Noa E. Kr-Dahav and Moshe Tennenholtz. Technion

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1 Multi-Agent Belief Reviion Noa E. Kr-Dahav and Mohe Tennenholtz Faculty of Indutrial Engineering and Management Technion Haifa 32000, Irael 1

2 Multi-Agent Belief Reviion Noa E. Kr-Dahav and Mohe Tennenholtz Faculty of Indutrial Engineering and Management Technion { Irael Intitute of Technology Haifa 32000, Irael Abtract We introduce a baic framework for multi-agent belief reviion in heterogeneou ocietie where agent are required to be conitent in their belief on hared variable. We identify everal propertie one may require a general multi-agent belief reviion operator to atify, and how everal baic implication of thee requirement. Our work reveal the connection between multi-agent belief reviion and the theory of ocial choice, and attempt to provide ome initial undertanding of the multi-agent belief reviion proce. 1 Introduction Conider an agent, a et of belief it adopt, and a new obervation it make. Taking the new obervation a a fact that hould be incorporated to it belief, the agent may need to revie it belief. Belief reviion i a baic reearch topic in AI [Gar88, KM91, FH94, ds92, Gro88, Doy91], and reearch on the theory of belief reviion eem to converge to a well-etablihed theory [Gar92]. Mot of the theory developed in thi area ha concentrated on ytem coniting of a ingle agent. Our aim in thi paper i to ue the current undertanding of the ingle agent belief reviion proce in order to dicu the more general proce of multi-agent belief reviion. Conider a et of agent operating in a hared environment. The agent may be robot, databae, or other articial entitie. Each of them may have it own perpective on the world, but uually there will be everal element that are hared among the agent. For example, in the context of ditributed

3 databae, dierent agent may have dierent data item they refer to, but there are alo common data item that everal agent may refer to. Auming that the perpective of an agent i captured by a et of belief, and that each agent may encounter new obervation, the agent may need to revie their belief. Mot previou workonmulti-agent belief reviion ha emphaized the reviion of an agent' model of other agent' model. For example, in [vdm94] Meyden conider the cae where new obervation are oberved by all agent imultaneouly, and a a reult of uch new obervation agent revie their belief about the belief of other agent. Other reearcher (e.g., Gapar [Ga91] and Gallier [Gal92]) conider ituation where obervation made by an individual agent maychange it belief and it belief about the belief of other agent, a well a eect the content of meage the oberver may wih to end to thee agent. In thi paper we wih to take another perpective onmulti-agent belief reviion. Thi paper initiate reearchon multi-agent belief reviion in heterogeneou ytem, where each agent may have it own perpective of the world but the agent need to coordinate (i.e., agree on) their belief on hared element. Unfortunately, excluding ome work on algorithmic apect of multi-agent belief reviion [HB91], no work ha addreed thi general heterogeneou multi-agent belief reviion proce. However, the latter i crucial for many application, uch a in the context of ditributed databae. In thi work we introduce a baic framework for multi-agent belief reviion, dicu everal requirement one may require a atifactory multi-agent belief reviion to atify, and preent ome baic implication of thee requirement. In particular, our work reveal a connection between multi-agent belief reviion and the theory of ocial choice, a connection we believe to be of ignicant importance. The tructure of thi paper i a follow. In Section 2 we briey dicu part of the theory regarding ingle agent belief reviion which we will make ue of in later ection. In Section 3 we introduce the multi-agent belief reviion etting. Section 4 lit our et of requirement from the multi-agent reviion proce. We conider conitency and rationality requirement, ocial rationality requirement, and additional emantic requirement. While thee requirement are not ultimate, we believe they capture the type of the propertie a atifactory multi-agent reviion operator hould have. In Sec-

4 tion 5 we preent the implication of our requirement. In particular, we preent everal baic reult regarding the exitence and uniquene of a atifactory multi-agent belief reviion operator. In Section 6 we further dicu our etting and reult, a well a ome related work. 2 Preliminarie The dicuion in the following ection will make ue of ome baic notion and reult of the theory of ingle agent belief reviion. Thi ection preent a brief introduction to thee notion and reult. Denition 2.1: Let = f' 1 ::: ' n g be a et of primitive propoition. Let L be the cloure of under : _. A belief iaentence KB 2L. Alternatively, a belief KB 2Li aociated with it et of truth aignment. 1 Given a current belief KB and a new obervation O, abelief reviion i a tranformation from KB 2Lto KB 0 2Lwhere all truth aignment of KB 0 are truth aignment of O. A belief reviion i therefore a proce in which an agent change it current belief in a way that the et of truth aignment aociated with the new belief will alo be truth aignment of the new acquired obervation. Naturally, not all belief reviion operator are atifactory. A atifactory (ingle agent) belief reviion operator i aumed to obey the AGM potulate [AGM85, Gar88]. Thee potulate aim to capture tability propertie, eliminating unneceary perturbation to the databae. For example, one of thee potulate tate that if a new acquired obervation i conitent with the agent' belief, then the agent' new belief i obtained by adding the new obervation to the current belief. In the cae of a propoitional knowledge bae, it ha been hown [KM91] that the AGM potulate coincide with the following condition (the Katuno and Mendelzon condition), where KB i the current belief,o i the current obervation, and KB O i the reult of reviing KB by O: 1 For a dicuion of variou approache to modeling belief and belief change the reader may conult [Gro88].

5 1. KB O implie O 2. If KB ^ O i atiable then KB O KB ^ O. 3. If O i atiable then KB O i alo atiable. 4. If KB 1 KB 2 and O 1 O 2 then KB 1 O 1 KB 2 O 2 5. (KB O) ^ O 0 implie KB (O ^ O 0 ) 6. If (KB O)^O 0 i atiable then KB(O ^ O 0 ) implie (KB O)^O 0 In the equel we will refer to the above condition a the KM condition. The above condition capture baic propertie required from a atifactory belief reviion operator. In addition, Katuno and Mendelzon upply the following repreentation theorem: Theorem 2.2 : A belief reviion operator atie the Katuno and Mendelzon condition if and only if there iaperitent aignment 2 which map each belief KB to a total pre-order KB over the poible aignment to o that the truth aignment to KB O are the minimal truth aignment of O with repect to KB. In thi paper we will not provide the detail of the above baic reult. However, the meage of thi reult, a well a of other reult which followed and extended it (e.g., [FH94]) i both imple and powerful. Roughly peaking, it can be ummarized a follow. A atifactory belief reviion operator can be viewed a an operator which generate a ranking (i.e., a total pre-order) over the et of poible aignment to a a function of the current belief, where the et of minimal element in thi ranking coincide with the current belief. Given a new obervation O the operator will chooe the minimal aignment in thi ranking that atify O to be the new (revied) belief. A a reult, we can view a ingle agent belief reviion operator a a mapping (R old O)! R new where R old and R new are ranking over the et of poible aignment to. 2 The denition of peritent aignment and other detail regarding thi reult are omitted from thi paper. For full detail ee [KM91].

6 3 Multi-Agent Belief Reviion In thi ection we dene a baic etting where multi-agent belief reviion can be invetigated. We treat multi-agent belief reviion a a two-phae proce and dicu our trategy for dening atifactory multi-agent belief reviion operator. In thi paper we will aume that there are only two agent in our ytem. 3 Denition 3.1: Let A = f1 2g be a et of agent functioning in a partially hared environment. The environment i decribed by a et of primitive propoition = f' 1 ::: ' n g. For each agent i there i a et i of primitive propoition that decribe agent i' view of the environment. A propoition ' 2 iaprivate propoition of agent i if ' 2 i n j for j 6= i. A propoition ' 2 iahared propoition if ' 2 i \ j for j 6= i. The private domain of agent i conit of all the private propoition of thi agent, while the hared domain of the agent conit of all of their hared propoition. An obervation by agent i i a et of poible truth aignment to a ubet of it private propoition or to a ubet of the hared propoition. The above denition capture a general etting where each agentha private propoition a well a other propoition which are hared with other agent. The hared domain dene alo the communication language for the agent. Agent will be able to communicate only about element of the hared domain. We make the aumption that an obervation refer either to private or to hared propoition, and that the hared domain include at leat three primitive propoition. We will alo make the aumption that only one agent canmake an obervation at a given point. 4 Given a new obervation the agent would need to incorporate it into their belief. Our main quetion i: what would be the tructure of a belief reviion proce in a multi-agent etting? 3 Our reult can be extended to ome n-party etting, where n>2. We will return to thi point in Section 6. Thi point i dicued in detail in [KD96]. 4 Similar etting can be dened for the cae where agent may have imultaneou obervation, or obervation about a mixture of private and hared propoition. Although the belief reviion operator which the agent ue may be dened for thee cae a well, our multi-agent etting aume the above-mentioned retriction.

7 Given the above etting we view the multi-agent belief reviion a a twophae proce. In the rt phae, if the new obervation contain information regarding i then agent i perform an individual belief reviion. We aume that an obervation about the hared domain i communicated by the oberver to the other agent hence, in the latter cae both agent would make an individual belief reviion. In the econd phae the agent may wih to coordinate their belief. The rt phae i in fact a ingle-agent belief reviion proce a invetigated in previou work the econd phae i till an unexplored tep which hould be carefully invetigated. Given the above dicuion we can dene: Denition 3.2: A multi-agent belief reviion i a mapping MULTI ; REV :(KB old 1 KBold 2 O a)! (KBnew 1 KB new 2 ) where KBi old i the old belief of agent i, KBi new i the new belief of agent i, O i an obervation, and a i the identity of the agent who made the obervation. The individual belief reviion phae i the rt phae of the multi-agent belief reviion proce, and it i a mapping Step 1 :(KB old 1 KBold O a)! 2 (KB I 1 KBI), where 2 KBI i = KB old i O if the value of a i i or O refer to the hared domain, and KB I i = KB old i otherwie. Auming the Katuno and Mendelzon condition hold for the individual belief reviion phae, we get that thi phae can be treated a a mapping Step 1 :(R old 1 Rold O a)! 2 (RI 1 RI 2 ), where Rold i and R I i are the ranking aociated with agent i before and after the individual belief reviion phae repectively. Given the above dicuion we are till left with a degree of freedom relative to the econd phae of the multi-agent belief reviion proce. Uing the terminology of ranking, we can treat the econd phae of the multi-agent belief reviion proce a a mapping Step 2 :(R I 1 RI 2 O a)! (Rnew 1 R new 2 ), where R I i i a dened above andr new i i the ranking baed on which individual belief reviion will be performed on the next belief reviion iteration. Notice that the above dicuion doe not aume that an agent need to manipulate explicit ranking. All we aume i that agent perform a twophae belief reviion proce, where the rt phae i a ingle agent belief

8 reviion proce which obey claical belief reviion potulate. In order to have a full characterization of multi-agent belief reviion, we would need to add a lit of requirement from Step 2 of the multi-agent belief reviion proce. Thi will be the topic of the following ection. 4 The Multi-Agent Belief Reviion Requirement In thi ection we dicu the econd phae of the multi-agent belief reviion proce that we preented in the previou ection. Motivated by the work on ingle agent belief reviion, our aim i to ugget an initial et of requirement from the multi-agent belief reviion proce and to invetigate what do thee requirement entail. For technical reaon, and for eae of expoition, we will preent the et of requirement in four tage. In the equel we will refer only to the requirement from the econd (joint) phae of the multi-agent belief reviion proce. Recall that thi proce can be decribed by the mapping Step 2 dened in the previou ection. 4.1 Conitency In thi work we are concerned with heterogeneou ytem where agent wih to coordinate their activitie. In particular, they would need to be conitent about their belief on the hared domain. Without uch an agreement, the ytem will lack ucient coordination. Denition 4.1: Let KB i be the belief, interpreted a a et of truth aignment, of agent i. Let u denote by KB i ( 1 \ 2 ) the projection of KB i on the hared domain. Thi projection i dened a the union of the projection of the truth aignment of KB i on 1 \ 2. Requirement 1 (Conitency): Let KBi new be the belief of agent i bytheendof the econd phae of a multi-agent reviion proce. Then, KB new 1 ( 1 \ 2 )= KB new 2 ( 1 \ 2 ).

9 The above requirement tate that the multi-agent belief reviion proce hould enure that agent will be conitent in their belief on the hared domain. For eae of expoition we aume that the belief of the agent are initially conitent. Given the conitency requirement we can view Step 2 a a two-tep mapping: Step 2:1 :(R I 1 RI 2 O a)! Step 2:2 :(R I 1 RI 2 O a)! (Rnew 1 R new 2 ) In the above mapping denote a et of aignment to the hared domain which the agent agree upon. We require that KB new 1 ( 1 \ 2 )= KB new 2 ( 1 \ 2 )=. Notice that Step 2:2 i, yet again, a ingle-agent belief reviion proce where agent need to revie their belief to be conitent with.however, Step 2:1 i a ubtle tep which hould be carefully dicued. 4.2 Rationality By now we have a general etting of multi-agent belief reviion where agent are required to be conitent in their belief on the hared domain. However, conitency doe not uce to obtain coordinated behavior. In particular, it i only natural to require that the agent will behave a a ingle agent a far a belief reviion baed on obervation about the hared domain i concerned. 5 Uing the machinery of Section 2 it i eay to ee that thi common-enical requirement can be tranlated into the following pair of requirement, which we call rationality requirement. Notice that thee requirement enforce a light change in the denition of Step 2:1. 5 The requirement from ingle-agent belief reviion implie that a ingle ranking will be aociated with each belief of an agent however, thi i an unnatural requirement when we conider only belief over the hared domain. Hence, in our etting the agent will need to behave a an agent with a particular belief reviion operator about the hared domain (which they need to implicitly agree upon at each point a we decribe), but thi belief reviion operator may change in the following iteration. Thi point i further dicued in [KD96] and in the full paper. We will return to thi point at Section 6

10 Requirement 2: The output of Step 2:1 i a joint ranking over the poible aignment to 1 \ 2. Requirement 3: If the obervation made i about the hared domain, then the belief on the hared domain will be revied baed on the previouly agreed upon ranking over the poible aignment to 1 \ 2. Notice that the above requirement are again a traightforward interpretation of the claical ingle-agent belief reviion theory, when applied to the hared domain. Given the above requirement, we can redene the multiagent belief reviion proce to conit of the following tep: Step 1 :(R old 1 Rold 2 Rold O a)! (R I 1 RI 2 ) Step 2:1 :(R I 1 RI 2 Rold O a)! (R new ) Step 2:2 :(R I 1 RI 2 Rnew O a)! (R new 1 R new 2 R new ) In the above mapping R old and R new denote the previouly agreed upon ranking over the hared domain, and the new agreed upon ranking over the hared domain, repectively. Initially, thee ranking are undened. Notice that the et of minimal element in R new and the et determined by projecting the minimal element in R new 1 and R new 2 on the hared domain, hould be identical. In a cae where the obervation O i about the hared domain, Step 2:1 would require that the minimal element in R new would coincide with the minimal element i R old that atify O. We would like to emphaize again that what we obtained here i a characterization of multi-agent belief reviion by mean of mapping between ranking. Thee however may be implicit ranking the key point ithatwe can view the multi-agent belief reviion proce a if it conit of a mapping between ranking a dened above. Notice that, given the above tructure, the multi-agent belief reviion proce i an iterative proce which i initiated eachtimeanewobervation i made. Each iteration of thi proce can be viewed a if it conit of the previouly mentioned tage. For eae of preentation, we will aume that the rt obervation made (when the ytem i initiated) i about the private domain of one of the agent.

11 4.3 Social Rationality Our previou dicuion left u with a degree of freedom relative tostep 2:1. Thi tep hould determine the new joint ranking of the agent over the hared domain. Naturally, thi new ranking hould be a function of the individual ranking of the agent over the hared domain in the beginning of Step 2:1. The individual ranking of agent i over the hared domain, R I S i,canbe extracted from it individual ranking R I i a follow. Let W denote the et of poible aignment to i, and let W denote the et of poible aignment to 1 \ 2. For every w 2 W, let G(w ) 2 W be a minimal element according to R I i which project w on the hared domain. We cannow dene the ranking R I S i by dening w w 0 to hold if and only if G(w ) G(w 0 ) in R I. i The mapping dened in Step 2:1 hould take into account the individual ranking of the agent, ince thee ranking capture the private belief of the agent. Therefore, we hould require ocial rationality about the way R new i built given R I S 1 and R I S 2. Requirement for ocial rationality can be found in the ocial choice literature [LR57, Arr63]. Thee requirement were already ued in the AI literature in a omewhat imilar context [DW89]. We will adopt the following baic ocial rationality requirement. Requirement 4 (Generality): The mapping dened in Step 2:1 hould be de- ned for all R I S 1 and R I S 2. Requirement 5 (Weak Pareto Optimality): For every x y 2 W,ifx<yboth in R I S 1 and in R I S 2 then x<yhould alo hold in R new Requirement 6 (Independence of Irrelevant Alternative): For every x y 2 W, the relation between x and y in R new depend only on their relation in R I S 1 and R I S Semantic Requirement The lat et of requirement relate to the fact the proce dened above hould be a fair, adequate, and cautiou belief reviion proce.

12 Requirement 7: The reult of the multi-agent belief reviion proce hould be independent of the name of the agent (i.e., 1 and 2). Requirement 8: If an obervation about the private domain of agent i ha changed it ranking over the hared domain then R new hould be dierent from R old. Requirement 8 may eem a bit technical, but it capture the deire that change caued by private obervation cannot be imply neglected. Notice that the need to conider hared obervation ha already been addreed by other requirement. Requirement 9: Let R new1 and R new2 be two candidate for R new in a particular iteration of a multi-agent belief reviion proce which atie requirement 1{8. Aume that R new1 the k-th rank of R new1 and R new2 proce hould prefer to chooe R new1 coincide until the k'th rank, but. Then, the rather than R new2 trictly contain the k-th rank of R new2. Requirement 9 capture the need to be a cautiou a poible in ignoring ome of the poible aignment, which iatypical aumption in nonmonotonic reaoning and belief acription context [BT94]. Although thi requirement i natural in many domain, one can conider ituation in which thi requirement i not eential for dening atifactory multi-agent belief reviion operator. Reult imilar to the one preented in the following ection can be obtained if we drop requirement 9 or replace it by imilar requirement. 5 Implication of the Multi-Agent Belief Reviion Requirement The previou ection dened a et of requirement from the multi-agent belief reviion proce. Although we believe that thee requirement are not ultimate one, we think they capture the type of requirement a atifactory multi-agent reviion operator hould atify. Hence, it may be of coniderable importance to tudy the implication of thee requirement on the identity of a atifactory multi-agent belief reviion operator. Firt, we can how that:

13 Theorem 5.1 : Any mapping of R I S 1 and R I S 2 to R new requirement 2,4-6, and 9, would have the property that R new = R I S R new 2. which atie = R I S 1 or The above theorem implie that given two individual ranking over the hared domain in Step 2:1, a mapping atifying our requirement would output one of thee ranking a R new. Thi reult i mainly an implication of Arrow' Impoibility Theorem [Arr63]. Notice that the proce we conider conit of a equence of iteration, where in each iteration a multi-agent belief reviion operator i applied. The above theorem implie that given a particular iteration of the multi-agent belief reviion proce, if we denote the individual ranking of the agent over the hared domain in Step 2:1 by Rank 1 and Rank 2 repectively, then R new will be either Rank 1 or Rank 2. Given thi we can how that: Theorem 5.2: Auming the agent have xed yntactic name, there i no multi-agent belief reviion proce atifying requirement 1{9. Moreover, there i no multi-agent belief reviion proce atifying requirement 1{6 and 8{9. The above theorem refer to the cae where we ue xed yntactic name for the agent. However, one may think of emantic name, where the name of the agent are not xed, but are determined by the ytem' hitory. For example, one may wih to ay that a far a joint deciion are concerned, agent 1 i taken to be the agent who oberved the latet obervation. The yntactic name of the agent will then play no role in joint deciion. Such emantic name can be captured by anaming function which map R I S 1 and R I S 2 to Rank 1 and Rank 2 baed on the previou hitory of the agent. 6 The requirement about the connection between R I S 1 and R I S 2 to the new joint ranking, will be replaced by imilar requirement about the connection between Rank 1 and Rank 2 to the new joint ranking. The cae where Rank i = R I S i for i =1 2, independently of the previou hitory, give u the cae of xed yntactic name. 6 Naturally, wemay wih to conider only the part of the hitory which both agent can refer to.

14 In the equel, the hitory of agent will include the equence of previou belief on the hared domain and obervation made. The naming function will be applied at each iteration to the agent' joint hitory. Formally, thi hitory i a equence (B 0 Ob 1 B 1 Ob 2 B 2 ::: B n;1 Ob n ), where B i i the agent' belief on the hared domain after the i` obervation, and Ob i i the identity of the agent who made the i-th obervation, where Ob n refer to the identity of the lat agent to initiate a belief reviion iteration the hitory alo tore the type (i.e., private or hared) of each Ob i for each Ob i (0 <i<n) which refer to an obervation about the hared domain, the hitory may alo tore the value of the correponding obervation. Naturally, the naming function will not refer to the content of private obervation but only to the identity of their oberver. Recall that in our tudy we refer to a equence of iteration where each iteration conit of a ingle multi-agent belief reviion proce. The naming function i the tool by which fairne requirement and other emantic conideration can be better incorporated into the ytem. Given that there i an agent who would dictate the new ranking in a given iteration, the naming function can determine the identity of the dictator. We can how that: Theorem 5.3: Let Rank 1 = R I S i where i i the agent who made the lat obervation. Then, there i no multi-agent belief reviion proce atifying requirement 1-9. Theorem 5.4: Let Rank 1 = R I S i where i i the agent who made the lat private obervation (i.e., obervation about a private domain). Then, there exit a multi-agent belief reviion proce atifying requirement 1-9. Thi multi-agent belief reviion atie that R new = Rank 1 at each iteration. The above theorem how that if the naming function alway elect a agent 1(oralway elect a agent 2) the agent whowa the lat to make an obervation, then we will not get a atifactory multi-agent belief reviion however, if the naming of agent will change only after having private obervation then we will get the exitence of a atifactory multi-agent belief reviion proce. Moreover, we can how that the above naming function i the only one which will yield a atifactory multi-agent belief reviion proce:

15 Theorem 5.5: The naming function dened intheorem 5.4, i the only naming function under which requirement 1{9 are atied. The above theorem ugget to u a particular atifactory multi-agent belief reviion operator. According to thi operator the reviion proce will be carried out by the agent which wa the lat to make a private obervation, uing the original belief reviion operator of thi agent. The outcome of the reviion made by thi agent will be announced to the other agent, which will then revie it belief appropriately. Notice that thi multi-agent belief reviion operator i eaily implementable. Each agent will need to keep track of which agent ha made the lat private obervation, but the reviion itelf will be carried out uing the ingle agent belief reviion operator the agent are occupied with. 6 Dicuion In the previou ection we have introduced a baic multi-agent belief reviion etting, and preented ome baic requirement from a multi-agent belief reviion proce. We have hown everal implication of thee requirement thee include reult pertaining the exitence and uniquene of a atifactory multi-agent belief reviion operator. In thi paper we made ome aumption which can be relaxed. For example, one may conider dierent variant of the rationality requirement, and of the pareto-optimality aumption. Neverthele, imple variant of thee requirement can be hown to lead to imilar reult. However, it i not our claim that our requirement are ultimate. Our belief however i that our requirement and reult upply an initial rigorou etting that enable the initiation of reearch on multi-agent belief reviion in heterogeneou ocietie. The reader hould notice that we have not required that the agent will behaveaiftheyhave a xed belief reviion operator over the hared domain. At each iteration, a reviion of the joint belief over the hared domain will atify the KM condition however, if we conider the proce a a whole (i.e., require the KM condition regarding reviion that may be made in dierent iteration) then condition 4 of the KM condition may not be atied. Thi i implied by the fact that we view tep 2.1 of each iteration, a a deciion about

16 the joint belief reviion operator (over the hared domain) for the following iteration. Naturally, thi deciion may depend on the whole information the agent have, and not only on their deciion on joint belief over the hared domain. In thi paper we have retricted ourelve to the cae of two agent. One may conider a larger group of agent. Our reult can be extended to the cae where a xed number of agent hare a ingle hared domain (i.e., each primitive propoition i either aociated with only one of the agent or i aociated with all of the agent). In thi cae the multi-agent belief reviion proce and the correponding requirement are imilar to the one which we have ued in the cae of two agent, and lead to imilar reult. If dierent agent may have dierent hared domain then the problem i much more complicated and further tudy eem to be required. We mentioned ome related work in the body of thi paper. We hould emphaize that our work ha been inpired by and i complementary to everal line of reearch. Firt, it i related to work on applying choice theory to the AI context [DW89, Doy91]. Second, it i related to work on the general theory of multi-agent belief reviion for uncoordinated 7 agent [vdm94, Gal92, Ga91], and to the tudy of algorithmic apect of multiagent belief reviion [HB91]. Le directly, thiwork i related to work on multi-agent non-monotonic reaoning [Mor90] a well a to work on peech act [GS90, AK88]. Lat but not leat our work heavily relie on the trong foundation upplied by theorie of ingle-agent belief reviion [Gar88, KM91, FH94, ds92, GM88, Bou92, Gro88], which enable u to treat the ingle-agent reviion proce a a building block forthemulti-agent cae. Our contribution i the introduction of baic principle and reult for the general theory of multi-agent belief reviion where agent are required to be both rational and conitent in their reviion. Appendix: Sketch of Proof Proof of Theorem 5.1: 7 See our dicuion in the introduction of thi paper.

17 In order to atify requirement 2,R new hould be a joint ranking over the poible aignment to 1 \ 2. Requirement 4-6 coincide with the condition of Arrow' Impoibility Theorem [Arr63]. In Arrow' etting, one wihe to generate a joint ranking from a et of individual ranking in a way that will be general, (weak) pareto-optimal, and independent of irrelevant alternative. Arrow' Impoibility Theorem implie that any mapping from a et of ranking to a joint ranking, which atie thee requirement, will be a dictatorial mapping. In a dictatorial mapping there i an agent uch that for every pair (x y) of aignment to 1 \ 2 if thi agent trictly prefer x over y then x will be trictly preferred over y in the joint ranking. Thi implie that in any mapping from (R I S 1 R I S 2 ) toajoint ranking, which atie requirement 2,4-6, one of the agent' individual ranking (e.g., R I S 1 or R I S 2 ) will be the bai of the joint ranking I.e., every trict preference in thi ranking will be impoed on the joint ranking. At the mot, the mapping will allow the other agent to inuence the order of aignment among which the dictatorial agent i indierent. However, Requirement 9 implie that in generating the joint ranking we will prefer to adopt each rank of the dictator a i. Proof of Theorem 5.2: The fact that requirement 1{9 are not imultaneouly atied i an immediate corollary of Theorem 5.1. The proof of Theorem 5.1 alo implie w.l.o.g that agent 1 will dictate the joint ranking at each iteration. A a reult, a private obervation byagent 2 whichchange it belief on the hared domain will not change the joint ranking. Thi contradict requirement 8. Proof of Theorem 5.3: There are two poible mapping of (Rank 1 Rank 2 )tor new. If R new = Rank 1 then the dictatorial agent at each iteration i the agent who made the lat obervation. If R new = Rank 2 then the dictatorial agent ateach iteration i the agent who did not make the lat obervation. Conider the cae where R new = Rank 1. Aume that the rt obervation i a private obervation made byagent 1, and that the econd obervation i a hared obervation made by agent 2.Given that R new = Rank 1, agent 1 will determine the joint ranking of the agent at the end of the rt iteration, and agent 2 will determine the outcome at the end of the econd iteration. Notice

18 that Requirement 3 require that given an obervation about the hared domain the belief over the hared domain will be revied baed on R old. However, ince agent 1 i the dictator of the rt iteration, and it dictate only the belief in the beginning of the econd iteration (and not the agent' individual ranking in thi tage) we get that Requirement 3 can not in general be atied. Conider the cae where R new = Rank 2. In thi cae it i eay to ee that aprivate obervation will never aect the joint ranking and requirement 8i not atied. Proof of Theorem 5.4: In tep 2:2 the agent which i not the dictator hould revie it belief to coincide with the dictator' belief over the hared domain. Thi implie that Requirement 1 i atied. The outcome of tep 2:1 i R I S i where i i the agent who made the lat private obervation. R I S i i a ranking over the poible aignment to 1 \ 2 therefore, the outcome of tep 2:1 i a ranking over the poible aignment to 1 \ 2, and Requirement 2 i atied. In an iteration initiated by an obervation over the hared domain, the dictator i the dictator of the previou iteration. Without lo of generality, if agent 1wa the dictator of the previou iteration then in the previou iteration R new = R I S 1 and agent 1 need not perform additional reviion in tep 2:2.Given that no change of dictator can occur between that (previou) iteration and the current one, the new reviion will be made baed on the individual ranking over the hared domain of agent 1,which coincide with the agreed upon ranking of the previou tage. Hence, Requirement 3i atied. The mapping in tep 2:1 i a fully dictatorial mapping, which ue only a emantic conideration when deciding on the identity of the dictator. Hence, it i eay to check that Requirement 4-7 and 9 are atied. If agent i make a private obervation then it determine the joint ranking. Hence, the eect of the new obervation on the agent' view of the hared domain are fully reected in R new, and Requirement 8 i atied. Proof of Theorem 5.5:

19 The proof follow from the following pair of lemma. Lemma 1: Any multi-agent belief reviion proce that atie requirement 2-6 and 9 will be a proce in which the dictator under a hared obervation ha been alo the dictator of the previou iteration. Proof of Lemma 1: A we have hown in the proof of Theorem 5.1, if the proce atie requirement 2,4-6 and 9 then R new = Rank 1 or R new = Rank 2. Requirement 3 demand that if we are reviing according to a hared obervation, the minimal aignment according to R new will be identical to the minimal aignment according to R old which atify the new obervation. The only way to enure thi (without relating to the agent' current belief and obervation or to the complete previou ranking) i to have athe dictator the agent whoe individual ranking over the hared domain by the end of the previou iteration coincide with R old. Thi agent need to be the dictator of the previou iteration. Lemma 2: Any multi-agent belief reviion proce that atie requirement 2,4-6,8 and 9 will be a proce in which the dictator under a private obervation i the agent who made the current obervation. Proof of Lemma 2: If the proce atie requirement 2,4-6 and 9 then R new = Rank 1 or R new = Rank 2. Requirement 8 demand that an agent who made a private obervation will be able to inuence the joint ranking and will not be ignored. Any mapping that will allow inomepointfora non-oberver to dictate the joint ranking after a private obervation will be potentially blocking thi inuence. Given our aumption about the naming function and the need for atifying requirement8,wemut allow the oberver to dictate the joint ranking which reult from a private obervation. Reference [AGM85] C. Alchourron, P. Gardenfor, and D. Makinon. On the logic of theory change: Partial meet contraction function and their aociated reviion function. Journal of Symbolic Logic, 50:510{ 530, 1985.

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