Quantifying Relative Autonomy in Multiagent Interaction

Transcription

1 In the IJCAI-01 wokshop on Autonomy, Delegation, and Contol: Inteacting with Autonomous Agents, Quantifying Relative Autonomy in Multiagent Inteaction Sviatoslav Bainov State Univesity of New Yok at Buffalo 226 Bell Hall Buffalo, NY Abstact In the pape we intoduce a quantitative measue of autonomy in multiagent inteaction. We quantify and analyze diffeent types of agent autonomy with espect to (a) an agent s use, (b) the othe agents, and (c) the othe goups of agents. We also intoduce a measue of goup autonomy that accounts fo the degee with which one goup depends on anothe goup. We analyze the question of finding a multiagent goup with maximum oveall autonomy and we pove that this poblem is NP-complete. Theefoe, the poblem of finding the optimal goup o agent with whom to shae a task (o to whom to delegate a task) is computationally had in geneal. This pompts fo developing appoximation algoithms fo measuing and adjusting autonomy. 1. Intoduction The concept of autonomy plays an impotant ole in multiagent inteaction. It elates to an individual o collective ability to decide and act consistently without outside contol o intevention. Autonomy has been a subject of continuous inteest in diffeent eseach aeas including multiagent systems (Castelfanchi, 1995 and 2000, Hexmoo, 2000a; Hexmoo and Kotenkamp 2000), sociology (Dwokin, 1988), and philosophy (Mele, 1995; Schneewind, 1997). The notion of autonomy has been used in a vaiety of senses and has been studied in diffeent contexts. Autonomy could be elative to an individual o a goup. Autonomy can be in egads to eithe acting o decision-making. Depending on the context we can have autonomy with espect to the physical envionment o with espect to the social envionment. A usual way to look at autonomy is to see it as self-contol, which is an ambiguous concept, because it might be efeing to eithe the capacity o competency to contol oneself, o to the actual condition of self-contol, o to the authoity to contol oneself. Heny Hexmoo Compute Science & Compute Engineeing Depatment, Engineeing Hall, Room 313, Fayetteville, AR It is clea that the main poblem of autonomy is to explain how autonomous agents make decisions and how they act upon them. In this egad it is wothwhile to distinguish between action autonomy and decision autonomy. Action autonomy elates to the way an agent acts in the envionment. Fo example, an agent may have patial contol ove the envionmental events that affect the outcome of an action. A digital financial assistant may be authoized to buy a stock in an electonic exchange. The agent may, howeve, not have contol ove the tansaction pice. The pice may depend on the actions of othe financial assistants and on the netwok communication delay. Decision autonomy is concened with the ability of an agent to make consistent choices. A decision autonomous agent must have knowledge about the use s pefeences and the potential altenatives. We may be eluctant to classify a digital financial assistant as autonomous, if it keeps asking its use what stock to buy and at what pice. Thoughout this pape we assume a distibuted poblem-solving envionment. Thee is a single use inteested in a single task, which can be achieved by deploying one o moe agents. A multiagent inteaction, whee diffeent agents act on behalf of diffeent uses, is beyond the scope of this pape. In geneal, autonomy can be consideed to have the following constituents: The subject/agent of autonomy: the entity (a single agent o a goup of agents), which has to be consideed autonomous. The influence of autonomy: the entity that influences the autonomy of the subject./agent. It could be the human use, physical envionment, anothe agent o goup of agents. The scope of autonomy: the specific means by which the influence can affect autonomy. This can include actions, esouces, infomation, indiect means, etc. The object of autonomy: includes all elements with espect to which the subject/agent can be autonomous. This could be a single action, goal, task, etc.

2 The degee of autonomy: measues that captue the extent to which influence can affect autonomy. Thee may be seveal objects of autonomy and each object may contibute in a diffeent way to the oveall autonomy. At the same time thee could be seveal influences; each influence can have diffeent means and each means may affect one o moe objects of autonomy in diffeent ways. Depending on the influence we can have autonomy with espect to an agent s use (o uses), autonomy with espect to the physical envionment, and autonomy with espect to othe agents (both human and atificial). Autonomy in the context of agent-use inteaction captues the notion of an agent s ability to act efficiently without the use s intevention. An agent may not be autonomous with espect to its use if it needs pemission fo cetain actions. An agent that has full pemission may still not be autonomous, if it has patial knowledge about the use s pefeences and the ways in which these pefeences could be met. In the context of inteaction with the physical envionment, autonomy is one s ability to act independently in the physical envionment. This kind of envionmental autonomy usually pesupposes (a) some kind of contol o mastey ove the envionmental events and objects, and (b) impeviousness o libety fom uncetainties, and (c) obustness against envionmental changes. Autonomy in the context of social multiagent inteaction is concened with vaiations in an agent s ability when othe agents ae involved. It might be desiable fo an agent s pefomance to be invaiant to inteactions i.e., stable and independent of inteaction. In this pape we focus on the last constituent of autonomy: the degee of autonomy. The degee of autonomy measues to what extent the influence can affect the object of autonomy and how the change elates to the oveall autonomy. In ou case the object of autonomy is a task. It is evident, that the task is a complex object that is made up of seveal othe objects (actions, plans, etc.). In ode to simplify the analysis we conside the task as a single object and assume that duing the task execution agents always make optimal decisions. In ode to apply a qualitative measue of autonomy we need a scale and some citeion fo distinguishing between autonomous and nonautonomous behavio. Since the object of autonomy is a task and it is expected to be executed efficiently, we use an agent s pefomance as a scale. Since agents ae pefoming a task on behalf of othe agents, autonomy is elated to some standad of achievement that deives fom the function an agent seves. If an agent s pefomance is eopone and continuously fails duing a task, we may be eluctant to call it autonomous egadless of how self-diecting and independent it might be (Meyes, 1989). Autonomy depends on othes expectation and is elative to those expectations. Fo example, an agent may be autonomous with espect to one task and not autonomous with espect to anothe task. Moeove, it is possible fo an agent to be autonomous and non-autonomous at the same time with espect to the same task, if diffeent uses apply diffeent pefomance standads. The elativeness of autonomy becomes moe pominent in envionments whee the use cannot pedict all contingencies upfont. In ou pevious eseach we investigated efficiency as a basis fo teaming among agents and pesented a pefomance-based teaming algoithm (Hexmoo and Duchschee, 2001). In this pape we conside an agent s pefomance elative to its context as an indicato of autonomy. We conside autonomy as a elative notion. It is undestood in the context of the envionment that can be made up of events, object, and othe agents. In othe wods, in ode to evaluate the degee of an agent s autonomy we have to put the agent in touch with objects, events, and othe agents. If an agent can pefom in the pesence of othe agents at least as well as it pefoms in isolation, then the othe agents ae not esticting the agent s autonomy. By woking in the pesence of othe agents, we make no assumptions about explicit coopeation o coodination o othe inteagent attitudes. We also do not make any assumptions about psychological influences among agents. When in pesence of othe agents, these othe agents ae consideed as a distinguished pat of the envionment. Fo example, a factoy woke who gets pats fo a widget and assembles it may wok alone o alongside othe agents who do the same. This factoy woke might expeience gains o losses in its poductivity in the pesence of these othe wokes, who ae a special pat of its envionment. Autonomy is by no means identical to efficiency. Late in this pape we will pesent the autonomy-efficiency dilemma and we will show that (a) autonomous behavio could be inefficient, as well as (b) efficient behavio might not necessaily be found with autonomous

3 agents. Relative pefomance i.e., the pefomance in a context, howeve, could be a good indicato of the degee to which the context esticts o extends individual autonomy. The compaative analysis of autonomy allows us to define and diffeentiate between diffeent kinds of autonomy elationships: an agent in the context of a goup, a goup in the context of anothe goup, and a goup in the context of an agent. In this pape we emphasize elative autonomy in the context of a use, envionmental factos, and in a social setting. Babe and Matin (1999) poposed anothe quantitative measue of agent autonomy. They define the degee of autonomy as an agent s elative voting weight in decision-making. This appoach has seveal advantages. Fo example, it allows fo explicit epesentation and adjustment of the agents autonomy. To ou knowledge, it has been the fist attempt to descibe an agent s autonomy fom a decision-theoetic point of view. Seveal indexes of agents voting powe have been poposed (Banzhaf, 1965; Shapley and Shubik, 1954). The game-theoetic eseach, howeve, eveals that an agent s elative voting weight is not always a good measue of voting powe, since it does not take into account the fequency with which an agent s vote is pivotal (Banzhaf, 1965). The concept of autonomy is closely elated to the concepts of powe, contol and dependence (Bainov and Sandholm, 1999; Castelfanchi, 2000). An agent is autonomous with espect to anothe agent, if it is beyond the influences of contol and powe of that agent. In othe wods, autonomy pesupposes some independence o at least esticted dependence. Futhe exploation of the elationship between powe, contol, and autonomy is beyond the scope of this pape. The pape is oganized as follows. In the next section we analyze autonomy in the context of use-agent inteaction. We popose a measue of autonomy that indicates the degee to which an agent is independent of its use. Section 3 pesents autonomy in the context of envionmental factos and defines a coesponding autonomy measue. In Section 4, we analyze autonomy in social multiagent inteaction and intoduce quantitative measues of goup autonomy. We exploe the idea of finding an agent goup with maximum oveall autonomy and pove that this poblem is NPcomplete. 2. A Measue of Autonomy in the Context of Use-Agent Inteaction In this section we analyze autonomy with espect to an agent s use. We conside the use as the main agent who has the ight to monito and contol an agent s pefomance. The use takes the esponsibility fo the agent s pefomance and povides identification fo the agent. Wheneve the agent identifies itself, exchanges digital cetificates, o caies out a tansaction, it acts on behalf of the use. Unde the ight cicumstances we assume the use can activate o deactivate the agent at he will. The use is typically a human and typically the owne of the agent, which takes legal esponsibility fo the agent. Howeve, it is not necessay fo the use to be a human agent. Fo example, a mobile agent can spawn a new agent and act as a use with espect to that agent. Since an agent acts on behalf of its use, use-agent inteaction has geate pioity fo the agent than the inteactions with othe agents. Othe inteactions could be consideed as instumental with espect to the use-agent inteaction. An inteesting aspect of autonomy is an agent s ability to maintain a sense of self and identity i.e., an agent s ability to keep its elationship with the use. An agent s identity becomes impotant given the agent s code, sensitive infomation (financial infomation, fo example), access contol ights, passwods, digital cetificates can be accessed o alteed by malicious thid paties. An inteesting case aises when an agent simultaneously seves multiple uses. By multiple uses we mean uses with diffeent identities (the case when diffeent uses inteact with the agent shaing the same identity is consideed as a single use). If the uses do not have pedetemined pioities fo the agent, the agent may exhibit autonomy by following the fist-come fist-seved ule. The agent has to detemine allocation of esouces among competing uses. In this case we assume that thee is always a single administato among the uses with distinguished contol pemissions. An agent may complete the task with o without the use s supevision. In ode to measue the agent s autonomy with espect to its use, we have to know the extent to which the use s supevision is helpful fo the agent. We assume that the agent s pefomance can be measued by some citeion of pefomance ν. The citeion ν may be thought of as a citeion of patial success, optimization function, index

4 of satisfaction, utility function, etc. The use detemines the citeion of pefomance ν. Fo the same task, diffeent uses may use diffeent pefomance citeia. With evey agent i we can associate at least two pefomance measues 1. The fist measue ν i is agent i s pefomance in the case whee it acts autonomously, i.e., without the use s supevision. The second measue ν i U is agent i s pefomance with the use s supevision. ν i U does not measue the pefomance of the use and the agent collectively. Intuitively, this is the agent s own pefomance with the use s supevision. Howeve, we make no assumptions about impoved pefomance and, in fact, pefomance degadation is quite possible. We follow the standad assumption of keeping all othe things equal. That is, the effect of othe agents o the envionmental events is the same fo both measues ν i U and ν i. Definition 1. By a degee of individual autonomy A i (autonomy with espect to the use) we mean the atio i ν. ν i U The degee of individual autonomy indicates the extent to which an agent may act well independently of the use i.e., what pat of an agent s pefomance must be attibuted only to the agent s capabilities. In geneal, individual autonomy vaies between - and +. The degee of individual autonomy can be intepeted as the degee of independence fom the use s supevision. The combined use and agent pefomance is not necessaily the maximal pefomance. Fo example, if the use is not competent enough, the agent may be moe efficient by acting autonomously. Definition 1 chaacteizes autonomy as a elative concept. In ode to evaluate an agent s autonomy the use must have some citeion of acceptable behavio o some expectation about the agent s behavio. Since diffeent uses may have diffeent equiements fo a task accomplishment, autonomy estimates may vay acoss diffeent uses. This means that diffeent uses could conside a patten of behavio as eithe autonomous o non-autonomous. Suppose, fo example, that an agent autonomously fulfills only 90% of a given task. A use may conside a 90% accomplished task as a success, and may be 1 In the next section we will intoduce a complete desciption of elative pefomance. willing to classify the agent s pefomance as autonomous. At the same time, anothe use may conside the same pefomance as a failue, and may be eluctant to egad the agent as autonomous. 3. A Measue of Autonomy in the Context of Envionment Inteaction In this section we analyze an agent s autonomy with espect to a set of envionmental factos, which may contain uncetainty o uneliability. These envionmental factos might be tools, instuments, electo-mechanical devices, o peishable esouces. Let s imagine the envionmental factos have a known pobability of eliability o uncetainty. In geneal, device eliabilities ae epesented as pecentages ove eliability anges. Fo example, a light bulb might be 90% of the time 99% eliable and 10% of the time uneliable. This can be extended to seveal anges, say 90% of the time 95% eliable (i.e., faily eliable), 5% of the time 99% eliable (i.e., highly eliable), and %5 of the time 10% eliable (i.e., uneliable). The pobabilities add up to 1.0 but the anges ae open. Afte access to the knowledge of these pobabilities of eliability, the agent may o may not decide to use the envionmental factos o make a decision about the envionmental factos. Let s imagine n anges each with α i pobabilities. The agent s ability to act and decide is contasted in each ange of eliability of the envionmental factos in light of the known pobabilities. ν i1 is the agent i s pefomance in the case whee it acts without the use of a set of envionmental factos knowing that they ae the most eliable (i.e., the best). ν i2 is the agent i s pefomance in the case whee it acts without the use of a set of envionmental factos knowing that it may have access to second best envionmental factos. Continue this until ν in whee the agent i s pefomance in the case whee it acts without the use of a set of envionmental factos knowing that it is has the least eliable (i.e, the wost) envionmental factos. ν i1 is agent i s pefomance with the use of the most eliable (i.e., the best) set of envionmental factos. ν i2 is agent i s pefomance with the use of the second best eliable set of envionmental factos. Continue this until ν in whee the agent i s pefomance with the use of the least eliable (i.e., the wost) set of envionmental factos. We make no assumptions about impoved

5 pefomance and in fact pefomance degadation is quite possible. Definition 2. By a degee of t autonomy with espect to an uneliable envionmental element we mean the atio α i ν / α ν i. The degee of envionment autonomy indicates the extent to which an agent may act well independently of the envionmental factos i.e., what pat of an agent s pefomance must be attibuted only to the agent s capabilities. In geneal, envionment autonomy vaies between - and +. Envionmental uncetainty implies that an agent has to decide between elying and not elying on the envionmental factos. The choice has to be made in complete ignoance about the actual level of eliability. If an agent chooses to ely, its expected pefomance will be α i ν. If it does not ely, then the expected pefomance is α ν i. The atio between the expected pefomances measues agent s i autonomy with espect to an uncetain envionment. In Definition 2 we assume that an agent is fee to decide whethe to use the envionmental factos. This implies that the agent has at least patial contol ove the envionment. In many situations, howeve, an agent cannot go aound the envionment factos and have to use them. In this case we view an agent s autonomy as the ability to choose the most favoable envionmental factos. Definition 3. By a degee of autonomy with espect to an uncetain envionmental element we mean the atio α ν i / ν i1. Accoding to definition 3 the degee of autonomy is the atio between the aveage (o expected) pefomance α ν i and the most successful pefomance ν i1. Since the agent does not have contol ove the envionmental factos, it cannot choose among them. If it had contol, it would choose the most favoable ones. Conside the following example. Both the use and the agent ae uncetain about the cuent conditions in the envionment. Howeve, they shae thei levels of uncetainty as common knowledge. With pobability 2/3 the conditions ae favoable and with pobability 1/3 they ae inauspicious. The agent gets 6 if the conditions ae favoable and 3 if they ae inauspicious. The agent expected pefomance is 5 and the most desiable outcome is 6. Theefoe, the agent s degee of autonomy is 5/6. The situation is shown in Fig.1 Natue s selection 2/3 1/3 Fig.1: Individual autonomy in an uncetain envionment. 4. Goup Autonomy In multiagent inteaction whee the agents actions intefee with one anothe, an agent may affect the autonomy of othe agents both diectly and indiectly. Indiect inteaction usually occus as a side effect of an agent s behavio. An agent s action may estict o extend the autonomy of othe agents by affecting the envionmental conditions, the set of feasible goals, etc. In some cases the effects could be even moe indiect. Fo example, an agent can affect anothe agent, which in tun may affect the autonomy of a thid agent. This pompts fo a quantitative measue of the degee of autonomy that takes into account vaious aspects of multiagent inteaction: an agent in the context of a goup, a goup in the context of anothe goup, and a goup in the context of an agent. We assume that agents may affect one anothe once they have been deployed in the envionment. Theefoe, it is not possible to devide the envionment into diffeent mutually independent goups of agents such that agents can affect one anothe if and only if they belong to the same goup. This is a natual assumption, since we cannot peclude agents fom intefeing with one anothe (in a positive o negative way) once they have been bought togethe. Then, the question is which agents to deploy? In othe wods, which subset of agents achieves maximum autonomy, maximum efficiency o some combination of them? This question is diffeent fom the poblem of finding the optimal coalition stuctue (Sandholm et al., 1999). A coalition envionment implies that agents can be devided into elatively independent goups called coalitions. Each coalition has its own 6 3

6 pefomance measue (value of the coalition). The poblem is to find a coalition patition that maximizes the sum of coalitions pefomances. In ou case all agents pefom the same task and thee is no eason to sepaate them into diffeent coalitions. That is, we assume that the gand coalition (involving all active agents) always foms. In othe wods, the use always deploys one coalition of agents. The poblem is which coalition to deploy. In ode to evaluate how well an agent is doing in the company of othe agents we need some indicatos of elative pefomance. With evey agent i we associate a vecto of elative pefomance 2 (ν i, ν i j, ν i k, ν i jk). Hee ν i epesents agent i s pefomance acting alone i.e., by acting autonomously. ν i j is agent i s pefomance in the company of agent j. In this case agents i and j can intefee with each othe eithe negatively o positively. ν i j could be geate o smalle than ν i depending on the type of intefeence. Fo example, if agent i depends positively on agent j, then ν i j ν i. ν i k is agent i s pefomance in the pesence of agent k. ν i jk measues agent i s pefomance if it acts concuently with agents j and k. In the case of 3 agents the length of the vecto of elative pefomance is 2 2. In geneal, the vecto s length is 2 n-1, whee n is the numbe of agents. The elements of the elative pefomance vecto should be intepeted as guaanteed pefomance values. Fo example, agent i can always get ν i j in the company of agent j. The actual pefomance may diffe depending on agent j s behavio, but it is always geate o equal to ν i j. In othe wods ν i j is the minimax pefomance that agent i can obtain in the company of agent j i.e., no matte how agent j behaves, agent i always gets at least ν i j. The following definition intoduces the concept of autonomy with espect to anothe agents. Definition 4. The degee of agent i s autonomy with espect to agent j is i ν j A(i/j) = i ν The degee of agent i s autonomy with espect to agent j is the atio of agent i s elative 2 Fo the sake of simplicity we constain ou attention to the case of thee agents i, j and k. The esults can easily be genealized to an envionment with an abitay numbe of agents. pefomance to its individual pefomance. In othe wods, the degee of autonomy indicates how well agent i pefoms in the pesence of agent j. It is 1 when agent j does not affect agent i. It could also be 0, if agent j completely blocks agent i. In geneal, it vaies between - and +. Goup pefomance is highly affected by intefeence among agents. The intefeence might eithe poduce positive o negative pefomance. The following dilemma states the poblem with the intefeence. Definition 5: Autonomy efficiency dilemma aises when we compose a goup of agents subject to the highest oveall goup pefomance with two classes of agents: (a) agents with low efficiency and high autonomy invaiance agents (agents that ae impevious to intefeence fom othe agents), and (b) agents with high efficiency and high autonomy vaiance (agents whose pefomance is highly susceptible to intefeence with othe agents). The following two-agent example illustates the autonomy-efficiency dilemma. Suppose that we have two agents i and j whose vectos of elative pefomance ae (5,1) and (2,2) espectively. These agents have diffeent levels of individual autonomy. By acting alone agent i gets 5, while agent j gets 2. If the agents ae bought togethe, then agent i gets 1 and agent j gets 2. Theefoe, the autonomy of agent i with espect to agent j, A(i/j), is 1/5. This indicates that agent j affects negatively agent i by educing agent i s pefomance 5 times. On the othe hand, agent j is autonomous with espect to agent i. Agent j s pefomance does not depend on agent i and it is always 2. If we ae looking fo maximum invaiance in autonomies, then we have to deploy only agent j. This, howeve, is not an efficient solution since agent j has low pefomance. Completely autonomous, agent j is not as efficient as agent i is. Agent j gets 2, while agent i achieves 5. Theefoe, if we ae looking fo maximum efficiency, we have to deploy only agent i. The dilemma autonomy-efficiency aises fom the fact that efficient agents may be highly susceptible to intefeence fom othe agents autonomous and vise vesa; agents with autonomies unaffected by othe agents may not be vey efficient. To alleviate this dilemma, we suggest the following assumption that all agents have the same individual autonomy. The following definition gives this assumption a name.

7 Definition 6: Equally-competent agents is the assumption that all agents have the same individual pefomance. That is, ν i =ν, fo all agents i. Unde equally-competent agents assumption, each agent i by acting alone can achieve the same standad of pefomance ν. Since all agents ae equally competent, the dichotomy of autonomy-efficiency disappeas. The following definition intoduces the concept of autonomy with espect to a goup of agents. In this pape by a goup of agents we mean any set of agents that act concuently. Definition 7. The degee of agent i s autonomy with espect to a goup of agents (j,k) is: ν i jk A(i/jk) = ν The degee of autonomy with espect to a goup measues to what extent the goup can estict o extend an agent s autonomy. A degee of 1 means independence fom the goup. A degee lage than 1 signals fo a synegetic inteaction. Conside the following example. Suppose that agent i s vecto of elative pefomance is (4,6,4,8). That is, ν=4, ν i j=6, ν i k=4, and ν i jk=8. This implies that agent i depends positively on agent j (A(i/j) = 6/4 = 1.5). At the same time it is autonomous with espect to agent k (A(i/k) = 4/4 = 1), and depends positively on the goup of agents j and k (A(i/kj) = 8/4 = 2). In the next definition we intoduce the concept of goup autonomy. It measues how well agents ae doing in a goup. Definition 8. The degee of goup autonomy of the goup of agents (i,j) unde the equallycompetent agents assumption is: i j ν j + νi A(ij) = ν The degee of goup autonomy compaes individual pefomance with goup pefomance and indicates whethe it is wothwhile to put the agents togethe. If the agents ae deployed in a goup, the esult is ν i j+ν j i. If only one agent (eithe one) is deployed, the pefomance is ν. Poposition 1. Goup autonomy unde the equally-competent agents assumption equals the sum of individual autonomies. That is, A(S) = A (i /S {i}) i S Whee S is a set of agents, and S-{i} is the set of agents S excluding agent i. Poof. Follows immediately fom Definitions 7 and 8. If we apply Poposition 1 to the goup of agents (i,j,k), we will get A(ijk)=A(i/jk)+A(j/ki)+A(k/ij) It is woth noting that Poposition 1 does not hold in geneal. The poposition depends on the equally-competent agents assumption, i.e, that all agents have the same level of individual autonomy. In geneal, goup autonomy is not linea with espect to individual autonomy. Definition 9. The degee of autonomy of the goup of agents i and k with espect to agent k unde the equally-competent agents assumption is: i j ν jk + νik A(ij/k) = i j ν j + νi The numeato in Definition 5 measues the goup pefomance of agents i and j in the company of agent k. The denominato is the autonomous pefomance of the goup. If we apply Poposition 1 to Definition 9, we will obtain the following poposition. Poposition 2. A(S/k) = i S A(i /S {i} + {k}) A(S) whee S-{i}+{k} is the goup S excluding agent i and including agent k. Poposition 2 says that the elative goup autonomy (with espect to a thid agent) depends positively on elative individual autonomies A(i/S-{i}+{k}) and negatively on the goup autonomy A(S). To illustate all these notions, conside the following example. Let s assume we can deploy up to thee agents i, j and k with the following vectos of elative pefomance: i s elative pefomance (ν i, ν i j, ν i k, ν i jk) = (4, 3, 4, 3) j s elative pefomance (ν j, ν j i, ν j k, ν j ik) = (4, 4, 2, 1) k s elative pefomance (ν k, ν k i, ν k j,ν k ij) = (4, 4, 5, 5) In this situation agent i is autonomous with espect to agent k, and depends negatively on agent j. Agent j is autonomous with espect to agent i, and depends negatively on agent k. Finally, agent k is autonomous with espect to agent j and depends positively on agent i. The dependence gaph is depicted in Fig. 2.

8 i j Fig. 2: Dependence gaph. This example shows that since A(ik/j)= (ν i jk + ν k ij) / (ν i k + ν k i) = (3+5) / (4+4) = 1, the goup of agents i and k is independent fom agent j. Moeove, the goup autonomy of agents i and k is A(ik) = A(i/k) + A(k/i) = 4/4 + 4/4 = 2.0. That is, by acting togethe they can incease thei individual pefomance 2 times. It is easy to check that the maximum goup autonomy A(ijk) = A(i/jk) + A(j/ik) + A(k/ij) = 3/4 + 1/4 + 5/4 = 2.25 is achieved when all agents ae bought togethe. This is not appaent fom the initial statement of the poblem, since agent j elates negatively to agents i and k. The poblem of finding the goup with maximum autonomy is of significant impotance fo multiagent inteaction. Wheneve a goup of agents ae deployed fo solving a paticula task, we have to know which goup of agents has the maximum autonomy. Along the same line of easoning, if an agent decides to shae o delegate its task to othe agents, it has to find the goup with the most desiable autonomy. A elated issue is finding a goup of agents with minimum vaiance in thei autonomy. This is impotant fo fault toleance easons since if agents wee allowed to come and go at will, we would not want the goup s pefomance to be significantly affected. We have looked at this poblem peviously [Hexmoo, 2000b]. Howeve, this is a diffeent poblem than seeking agents with maximum autonomy. Accoding to the following poposition the poblem of finding the goup with maximum autonomy is computationally had. The poblem is even moe difficult if we have to account fo the autonomyefficiency dilemma. Poposition 3. Finding a goup with maximum autonomy is NP-complete. Poof. The decision poblem can be defined as follows. Given elative pefomance vectos, fo some eal numbe N, does thee exist a goup of agents whose goup autonomy is N? The poblem is in NP because veifying the degee of autonomy fo a given goup can be done in polynomial time. It involves summing k the agents elative pefomance measues and dividing the esult by the individual pefomance measue. What emains to be shown is that the poblem is NP-had. We pove this by educing the subset-sum poblem to ou poblem. The subset-sum poblem is the following: given a finite set of natual numbes S and a numbe K, is thee a subset S, S S, whose elements sum to K? This is a classic NP-complete poblem (Comen at al., 1990). We use the following eduction. Let N=K. Let the S be the set of all agents. We associate evey agent i with some natual numbe ν i. Let the elative pefomance vecto of agent i be (1, ν i, ν i, ν i,.). That is, agent i s individual autonomy is 1 and its elative pefomance is always ν i. Now, the elements of a set of numbes S sum to K if and only if the set of agents S that has a goup autonomy K. Thus, ou poblem is NP-had. 4. Conclusions In this pape we intoduced seveal quantitative measues of elative autonomy. The fist measue defines individual autonomy with espect to use-agent inteaction. The second measue elates to autonomy with espect to envionmental factos. The thid measue defines autonomy among goups and individuals. Ou measues ae domain independent and do not ely on specific inteaction potocols. We also analyzed the question of finding a multiagent goup with the maximum autonomy. We poved that this poblem is NP-complete. Theefoe, the poblem of finding the optimal goup o agent with whom to shae a task (o to whom to delegate a task) is computationally had in geneal. This suggests development of appoximation algoithms fo measuing and adjusting autonomy. Ou futue wok includes looking into the elationship between maximum goup autonomy and least vaiance in goup autonomy. Refeences Banzhaf, J Weighted Voting Doesn t Wok: A Mathematical Analysis. Rutges Law Review 19: Babe, S., Matin, C Agent Autonomy: Specification, Measuement, and Dynamic Adjustment. In Poceedings of the Autonomy Contol

9 Softwae Wokshop, Agents '99, pp May 1-5, 1999, Seattle, WA. Bainov S., Sandholm T Powe, Dependence and Stability in Multiagent Plans. In Poceedings of AAAI 99, pp , Olando, Floida. Castelfanchi, C Social Powe. In Demazeau Y. and Mulle J.-P. eds. Decentalized AI - Poceedings of the Fist Euopean Wokshop on Modeling Autonomous Agents in a Multi-Agent Wold, pp Elsevie Science Publishes. Castelfanchi, C., 1995 Guaanties fo Autonomy in Cognitive Agent Achitectue. In N. Jennings and M. Wooldidge (eds.) Agent Theoies, Achitectues, and Languages, pp , Spinge-Velag. Castelfanchi, C Founding Agent s Autonomy on Dependence Theoy, In poceedings of ECAI 01, pp , Belin. Comen, T., Leiseson, C., Rivest, R Intoduction to Algoithms. MIT Pess. Dwokin, G The Theoy and Pactice of Autonomy. Cambidge. Hexmoo, H. 2000a. A Cognitive Model of Situated Autonomy. In Poceedings of PRICAI-2000 Wokshop on Teams with Adjustable Autonomy, Austalia. Hexmoo, H. 2000b. Towads Empiical Evaluation of Tadeoffs between Agent Qualities, In PRIMA-2000, (C. Zhang and V. Woo, eds), LNAI Volume 1881, Austalia. Hexmoo, H., Duchschee, H Efficiency as Motivation fo Teaming, In Poceedings of FLAIRS 2001, AAAI pess. Hexmoo, H., Kotenkamp, D Autonomy Contol Softwae, An intoductoy aticle of the special issue of Jounal of Expeimental and Theoetical Atificial Intelligence, Kluwe. Jennings, N., Sycaa, K., Wooldidge, M A Roadmap of Agent Reseach and Development. Autonomous Agents and Multi-Agent Systems, 1:7-38. Meyes, D. Self, Society and Pesonal Choice. Columbia Univesity Pess, Mele, A Autonomous Agents: Fom Self- Contol to Autonomy. Oxfod Univesity Pess. Sandholm, T., Lason, K., Andeson, M., Schehoy, O., Tohme, F Coalition Stuctue Geneation with Wost Case Guaantees. Atificial Intelligence, 111: Schneewind, J. B., The Invention of Autonomy: A Histoy of Moden Moal Philosophy, Cambidge Univ. Pess. Shapley, L., Shubik, M A Method fo Evaluating the Distibution of Powe in a Committee System. Ameican Political Science Review, 48: