Collaborative and competitive scenarios in spatio-temporal negotiation with agents of bounded rationality

Transcription

1 Collorative and copetitive scenarios in spatio-teporal negotiation with agents of bounded rationality STRCT Yi Luo School of Electrical Engineering and Coputer Science University of Central Florida Orlando, Florida In spatio-teporal negotiation evaluating an offer for feasibility or utility often requires coputationally expensive path planning, thus practical negotiation strategies can evaluate only a sall subset of the possible offers during offer foration. s equilibriu strategies are not practically possible, we are interested in strategies with bounded rationality, which achieve good perforance in a wide range of practical negotiation scenarios. Naturally, the perforance of a strategy is dependent on the strategy of the opponent and the characteristics of the scenario. The utility of a deal alone for a particular agent is not a good easure of the quality of the negotiation strategy; we also need to consider whether better deals were overlooked or whether the agent had outsarted the opponent, by convincing it to accept a lesser deal. We also have an intuition of collorative scenarios (where the agents interests are closely aligned) versus copetitive scenarios (where the gain of the utility for one agent is paid off with a loss of utility for the other agent). Using the Children in the Rectangular Forest (CRF) gae as a canonical odel of spatio-teporal negotiation, we develop a series of quantitative etrics for the characterization of deals in relation to the possibilities of the scenario and the interest of the other agent. We also develop a etric for the collorativeness of the scenario. Through an experiental study involving three negotiation strategies of increasing coplexity, we show that the proposed etrics atch our intuition out the scenarios and can serve as a tool in analyzing and developing strategies as well as in designing negotiation echaniss prooting cooperative behavior.. INTRODUCTION Colloration between ebodied agents often requires the teporal and spatial collocation of the agents. gents need to coordinate their oveents, agree on eeting points, tie, coon path and speed, as well as locations where they split and start oving on independent trajectories. Such probles appear as sub-probles in any practical applications such as transportation and disaster res- CN-08 May 2 6, 2008, Estoril, Portugal. Ladislau ölöni School of Electrical Engineering and Coputer Science University of Central Florida Orlando, Florida lboloni@eecs.ucf.edu cue. In previous work [6] we have identified five differentiating features of the spatio-teporal negotiation probles: () heterogeneous issues (which include spatial locations and tie points), (2) non-onotonic valuation of issues, (3) an evolving environent, (4) offers need to be verified for feasibility (usually by both parties) and (5) there is an interaction between the negotiation tie and the physical tie. s this category of probles can not be conveniently represented through the split the pie odel of negotiation, we proposed a odel, Children in the Rectangular Forest (CRF), which captures the ain features of these probles and can be used as an canonical proble for the study of spatio-teporal negotiation strategies. The evaluation (and creation) of offers in spatio-teporal negotiation probles is coputationally expensive, as it often involves path planning. s the negotiation happens in real physical tie, agents can not afford to evaluate a large nuber of offers for feasibility and utility. s equilibriu strategies are not practically possible, we are interested in developing strategies with bounded rationality, which achieve good perforance in a wide range of practical negotiation scenarios. Naturally, the perforance of a strategy is dependent on the strategy of the opponent and the characteristics of the scenario. The utility of a deal alone for a particular agent is not a good easure of the quality of the negotiation strategy; we also need to consider whether better deals were overlooked or whether the agent had outsarted the opponent, by convincing it to accept a lesser deal. We also have an intuition of collorative scenarios (where the agents interests are closely aligned) versus copetitive scenarios (where the gain of the utility for one agent is paid of with a loss of utility for the other agent). To show the intuition behind collorativeness in a negotiation scenario, let us first consider the sipler scenario of the split the pie gae. Here two agents are negotiating over the partitioning of a pie. s the parts allocated to one agent are lost for the other agent, the single pie gae is fully copetitive. This is true even for the cases when we are partitioning over ultiple pies. Note that although all zero-su gaes are fully copetitive, not all fully copetitive gaes are zero su. For instance, a split ultiple Soe gae theory texts, such as [7] equate fully copetitive with zero su, by aking the assuption that the utility function is just a convenient expression of the preference ordering. In our case, however, the utility has the diensionality of tie, and it can not be arbitrarily scaled. There is a difference between a scenario where a second utility decrease fro one agent gives second utility gain to the other agent, and the scenario where second utility decrease gives 00 seconds utility gain for the opponent. In our language the first scenario is fully copetitive and zero su, while the second

2 pies gae where the agents value the different pies with different, positive values, is still fully copetitive, but not zero su. On the other hand, a fully cooperative gae is one where there is a possible agreeent which is individually optial for both agents. n exaple of a fully collorative gae is a split the ultiple pie gae with two pies P and P 2, where agent values P positively and P 2 negatively, while agent values the the other way around. In this case the agents can easily agree on a partitioning where agent gets the pie P, while agent gets the pie P 2. For spatio-teporal negotiations siilar considerations apply; however both the definition of collorativeness and the calculation of the optial deal is ore difficult. 2. RELTED WORK While autoated negotiation [5] generated a lot of interest in recent years, negotiation out spatio-teporal issues in ebodied agents has received relatively little attention. Nevertheless, any research results in ulti-issue negotiation or collorative robotics have relevance to our work. Sandhol and Vulkan [9] analyze the proble of negotiating with internal deadlines where the deadlines are private inforation of the agents. The negotation proble is a split a single pie, zerosu negotation. They find that for rational agents, the sequential equilibriu is a strategy which requires agents to wait until their deadline, and at that oent, the agent with the earliest deadline concedes the whole cake. Fatia, Wooldridge and Jennings [2] extensively study the proble of ulti-issue negotation under deadlines. The proble considered is the split ultiple pie proble where the pie is assued to shrink after every negotiation round, under both coplete inforation and incoplete inforation assuptions. The authors copare three negotiation procedures: the package deal procedure where all the issues are discussed together, the siultaneous procedure where issues are discussed independently but siultaneously, and the sequential procedure where issues are discussed one after another. The authors show that the package deal is the optial procedure for both agents. Golfarelli et al. [3] considers the case of robotic agents which are assigned a set of tasks which are attached to physical locations. The tasks carry precedence constraints (execute one specific task earlier than the other) and object constraints (fetch the object in order to execute the task). gents need to deterine, on a network of places and routes, a sequence of places to be visited in order to carry out a set of tasks. Through swapping tasks based on announceent-bidaward echanis, the agents can decrease their tasks execution costs in the ap. n extended version of this work [4], allows the agents to exchange clusters of tasks to avoid being stuck in local inia. To cluster siilar tasks, the authors calculate spatial distance and teporal distance of tasks, and apply thresholds to differentiate between near and far tasks. Saha and Sen [8] discuss the proble of negotiating efficient outcoes in a ulti-issue negotiation where soe of the paraeters of the agent are not coon knowledge. The distributive and integrative scenarios proposed by the are the equivalents of the copetitive and collorative scenarios we define for the spatio-teporal negotiation proble. Crawford and Veloso [] applied the experts algorith to solve the ulti-agent scheduling proble. In this algorith the agent is helped by a nuber of experts, but it needs to decide which experts advice it should follow. The learning agent can dynaically change its strategy according to its opponents behavior. The perscenario is fully copetitive but not zero su. S S L Rectangular forest L s Figure : The Children in the Rectangular Forest proble. The trajectories associated with the conflict deal are shown with an interrupted line, while the trajectories corresponding to a possible agreeent are shown with a continuous line. forance of each algorith is easured in ters of total utility achieved over each of the trials. 3. METRICS FOR THE CHRCTERIZ- TION OF THE CRF PROLEM 3. The CRF proble The Children in the Rectangular Forest (CRF) proble considers a world in which two children and are oving fro source points S and S on one end of a rectangular forest to their respective destination points D and D on the other side of the forest (see Figure ). Traveling alone, the children can not enter the forest and they need to go around it. However, if they join together in a coalition, the children can traverse the forest, thus possibly shorten their trip. The children travel at different velocities v and v ; when traveling together, they ove with the velocity of the slower child. The selfish objective of each child is to reach his or her respective destination as early as possible. The two children are using negotiation to agree on the paraeters of the deal. This is a 4-issue negotiation, with two issues being spatial locations (eeting point L and split point L s ), and two issues tie points (eeting tie t and split tie t s ). If the negotiation is unsuccessful, the agents are taking the conflict deal; that is, they go around the forest. Several properties of rational deals in the CRF proble, are suarized below: The optial trajectories of the conflict deal and the collorative deal are sequences of straight segents. However, the trajectory of an agent before aking a deal ight be curvilinear, reflecting, for instance, the evolving estiation of the agent out the likelihood of the deal. The eeting point and the split point of any Pareto optial deal is at the edge of the forest in non-degenerate cases. In degenerate cases, there is a deal with the sae tie to destination for both agents where the points are the edge of the forest. The four negotiation issues are not copletely independent. For instance, if we know the axiu velocity of both agents, the split tie t s can be calculated fro L, L s, and t. Siilarly, if all inforation is known out the current location and speed of the agents, the Pareto optial value of t can be calculated, knowing L. We can ake a ore general observation concerning negotiation probles which involve eeting and splitting locations and ties of convoys. If (a) the utility of all participants depends only on the tie to reach the destination and (b) the negotiation is full D D

3 knowledge, then the negotiation can be reduced to the spatial coponents, as the teporal coponents can always be calculated by considering the tie it takes for the last participant to reach the given point. However, if the negotiation is not full-knowledge, then the agent ight find it ipossible to for an offer where the teporal values are guaranteed to be feasible for both agents. In the siplest case, an agent ight not know the current location of the negotiation partner, and thus, naturally, it cannot calculate the tie at which it can reach a certain point. We call a fully specified offer a quadruple O = {L, t, L s, t s } which specifies both the spatial and teporal coponents of an offer. spatially specified offer specifies only the spatial coponents of the offer: O = {L,?, L s,?}. n agent can coplete a spatially specified offer by calculating the tiepoints t () and t () s which are the earliest feasible ones for the agent. The resulting offer is the best tie copletion for of the spatially specified offer O: T (O) = T ({L,?, L s,?}) = {L, t (), L s, t () s } () 3.2 Metrics Each of us has an intuitive feel for negotiation scenarios which are easy because the negotiation partners have a strong incentive to for a deal and for scenarios which are hard because a rational agreeent is difficult to find (or it ight not exist). lso, we have an intuition of certain negotiation scenarios where one of the participants has ore to gain fro an agreeent. Our objective is to develop etrics which atch well with these intuitions, while stract away the other paraeters of the gae (such as the location and destination of the agents). CRF scenario is defined by the ap of the CRF gae (the size of the forest), the source points of the two agents S and S, the destination points of the two agents D and D, and the axiu velocities of the agents v and v. The path of the agents are series of segents together with the velocities of the vehicle on the different segents. We call tie to destination (O) of agent for a particular offer O = {L, t, L s, t s } the tie it takes for the agent to reach its destination if it accepts the offer and follows the trajectory. The lower the tie to destination, the ore desirle is the offer for the agent. The tie to destination is coposed of three coponents: the tie it takes for both agents to reach the eeting location, the tie for traveling together in the forest, and tie fro the split location to the agent s destination. ( S, L (O) = ax, S, L v v ) + L, L s in (v, v ) + L s, D v (2) The tie to destination of the conflict deal conflict is the tie for the agent to reach its destination if it does not ake any deal. This value of the baseline of the negotiation; a rational agent will not accept an offer which will yield a tie to destination later than the conflict deal. DEFINITION. The utility of an offer O for agent, denoted with P (O), is the tie the agent saves accepting the offer copared to the conflict deal. U () (O) = conflict C() (O) (3) DEFINITION 2. We define the solute best tie to destination for agent the tie it would take it to reach the destination assuing an ideally perforant and ideally collorative negotiation partner. For the CRF proble, the trajectory associated to the solute best tie to destination is a straight line fro the source to destination traversed by the agent with its axiu velocity. = S, D v (4) This assues that there is an ideal negotiation partner, who is (a) willing to accept any geoetric location for eeting and splitting points proposed by the agent, (b) its velocity is greater than or equal of the current agent and (c) its current position is such that it can reach the eeting point at a tie earlier or equal with the tie it takes agent to reach it. Note that for a practical scenario, the solute best tie to destination ay not be feasible, even for an ideally cooperative negotiation partner. DEFINITION 3. We define the ility constrained best tie to destination,{} acb, of an agent negotiating with an agent, the tie can reach the destination assuing an ideally collorative agent. The ility constrained best tie takes into account the physical liits of the negotiation partner and the scenario. The eeting and split point of the offer associated with the ility constrained best tie ight not be the one situated on the intersection of the straight line to destination with the forest. The offer(s) associated with,{} acb ight not be rational for agent. DEFINITION 4. The rationality constrained best tie to destination U (),{} rcb for agent negotiating with agent is the tie to destination of agent which can be obtained assuing that agent will accept any offer, as long as it is rational for. s,{} acb we have: and U (),{} rcb introduce successive restrictions over, conflict C(){} rcb {} acb (5) Each of these tie to destination values define a set of one or ore concrete offers which actually achieve the. Thus we define a rationality constrained best offer of to be an offer O (),{} rcb such that C ( ) () O (),{} rcb = C (){} rcb (6) The etrics introduced until now characterize the scenario fro the point of view of one of the agents. Let us now develop a etric which quantifies the desirility of a certain offer O fro the point of view of the social good. DEFINITION 5. We call the social cost of the offer O any function C social (O) = C social ( (O), C () (O) ) which is onotonically increasing both with and with C () : C (), 2 C social (, C() ) C social ( 2, C() ), C () C () 2 C social (, C () ) C social(, C () 2 ) (7) We call denote with O social the set of offers which iniize the social cost: O social = argin (C social (O)) (8) O Within the constraints of this definition, there are any possible functions which can serve as the social cost function. The choice of a specific function depends on the policy of the supervisor. One siple choice is to define the social cost as the su of the individual costs. C social (O) = C + (O) = (O) + C () (O) (9)

4 Note however, that a social best offer ight not be rational for both agents. We can define a rationality constrained social cost, which assues a cost of plus infinity for the offers which are not rational for one of the agents: C rcsoc (O) = + C social (O) ( ) ( ) (O) > conflict C () (O) > C () conflict otherwise (0) ased on this definition, we can define the set of rationality constrained social best offers O rcsoc as: O rcsoc = argin (C rcsoc (O)) () O offer(o ) offer(o 2 ) offer(o 3 ) offer(o n ) accept(o n ) offer(o ) eval(o ) offer(o 2 ) eval(o 2 ) offer(o 3 ) eval(o n- ) offer(o n ) eval(o n ) accept(eval(o n) )) DEFINITION 6. We define as the collorativeness of the scenario fro the point of view of agent, negotiating with agent, the ratio of the utility of the rationality constrained social best deal to the axiu rationally obtainle utility: Ξ (),{} = C() conflict C(),{} rcsoc conflict C(),{} rcb (2) Let us verify that this definition satisfies our intuition out the collorativeness of a scenario. In a fully copetitive scenario, there is no rational deal possible, thus the cost of the rational deal will be the conflict deal, thus we have Ξ (),{} = 0. On the other hand, we say that a scenario is fully cooperative fro the point of view of agent if the rationality constrained social best offer is also the rationality constrained best offer for agent. In this case Ξ (),{} =. DEFINITION 7. We define the relative utility of an offer for agent as the ratio of the utility of the offer to the axiu rationally obtainle utility: U (),{} rel (O) = C() conflict C() (O) conflict C() rcb, {} (3) The relative utility of the agent can range fro 0 to. Notice that the relative utility of a deal does not tell us whether the agent has negotiated better than the negotiation partner. There are situations when both agents can reach the axiu relative utility. DEFINITION 8. We define the copetitive utility of an offer for agent as the ratio of the utility of the offer to the utility of the rationality-constrained social best offer: U (),{} cop (O) = C() conflict C() (O) conflict C() (),{} rel (O) rcsoc, {} = U (4) Ξ (),{} The copetitive utility can range fro 0 to >. Intuitively, U cop (),{} (O) = eans that the agent obtained the social Ξ (),{} deal. If U cop (),{} (O) < and U cop (),{} (O) > it can be interpreted that agent outsarted agent in the negotiation. If is not possible that both copetitive utilities to be ove, as the specific offer would becoe the new social deal. However, it is possible that both values are below, which eans that the negotiating agents agreed on a deal which is not Pareto optial. s finding a Pareto optial deal in spatio-teporal probles is a non-trivial collorative search task, this can happen quite often for real-world negotiations. Figure 2: Exaple runs of negotiation protocols. (left) Exchange of binding offers (EO). (right) Exchange of offers with andatory, non-binding evaluations (EOMNE). 3.3 Negotiation protocols for CRF For ost negotiation settings, it is assued that the coplexity of the negotiation lies in the strategy, while the protocol is a relatively trivial alternating exchange of offers by the two parties. Such a siple protocol would still work well for the CRF gae with full knowledge. In the case of incoplete knowledge, however, the difficulty of foring a feasible offer as well as evaluating whether a given offer represents a concession or not, ake siple offer-exchange protocols little better than rando search. The siple protocol can be enhanced by schees in which the agents add additional inforation of the negotiation flow to aid the negotiation partner in the offer foration. In the following we illustrate the design space for the CRF negotiation protocols through several exaples. Siple exchange of binding offers (EO). In this siplest negotiation protocol, the agents are alternating in aking fully specified offers in the for O = {y, t, y s, t s }. The offers are binding for the agents who ade the offer, in the sense that once ade by an agent and accepted by the other agent, the offer will be the outcoe of the negotiation. n exaple run of this protocol is illustrated in Figure 2-left. Exchange of binding offers with andatory, non-binding evaluations (EOMNE). In this protocol the agents are exchanging pairs of offers and evaluations. gent first chooses a spatially specified offer O = {y,?, y s,?}, and coputes the associated best tie copletion T (O) = {y, t (), y s, t () s }. This is the offer which will send to agent, which is guaranteed to be feasible for and is binding for. gent will calculate its own best tie copletion TC () (O) = {y, t (), y s, t () s } for the sae spatially specified offer. Using the two best tie offers, will for an evaluation of the initial offer E(O) = { y, ax(t (), t () ), y s, ax(t () s, t () s ) } (5) This evaluation has the for of an offer which is feasible for both agents, but it is not binding for the evaluating agent. Rather it represents a critique of the original offer, and such it helps the other agent in the foration of feasible offers. Note that the evaluation provides soe inforation out the evaluating agent, but it does not iediately disclose the source, destination or axiu speed (for instance, it is providing less inforation then the exchange of the TC would). lso, if the evaluation aounts to an offer which is not feasible for the evaluating agent, an epty evaluation will be returned instead.

5 In practice, agents would couple the evaluation of the received offer with the sending of new offer. n exaple run of this protocol is illustrated in Figure 2-right. Exchange of binding offers with optional, non-binding evaluations (EOONE). variation of the previous protocol reoves the requireent that the agents evaluate every received offer. The advantage of this protocol is that agents would not be required to disclose inforation in response to offers which they would not consider. n agent would norally evaluate only offers which are satisfactory fro the point of view of the spatial coponents. y s y s () O 7 O 5 Deal paraeters O 9 O O 2 O 0 O 8 O 6 O 4 O 2 O 3 Other cobinations are also possible. For instance, exchange of offers with optional but binding evaluations is the (near) equivalent of a siple exchange of offers strategy where one of the agents is choosing as its next offer the evaluation of the opponents offer. There is an interdependence between the negotiation protocol and the strategies of the agents. negotiation strategy created for the EO protocol can be trivially extended to the EOMNE as the evaluation can be created autoatically - but the strategy would not take advantage of the inforation contained in the evaluations. The sae strategy can be also trivially extended to EOONE, by choosing not to send any evaluation. It is ore difficult to downgrade strategies which rely on inforation fro evaluations such as in EOMNE protocol, to protocols where this inforation ight not be availle, such as EO. The protocols described in the next section are expected to operate in the EOMNE protocol, although the first protocol presented does not take advantage of the evaluations and it would work siilarly under the EO protocol. 4. NEGOTITION STRTEGIES In the following we briefly introduce three negotiation strategies for the CRF proble. The goal is to illustrate a broad range of strategies which the agents ight deploy. For all these strategies the assuption is the no initial inforation setting - that is the agents start with no inforation out the paraeters of the negotiation partner. The negotiation protocol is assued to be the exchange of binding offers with andatory, non-binding evaluations (EOMNE); however, not all protocols are taking advantage of the inforation in the evaluations. Due to lack of space, the description of the strategies will be inevitly cursory. 4. Monotonic concession in space (MCS) lthough onotonic concession is one of the basic strategies in ost negotiation settings, for the CRF gae with incoplete inforation, onotonic concession is not possible. One coproise is to liit the concession to the spatial doain. This will usually, but not always represent a concession in ters of the utility function of the opponent. The onotonic concession in space agent is paraeterized by the pair (c, c s ) representing the concession rate in the eeting point and splitting point respectively. The agent will start its negotiation by proposing an offer corresponding to its solute best O (), = O () = {y (),, t (),, y (), s, t (), s }. In response to this, the agent will receive an evaluation (which will contain the offer corresponding to the ility constrained best O () (acb) and the counteroffer of agent, O (), = {y (),, t (),, y (), s, t (), s }. y s () O y () y () Figure 3: negotiation trace between the a onotonic concession in space agent and an internal negotiation deadline agent. The next offer of agent is described by the following values: y (),n c if y (),n < y (),n c y (),n+ = y (),n y (),n if y (),n if y (),n y (),n + c if y (),n < y (),n > y (),n > y (),n + c y (),n + c y (),n c y (6) with a siilar expression for y (),n+ s. Using the resulting spatially specified offer {y (),n+,?, y (),n+ s,?}, the agent will calculate the best tie copletion. The resulting fully specified offer will be evaluated for rationality. If the offer is not rational, the agent will break the negotiation. The agent will accept the opponent s offer if it evaluates to a utility which is higher than the next counteroffer the agent is out to ake. 4.2 Internal negotiation deadline (IND) In the internal deadline negotiation strategy the agent sets to itself a deadline (expressed as a nuber of negotiation rounds) and adapts the speed of concession in function of the reaining negotiation rounds and the difference between the current offers. If the nuber of rounds have expired without an agreeent being reached the agent breaks the negotiation. The negotiation strategy is paraeterized by the negotiation deadline. Siilarly to the MCS strategy, the agent starts by offering the solute best O (). t every step the agent will calculate the relative utility of its own previous offer based on the evaluation ade by the opponent agent. Then it adapts the concession rate based on its own and the opponents previous concession such that the deal will be reached at the negotiation deadline. Naturally, a deal can be reached sooner if the opponent agent accepts an offer. The agent will insist on an offer (by repeating it without change), if the offer is evaluated to be rational by the opponent while the opponents offer is not rational for the agent. Figure 3 shows a negotiation trace between a MCS and IND agent. Note the adaptation of the concession speed by the IND agent. 4.3 Estiation of the opponents paraeters (EOP) The ain proble with the previous two strategies is that it is difficult to create attractive offers, because the agents do not know whether a particular offer represents a concession. Furtherore, the offering agent can not even evaluate the utility of a spatially spec-

6 ified offer for itself, as the offer needs to be ility and rationality constrained by the opponent agent. The EOP strategy tries to iprove its offer foration by estiating the opponent s speed and current location based on the offers and evaluations ade by the opponent. With this estiate, the agent can calculate the ility constrained t and t s values for a spatially specified offer. The EOP strategy aintains an offer pool which contains a set of pre-calculated offers, which are believed to be rational for both agents. The agent will ake offers fro the pool in the decreasing order of its own utility. The agent accepts the opponents offer if it is higher than the next offer in the pool, and breaks the negotiation if the offer pool is epty. Naturally, the offer pool needs to be updated whenever the estiates change. Note that will all these negotiation strategies it is possible that two agents will not reach an agreeent despite the fact that a utually rational offer exists source area setting source area setting 2 source area setting 3 Forest destination area setting destination area setting 2 destination area setting Figure 4: Three settings for the distribution of the source and destination areas for the study of the distribution of collorativeness aong scenarios. 5. EXPERIMENTL RESULTS In the following we describe the results of a series of epirical studies which study the influence of the collorativeness of the scenario on the results of negotiations. In particular, the three ain questions we plan to answer is: What is the distribution of collorativeness in scenarios? How is the efficiency of the negotiation process affected by the collorativeness of the scenario? How does the perforance of a negotiation strategy affected by the collorativeness of the scenario? The negotiation protocol considered was exchange of offers with andatory, non-binding evaluations (EOMNE). We have used three agents: (a) onotonic concession in space (MCS), with concession size (2, 2), (b) internal negotiation deadline (IND) with internal deadline 30 and (c) estiation of the opponents paraeters (EOP), with the offer generation resolution of The distribution of the collorativeness The distribution of the collorativeness provides the answer to the question: if we pick a rando scenario, is it going400 to be copetitive or collorative? Naturally, the distribution of the collorativeness depends on the distribution of the source and destination 200 locations of the scenarios, as well as the distribution of the speed of the agents. Let us assue that the source and destination are distributed uniforly in rectangular areas situated iediately 0 on the left and right side of the forest. To study a variety of possible distributions we consider three settings corresponding to the source and destination areas shown in Figure 4. For each setting, we generate 000 scenarios by choosing the source and destination according to a unifor spatial distribution fro the corresponding source and destination rectangles. We calculate the value of collorativeness according to Equation 2, and asseble the values in a 0-bucket histogra. The three resulting histogras are shown in 5. We can ake the following observations: Setting : has the source and destination areas a square of the sae height as the forest. The histogra shows a U-shape, with higher nuber of scenarios falling at the higher and lower extrees of collorativeness. Setting 2: has the source and destination areas rectangles of the sae height as the forest but a width of half as uch. The corresponding histogra shows a siilar U-shape like in the previous case, but it is shifted towards the higher collorativeness. We conclude that the closer is the forest to the source and destination, the higher the probility that foring a coalition to traverse the forest will be advantageous. nuber of scenarios 600 setting collorativeness Figure 5: The coparison of collorativeness distributions in three cases of restricted areas. Setting 3: has the source and destination areas square and half the height of the forest. We find that the distribution of the collorativeness is weighted toward the higher values. This result atches our intuitive expectations. For instance, citizens in tightly packed cities such as New York and San Francisco rely ore setting on 2 public transportation, setting 3 as their source and destination locations are frequently 600 correlated. In cities spread over large areas such as Orlando or Phoenix, the transportation interests are rarely collorative. 400 eyond atching intuition, however, our approach allows us to put a nuerical value on the distribution, which is useful for both an acting guide200 for individual agents 5.2 Copetitive (relative) utility in function of collorativeness collorativeness collorativeness In our next set of experients we study the relative utility achieved by specific strategies against specific opponents under in scenarios with various levels of collorativeness. Naturally, if a deal falls through, the relative utility is zero. One of the difficulties of our study is that we cannot generate directly rando scenarios with predefined collorativeness level. Thus we rely on rejection sapling, a technique borrowed fro Monte Carlo siulation ethods: we generate scenarios by picking source and distribution points according to a unifor distribution, calculate their collorativeness, group the into 20 buckets, and then randoly reject scenarios fro the buckets which are too full until they are uniforly filled with 500 scenarios each. The resulting collection of 0,000 scenarios was used in siulations. The agents are using the EOMNE negotiation protocol. Once the negotiation terinated (either with an agreeent or a conflict), we easured the relative utility of the deal. To present both the variility in the results of negotiation, as well as the underlying trends, we choose to superipose a scatterplot of the siulation results with a plot of the average values

7 Figure 6: The scatterplot and average values function of collorativeness for two onotonic conceding in space (MCS) agents with identical paraeters negotiating with each other. Figure 8: The scatterplot and average values function of collorativeness for two estiating the opponents paraeters (EOP) agents with identical paraeters negotiating with each other. relative utility Figure 7: The scatterplot and average values function of collorativeness for two internal negotiation0.4 deadline (IND) agents with identical paraeters negotiating with each other calculated on a per-bucket basis. Plotting only the average value Figure 9: The coparision of average relative profits both 0.7 would be isleading, as the spread of the siulation 0 values is not agents get when deadline agent negotiates with the estiating accidental, but 0.8 an0.9 intrinsic property which would not disappear if agent in different scenarios. collorativeness collorativeness we would, for instance, run a larger nuber of experients. 0.5 In the previous exaples, we always had the sae types of Figure 6 shows the scatterplot and average values of relative utility for the case of two MCS agents negotiating against each other. 0.4 agents negotiating each other. Pitting agents of different types 0.3 against each other can give as insight into their relative perforance. Figure 9 EOP vs IND shows the average relative utility function of We find that, as expected, the relative utility is increasing with the collorativeness of the scenario. However, the scatterplot shows 0.2 IND vs EOP collorativeness graphs an IND agent negotiating with an EOP that the results were spread over a large range of relative utilities. IND vs IND 0. agent. The two plots represent the results for the sae series of Even for scenarios with very high collorativeness levels, there 0 experients seen fro the point of view of the two agents (that are any negotiations which end without an agreeent. In general, is the U (IND),{EOP} this reflects weaknesses of the negotiation strategy, as the negotiation breaks down while there were possible deals which would have rel collorativeness and the U (EOP),{IND} rel values). For reference, we also added the plot of two IND agents negotiating each other. The overall shape of the curves is what we expected, the relative utility been utually acceptle. increases with the collorativeness. However, the agent using the The second experient uses the internal negotiation deadline EOP strategy is le to consistently achieve higher utility values (IND) strategy for both agents. The results are shown in Figure than the IND agent, which shows the (expected) superiority of the 7. The shape of the graph is roughly siilar to the MCS vs. MCS EST strategy in this setting. graph, however, the strategy shows better results for very low col scenarios. 0.9 For instance, for a collorativeness of n interesting observation can be ade coparing the results of lorativeness the IND agent versus the EOP agent and versus another IND agent. collorativeness the IND vs. IND settings obtains an average relative utility For low collorativeness levels the IND strategy perfors worse of 0.295, copared to 0.09 for the MCS vs. MCS case. against the EOP strategy than against the EOP strategy, which we In the third experient, we plot the sae values for two agents can interpret that EOP outsarted its less sophisticated opponent, using the estiating the opponents paraeters (EOP) strategy negotiating with each other. The results are shown in Figure 8, and and pushed it into less advantageous deals. For high collorativeness levels, however, the situation is reversed; for these scenarios the overall trend reains the sae as in previous cases: the relative the two agents have largely aligned interests, thus the efforts of utility grows with the collorativeness of the scenarios. We find EOP to find better deals for itself also iproves the deals for the that the overall relative utility of the EOP agents is uch higher IND agent. than the other two cases. Looking at the scatterplot, we can see that this was accoplished both through higher utility values for deals, as well as the reduction of the scenarios which ended in conflict. 6. CONCLUSIONS relative utility

8 The contribution of this paper is the introduction of a set of etrics which allows us to easure the perforance of a negotiation strategy in coparison to its peers, as well as to evaluate, without assuing any particular negotiation strategy whether a negotiation will be easy or difficult. Measuring the perforance of a negotiation strategy against a particular type of partner is not trivial. weak perforance in a particular scenario does not necessarily ean that the strategy is weak; it is possible that in the particular scenario the odds were against the agent. The relative utility and copetitive utility etrics allows us to evaluate the perforance fro a sall nuber of negotiation runs in the cases when we are interested in the solute value of gained utility or in the copetitive advantage over the negotiation partner. The collorativeness etric we introduced allows us to put a quantitative value on the intuition of easy vs. hard negotiation scenarios. This easureent is independent of the strategy, but, naturally, soe strategies ight be better in exploiting the advantage of collorative scenarios, or to retain acceptle utility in copetitive scenarios. We had also seen soe epirical evidence for the clai that negotiating against a sophisticated strategy leads to lower relative utility for scenarios with low collorativeness levels, but it actually becoes an advantage when negotiating in scenarios with high collorativeness levels. One can envision strategies which perfor a collorativeness analysis on the scenarios they encounter and ake a prediction of the success of the negotiation. One way to use this inforation ight be agents which can act while negotiating. If an agent is alost sure that a deal will be reached, it ight start to ove towards the predicted location of the deal while the negotiation is in progress. lternatively, if an agent is le to choose which agent it will launch into negotiation for colloration, the agent ight pick the negotiation with the highest collorativeness value as the negotiation ost likely to succeed and reach a high relative utility. ppendix: Generalized CRF n iportant question which ight be asked is whether the results in this paper can be applied to practical applications. Naturally, the CRF gae itself does not ap one-to-one to any practical application. Let us now introduce a generalized version of the CRF gae. s before, we assue that two agents and are oving fro source points S and S to their respective destination points D and D. Their velocities in vector for are v = (v,x, v,y ) and v = (v,x, v,y ) respectively. They can for a coalition, in which case they have the coon velocity v +. The velocity of the agents is subject of restrictions of the for R (v, x, y ), R (v, x, y ) and R + (v +, x +, y + ) The CRF gae is a special version of the generalized CRF where: v c if (x, y ) forest R : (7) v = 0 if (x, y ) forest R + : v + in(v, v ) (8) Notice, that this iediately generalizes to situations where the forest is of arbitrary size and position. It can also generalize to ultiple obstacles, including those which produce only a slowdown as opposed to acting as a barrier. The approach also generalizes to cases where the agents are of different type. For instance, if is a huan, and is a vehicle which the huan can board, R + is changes as follows: R + : v + v (9) We can also develop restrictions for cases when, for instance, is a ferry, while is a vehicle. In this case and the coalition of + has obility on water, while has obility on solid ground. These exaples all have restrictions on the scalar coponent of the velocity; this, however, ight not always be the case. We can odel roads, where lateral oveent is ipossible or slower, or rivers where the speed of oveent is greater in the direction of the flow. The question is, how uch of the results of this paper reain applicle to the generalized CRF? The etrics defined in Section 3.2 are applicle without odification. Siilarly, the proposed negotiation protocols 3.3 reain valid and functional. The negotiation strategies, however, need to be adapted for the specific restriction odels. In a generalized setting it is even ore difficult to define what constitutes a concession. With a ore coplex ap, offer generation becoes ore expensive, which liits how any offers an agent can aintain in its offer pool. Nevertheless, the general outline of negotiation strategies will be still functional; it will need, however, to be adapted to cover the specifics of the restrictions. cknowledgeents This work was partially funded by the NSF Inforation and Intelligent Systes division under award This research was sponsored in part by the ry Research Loratory and was accoplished under Cooperative greeent Nuber W9NF The views and conclusions contained in this docuent are those of the authors and should not be interpreted as representing the official policies, either expressed or iplied, of the ry Research Loratory or the U.S. Governent. The U.S. Governent is authorized to reproduce and distribute reprints for Governent purposes notwithstanding any copyright notation hereon. 7. REFERENCES [] E. Crawford and M. Veloso. n experts approach to strategy selection in ultiagent eeting scheduling. utonoous gents and Multi-gent Systes, 5():5 28, [2] S. S. Fatia, M. Wooldridge, and N. R. Jennings. 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