An Exploration-Exploitation Model of Negotiation
Laurie R. Weingart David A. Tepper School of Business Carnegie Mellon University Pittsburgh, PA 15213
[email protected]
Michael J. Prietula Goizueta Business School Emory University Altanta, GA 30322-2710
[email protected]
Paper accepted for presentation at the Annual Meeting of the Academy of Management, August 5-10, Honolulu, Hawaii
Available as: Working Paper, Tepper School of Business Carnegie Mellon University business.tepper.cmu.edu
Working Paper, Goizueta Business School Emory University gbspapers.library.emory.edu
Acknowledgement. We thank Jeanne Brett of Northwestern University for comments on an earlier version of this paper. This project was partially funded by the Dean’s Research Fund, Tepper School of Business.
An Exploration-Exploitation Model of Negotiation Abstract We propose a model of negotiation that may better explain how the specific progression of offer traces emerge in a dyadic negotiation and how those traces converge to a solution. Negotiation is viewed as a type of ill-structured collaborative problem that negotiators attempt to resolve by jointly searching the offer space, which is an abstract representation of the entire set of potential solutions to the problem. We suggest that boundedly rational negotiators simplify the task by systematically coordinating searches of small subsets of the offer space. These coordinated searches define a common ground of understanding of the regions of the space that may contain a solution. Furthermore, these regions are defined more by their structural (rather than value) properties of offers. Consequently, we propose that the early phase of negotiation is described by search processes that explore the offer space to discover regions of potential (joint) interest. When a promising region is found, a second “final path” phase of search begins whereby that region is exploited in order to achieve a final solution. Both phases are revealed by examining sequential offers which afford a trace of the paths taken to an agreement. We describe a method of representing sequences of offers that categorizes exploration and exploitation patterns. We argue that by incorporating these concepts in analyses, insight into the underlying dynamics of negotiation can be better understood. To illustrate, we apply these concepts to an illustrative set of dyadic negotiation data.
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Negotiation is a ubiquitous process in organizations. It occurs whenever two parties with differing or conflicting interests attempt to find a mutually acceptable agreement. Negotiation can be used to resolve conflict or complete a transaction that involves conflict of interest. In the case of transactions where goods and services are being exchanged, the most common negotiator choices involve those that maximize one’s own outcome, as well as those that are acceptable to the other party so that agreement can be reached. However, even apparently simple negotiations can quickly task the cognitive resources of a negotiator and the outcomes will likely diverge substantially from optimal. The intent of this paper is to present a descriptive model of the processes that account for the underlying behaviors that lead to a solution, whatever that solution may be. In any given negotiation, negotiators must jointly consider several solutions (i.e., offers) in order to find an acceptable, if not optimal, agreement. However, the total number of possible solutions that could be proposed for even a simple negotiation can be a large – much larger than the number of offers we see in negotiation tasks. Thus, given the apparent complexity of the task as defined by the number of potential solutions, this begs two questions: How do negotiators select the offers they propose? How can two negotiators reach a joint agreement? To answer these questions we take a distinct problem solving perspective, whereby negotiation is seen as a joint problem solving activity (Carroll, Bazerman & Maury, 1988; Prietula & Weingart, 1994). Accordingly, we employ the robust “standard model” of information processing psychology as this provides the theoretical foundation for defining problem solving as search through a problem space (or set) of possible alternatives (e.g., Simon & Kaplan, 1989; Klahr & Kotovsky, 1989; Newell & Simon, 1972; Simon, 1978). When problem spaces get sufficiently large or complex such that a problem solver is unable to search (consider) all of the alternatives, then methods must be engaged to reduce the complexity, as humans are indeed
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boundedly rational (Simon, 1956, 1982). As problem solvers, negotiators often cannot consider all possible solutions in the offer space (i.e., the problem space we are defining for the negotiation – all possible offers). In addition, they are unable to engage methods that result in the best possible solution, but rather engage heuristic methods that accommodate their cognitive constraints, their knowledge of the situation, and the demands of the task (Bazerman & Neale, 1983; Gigerenzer & Goldstein, 1996; Newell, 1990; Simon, 1990). Therefore, if we discover how negotiators simplify the task, we will discover how they generate and agree upon offers. We view negotiation as a dynamic process of joint search, where negotiators exchange offers as a means of discovering mutually acceptable agreements. Thus, the form, sequence, and patterns of offers define joint search. We argue that often “offers build on offers” and that the mere articulation of an offer serves as signal to the other party as to what agreements are possible. Specifically, we suggest that joint search is conducted and coordinated by the exchange of offers between the parties, and these exchanges implicitly partition the total set of possible solutions into smaller regions. Thus, while negotiators do keep in mind the total value of an offer on the table, we suggest they are substantially influenced by the emerging structure of the agreement itself. Furthermore, research on individual problem solving has revealed that problem solvers often go through a 2-phase process, whereby the first phase is exploratory (and sometimes lengthy and error prone), while the second phase represents “final path” movement rapidly converging toward a solution (Kotovsky, Hayes & Simon, 1985). We suggest that in characterizing negotiation problem solving as search, a similar 2-phase process is engaged, which we refer to as an exploration-exploitation model (March, 1991), which we adapt to more reflect the joint nature of the task. Conceptually, the general model is simple – given a restricted resource (e.g., time, effort, attention, information), negotiators repeatedly select between exploiting a particular choice for potential gain or exploring the environment for other (presumably more
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valuable) choices to subsequently exploit. This distinction provides a simple scheme of categorizing and interpreting negotiation search behaviors, as revealed in offers and counteroffers. In particular, we suggest that exploration reflects the more uncertain and speculative component of joint search between the two negotiators in which they attempt to more precisely define, refine, and align their understanding to establish a “common ground” (i.e., required knowledge to be held in common for a successful dialogue; Clark & Marshall, 1981; Pickering & Garrod, 2004) to be jointly searched. Thus, we define exploration as a process of broadly sampling potential offers. On the other hand, we suggest that exploitation is a distinctly more coordinated activity emerging as the common ground is aligned and begins when the two parties start to converge to an understanding of what constitutes a possible solution. In this paper we present a parsimonious model that we apply to a type of negotiation in which key characteristics of the situation as understood by either party (i.e., their representations) are assumed to be equivalent – the number of issues, the number of options, the (induced) utilities of each option, the requirements for solution, and the conditions for dissolution. We then propose that it is unlikely for negotiators to consider all possible solutions to the particular negotiation problem – the offer space; rather, they would likely consider a small fraction of that total number. Furthermore, we suggest that to solve the problem, the two negotiators have to align their representations, where alignment means that their respective aspirations have to result in agreements on issue options. The process of alignment is revealed by search of the offer space that is characterized by two processes: exploration and exploitation. Exploration search is less coordinated and more oriented toward mutual calibration of aspirations. Exploration turns into exploitation, as common ground is established. Exploitation sequences are generally based on the structure of offers, suggesting a bias toward structural similarity and negotiators tend to resist significant changes to the common ground.
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The paper is organized as follows. We first describe our general approach of viewing negotiation as problem solving. Here we are not developing a theory of problem solving, but applying a theory of problem solving in order to develop a descriptive, process model of negotiation problem solving. We then briefly consider the primary literature in negotiation that has relied on the examination of offer patterns, for it is toward an explanation of these types of patterns of behavior that our work is directed. We then characterize the offer space (i.e., the set of all possible offers) in terms of a typical task used in basic negotiation research (three issues with integrative potential). This supposedly simple task has a surprisingly complex offer space (see Mumpower, 1991). We use illustrative data from this task to demonstrate how we code and interpret exploration and exploitation phases of negotiation. In the concluding section, we build on these concepts and show how they can serve as heuristics for offer generation as well as the foundation for description and analysis of negotiation dynamics. Negotiation as Problem Solving According to information processing psychology, problems are defined in terms of internal (cognitive) representations of a task, which includes a goal (or set of goals) to be achieved and the allowable actions that can be taken. A goal is achieved through a search process, whereby alternative solutions are considered, adopted or discarded (e.g., Newell, 1990; Newell & Simon, 1972). The nature of the representation and search are largely influenced by characteristics of the task environment (the task as defined and understood by the problem solver, the other negotiator, etc.) and how the problem solver integrates the information gained while performing the task into his/her own representation (Newell, 1990; Newell & Simon, 1972; Suchman, 1987). Each problem solver’s representation and search both influences and is influenced by the dynamics of interaction. Furthermore, the communication process, in terms of offers and counter-offers, reflects the view that language embodies joint action by individuals who
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cooperate and coordinate to achieve goals (Clark, 1996) and reveals insight into the problem solving processes of the problem solvers (Ericsson & Simon, 1993, 1998). Thus, search is the process by which negotiations are accomplished and the offers made during a negotiation contain much information about search. If an offer is made and accepted, the negotiation is done; otherwise, an offer rejection results in “the dance”, with subsequent offers and counter-offers being made (in addition to other types of information exchange). By examining the nature and progression of these offers, it is both useful and insightful to interpret the progression as traces of a collaborative search effort to achieve a solution to a commonly held, and commonly defined, problem – a negotiation. By examining these traces in the context of the exploration and exploitation model, we gain better insight into how negotiators simplify and coordinate that search. Offers and Patterns Negotiators exchange information in their attempts to reach agreement. They provide information about their preferences and priorities, argue their positions, and exchange offers (Olekalns, Smith, & Walsh, 1996; Putnam & Wilson, 1989; Weingart et al., 1990; Weingart, Hyder, & Prietula, 1996). Offers are proposals made to the other party that reflect an acceptable alternative to the source. Offers provide information to the other party about one’s own desires, and the progression of offers over time provides information about one’s willingness to concede (Komorita & Brenner, 1968; Kwon & Weingart, 2004). Offers can include single or multiple issues. While single issue offers reflect desires on a given issue, multi-issue offers combine preferences on issues such that the issues can be packaged together or traded off across the parties. Consequently, much information is contained in offers and in offer sequences and in negotiation, perhaps the most important communication is indeed the exchange of offers and counter-offers (Tutzauer, 1992).
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In negotiation research, offer patterns have been studied in distributive negotiations where two parties are attempting to reach agreement on a single issue. In this literature, offers are framed in terms of the concessions they represent and analyzed in terms of the amount and timing of the concessions (Allen, Donohue, & Stewart, 1990; Komorita & Brenner, 1968; Smith, Pruitt, & Carnevale, 1982). In multi-issues negotiations that are characterized by integrative potential (such that both parties can increase their outcomes simultaneously), offer patterns have been studied in terms of “systematic concessions” (Siegel & Fouraker, 1960; Kelley & Schenitzki, 1972) or “heuristic trial and error” (Pruitt, 1981) as well as in terms of the “dance of packages” or joint construction (Raiffa, 2002). While both have provided insight into the way offers might be constructed, neither stream of research has provided sufficient insight into detailed processes accounting for joint offer patterns. Raiffa (2002) describes two (extreme) patterns of offers that more closely resemble our theoretical approach: a dance of packages that reflects evidence of joint exploration and joint construction of a compromise contract that resembles processes of joint exploitation. The first pattern is the “dance of packages” whereby the two parties exchange packaged offers as a consequence of strategic choice and in response to the other party. Both parties negotiate at the “package level” which may (or may not) result in either systematic behaviors or reasonable results. Raiffa (2002) characterizes this dance of packages as moving in and out of the feasibility domain and being somewhat “disorderly” at times (p. 275). Raiffa (2002) describes the second pattern as a “joint construction of a compromise contract” whereby a negotiation is resolved as the result of a series or succession of agreements on individual issues or subsets of issues. In this pattern, offers built on issue sub-agreements are tentative, for “agreements made early on are not treated as irrevocable, but are reviewed later as the package evolves and gains in complexity” (p. 277). One would expect less discontinuity in this type of pattern and more (in Raiffa’s
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terminology) migration. Central to this theoretical approach (but not assumed by models of systematic concession making and/or heuristic trial and error which consider offer formation solely from the focal negotiator’s perspective) is the assumption that negotiators incorporate information obtained from the other party in crafting their own responses, and negotiators are boundedlyrational actors who do not exhaustively define and consider all potential solutions at every move. In support of these assumptions, research on negotiation processes has demonstrated the temporal interdependencies of negotiator behavior (Putnam & Jones, 1982; Weingart, Prietula, Hyder, & Genovese, 1999) as well as substantial bounds on negotiator cognition (Neale & Bazerman, 1991). If offers do respond to the other party’s behavior, then we would expect offers to be influenced by prior offers. If negotiators cannot exhaustively search all possible solutions, then we would expect some evidence of pruning the offer space. But what would be the nature of such pruning? Before considering how offers progress, we first must understand the nature of that space through which they travel. Characterizing the Offer Space Let us consider a typical negotiation task of the type used in many laboratory studies – a three-issue, role-playing negotiation conducted between two individuals (described in Weingart, Thompson, Bazerman, & Carroll, 1990). Two college students are negotiating typing services. There are three issues to be negotiated: completion time of the manuscript (Speed), editing effort to detect errors (Editing), and appearance of the manuscript as specified by the type of output device (Appearance). Each role (Buyer, Typist) has a table depicting preference values for alternatives (also called a payoff schedule), but the table is held as private information (both tables are collapsed together and are shown in Table 1). Offers are made by specifying the option (integer from the left most column) of a particular issue (e.g., speed, edit, appearance), where
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offers can be single issue offers (e.g., “3 days for speed”), two issue offers (e.g., “3 days for speed and maximum 5 errors for edit”), or three issue offers (e.g., “3 days for speed and maximum 5 errors for edit and dot matrix for appearance”). The utility for an offer (and therefore to the individual) is calculated by summing the associated values for each issue named at the specified option using the table for the particular role. The two parties attempt to reach a joint agreement on these issues and exchange information about what they would like and what they can do. Insert Table 1 about here As suggested above, the offer space is the set of possible package offers in a two-party negotiation and can be (and often is) efficiently displayed as a two-dimensional graph, where one axis represents the total value of possible offers (summed over all the issues) for Party 1 (e.g., the Buyer) and the other axis represents the total value of possible offers to Party 2 (e.g., the Seller), and the plot represents the sum of the two values comprising the negotiation solution if agreed upon (see Figure 1). Insert Figure 1 about here For the typing services problem, there are 729 possible (and discrete) three-issue (i.e., comprehensive) offers, as each of the three issues has 9 possible choices of options. When the unacceptable outcome sets are removed (in this case, alternatives that are valued as a loss to one of the parties), the remaining offers are sometimes called the zone of possible agreement (Raiffa, 1982) and represent the set of plausible solutions for the negotiation as any solution in this set represents a gain over a failed negotiation or over the value of their best alternative (i.e., BATNA, Fisher & Ury, 1981). Solutions within this set essentially reflect the negotiation alternatives when the problem solvers playing properly, but not necessarily playing effectively. In this task, there are 521 such possible solutions remaining. From this type of representation, measures of performance, such as joint outcome for
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given offers, have been discussed (e.g., Clyman, 1995; Tripp & Sondak, 1992) and, visually, this representation suggests obvious and intuitive concepts of distance (between offers, or to the Pareto frontier) as well as density (e.g., the number of points, or offers, in a given sector) for the entire set of solutions. Albeit useful, we suggest that landscape depicted by Figure 1 is not necessarily (or even likely) the landscape defined by (or considered by) the negotiators, and it is a depiction of the solutions from a static and summarized representation of the task. Offers as a Trace The patterns of offers made during a negotiation provide a trace of the negotiators’ search through the space of solutions. Consequently, one key to understanding the processes of negotiation is to understand the nature of the offer patterns that emerge as the negotiation unfolds. From such an analysis we can gain insight into the patterns underlying the progression of offers that lead to agreements and gain insight into the negotiation processes that underlie the patterns themselves. Offer/counter-offer pairs (and their sequences) are the primary units of analysis for our model. We adopted this approach based on three theoretical positions. First, the offer/counteroffer is consistent with the “presentation-acceptance” model of discourse contribution (Clark & Schaefer, 1989). Second, taking turns in discourse is fundamental to conversations (Sacks, Schegloff & Jefferson, 1974; Schegloff & Sacks, 1973). Third, language (in the context of problem solving and game playing) reflects not only the existence of joint action, but embodies the mechanisms to coordinate that action (e.g., Clark, 1996; Schelling, 1960). Furthermore, joint action serves to define and refine elements of the common ground, and the offer/counter-offer pairs reflect key changes in that common ground state (the structure of a proposed solution). Exploration and Exploitation via Offers When viewing the sequences of offer and counter-offers that emerge from actual dyadic
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negotiations, the resulting paths are often irregular, seemingly random, and sometimes quite complex movements toward an agreement. We suggest that the apparent complexities of the offer traces are explained by simple and systematic search mechanisms conducted by the negotiators. One mechanism, exploration, is less coordinated and more individualistic. The other mechanism, exploitation, demands more coordination but rapidly defines a smaller region of the offer space to which joint attention is paid, resulting in a more focused search. In that exploration is an uncertain and speculative component of joint search, we define it as a process of broadly sampling potential offers such that offer sequences do not reflect agreement on any issue. To illustrate exploration, we employ dyadic negotiation data (n = 8 dyads) from a previously published study that used the negotiation task summarized in Table 1 (Weingart et al., 1990). The Seller (S) and the Buyer (B) make a series of offers, where each offer is depicted as (i, j, k) reflecting the ith option of Speed, the jth option level of Edit and the kth option of Appearance (as defined in Table 1). We can define exploration in terms of the structure of the offer sequence. Exploration. Any two sequential offers from two (N)egotiators, N1(i1, j1, k1) and N2(i2, j2, k2) are said to represent an exploration sequence if i1 ≠ i2 and j1 ≠ j2 and k1 ≠ k2. Therefore, if an adjacent pair of offers (offer/counter-offer) does not have any options in common, we assume that the exploration of the offer space reflects no refinement of the common ground. Consider the following initial offer sequence from Group 4: 1. B(449) Æ 2. S(667) Here the Buyer initiates the negotiation with an offer indicating preferences for a 4-day turnaround (Speed), a maximum of three typographical errors (Editing), and will accept a low quality typewritten document (Appearance). The Seller counters with an offer that differs on all three issues; therefore, we define this as exploration.
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Exploration, as we have suggested, is generally not a primary strategy for reaching an agreement, given the small likelihood that any unique offer will be unequivocally accepted by the other party. Rather, negotiators will build off of issues and exploit the restricted offer space by varying less than three issues. Thus, we define exploitation is a distinctly more coordinated activity that begins when the two parties start to converge to an understanding of what constitutes a possible solution.
Exploitation. Any two sequential offers from two (N)egotiators, N1(i1, j1, k1) and N2(i2, j2, k2) are said to represent an exploitation sequence if i1 = i2 and/or j1 = j2 and/or k1 = k2. Thus, if an adjacent pair of offers (offer/counter-offer) has at least one option in agreement, we assume that there convergence on some common ground that can be exploited. From the same example, consider the next sequence of offers. 2. S(667) Æ 3. B(467) Here the Buyer (offer 3) accepts both the options of Editing and Appearance set by the Seller in the prior offer (offer 2). Conceptually, these negotiators agree on two of the required three issues, thus narrowing the possible solutions remaining to be searched. Exploitation should dominate negotiations, as this reflects attempts at reducing the complexity of the task and achieving an understanding that may lead to convergence to a solution. Examination of the 8 dyads in our illustrative data shows this to be the case. On average, 94.2% of offer sequences were classifiable as exploitation (range = 77.8% to 100%) as opposed to exploration. The dominance of exploitation sequences suggests that negotiators may implicitly reduce the offer space by incorporating issue options from immediately preceding offers. Therefore, to gain insight into the paths of offers and how they lead to solutions, we can examine the details of exploitation. This includes an examination of how the landscape of the
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reduced offer space is determined and how the characteristics of the landscape influences the search for solutions. Structure and Influence We will first take a second look at the landscape for the problem. Recall that we had shown a typical joint-sum plot for this problem in Figure 1; however, in actuality only 165 of the 521 possible solutions are “visible”. This is because the plotted offers reflect the 165 uniquelyvalued offers, but not the 521 uniquely-configured offers. Thus, several of the solutions (as points) in Figure 1 represent multiple offer configurations of the same value (to each negotiator) and are a type of equivalence set we call the solution equivalence set. This is an important, but rarely identified, structural feature of the solution space. Solution Equivalence set. Given an solution of total value Ti to one negotiator and Tj to the other negotiator, there can be a set of solutions in the offer space, called the solution equivalence set, whose members yield the same total value to each negotiator, but differ in their structural configuration. From an omniscient observer’s perspective, solution equivalence produces a “third-dimension” to the joint-sum density plot that indicates the number of equivalent solutions for a particular point (here ranging from 1 to 5). This is shown in Figure 2. Consequently, the offer space is somewhat more complicated than it is typically depicted and further suggests the necessity for mechanisms to systematically reduce the complexity of search. Insert Figure 2 about here When an offer is made, it serves as a signal to the other negotiator as to what may constitute an agreement. As noted, during exploration, sequential offers do not signal acceptance of any issue option of the previous offer, seemingly ignoring the content of the prior offer. Instead, we suggest that exploration offers are most likely determined by individualistic value
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search, as negotiators configure offers in unique ways to satisfy their own value aspirations and at least minimally satisfy the other party. In contrast, exploitative offer configurations are influenced by structural components of prior offers (primarily the other party’s but also their own) and evaluated within that context. The structure of offers constrains the values of the offers and, consequently, the value of the agreement. The influence of structure. Negotiators are certainly aware of the value to themselves of the individual issue options, but this awareness does not necessarily lead to the selection of globally optimal combinations. When a negotiator generates an offer, we assume that the offer reflects the results of a limited search of a set of possible alternatives from which the generated offer appears to best satisfy the aspirations of that negotiator. The negotiator may have selected the best offer of the ones that were considered, but which ones were considered? We turn to the concept of an offer equivalence set to address this question. We assume that negotiators attempt to satisfy their value aspirations during a negotiation, and these value aspirations can be met with a variety of structurally different agreements. An offer equivalence set includes the set of potential agreements that represent the same value to the negotiator making the offer. Recall the sequence 2. S(667) Æ 3. B(467). With two issues in agreement, only the first, Speed, needs to be determined. Based on our assumption that the value of the offer represented by the structure B(467) is considered acceptable to the Buyer (in this case, 1800 points), the Buyer should be indifferent to any alternative offer that, while structured differently, provides the same amount of value (1800 points). An analysis of the offer space shows that the Buyer has a choice of 15 different issue option configurations for this particular value, and we call this group the offer equivalence set. Stated a little more formally: Offer Equivalence set. Given an offer, Ni(ii, ji, ki), the set of possible negotiation solutions in the offer space that yield the same total value to a single negotiator as Ni(ii, ji, ki), is said to
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be the equivalence set for that offer. Thus, an offer equivalence set is not from the perspective of an omniscient observer, but specifically from the perspective of a negotiator examining options within the context of a specific offer. An offer equivalence set provides information about the context within which offers are configured. It recognizes that at any given aspiration level there are multiple packaged offers that can be made, but that these offers are structured differently. The question of the influence of structure may be partially answered by examining how “close” the structure of offers are to prior offers in the context of their offer equivalence sets. We can use the notion of an offer equivalence set to determine the level of interdependence between sequential offers. For example, when comparing the structure of the offers in the Buyer’s equivalence set (B(467), n = 15 equivalent offers) with the prior offer from the Seller (667), the Buyer’s response had an implicit agreement on two issues (Editing = 6, Appearance = 6). Of the 15 possible responses that were equivalent in value, the one selected by the Seller, B(467), was the one that was most similar to the prior offer in the set. It appears that the Buyer’s offer, although probabilistically unlikely (1/15 = 0.06%), was dependent on the structure of the prior offer.1 We quantify this similarity using the metric of structural distance, DS. We differentiate between simple structural distance, the distance between two offers, and average structural distance between an equivalence set and a prior offer. We score simple structural distance between two offers by counting the number of issue option changes (structural adjustments) it would take to transform an offer from one negotiator, N1(I1,I2,I3), into the subsequent offer from the other negotiator, N2(I1,I2,I3) (see Hamming, 1980). 1
Note that it also may be the case that the response to an offer is (also) influenced by the structure of the same negotiator’s previous offer – a negotiator is “being firm.” [Given equivalence sets, “being firm” may reflect either retaining structural similarity (letting the value vary) or value similarity (letting the structure vary).] A similar analysis of the Buyer’s equivalence set shows that none of the possible configurations has two issues that match the Buyer’s previous offer. We may speculate that, in this case, the Buyer was more structurally influenced by the Seller’s current offer than the Buyer’s previous offer.
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DS = ∑ N1 ( I i ) − N 2 ( I i ) i =1
Using this metric we can calculate the structural distance from the prior offer made by the other party (a “between” offer sequence measure), the structural distance from the prior offer made by the same party (a “within” offer sequence measure). The average structural distance between the offer equivalence set and the prior offer is calculated as follows. For each member of the Offer2 equivalence set (i.e., each offer of equal value), we evaluate the structural distance (DS) from the preceding offer (Offer1) and obtain an average structural distance of the equivalence set. This average quantifies the context within which we can evaluate the dependency among actual offers. We do this by comparing the simple structural distance between Offer1 and Offer2 to the average structural distance between Offer1 and Offer2’s equivalence set. The greater the difference between the two structural distances, the greater the influence of the structure of Offer1 on Offer2 (assuming the simple structural difference is smaller than the average structural distance). Simple structural and average structural distances were computed from the illustrative data, averaged within group, and the results are shown in Table 2 along with the average size of the equivalence sets. For example, in the negotiation of Group 1, the average offer differs from the prior offer of the other party by 2.3 options and differs from the prior offer made by the same party by 1.6 options. In responding to an offer, the negotiator has an average set of 12.5 offers of different structure that are of the same value from which to choose. The average difference from the prior offer of the other party is 5.6 options and from their own prior offer is 5.6 options. Thus, Group 1’s offers appear to have been influenced by the structure of prior offers (e.g., between comparison: 2.3 < 5.6), and the same could be said across all the groups (e.g., between comparison: 3.4 < 6.4). This illustrates what we call the structural similarity bias
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that will tend to occur as the number of alternative solutions in an offer equivalence set increases. Structural Similarity Bias. Negotiators will select offers from their offer equivalence sets (i.e., sets of offers whose values are equivalent) that will tend to be structurally similarity to the prior offer. Insert Table 2 about here The structure of influence. We argued that negotiators craft their offers in response to other parties’ offers and this process reduces the search of the offer space. We can identify this reduced region of search by focusing on the region left unconstrained by sequential offers during exploitation. As defined earlier, during exploitation there is tentative agreement on at least one issue, constraining the number of offers potentially under consideration. The set of possible offers that remain within that constraint define the restricted region of the offer space that is the focus of joint attention. We refer to these restricted regions of the offer space as residual sets – the set of possible offers that remain after one or more issue options have been (perhaps tentatively) agreed upon. Residual sets are not explicitly defined by negotiators and negotiators are probably not even directly aware of them; however, they do reflect an alignment of the common ground between the two negotiators. Residual sets are thus an artifact of the negotiators’ interaction with the task and with each other, revealing the set of solution possibilities given a joint agreement on less than all of the issues under negotiation. Residual sets are defined only in the context of exploitation. Residual Set. Given any two sequential offers from two (N)egotiators, N1(i1, j1, k1) and N2(i2, j2, k2), where at least one of the following holds: i1 = i2 or j1 = j2 or k1 = k2, the residual set (for this pair) is defined as the inclusive set of possible offers denoted by the structural range of options of the issue or issues specified, but unresolved. We do not define residual sets in terms of two sequential offers from the same negotiator,
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because residual sets by definition reflect joint articulations between two negotiators within the offer space. Thus a residual set is the set of possible offers that are encompassed by the structural differences between two sequential offers from opposing parties. We also distinguish between search influenced by alternative structures of offers (i.e., choices of options within issues, and between issues) and search influenced by alternative values of offers. Of course, these are not mutually exclusive, but we believe that in complex, multi-issue negotiations, a search by structure (then evaluated by value) is substantially easier than a search by value (possibly accounting for the structural similarity bias noted earlier). In that alternative offers are characterized by explicitly differing issue options, it is cognitively less expensive to represent offers in terms of combinations of issue options than to calculate their underlying combined value each time a new offer is made. Search by value requires values or value ranges to be addressed (e.g., the Buyer wants “2600 points” or “something better than 1900 points”) and the particular configuration(s) of issue options that can satisfy that goal must be determined. As a residual set contains more than one possible solution, it is informative to determine how many solutions comprise the set. That is, it is informative to determine the size of the residual set. The size tells us how large of a space the negotiators are jointly considering. The size of a residual set, R, defined by two offers is given as:
R = ∏ ( N 1 ( I i ) − N 2 ( I i ) + 1) 3
i =1
Returning to our example from Group 4 of the illustrative data, consider the third and fourth offers in the sequence: the Buyer proposes option 4 on Speed (turnaround in four days), option 6 on Editing (a maximum of 5 errors), and option 7 on Appearance (near letter quality), while the Seller prefers option 6 on Speed (turnaround in six days), option 5 on Editing (a maximum of 4 errors), and agrees with the Appearance issue.
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3. B(467) Æ 4. S(657) The residual set is comprised of the three (inclusive) offers that reside within those bounds for Speed (options 4, 5, and 6) and the two issues of Editing (options 5 and 6). Collectively, these describe six possible offers (Speed options 4, 5 and 6 crossed with Edit options 5 and 6). We can plot the values of pairs of offers and their residual sets to get a visual representation of the search space. For example, the three possible offers comprising this residual set are represented by open squares in Figure 3. We elaborate on the plotting of residual sets next. Over the course of the negotiation for Group 4, 12 offer exchanges occurred in exploitation sequences (defining 12 residual sets) before an agreement was reached. Figure 3 is a value plot of the set of the 10 explicit unique offers made (there are fewer unique offers than offers exchanged because duplicate offers were made) (open circles) and the 12 implicit unique solutions contained in the residual sets implied by those offer exchanges (dots). Collectively, the residual sets defined 40 solutions over the course of the negotiation, but there was 70% redundancy of offers across these residual sets. The first two offers in this example are exploratory, 1. B(449) and 2. S(667), but as shown above the third offer, 3. Buyer (467), builds off of the second defining the initial residual set {4-6, 6, 7} (the open squares). The average size of a residual set for this group was 4.5 offers. As the figure suggests, the residual sets in this group describe a small portion of the offer space to which attention was likely paid. Placed in perspective for our illustrative data, the average size of any given residual set (7.4) represents about 1.3% of the offer space (see Table 3). However, considering the average redundancy across residual sets within a given negotiation is rather substantial (37.9%), we can estimate that, through residual sets, two negotiators jointly consider an upper level estimate of only 3.4% of the offer space.
20
Insert Figure 3 and Table 3 about here In our framework, only comprehensive package offers (offers that explicitly or implicitly specify values for all the issues) are considered because only comprehensive offers constitute a potential binding agreement. However, negotiators can, and do, make non-comprehensive (or partial) offers during the negotiation. For example, the parties might discuss an offer of 5 days for speed and then move on to discuss laser printer appearance. We see partial offers as potentially relevant to the search process for, as we noted, partial offers reflect search in, and a partitioning of, the offer space. That search produces knowledge that negotiators use when conducting searches on other issues. Therefore, we assume that negotiators’ search on a current issue incorporates knowledge from recent searches on other issues. Thus, recent offers on other issues can carry forward, as can offers on other issues that have been explicitly agreed upon – multi-issue offers can (although not necessarily) form via accumulation over time. However, only the multi-issue offer, once formed, is considered in the trace of offers that define a residual set. For example, if two negotiators tentatively agree on an issue option, we assume that this would be carried forward and be incorporated into the definition of the residual set. On the other hand, if a proposed issue option is explicitly rejected (without counter discussions), then we would assume that it is not carried forward and remains as an unconstrained issue within the residual set. If the offer is ignored, rather than accepted or rejected, it would drop out of the negotiation rather quickly due to the negotiators’ shift in attention and limited information processing capacity. In conclusion, sequential offers impact the segment of the offer space that is likely searched for solutions. We have shown how offers can suggest exploitation strategies and define residual sets that constrain the offer space and solutions achievable in the negotiation. We have suggested that there can be a distinctive structural bias in how adjacent offers are configured
21
during exploitation sequences. Armed with these concepts, we can now turn to the investigation and interpretation of the dynamics of negotiation – tracing and interpreting the proximity and progression of offers and residual sets. We argue that the residual sets of exploitation are best considered as dynamic rather than static, that they will influence future offers made, and that they will improve our ability to predict where, in the offer space, that final agreements might occur. Residual Sets, Offers, and Proximity Thus far we have depicted a rather static representation of offers, focusing on ways to characterize the search space, offers, size, and proximity. However, as multiple offers generate multiple residual sets over time, we can take a more dynamic approach and track the shifting of residual sets and the proximity of offers in offer space relative to residual sets. Referring back to Figure 3, the next offer subsequent to those defining the initial residual set, S(667) and B(467), could have come from within that set of three potential offers and, most likely, the negotiation would rapidly converge to an agreement. However, negotiations rarely are so deterministic. Negotiators have the capacity to reconstitute offers by altering previously suggested issue options, reflecting changed aspirations, newly discovered alternatives, or strategic moves to influence play. Thus, additional offers can be made that are outside the initial residual set, defining a new residual set. The change in residual set reflects a potential change in the focus of attention within the context of the offer space. To illustrate this, we can consider offers from a different group, Group 2. The sequence is as follows: 1. B(117) Æ 2. S(297) Æ 3. B(249) Æ 4. B(249) Æ 5. S(169) The simple path of this offer sequence is shown in Figure 4. There does seem to be an apparent reciprocation of concessions and Pareto improvement between these negotiators. This becomes more evident and explainable when the underlying residual sets are included in the
22
graph (Figure 5). The first two offers define the initial residual set (comprising 18 offers) which is depicted in Figure 5 as open squares. The second residual set is derived from offers 2 and 3, and is depicted in Figure 5 as open circles. This residual set is also comprised of 18 solutions; however, it is highly redundant with the prior residual set (10 offers or 55.5% overlap). As additional offers are made, the specific combinations of options extend the residual set into new areas of the offer space. The final residual set for this dyad is defined by offers 3 and 4, and is shown as open triangles in Figure 5. It is smaller than the prior residual sets (6 potential solutions), has 50% overlap with the previous residual set, and contains the final agreement. We can demonstrate the influence of residual sets on final agreements by examining their structural distances (Ds). Not surprisingly, the final agreement is very proximal to the penultimate residual set. The final agreement is one structural adjustment from the nearest member of the residual set established immediately prior to the agreed upon offer. Interestingly, the final agreement is also quite proximal to the initial residual set, only differing by two structural adjustments. In fact, an examination of all the illustrative data demonstrates the structural distance agreements have to initial and penultimate residual sets within the negotiation (see Table 4). As can be seen, the average structural distances between agreement and residual sets are, for the most part, low. What this illustrates is not only the influence of structure on negotiation offers, but also the structural proximity maintained between offers and residual sets. We suggest that this reflects a structural proximity heuristic that facilitates the coordination of the negotiation search activity. Structural proximity heuristic. Residual sets (as common ground components) tend to evolve by relatively small changes in proximity (via structural changes in offers) that serves to maintain the continuity of joint attention and to reduce disturbance to the current state of the common ground.
23
Insert Figures 4, 5 and Table 4 about here Residual Sets and Offer Dynamics In that residual sets influence subsequent offers and final agreements, it is important to consider the patterns of search as represented by changes in residual sets. We have already distinguished between broad search via exploration and more local search via exploitation. We further identified residual sets as traces of exploitation. In this section we consider the ways in which residual sets shift over time. The patterns of search. We describe four primary patterns of search in negotiation dynamics revealed by changes in offers and residual sets, where the first three reflect exploitation and the fourth reflects exploration. Taken together, they vary from highly constrained and local search to completely unconstrained, broad search. Any given negotiation may evidence any or all of these patterns. First, search can exhibit a pattern of low (or zero) growth and include high regional activity (i.e., offers that are generated from the residual set) within the offer space. This is the case when offers made between negotiators are generated from current or past residual sets, or current or past residual sets are modified via reconstituted offers. Thus, existing (though not necessarily articulated) alternatives are revisited. For example, consider the negotiation of Group 4 (prior Figure 3). Within that negotiation, there is an exploitation sequence when the Buyer makes an offer, B(467) and the Seller responds, S(459), defining a residual set {4, 5-6, 7-9} that is comprised of six potential solutions. Over the subsequent exchanges, eleven offers are made and five of those, including the eventual agreement, originate from that set. Second, search can exhibit a systematic extension into neighboring regions of the offer space not covered by existing residual set, but is proximate to the residual set. In this case, the emergence of offers and residual sets are tightly intertwined such that offers are influenced by the
24
active residual set. Thus, the progression of offers is strongly influenced by the most recent offers and active residual sets. This results in a systematic and constrained pattern of extension into the offer space within a relatively narrow region. Consider the example of Group 2 (prior Figure 5). The first two offers defined the initial residual set and the third offer came from outside (but near) that set and redefined the residual set closer to the Pareto frontier. The final offer is resolved via a within-residual set search of the third residual set. Third, there can be a reorientation of the residual set. This involves what can be described as pivot that reorients a residual set according to the range of joint value to the negotiators. There can be one or more solutions common to the reoriented residual set (few rather than more), but the set itself undergoes a substantial change in average value to negotiators. Consider the negotiation of a new group, Group 7, shown in Figure 6. The initial residual set is depicted as open squares as defined by the two offers: B(583), S(588). For this residual set, the average value to the Buyer is 850 (with a low standard deviation of alternatives, as can be induced by the low range the set covers on the x-axis), while the average value to the Seller is 2950 (with a higher standard deviation of alternatives). The Buyer then poses a counter offer, B(388), that reorients the residual set (as shown by open circles), by agreeing on the Editing issue (option 8), but altering the proposed options on the other two issues. The effect of this reorientation is to radically reduce the number of alternatives in the joint search for both negotiators. If the resulting values are acceptable to both parties, this can be effective in achieving a solution. Finally, search can involve a discontinuous shift to a new residual set via exploration. This is more likely to occur early in negotiation, if at all (see prior Table 2), as exploration tends to quickly shift to exploitation. However, occasionally a new offer is made that has no issue options in common with the active or prior residual sets reflecting Raiffa’s dance of packages. Consider the same example, Group 7, shown in Figure 6. The Buyer shifts from an exploitative process to
25
an explorative process, via an offer that is worth substantially more for the Buyer, but less (in fact below the reservation value) for the Seller, B(332). The Seller responds with an explorative offer, S(268) and, from that offer, the next sequences define the final residual set that contains the agreement that is determined by local search. In this case, the initial residual sets were not acceptable as defining regions in the offer space that could support joint search. However, the discontinuities afforded by exploration did result in a residual set that accommodated the joint interest of the parties. Insert Figure 6 about here Additional Considerations There is an interesting, albeit speculative, connection between recent theories of dialogue/language comprehension and the dynamics of offer exchanges. For effective communication, two conversational partners need to align their situation models (internal representations of the situation) through alignment at other linguistic levels (i.e., semantic, lexical, syntactic, phonological, phonetic) on a continual basis. Pickering and Garrod (2004) propose that much of this alignment is automatic, explaining why conversation is so easy (Garrod & Pickering, 2004). This explains, for example, why speakers efficiently reuse representational structures that they have just interpreted as listeners, leading to sequential effects (in negotiation, such as those found in Weingart et al., 1999), and as the dialogue continues, the production and comprehension become interdependent as stress for alignment pressures behavior toward efficient and dynamic routinization. Could we view the problem representations generated by the negotiators as a type of situational model as defined in comprehension? Would an agreement at that level constitute a form of alignment between the situation models? If so, there should be more exploitation (as movement toward alignment) than exploration, and the illustrative data suggests. In thinking about the how this model advances the field of negotiation, we need to
26
consider how it relates to prior research on negotiation processes. Earlier we mentioned research on distributive bargaining offer patterns and systematic concession making and considered how our approach extended that work. Whereas previous theories of offers took a more individualized approach, our model considers the co-construction of knowledge and search via offers. This approach implies different types of predictions for the effects of offers on negotiated agreements. Rather than focusing on the individual, our model focuses on the social interaction. Rather than focus on concession rate or systematic search, our model considers proximity of offers, exploration versus exploitation, and proximity or size of residual sets. We can also consider this model as it relates to two negotiation phenomena – reference points and tactical behavior. Reference points are relevant comparators that can serve as cognitive anchors for subsequent offers and outcomes (Blount, Thomas-Hunt, & Neale, 1996; Galinksy, Mussweiler, & Medvec, 2002). Both internal (e.g., reservation prices, aspirations, or opening offers) and external (e.g., market value) reference points have been examined in the context of two-party price negotiations (Galinsky et al., 2002; Kristensen & Garling, 1997, 2000; Van Poucke & Buelens, 2002) and the relative weighting of these reference points depends on the context of the negotiation (Blount et al., 1996). Whereas this literature considers the impact of fixed reference points on initial offers and outcomes, as well as on satisfaction with outcomes, our model takes a more dynamic approach by framing all offers as potential cognitive anchors from which negotiators will formulate their responses. Thus, our model picks up from where the reference point literature leaves off by considering how negotiators update their framing of the negotiation in response to subsequent offers. Our model also complements extant research on negotiation processes. Research on negotiation process using tasks with integrative potential has largely focused on the use of integrative and distributive tactical behavior – e.g., information exchange, offers, argumentation,
27
threats – in terms of their frequency, sequences, and phases (Weingart & Olekalns, 2004). Aside from the prior work on heuristic trial and error/systematic concession making, no one has considered the role of the content of the offers in the negotiation process more generally.2 Instead, negotiation process research differentiates between single versus multi-issue offers and considers how often and when they, and other tactical behavior, occur. Single-issue offers are typically treated as distributive tactics designed to claim value whereas multi-issue offers are integrative tactics that can be used to discover tradeoffs and create value. Results of that literature reflect the ways negotiators tactically respond to one another during the course of the negotiation (e.g., are offers reciprocated?) and whether negotiations pass through predictable phases (e.g., when do offers occur). Our model complements this work by considering the content of offers and how negotiators alter the content of offers over time in their search for mutually acceptable agreements. Perhaps one of the most important extensions to the model will be the understanding of how offer content, in conjunction with other tactical behavior, influences quality of agreement and the discovery of Pareto optimal solutions. Offers are not the only mechanism for search and discovery; information exchange, a foundational tactical behavior, also serves a search function. Interesting questions arise when considering the two approaches simultaneously. How do information exchange and offer content play off one another? Do negotiators use information exchange to “fill in the blanks” between offers – that is, will simultaneously capturing information exchange help us predict the ways in which offers are altered and when negotiators might engage in exploration as opposed to exploitation. If a negotiating group relies on one mechanism (offers or direct information exchange) over the other, will that influence their ability
2
An exception can be found in the communication literature that qualitatively tracks the development of issues (rather than offers) over time (Putnam & Holmer, 1992). This approach differs markedly from the current theory which takes a more quantitative approach.
28
to discover an optimal agreement? Are there situations where exploration with offers is more effective than information exchange-as-search (or vice-versa)? In that search is goal directed behavior, our model should also be extended to consider the role of social motives (e.g., whether negotiators are more cooperative or individualistic in their orientation toward a negotiation). Social motives are typically defined in terms of goal maximization – individualists work toward a goal of maximizing their own outcomes whereas cooperatives work toward maximizing their own and others’ outcomes simultaneously. We might expect individualists to be more exhaustive than cooperatives in their equivalence set search, being reluctant to sacrifice their own value, but less concerned with their opponent’s outcome. They might also be less likely than cooperatives to incorporate information from their opponent’s offers into their own, such that their offer is more similar in structure to their own prior offer than to the other party’s prior offer. Our model focuses on a two-party situation; extensions into multi-party settings present an interesting challenge. The current approach maps residual sets and offer patterns in terms of value to each party. As we add more parties into the mix, the mapping would be in n-dimensional space. While visualization may be difficult, calculation of exploration, exploitation, equivalence sets, structural distance, and residual sets would largely be the same because those measure tap into structural change (offer content) rather than offer value change to each of the parties. Similar extensions can be made by increasing the number of issues. In either case, would exploration sequences be extended? Would exploitation and common ground be more resistant to change? Is the 2-issue exploitation definition invariant and a consequence of bounded rationality? Another extension of the model is needed to consider the treatment of issues that are continuous in nature, like price, when capturing offer movement and residual sets. Our formulae for defining exploration, exploitation, equivalence sets, structural distance, and residual sets all
29
presumed issues whose options were discrete, but ordered in terms of utility to oneself. Additional work is needed to expand the conceptualization to include negotiations that include issues whose options are continuous. In conclusion, Raiffa (2002) notes that in the real world, negotiators often exchange packages and advises that the worst thing a negotiator can do is to “go into the details of AAA’s package and propose amendments to it.” (p. 274). We agree that this approach will be counterproductive for a negotiator when those details anchor the negotiator in terms of offer value. However, negotiators must, and we believe do, pay attention to the information embedded in those offers to help them find agreements that are mutually acceptable, if not Pareto optimal. This is the role of offers as search. We believe our model moves us toward understanding how negotiators use offers to search via exploration and exploitation of the potential set of agreements. A more complete understanding of the progression of offers during a negotiation can move us closer toward mapping the routes through which agreements are reached.
30
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Pickering, M. & Garrod, S. (2004). Toward a mechanistic psychology of dialogue. Behavioral and Brain Sciences, 27, 169-226. Prietula, M. J., & Weingart, L. R. (1994). Negotiation as problem solving. In J. Meindl, J. Porac, & C. Stubbart (Eds.), Advances in managerial cognition and organizational information processing. Greenwich, CT: JAI Press. Pruitt, D. (1981). Negotiation behavior. New York, NY: Academic Press. Putnam, L. L. & Holmer, M. (1992). Framing, reframing, and issue development. In L. L. Putnam & M. E. Roloff (Eds.), Communication and negotiation. Newbury Park, CA: Sage. Putnam, L. & Jones, T. (1982). Reciprocity in negotiations: An analysis of bargaining interaction. Communication Monographs, 49, 171-191. Putnam, L.L., & Wilson, S.R. (1989). Argumentation and bargaining strategies as discriminators of integrative outcomes. In M.A.Rahim (Ed.), Managing conflict: An interdisciplinary approach (pp 121-141), New York: Praeger. Raiffa, H. (1982). The art and science of negotiation. Cambridge, MA: Harvard University Press. Raiffa, H. (2002). Negotiation analysis: The science and art of collaborative decision making. Cambridge, MA: Belknap. Sacks, H., Schegloff, E. & Jefferson, G. (1974). A simplest systematics for the organization of turn-taking in conversation. Language, 50 (4, Part 1), 696-735. Schegloff, E. & Sacks, H. (1973). Opening up closings. Semiotica, 8, 289-327. Schelling, T. (1960). The strategy of conflict. Cambridge, MA: Harvard University Press. Siegel, S., & Fouraker, L. E. (1960). Bargaining and group decision making: Experiments in bilateral monopoly. New York: McGraw-Hill. Simon, H. (1956). Rational choice and the structure of the environment. Psychological Review, 63, 129-138. Simon, H. (1978). Information-processing theory of human problem solving. In W. Estes (Ed.), Handbook of learning and cognitive processes: Volume 5. Hillsdale, NJ: Erlbaum. Simon, H. (1982). Models of bounded rationality, Vol. 1-2. Cambridge, MA: MIT Press. Simon, H. (1990). Invariants of human behavior. Annual Review of Psychology, 41, 1-19. Simon, H. & Kaplan, C. (1989). Foundations of cognitive science. In M. Posner (Ed.), Foundations of cognitive science (pp. 1-47). Cambridge, MA: MIT Press. Smith, D. L., Pruitt, D.G., & Carnevale, P. J. (1982). Matching and mismatching: The
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Table 1 Preference point value table (from Weingart et al., 1990) SPEED Optio
Done
n 1
Tomorr
Buye
Selle
r
r
3000
-200
EDITING
APPEARANCE
Errors
Printer
Buyer
Seller
Buye Selle r
r
None
1800
-600
Typeset
600
-1000
ow 2
2 days
2500
-100
Max 1
1500
-300
Laser
500
-500
3
3 days
2000
0
Max 2
1200
0
400
0
4
4 days
1500
100
Max 3
900
300
300
500
5
5 days
1000
200
Max 4
600
600
Letter Quality IBM Selectric Spinwriter
200
1000
6
6 days
500
300
Max 5
300
900
Ink Jet
100
1500
7
7 days
0
400
Max 6
0
1200
0
2000
8
8 days
-500
500
Max 7
-300
1500
Near Letter Quality Dot Matrix
-100
2500
9
9 days
-1000
600
Max 8
-600
1800
SmithCorona
-200
3000
35
Table 2 Mean distances (DS) between offers in exploitation sequences
Simple Structural Distances (Between Offers)*
Average Structural Distances (Offers to Equivalence Sets)
Between
Within
Between
Within
1
2.3
1.6
5.6
5.6
12.5
2
6.0
6.0
7.4
6.5
8.0
3
3.0
1.9
6.0
7.5
21.0
4
2.5
2.3
5.6
5.5
11.1
5
2.0
6.0
5.7
7.3
10.0
6
4.5
4.1
6.9
5.7
9.3
7
5.4
4.9
8.1
6.5
9.0
8
2.0
1.8
6.3
5.7
8.3
Ave
3.4
4.2
6.4
6.2
11.1
Group
Average size of Equivalence Set **
*
Number of issue options that differ between adjacent offers within exploitation search sequences. ** Average number of unique issue option combinations in the equivalence set.
36
Table 3 Basic properties of residual sets for the illustrative data Group 1 2 3 4 5 6 7 8 Average
Average Size 3.2 13.0 4.0 4.5 4.6 14.8 12.0 3.3 7.4
Average Size as % of Offer Space*
Percent Redundancy**
0.6 2.4 0.7 0.8 0.8 2.8 2.3 0.6 1.3
68.0 30.9 0.0 67.5 15.3 25.0 57.1 40.0 37.9
Percent of Offer Space Searched*** 1.5 5.5 1.5 2.4 2.1 4.6 7.4 2.4 3.4
*
This is the percent of offer space covered by the average size of the residual set, based on the adjusted 521 possible solutions within the zone of agreement for this problem. ** Percent redundancy is over all residual sets of that negotiation. *** Percent of offer space covered over the entire negotiation by number of unique offers generated over all residual sets, based on the adjusted 521 possible solutions within the zone of agreement for this problem.
37
Table 4 Distance DS of Agreement from Residual Sets Group 1 2 3 4 5 6 7 8 Average
DS from initial residual set [Average DS from set] 0 2 1 3 2 2 7 0 2.1
[1.4] [4.8] [2.5] [4.0] [4.0] [7.8] [9.5] [1.0] [4.3]
DS from penultimate residual set [Average DS from set] 0 [1.4]* 1 [3.5] 0 [1.4] 0 [1.0] 2 [2.0] 4 [7.0] 2 [2.5] 0 [1.0] 1.1 [2.4]
Numbers represent minimal distance of the agreement to that set, numbers in parentheses represent the average distance from each member of that set to the final agreement.
38
Figure 1 Joint sum plot of discrete 3-issue offer space
6000 1
5000
a b
Total Value to Seller
4000
6 8
3000
c
3
7
2000 5
1000 0
d
4
Seller's reservation value
Buyer's reservation value
-1000 -2000 -2000
2
-1000
0
1000
2000
3000
4000
5000
6000
Total Value to Buyer
39
Figure 2 Joint sum density plot of discrete 3-issue offer space (all solutions)
6000
Total Value to Seller
5000
1
4000
2 3
3000
4 5
2000
1000
0 0
1000
2000
3000
4000
5000
6000
Total Value to Buyer
40
Figure 3 Offers (open circles) and Residual Sets (open squares, closed circles) for Group 4
6000
Initial Residual Set, {4-6,6,7}
Pareto frontier
5000 Final Offer (and agreement) by Seller
Value to Seller
4000 Initial (Explorative) Offer by Buyer, 1:B(449)
3000
2000
Subsequent (Explorative) Offer by Seller, 2:S(667)
Exploitation Offer by Buyer, 3:B(467)
1000
0 0
1000
2000
3000
4000
5000
6000
Value to Buyer
41
Figure 4 Offer trace for Group 2 negotiation 6000 Pareto frontier
5000 Subsequent Offer by Seller, S(297)
Value to Seller
4000
Final Offer (and agreement) by Seller
3000
Initial Offer by Buyer, B(117)
2000
1000
0 0
1000
2000
3000
4000
5000
6000
Value to Buyer
42
Figure 5 Offer trace showing underlying residual sets for Group 2 negotiation
6000 Pareto frontier
5000 Subsequent Offer by Seller, S(297)
Value to Seller
4000
Final Offer (and agreement) by Seller, S(169)
3000
2000
Initial Offer by Buyer, B(117) Initial Residual set: {1-2, 1-9, 7}
1000
0 0
1000
2000
3000
4000
5000
6000
Value to Buyer
43
Figure 6 Partial offer trace and residual sets for Group 7
6000
Subsequent Offer by Seller, 2:S(588)
Counter Offer by Buyer, 3:B(388) Agreement
Value to Seller
5000 Final Residual Set {1-2, 2-6,9}
4000
3000 Explorative Offer by Seller, 5:S(268)
2000
Explorative Offer by Buyer, 4:B(332)
1000 Initial Offer by Buyer, 1:B(583)
0 0
1000
2000
3000
4000
5000
6000
Value to Buyer
44
