic ehaviour of agent y treating the information they have a common knowledge. In practice however, the information that a player ha aout it opponent i motly acquired through learning from previou encounter. An agent elief aout it opponent will therefore not e known to it opponent. Secondly, thee model do not conider agent deadline. We overcome thee prolem y conidering each agent to have it own deadline and y treating each agent information tate a it private knowledge. In thi cae we otain the connection etween thi private knowledge and the exitence of equilirium for ingle iue negotiation. We then extend thi model for multi-iue negotiation and tudy the propertie of the equilirium olution. To provide a etting for our negotiation model, we conider the cae in which negotiation need to e completed y a pecified time, which may e different for different partie (ince thi i the mot realitic cae). Apart from the agent repective deadline, the time at which agreement i reached can affect the agent (patient or impatient) in different way [7]. To thi end, Fatima et al [7] preented a ingle-iue model for negotiation etween two agent under time contraint and in an incomplete information etting y conidering the agent information a it private knowledge. Within thi context, they determined optimal trategie for agent ut did not addre the iue of the exitence of equilirium. Here we adopt thi framework and prove that mutual trategic ehavior of agent, where oth ue their repective optimal trategie, reult in equilirium. We then extend thi framework for multi-iue negotiation. The order in which iue are argained over and agreement are reached i determined y the equilirium trategie. The time of equilirium agreement may not e equal for all the iue. Conequently, the outcome of multi-iue negotiation can e implemented in two way: equentially or imultaneouly. We then determine condition under which agent have imilar, a well a conflicting, preference over the implementation cheme. Finally, we tudy the propertie of the equilirium olution. 4

5 Thi work extend the tate of the art in multi-iue negotiation y preenting a more realitic negotiation model that capture the three apect, mentioned aove, that are aociated with many real-life argaining ituation. Firtly, it i a model for negotiating multiple iue. Secondly, it take the time contraint of argainer into conideration, oth in the form of agent deadline and their dicounting factor. Thirdly, it allow agent to have incomplete information aout each other, and each agent information i treated a it private knowledge. Although we tudy argaining in which agent have one pecific information tate and the agenda i endogenou, our negotiation framework i general and can e ued for exploring a wide range of negotiation environment y changing the agent information tate or the way in which the player manipulate the agenda. The paper i organied a follow. Section 2 decrie the component that make up a negotiation model. In Section 3, we decrie the ingle iue negotiation model. Section 4 extend thi for negotiating multiple iue. Section 5 dicue related work. Finally, Section 6 give ome concluion and ugget ome topic for future work. Appendix A provide a ummary of notation employed throughout the paper. 2 Component of a negotiation model The four component of a negotiation model are a follow [31]: (1) The negotiation protocol. (2) The negotiation trategie. (3) The information tate of agent. (4) The negotiation equilirium. The protocol pecifie the rule of encounter etween the negotiation participant. That i, it define the circumtance under which the interaction etween agent take place, what deal can e made and what equence of offer are allowed. In general, agent mut reach agreement on the negotiation protocol to ue efore negotiation proper egin. A negotiation protocol can e deigned for handling a ingle iue or multiple iue. Within the cla of multi-iue negotiation, we can have protocol that negotiate on all the iue together or one y one. An agent negotiation trategy i a pecification of the equence of action (uually offer or repone) the agent plan to make during negotiation. There will uually e many trategie that are compatile with a particular protocol, each of which may produce a different outcome. For example, an agent could concede in the firt round or argain very hard throughout negotiation until it timeout i reached. It follow that the negotiation trategy that an agent employ i crucial with repect to the outcome of negotiation. It hould alo e clear that the trategie which perform well with certain protocol will not necearily do o with other. The choice of a 5

6 trategy to ue i thu a function not jut of the pecific of the negotiation cenario, ut alo the protocol in ue. An agent information tate decrie the information it ha aout the negotiation game. Von Neumann and Morgentern [26] introduced the fundamental claification of game into thoe of complete information and thoe of incomplete information. The former category i aic. In thee game the player are aumed to know all relevant information aout the rule of the game and player preference that are repreented y utility function. In the latter category, information may e lacking aout a variety of factor in the argaining prolem. Thu each player may have ome private information aout hi own ituation that i unavailale to the other player, while having only proailitic information aout the private information of other player. Following Haranyi [14,15], model of game of incomplete information proceed from the aumption that all player tart with the ame proaility ditriution on thi private information and that thee prior are common knowledge. Thi i modelled y having the game egin with a proaility ditriution, known to all player. Thu player not only have prior over other player private information, they alo know what prior the other player have over their own private information. Strategic model of incomplete information thu include an extra level of detail, ince they pecify not only the action and information availale to the other player in the coure of the game, ut alo their proaility ditriution and information prior to the tart of the game. A negotiation mechanim conit of a negotiation protocol together with the negotiation trategie for the agent involved. A negotiation mechanim ha to e tale (i.e., a trategy profile mut contitute an equilirium), the earliet concept of which wa the Nah equilirium for game of imultaneou offer [25]. Two trategie are in Nah equilirium if each agent trategy i a et repone to it opponent trategy. Thi i a neceary condition for ytem taility where oth agent act trategically. For equential offer protocol, the Nah equilirium concept wa trengthened in everal way y requiring that the trategie tay in equilirium at every tep of the game [39]. In ummary, rationality, a undertood in game theory, require that each agent will elect an equilirium trategy when chooing independently. Given thi, game theory precrie the following main criteria [28] for evaluating the equilirium outcome: (1) Uniquene. If the olution of the negotiation game i unique, then it can e identified unequivocally. (2) Efficiency. An agreement i efficient if there i no wated utility (i.e., the agreement atifie Pareto-optimality). An outcome i Pareto-efficient if there i no other outcome that improve the payoff of one agent without making another agent wore off. All other thing eing equal, Pareto-efficient olution are preferred over thoe that are not. (3) Symmetry. A argaining mechanim i aid to e ymmetric if it doe not treat the player differently on the ai of inappropriate criteria. Exactly what con- 6

7 titute inappropriate criteria depend on the pecific domain. For example, if the argaining outcome remain the ame irrepective of which player tart the proce of argaining, then it i aid to e ymmetric with repect to the identity of the firt player. (4) Ditriution. Thi property relate to the iue of how the gain from trade are plit etween the player; doe the outcome plit the gain equally etween the trader or doe it favour one agent more than the other? In thi paper, our aim i not to deign a negotiation mechanim that divide the gain fairly among player ut to tudy the outcome that reult when oth agent are elf-intereted. With thee road guideline in mind, many different model can e deigned. Below, we report on the development of a new model aed on negotiation deciion function (ee Section 3.2) for argaining over multiple iue. We firt decrie the ingle iue model and tudy it equilirium trategie and outcome. We then extend thi model for multi-iue negotiation and tudy it equilirium propertie. 3 The Single-Iue Negotiation Model We firt decrie the ingle iue negotiation protocol and otain the agent optimal trategie. We then prove that the optimal trategy profile form equential equilirium. 3.1 The Negotiation Protocol Here we adopt what i aically an alternating offer protocol [28]. Let denote the uyer, the eller and let [ÁÈ ÊÈ ] denote the range of value for price that are acceptale to agent, where ¾. We let denote agent opponent. A value for price that i acceptale to oth and (i.e., the zone of agreement) i the interval [ÊÈ ÊÈ ] and (ÊÈ ÊÈ ) i known a the price-urplu. The uyer initial price, ÁÈ, ha a value le than the eller reervation price. Similarly, the eller initial price ha a value greater than the uyer reervation price. In other word, oth ÁÈ and ÁÈ lie outide the zone of agreement. The agent alternately propoe offer at time in Ì ¼ ½. Each agent ha a deadline. Ì denote agent deadline where Ì ¾ Ì. Let Ô Ø denote the price offered y agent at time Ø. Negotiation tart when the firt offer i made y an agent. The agent who make the initial offer i elected randomly at the eginning of negotiation. When an agent, ay, receive an offer from agent at time Ø, i.e., Ô Ø, it rate the offer uing it utility function Í. If the value of Í for Ô Ø at time Ø i greater than the value of the counter-offer agent i ready to end in the next 7

8 time period, Ø ¼, i.e., Í Ô Ø Ø¼ ص Í Ô Ø¼ µ for Ø ¼ Ø ½, then agent accept the offer at time Ø and negotiation end uccefully in an agreement. Otherwie a counter-offer i made in the next time period, Ø ¼. Thu the action,, that agent take at time Ø, in repone to the offer Ô Ø, i defined a: Ø Ô Ø µ Quit if Ø Ì Accept Offer Ô Ø¼ at Ø ¼ if Í Ô Ø Ø¼ µ Í Ô µ otherwie Agent utilitie are defined with the following two von Neumann-Morgentern utility function [17] that incorporate the effect of time dicounting. Í Ô Øµ Í Ô ÔµÍ Ø Øµ (1) ÍÔ and ÍØ are unidimenional utility function. Here, preference for attriute Ô, given the other attriute Ø, do not depend on the level of Ø. ÍÔ i defined a: Í Ôµ ÊÈ Ô for the uyer Ô ÊÈ for the eller ÍØ i defined a ÍØ Øµ Æ µ Ø, where Æ i the dicounting factor. Thu when Æ ½µ the agent i patient and gain utility with time and when Æ ½µ the agent i impatient and loe utility with time. The utility from conflict i lower than the utility from any of the poile agreement for oth agent. Each agent prefer to reach an agreement, rather than diagree and not reach any agreement at all, ince the utility from an agreement i alway higher than conflict utility. Conequently, it i optimal for agent to offer ÊÈ at the latet y it deadline, if it ha not done o earlier, and avoid a conflict (ee Section 3.5 for detail on an agent optimal trategy). Agent are aid to have imilar time preference if oth gain on time or oth loe on time. Otherwie they have conflicting time preference. 3.2 Counter-Offer Generation The tactic for generating offer and counter-offer are defined a follow. Since oth agent have a deadline, we aume that they ue a time dependent tactic (e.g. linear (L), Boulware (B) or Conceder (C) [3]) for generating offer. In thee tactic, the predominant factor ued to decide which value to offer next i time Ø. Thee tactic vary the value of price depending on Ø and Ì. The initial offer i a point in the interval [ÁÈ, ÊÈ ]. The contant multiplied y the ize of interval determine the price to e offered in the firt propoal y agent. The offer made y agent 8

9 Price (Conceder) (Linear) IP (Boulware) t/t Fig. 1. Negotiation deciion function for the uyer. to agent at time Ø (¼ Ø Ì ) i modelled a a function depending on time a follow: Ô Ø ÁÈ Øµ ÊÈ ÁÈ µ for ÊÈ ½ صµ ÁÈ ÊÈ µ for. A wide range of time dependent function can e defined y varying the way in which ص i computed (ee [3] for more detail). However, function mut enure that ¼ ص ½, ¼µ (where lie in the interval ¼ ½ ), and Ì µ ½. That i, the offer will alway e etween the range ÁÈ ÊÈ, at the eginning it will give the initial contant and when the deadline i reached it will offer the reervation value. The initial offer i ÁÈ if ¼, lie etween ÁÈ and ÊÈ for ¼ ½, and i ÊÈ for ½. Thu y varying etween ¼ and ½, the initial price that i offered can e varied etween ÁÈ and ÊÈ. Since we want ÁÈ to e the initial offer, we et to 0. Function ص i called the negotiation deciion function (NDF) and i defined a follow: ص ½ µ Ø ½ Ì Thee NDF repreent an infinite numer of poile tactic, one for each value of (ee [3] for more detail). However, depending on the value of, three extreme et how clearly different pattern of ehaviour (ee Figure 1). (1) Boulware (B) [30]. For thi tactic, ½ and cloe to zero. The initial offer i maintained till time i almot exhauted, when the agent concede up to it reervation value. Figure 1 how the Boulware function for ¼¼¾. (2) Conceder (C) [29]. For thi tactic, ½. The agent goe to it reervation value very quickly ½ and maintain the ame offer till the deadline. Figure 1 (2) ½ A increae(decreae) ecome more Conceder(Boulware). At very high(low) val- 9

10 how the Conceder function for ¼. (3) Linear (L) Finally, when ½, price i increaed linearly. The value of a counter offer depend on the initial price (IP) at which the agent tart negotiation, the final price (FP) eyond which the agent doe not concede, the time Ø at which it offer the final price, and. Thee four variale form an agent trategy. Definition 1 An agent trategy Ë i defined a a quadruple whoe element are the initial price ÁÈ µ at which the agent tart negotiation, the final price È µ eyond which the agent doe not concede, time (Ø ) at which the final price i offered, and. Thu Ë ÁÈ È Ø Agent ue it trategy, Ë, to generate an offer, Ô Ø, for Ø Ø. Different trategie can e defined for different value of thee four element. For example, when tart making offer at reervation price, and offer it own reervation price at a time, ay Ì, and ue an extreme Boulware NDF, then Ë i defined a Ë ÊÈ ÊÈ Ì. Note that the in Ë i a value for that give the Boulware function. In general, we ue,, and Ä to indicate a value for that give the Boulware, Conceder, and Linear NDF repectively. When oth agent ue trategie of thi form, negotiation can end either in an agreement or a conflict, depending on the four element that contitute each agent trategy. Definition 2 The negotiation outcome ǵ i an element of Ô Øµ. The pair Ô Øµ denote the price and time of agreement where Ô ¾ ÊÈ ÊÈ and Ø ¾ ¼ ÑÒ Ì Ì µ. denote the conflict outcome. A an illutration, when agent trategy i defined a Ë ½ ÁÈ ÊÈ Ì and agent trategy i defined a Ë ½ ÁÈ ÊÈ Ì, the outcome (Ç ½ ) that reult i hown in Figure 2(a) (i.e. the point where Ë and ½ Ë ½ converge). A hown in the figure, agreement i reached at a price ÊÈ ÔÖ- ÙÖÔÐÙ ¾µ and at a time cloe to Ì. Similarly when the NDF in oth trategie i replaced with, then agreement (Ç ¾ ) i reached at the ame price ut toward the eginning of negotiation. If the agent trategie do not converge efore the deadline, then negotiation reult in a conflict. Thi i illutrated in Figure 2(), where oth agent ue the extreme Boulware NDF, ut offer the final price at different time, therey reulting in conflict. Since agent are aumed to e Von Neumann and Morgentern [26] expected utility maximizer, we need to determine the four element of each agent trategy ue of, i an extreme Boulware(Conceder). In our dicuion, Boulware alway refer to the extreme Boulware for which the function generate the initial price from the eginning until the time point jut prior to Ì, and the final price at time Ì. Similarly Conceder alway refer to the extreme Conceder. 10

11 IP Price (a) S 1 S 2 Price S 3 () O 2 O 1 Zone of agreement S 2 IP S 1 S 3 T T T Time T Fig. 2. Negotiation outcome for Boulware and Conceder function. (a) Agreement. () Conflict. that will give it maximum poile utility. An agent optimal trategy depend on the information it ha aout the negotiation parameter. We therefore define the information tate for each agent and then how how the optimal trategie are determined. 3.3 Agent Information State Each agent ha a reervation limit, a deadline, a utility function and a trategy. Thu the uyer and eller each have four parameter denoted ÊÈ Ì Í Ë and ÊÈ Ì Í Ë repectively. The outcome of negotiation depend on all thee eight parameter. The information tate, Á, of an agent i the information it ha aout the negotiation parameter. An agent own parameter are known to it, ut the information it ha aout the opponent i not complete. We define Á and Á a: and Á ÊÈ Ì Í Ë Ä Ô Ä Ø Á ÊÈ Ì Í Ë Ä Ô Ä Ø where ÊÈ, Ì, Í and Ë are the information aout the uyer own parameter and Ä and Ô Ä are it elief aout the eller. Similarly, Ø ÊÈ, Ì, Í and Ë are the eller own parameter and Ä Ô and Ä Ø are it elief aout the uyer. Ä Ø and Ä Ô are two proaility ditriution ¾ that denote agent elief aout agent deadline and reervation price. Ä Ø i an Ò-tuple of ordered pair of the form Ì «, where ½ Ò. The firt element in a pair, Ì, (where Ì ¾ Ì for ½ Ò) denote a poile value for the eller deadline and the econd element, «, denote the ¾ The difference etween thi model and [7,6] i that in the latter, agent have a inary proaility ditriution over their opponent reervation value and deadline wherea here we conider the more general cae y taking a proaility ditriution over Ò value. 11

12 proaility with which the eller deadline i Ì. In other word, the pair are poile time value for agent deadline and the aociated proailitie. One of the Ò poile value i agent actual deadline. The pair are aumed to e arranged in acending order of time value, i.e., Ì Ì ½ for ½ Ò ½. Ä i analogou to Ô Ä Ø and denote the uyer elief aout the eller reervation price. The element of Ä Ô are pair are denoted ÊÈ where ½ Ñ. The firt element i a poile value for the eller reervation price and i the aociated proaility. Similarly Ä Ø and Ä Ô are two proaility ditriution that denote the eller elief aout the uyer deadline and reervation price. The element of Ä Ø are of the form Ì «(where Ì ¾ Ì for ½ Ò) and the element of Ä Ô are of the form ÊÈ. For our preent analyi we conider the cae where ÊÈ ½ ÊÈ Ñ, i.e., the highet poile value for the eller reervation price i le than the lowet poile value for the uyer reervation price. We treat the agent elief a eing tatic and not changing during negotiation. Thu agent have uncertain information aout each other deadline and reervation value. Moreover, agent do not know their opponent utility function or trategy. In other word, an agent information tate model two parameter of the opponent: the opponent reervation price and it deadline. Each agent information tate i it private information that i not known to it opponent. 3.4 Negotiation Scenario On the ai of the relationhip etween agent deadline and their dicounting factor, we define ix negotiation cenario. An agent negotiate in one of thee ix cenario. The uyer elieve that with proaility «, the eller deadline i Ì. Thi give rie to three relation etween agent deadline. All the Ò poile eller deadline could e le than the uyer deadline, ome of them could e le and the other greater, and finally all of them could e greater than the uyer deadline. For each of the two poile realization of the uyer dicounting factor, thee three relation can hold etween agent deadline. In other word, negotiation can take place in any one of the ix cenario, Æ ½ Æ, lited in Tale 1. The et of negotiation cenario for the eller can e defined in the ame way. The cenario comination that are poile for the two agent to interact are lited Future work will deal with the ituation where ÊÈ ½ ÊÈ Ñ. Future work will deal with the ituation where an agent learn and change it elief during negotiation. An agent information tate may e different for different negotiation. Alo, a hown in [5], the information tate of agent trongly influence the negotiation outcome. It would therefore e intereting to tudy the negotiation proce y modelling other parameter of the opponent. Future work will deal with uch a tudy. 12

13 Negotiation cenario Relationhip etween agent deadline Dicounting factor Æ ½ Ì Ò Ì Æ ½ Æ ¾ Ì Ì Ì ½ for ½ Ò Æ ½ Æ Ì Ì ½ Æ ½ Æ Ì Ò Ì Æ ½ Æ Ì Ì Ì ½ for ½ Ò Æ ½ Æ Ì Ì ½ Æ ½ Tale 1 Poile negotiation cenario for agent. Agent Agent Æ ½ Æ ¾,Æ,Æ,Æ Æ ¾ Æ ½,Æ ¾,Æ,Æ,Æ,Æ Æ Æ ½,Æ ¾,Æ,Æ Æ Æ ¾,Æ,Æ,Æ Æ Æ ½,Æ ¾,Æ,Æ,Æ,Æ Æ Æ ½,Æ ¾,Æ,Æ Tale 2 Poile negotiation cenario for uyer-eller interaction. in Tale 2. For intance, when agent i in cenario Æ ½, Ì i le than Ì. In uch a ituation, agent can only e in one of the four cenario Æ ¾, Æ, Æ or Æ, ince Ì can e le then Ì in only thee four cenario. Recall that one of the poile value i the opponent actual deadline. Thi implie that when agent i in cenario Æ ½, agent can neither e in cenario Æ ½ nor in Æ. Thu in general when agent i in cenario Æ ½, agent may e in any one of the four cenario Æ ¾, Æ, Æ, or Æ. The remaining cenario comination, lited in Tale 2 can e otained uing imilar reaoning. Note that it i poile for the agent to have equal deadline in the following cae: when oth agent are in cenario Æ ¾, or when oth agent are in cenario Æ, or when agent i in cenario Æ ¾ and agent i in cenario Æ. For all the other comination, the agent have different deadline. 3.5 Optimal Strategie We decrie how optimal trategie are otained for player that are von Neumann- Morgentern expected utility maximizer. The dicuion i from the perpective of the uyer (although the ame analyi can e taken from the perpective of the eller). In order to implify the dicuion we firt aume that Ä Ô contain a ingle 13

14 element, which i the eller reervation price, and otain the optimal trategy. We then extend the analyi to the more general cae where Ä Ô contain Ñ element. Each agent optimal trategy i determined on the ai of it own information tate, i.e., the uyer optimal trategy i determined on the ai of Á and the eller optimal trategy i determined on the ai of Á. We then determine if thi mutual trategic ehavior of agent reult in equilirium Optimal trategie for the uyer when Ä Ô contain a ingle element Price (a) N Price 1 () N IP S S S S 1 j n 1 S IP j Time Time Price T T T T n T T T T 1 j Price 1 j k T T k+1 n (c) N 3 (d) N 4 IP Price S o IP Time Time T T T T T T 1 n 1 n Price (e) N 5 (f) N S o S o IP S o IP Time T T T T T 1 T k k+1 T j n T 1 T n Time Fig. 3. Buyer trategie in different cenario when Ä Ô contain a ingle element. In all the ix cenario, the trategie hould enure agreement y the earlier deadline (i.e., Ì if Ì Ì and Ì if Ì Ì ). Otherwie the agent with the earlier deadline quit and negotiation end in a conflict, a ituation which oth agent prefer to avoid. We egin with cenario Æ ½ where all the Ò poile value for the 14

15 3e e+16 Buyer Expected Utility 2e e+16 1e+16 5e Strategy j Fig. 4. Buyer EU for the poile trategie in cenario Æ ½. eller deadline are le than Ì. Since Æ ½ in cenario Æ ½, the uyer prefer to reach agreement at the latet poile time and at the lowet poile price. A Ì Ì in cenario Æ ½, the latet poile time for reaching an agreement i Ì Ò. The outcome of negotiation depend on oth agent trategie. Since oth agent ue a time dependent trategy, an agent alway play a trategy that offer it own reervation price at it deadline. The uyer doe not know the eller deadline, ut it ha a lottery Ä Ø µ over Ò poile value for the eller deadline. So the uyer know that if the eller deadline i Ì, then the eller will play a trategy, Ë, that offer ÊÈ at Ì. The proaility that the eller deadline i Ì i «, i.e., the eller will play trategy Ë with proaility «. From it lottery Ä Ø µ the uyer know that the eller can play Ò different trategie, and will play trategy Ë with proaility «. In other word, although the uyer doe not know the eller actual trategy, it know the poile trategie the eller can play and the aociated proailitie. Since the maximum poile value for the eller deadline i le than Ì, the uyer can minimize the price of agreement y waiting for the eller to offer ÊÈ. Thu the optimal price of agreement, denoted ÈÓ, i ÊÈ. A an agent utility alo depend on time, and Æ ½, the uyer trie to maximize the time of agreement. Since the uyer ha Ò poile value for the eller deadline, it ha Ò trategie to chooe from. At time Ø during negotiation, trategy Ë i defined a ÁÈ ÊÈ Ì for all Ø Ì. At all later time, (i.e. etween Ì and Ì Ò ) the trategy offer the price ÊÈ. Thu the earliet time at which agreement can e reached uing trategy Ë i Ì and the latet time i Ì Ò. If the eller actual deadline i le than Ì, then Ë reult in conflict. Thee trategie are depicted in Figure 3(a). Out of thee Ò trategie, the one that give the uyer the maximum expected utility ÍÓ µ i it optimal trategy ËÓ µ. Agent expected utility from trategy Ë, i: Note that the uyer doe not know the eller complete trategy. It only know the eller final price and the time at which the price i offered. 15

16 ܽ Ò ½ Í «Ü Í µ «Í ÊÈ Ì µ Ý ½ «Ý Í ÊÈ Øµ where Ì Ø Ì Ò (3) Thi i the general expreion for the uyer EU from different trategie. Here, the value of Ø depend on the opponent trategy. In Section we will explain how to otain the value of Ø. For the preent aume that thi value i known to u. For thi given value of Ø, the expected utility depend on the proaility ditriution («), the utility function (Í ), and. For example, the EU for different value of etween ½ and ½ i illutrated in Figure 4. In thi example, «wa defined a a Poion ditriution and Æ wa 1.6 (a value greater than 1). A een in the figure, Í i maximum at, indicating that the optimal time for entering the zone of agreement, denoted Ì Â, i Ì. The optimal trategy i therefore Ë. In thi figure, the time point are at uniform dicrete interval. However, thi i not neceary a long a the condition for convergence of agent trategie (lited in Section 3.5.3) are atified. For a higher value of Æ, Í i maximum at a higher value of. Lowering the value of Æ caue the peak of the curve to hift left. In other word Ì Â increae a Æ increae and Ì Â decreae a Æ decreae. For Æ ½, Í i at a maximum for ½. Thi happen ecaue the agent i indifferent to time. Higher value of reult in ome conflict ituation and thu give a lower utility. But when Æ ½, the agent gain utility with time and the maximum utility i otained for ½. The uyer optimal trategy for cenario Æ ½ i lited in Tale 3. Let ËÓ Øµ denote the price generated y the uyer optimal trategy at time Ø. The uyer action function for cenario Æ ½ i defined a follow: Ø Ô Ø µ Quit Accept Offer Ë Ó Ø¼ µ in the next time period Ø ¼ if Ø Ì if Ô Ø Ë Ó Øµ otherwie In the definition of an agent action given in Section 3.1, the opponent offer i accepted if the utility from the opponent offer at time Ø i greater than or equal to the utility of the offer the agent i willing to generate at time Ø ¼. But here, in order to decide when to accept the eller offer, the price offered y the eller at time Ø (Ô Ø ) i compared with the price generated y the uyer optimal trategy (Ë Ó) at time Ø. Thi i ecaue the eller actual deadline i not known to the uyer, and Ø could e the eller deadline, in which cae the eller quit and negotiation end in a conflict if the uyer doe not accept the offer at time Ø. So even though the 16

17 Negotiation cenario Time Ø during negotiation Optimal trategy Æ ½ Ø Ì Â ÁÈ ÊÈ Ì Â Ø Ì Â ÊÈ ÊÈ Ì Ò Ä Æ ¾ Ø Ì Â ÁÈ ÊÈ Ì Â Ì Â Ø Ì Ø Ì ÊÈ ÊÈ Ì Ä ÊÈ ÊÈ Ì Æ Ø Ì ÁÈ ÊÈ Ì Æ Ø Ì ¼ ÁÈ ÊÈ Ì ¼ Ø Ì ¼ ÊÈ ÊÈ Ì Ò Ä Æ Ø Ì ¼ ÁÈ ÊÈ Ì ¼ Ì ¼ Ø Ì Ø Ì ÊÈ ÊÈ Ì Ä ÊÈ ÊÈ Ì Æ Ø Ì ¼ ÁÈ ÊÈ Ì ¼ Ø Ì ¼ ÊÈ ÊÈ Ì Ä Tale 3 Optimal uyer trategie in different negotiation cenario when Ä Ô contain a ingle element. Ì ¼ denote the econd time period, i.e., if negotiation egin at time Ø, Ì ¼ Ø ½. uyer utility increae with time, it ha to accept the eller offer if Ô Ø Ë Ó Øµ and therey avoid the chance of a conflict. In cenario Æ ¾, the eller deadline can e either le than or greater than Ì. Since ome of the poile value for the eller deadline are le than Ì, the uyer optimal trategy would e to wait for the opponent to offer ÊÈ. If Ì Ì, the latet time y which the eller will offer ÊÈ i Ì. Thu until Ì, the uyer need not offer a price greater than ÊÈ. If an agreement i not reached y Ì, it implie that the eller deadline i greater than Ì and to avoid conflict, the uyer need to offer it reervation price ÊÈ at Ì. Thu agent hould enter the zone of agreement at the latet poile time (to enure that agreement i not reached earlier than that), remain at ÊÈ until Ì and then offer/accept it own reervation price, ÊÈ, at Ì. The poile time for entering the zone of agreement are Ì ½ Ì. Thee trategie are depicted in Figure 3(), where trategy Ë enter the zone of agreement at Ì. The expected utility for trategy Ë i: 17

18 3e e+08 "k=2" "k=4" "k=8" "k=14" Buyer Expected Utility 2e e+08 1e+08 5e Strategy j Fig. 5. Buyer EU for different trategie in cenario Æ ¾ when Æ ½. ܽ ½ Í «Ü Í µ «Í ÊÈ Ì µ «Ý Í ÊÈ Ø ½ µ Ò Ý ½ Þ ½ «Þ Í Ô Ø ¾ µ (4) where ÊÈ Ô ÊÈ and Ì Ø ½ Ì Ý and Ì Ø ¾ Ì A for cenario Æ ½, aume that the value of Ô, Ø ½, and Ø ¾, are known. In Section we will explain how to otain thee value. For the given value of Ô, Ø ½, and Ø ¾, the value of Í for different value of etween 1 and 15 and Æ ½ (where «i a Poion ditriution) are depicted in Figure 5. A een in the figure, the value of for which Í i maximum depend on the value of. For higher value of Æ, we get the ame pattern a in Figure 5 ut the peak of the curve hift to the right. Lowering the value of Æ hift the peak to the left. In other word, the optimal time (Ì Â ) for entering the zone of agreement increae a Æ increae and decreae a Æ decreae. Figure 6 how Í for Æ ½. A een from the figure, Í i maximum at ½. Thi happen ecaue, for Æ ½, the agent i indifferent to time. Higher value of reult in ome conflict ituation and thu give a lower utility. But when Æ ½, the agent gain utility with time and the maximum utility i otained for ½. The uyer optimal trategy for cenario Æ ¾ i lited in Tale 3. The uyer action function for cenario Æ ¾ i the ame a that for cenario Æ ½. In cenario Æ, the uyer gain utility with time Æ ½µ and Ì Ì ½. The uyer optimal trategy here i Ë Ó ÁÈ ÊÈ Ì. Thi trategy (hown in Figure 3(c)) enter the zone of agreement at the latet poile time, which i cloe to the earlier deadline Ì, and therey maximize the time of agreement. It alo optimize the price of agreement y offering ÊÈ only at Ì. 18

19 40 35 "k=2" "k=4" "k=8" "k=14" 30 Buyer Expected Utility Strategy j Fig. 6. Buyer EU for different trategie in cenario Æ ¾ when Æ ½. In the remaining three cenario, Æ to Æ, Æ ½ and the uyer loe utility with time. In cenario Æ (hown in Figure 3(d)), it i clear that the uyer can optimize oth the price and the time of agreement y offering ÊÈ right from the eginning of negotiation, until Ì Ò (ee Tale 3). Contrat thi with ËÓ of cenario Æ ½, in which the zone of agreement i entered at Ì Â, wherea here it i entered at the eginning of negotiation uing the Conceder function (ince Æ ½). In cenario Æ, the uyer optimal trategy i to offer ÊÈ from the eginning of negotiation until Ì. If Ì Ì, then negotiation end at the latet y Ì. Otherwie it continue eyond Ì. The uyer then ha to concede up to ÊÈ in order to enure agreement (ee Figure 3(e)). Thi trategy i lited in Tale 3. Finally, in cenario Æ, the uyer optimal trategy i to offer ÊÈ right from the eginning of negotiation until Ì (ee Figure 3(f)). Thi i ecaue when the uyer i in cenario Æ, the poile cenario for the opponent are Æ ½, Æ ¾, Æ or Æ. Since the eller alo ehave trategically, in none of thee cenario will agent concede eyond ÊÈ, until time Ì, uing it optimal trategy. Thu for the price ÊÈ, the time of agreement i optimized in trategy Ë Ó uing the Conceder function. Tale 3 lit the uyer optimal trategie in all the ix negotiation cenario. The uyer action function in all the cenario i the ame a that for cenario Æ ½ Optimal trategie for the uyer when Ä Ô contain more than one element Optimal trategie for the uyer when Ä Ô contain more than one element remain the ame a thoe otained in Section in ome, ut not all, negotiation cenario. Only thoe optimal trategie (lited in Tale 3) that depend on the opponent reervation price change, while the other remain the ame. More pecifically, the optimal trategie in cenario Æ and Æ remain the ame, while thoe in cenario Æ ½, Æ ¾, Æ, and Æ change when Ä Ô contain more than one element. We analyze 19

20 Price i m 1 m IP. S 1,1 S i,j S m,n T T T T T 1 2 j n 1 n S m 1,1 T Time Fig. 7. Poile uyer trategie in cenario Æ ½ when Ä Ô contain more than one element. each of thee four cenario elow. The uyer action function,, doe not depend on the numer of element in Ä Ô and therefore remain the ame a defined in Section for all the cenario. A mentioned in Section 3.3, the information tate of the uyer, Á, ha Ò poile value for the eller deadline and Ñ poile value for it reervation price. Alo, recall that agent elieve that i the proaility that the opponent reervation price i ÊÈ and that «i the proaility that the opponent deadline i Ì. The proaility that the eller ha the reervation price ÊÈ and deadline Ì i thu the product of and «, and i denoted. Conider cenario Æ ½ firt. The poile uyer trategie for thi cenario are depicted in Figure 7. The numer of poile trategie here i Ñ Ò. We ue Ë to denote the trategy that tart making offer at ÁÈ, offer ÊÈ at Ì uing the Boulware function, and doe not change the price thereafter. The trategy that yield the highet EU i the uyer optimal trategy. Let Á and  denote the value of and that give agent the highet utility. Here we need to find thee two value. Contrat thi with the cae where Ä Ô had a ingle element which required finding only the optimal value of, i.e., Â. The outcome of negotiation depend on oth the uyer and the eller trategy. The uyer doe not know the eller trategy, ut it ha two lotterie, Ä Ô and Ä Ø, over the eller reervation price and deadline. So if the eller reervation price and deadline are ÊÈ and Ì, then it play trategy Ë that offer ÊÈ at Ì. The proaility with which the eller play trategy Ë i. Thu although the uyer doe not know the eller actual trategy, it know that the eller can play Ñ Ò different trategie and the aociated proailitie. Conider the trategy ËÑÒ. Thi trategy reult in an agreement only if the eller 20

21 actual reervation price i ÊÈ Ñ and it deadline i Ì Ò. All the other value for the eller reervation price or deadline reult in a conflict. Thu the EU from trategy ËÑÒ i: Ñ ½ Í ÑÒ Ò Ü½ ݽ ÜÝ Í µ Ò ½ ܽ ÑÜ Í µ ÑÒ Í ÊÈ Ñ Ì Ò µ (5) In general, trategy Ë reult in conflict if either the eller reervation price i higher than ÊÈ, or it deadline i le than Ì. The utility from Ë i therefore: ½ Í Ò ½ ½ Í µ Í ÊÈ Ì µ Ñ Ý ½ ½ ½ Ò Í µ Ü Í ÊÈ Ø ½µ Ü ½ ½ ÝÞ Í µ Ý Í Ô ½ Ò Ì µ Þ½ Þ ½ where ÊÈ Ý Ô ½ ÊÈ and ÊÈ Ý Ô ¾ ÊÈ and Ì Ø ½ Ì Ü and Ì Ø ¾ Ì Þ ÝÞ Í Ô ¾ Ø ¾ µ In the aove expreion, the value of Ô ½ and Ô ¾ depend on two factor: the opponent trategy and the identity of the player that make a move at the earlier deadline. The value of Ø ½ and Ø ¾ depend only on the opponent trategy. Although the uyer doe not know the opponent actual trategy, it doe know that the opponent will alo ehave trategically. Thi trategic ehavior depend on the opponent cenario. Recall that when the uyer cenario i Æ ½, the eller can e in any of the four cenario: Æ ¾, Æ, Æ, or Æ. We know from Section that in cenario Æ, an agent optimal trategy i to offer it reervation price uing the Conceder function. Thu if agent i in cenario Æ, it optimal trategy i to offer ÊÈ uing the Conceder function. In addition to the eller trategy, the value of Ô ½ and Ô ¾ alo depend on who make an offer at the earlier deadline. The player that make an offer at the earlier deadline could e the uyer or the eller, depending on who made the initial offer. Conider the cae where it i the eller turn to make a move at the earlier deadline. The eller optimal trategy in cenario Æ i to offer ÊÈ uing the Conceder function. A per the uyer action function, the uyer accept the eller offer at Ì. We therefore get Ô ½ Ô ¾ ÊÈ Ý and Ø ½ Ø ¾ Ì. On the other hand, if it i the uyer turn to make a move at the earlier deadline, it offer. For Þ, ÊÈ ÊÈ, and the eller accept the uyer offer at time Ì. Thi make Ô ½ Ô ¾ ÊÈ and Ø ½ Ø ¾ Ì. Uing imilar analyi, it can e een that when agent i in any of the remaining three cenario (Æ ¾, Æ, or Æ ), we get Ô ½ Ô ¾ ÊÈ Ý, Ø ½ Ì Ü, and Ø ¾ Ì Þ if the eller make an offer at the earlier deadline; and Ô ½ Ô ¾ ÊÈ, Ø ½ Ì Ü, and Ø ¾ Ì Þ if the uyer make an offer at the earlier deadline. The uyer know who will make an offer at the earlier ÊÈ (6) 21

22 Price i m 1 m IP.. S i,j T T T T T 1 2 j k T T k+1 n 1 T n Time Fig. 8. The uyer trategy Ë in cenario Æ ¾ where Ä Ô contain more than one element. deadline, ince the deciion aout which player will make the initial offer i made at the eginning of negotiation and thereafter player take turn alternately at each ucceive time period. Since the uyer doe not know the eller cenario, we aociate equal proailitie with each of the four poile eller cenario, Æ ½, Æ, Æ, and Æ. Let Ù ½ denote the value of Equation 6 when the eller cenario i Æ ¾, Æ, or Æ. Alo, let Ù ¾ denote the value of Equation 6 when the eller cenario i Æ. The uyer EU therefore ecome: Í Ù ½ ½ Ù ¾ (7) The value of and for which Equation 7 i at a maximum are denoted Á and Â. The uyer optimal trategy for cenario Æ ½, in term of Á and Â, i lited in Tale 4. In cenario Æ ¾, the uyer ue a trategy Ë of the form depicted in Figure 8. Thi trategy tart at ÁÈ, offer ÊÈ at Ì uing the Boulware function, keep the price contant at ÊÈ until Ì, and thereafter ue the Boulware function again to offer ÊÈ at Ì. It i clear from Figure 8 that can vary etween 1 and Ñ and can vary etween 1 and. Thu there are Ñ poile trategie and the one that yield the maximum EU i the uyer optimal trategy. Let Á and  denote the value of and repectively that give the highet utility. Here we need to find thee two value. Contrat thi with the cae where Ä Ô had a ingle element, which required finding only Â. The uyer EU from trategy Ë i: Í Í ½ Í ¾ Í (8) Here, the term Í ½ denote agent EU if the eller actual reervation price i 22

23 higher than ÊÈ, Í ¾ denote it EU if the eller actual reervation price i equal to ÊÈ, and Í denote it EU if the eller actual reervation price i lower than ÊÈ. We otain each of thee three term elow. For Í ½ (i.e., for ÊÈ ÊÈ ), the eller deadline can e either le than or equal to Ì, or it can e greater than or equal to Ì ½ (ee Figure 8). If Ì Ì, then negotiation end in a conflict. Í ½ i therefore given y: ½ Í ½ ܽ ݽ ÜÝ Í µ Ò Ý ½ ÜÝ Í Ô ½ Ì µ where ÊÈ Ü Ô ½ ÊÈ µ (9) Note that the value of Ô ½ depend on the opponent trategy and the identity of the player that make an offer at the earlier deadline. The four poile eller cenario for the econd term of Equation 9 (i.e., Ì Ì ) are Æ ½, Æ ¾, Æ, or Æ. For each of thee cenario, the eller trategic ehavior give Ô ½ ÊÈ if the uyer make a move at the earlier deadline, and Ô ½ ÊÈÁ if the eller make a move at the earlier deadline. Note that in order to get thee value for Ô ½, the uyer and eller trategie need to converge efore the earlier deadline. The condition for convergence of agent trategie are lited in Section Alo note that the value of ÊÈÁ i preent in the eller information tate and i not known to the uyer. The uyer can therefore only take Ô ½ ÊÈ a the cloet approximation. The next term, Í ¾, i the uyer EU when ÊÈ i equal to ÊÈ Ü½ Í ¾ ½ Ü ½ Ü Í µ Í ÊÈ Ì µ Ü Í ÊÈ Ø ½µ Ò Ü ½ and i: Ü Í Ô ¾ Ø ¾ µ (10) where ÊÈ Ô ¾ ÊÈ µ and Ì Ø ½ Ì Ü µ and Ì Ø ¾ Ì µ The poile cenario for the eller for the third term of Equation 10 are Æ ¾, Æ, Æ, or Æ. Conidering the eller trategic ehavior, we get Ø ½ Ì if the eller cenario i Æ and Ø ½ Ì Ü otherwie. The poile cenario for the eller, for the fourth term of Equation 10, are Æ ½, Æ ¾, Æ, or Æ. Conidering the eller trategic ehavior, we get Ø ¾ Ì for all the four cenario. The value of Ô ¾ depend on the player that make a move at the earlier deadline. If the uyer make a move at the earlier deadline, we get Ô ¾ ÊÈ. On the other hand, if the eller make a move at the earlier deadline we get Ô ¾ ÊÈÁ. A for Ô ½, ince the uyer doe not know ÊÈÁ, it can only take Ô ¾ ÊÈ a the cloet approximation for all poile eller cenario. The lat term, Í (i.e., for the cae ÊÈ ÊÈ ) i a follow: 23

24 Í Ñ Ò Ü ½ Ý ½ ½ ÜÝ Í µ Ü Í Ô Ì µ ݽ ÜÝ Í Ô Ø µ Ý ½ ÜÝ Í Ô Ø µ where ÊÈ Ü Ô ÊÈ µ and ÊÈ Ü Ô ÊÈ µ and ÊÈ Ü Ô ÊÈ µ and Ì Ø Ì Ý µ and Ì Ø Ì µ (11) The poile cenario for the eller for the econd and third term of Equation 11 are Æ ¾, Æ, Æ, or Æ, while for the fourth term they are Æ ½, Æ ¾, Æ, or Æ. Conidering the eller trategic ehavior we get Ø Ì if the eller cenario i Æ, and Ø Ì Ý otherwie. For all the poile eller cenario Ø Ì. The value of Ô, Ô, and Ô depend on the identity of the player that make a move at the earlier deadline. Ô Ô ÊÈ Ü if the eller make a move at the earlier deadline, and Ô Ô ÊÈ if the uyer make a move at the earlier deadline. Finally, Ô ÊÈÁ if the eller make a move at the earlier deadline and Ô ÊÈ if the uyer make a move at the earlier deadline. Again, a for Ô ½, the uyer can only take Ô ÊÈ a an approximation. The uyer utility from trategy Ë, i the um of Í ½, Í ¾, and Í. Let Ù ½ denote the value of Equation 8 if the eller cenario i Æ, and Ù ¾ denote it value otherwie. A each of the four poile cenario for the eller i equally proale, Í ecome: Í ½ Ù ½ Ù ¾ (12) The value of and that give the uyer the maximum EU are denoted Á and Â. The uyer optimal trategy for cenario Æ ¾, in term of Á and Â, i lited in Tale 4. In the next cenario, i.e., Æ, the uyer optimal trategy doe not depend on the opponent reervation price. Thu the uyer optimal trategy when Ä Ô contain more than one element i the ame a it optimal trategy when Ä Ô contain a ingle element. Thi i alo true for cenario Æ. In negotiation cenario Æ, the uyer optimal trategy i to offer the opponent reervation price, ÊÈ, immediately after negotiation tart and continue to offer the ame price until negotiation end. The poile trategie that the uyer can ue when Ä ha more than one element are of the form Ô Ë, where ½ Ñ. Thi i hown in Figure 9(a). The uyer EU from trategy Ë i: 24