# belief_revision_games__915f1801.pdf

Belief Revision Games

Nicolas Schwind Transdisciplinary Research Integration Center National Institute of Informatics Tokyo, Japan email: schwind@nii.ac.jp

Katsumi Inoue National Institute of Informatics The Graduate University for Advanced Studies Tokyo, Japan email: inoue@nii.ac.jp

Gauvain Bourgne CNRS & Sorbonne Universit es, UPMC Univ Paris 06, UMR 7606, LIP6, F-75005, Paris, France email: Gauvain.Bourgne@lip6.fr

S ebastien Konieczny CRIL - CNRS Universit e d Artois Lens, France email: konieczny@cril.fr

Pierre Marquis CRIL - CNRS Universit e d Artois Lens, France email: marquis@cril.fr

Belief revision games (BRGs) are concerned with the dynamics of the beliefs of a group of communicating agents. BRGs are zero-player games where at each step every agent revises her own beliefs by taking account for the beliefs of her acquaintances. Each agent is associated with a belief state defined on some finite propositional language. We provide a general definition for such games where each agent has her own revision policy, and show that the belief sequences of agents can always be finitely characterized. We then define a set of revision policies based on belief merging operators. We point out a set of appealing properties for BRGs and investigate the extent to which these properties are satisfied by the merging-based policies under consideration.

Introduction In this paper, we introduce belief revision games (BRGs), that are concerned with the dynamics of the beliefs of a group of communicating agents. BRGs can be viewed as zero-player games: at each step of the game each agent revises her current beliefs (expressed in some finite propositional language) by taking account for the beliefs of her acquaintances. The aim is to study the dynamics of the game, i.e., the way the beliefs of a group of agents evolve depending on how agents are ready to share their beliefs. BRGs could be useful to model the evolution of beliefs in a group of agents in social networks, and to study several interesting notions such as influence, manipulation, gossip, etc. In this paper we mainly focus on the definition of BRGs, using formal tools coming from belief change theory, and investigate their behavior with respect to a set of expected logical properties. Let us introduce a motivating example of a BRG.

Example 1 Consider a group of three undergraduate students, Alice, Bob and Charles, following the same CS curriculum. Bob is a friend of both Alice and Charles, but Alice and Charles do not know each other. Alice, Bob and

Copyright c 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

Charles want to prepare the final exam of the Basics of programming course. Each student has some feelings about the topics which will be considered by their teacher for this exam. At start, Alice believes that Binary search will not be among the topics of the final exam, unlike Bubble sort ; Bob believes that Binary search will be kept by the teacher, and that if Bubble sort is kept then Quick sort will be chosen as well by the teacher; finally, Charles just feels that Binary search will not be considered by the teacher. Each pair of friends exchange their opinions by sending e-mails in the evening. Each student is ready to make her opinion evolve by adopting the opinions of her friends when this does not conflict with hers, and by considering as most plausible any state of affairs which is as close as possible to the set of opinions at hand (her own one plus her friends ones) in the remaining case. At the end of each day, Alice e-mails to Bob with her feelings, Bob to both Alice and Charles, and Charles to Bob. One is asked now about what can be inferred from this description. Some of the key questions are: (1) How beliefs must be updated? (2) Will agents always agree on some pieces of belief if they agree on it at the beginning of the game? (3) Will they eventually stop changing their beliefs? In the following, we present a formal setting for BRGs. Our very objective is to provide some answers to the questions above. Thus, we address question (1) by putting forward a set of revision policies which are based on existing belief merging operators from the literature and the induced belief revision operators. We identify a set of valuable properties for BRGs. They include unanimity preservation which models question (2) and convergence which models question (3). For each revision policy under consideration, we determine whether such properties are satisfied or not. An extended version (including proofs) is available at http://www.cril.fr/brg/brg-long.pdf.

Belief Revision Games Belief sets are represented using a propositional language LP defined from a finite set of propositional variables P and

Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence

the usual connectives. (resp. ) is the Boolean constant always false (resp. true). An interpretation is a total function from P to {0, 1}. The set of all interpretations is denoted W. An interpretation ω is a model of a formula ϕ LP if and only if it makes it true in the usual truth functional way. Mod(ϕ) denotes the set of models of the formula ϕ, i.e., Mod(ϕ) = {ω W | ω |= ϕ}. |= denotes logical entailment and logical equivalence, i.e., ϕ |= ψ iff Mod(ϕ) Mod(ψ) and ϕ ψ iff Mod(ϕ) = Mod(ψ). A profile K = ϕ1, . . . , ϕn is a finite vector of propositional formulae. Two profiles of formulae K1 = ϕ1 1, . . . , ϕ1 n and K2 = ϕ2 1, . . . , ϕ2 n are said to be equivalent, denoted K1 K2 if there is a permutation f over {1, . . . , n} such that for every i 1, . . . , n, ϕ1 i ϕ2 f(i). Let us now introduce the formal definition of a Belief Revision Game. Definition 1 (Belief Revision Game) A Belief Revision Game (BRG) is a 5-tuple G = (V, A, LP, B, R) where V = {1, . . . , n} is a finite set; A V V is an irreflexive binary relation on V ; LP is a finite propositional language; B is a mapping from V to LP; R = {R1, . . . , Rn}, where each Ri is a mapping from LP LP in(i) to LP with in(i) = |{j | (j, i) A}| the in-degree of i, such that for all formulae ϕ1 0, ϕ1 1, . . . , ϕ1 in(i), ϕ2 0, ϕ2 1, . . . , ϕ2 in(i), if ϕ1 0 ϕ2 0 and ϕ1 1, . . . , ϕ1 in(i) ϕ2 1, . . . , ϕ2 in(i) , then Ri(ϕ1 0, ϕ1 1, . . . , ϕ1 in(i)) Ri(ϕ2 0, ϕ2 1, . . . , ϕ2 in(i)), and such that if in(i) = 0, then Ri is the identity function. Let G = (V, A, LP, B, R) be a BRG. The set V represents the set of agents under consideration in G. The set A represents the set of acquaintances between the agents. Intuitively, if (i, j) A then agent j is aware of the beliefs of agent i in the sense that agent i communicates her beliefs to agent j during the game. The set B represents each agent s beliefs expressed by a formula from LP: for each i V , the formula B(i) (noted Bi for short) is called a belief state and represents the initial beliefs of agent i. Lastly, each element Ri R is called the revision policy of agent i. Let us denote Ci the context of i, defined as the sequence Ci = Bi1, . . . , Biin(i) where i1 < < iin(i) and {i1, . . . , iin(i)} = {ij | (ij, i) A}. Then Ri(Bi, Ci) is the belief state of agent i once revised by taking into account her own current beliefs Bi and her current context. It is assumed by definition that all beliefs are considered up to equivalence (i.e., the syntactical form of the beliefs does not matter) and that an agent s beliefs do not evolve spontaneously when she has no neighbor. Playing a BRG consists in determining how the beliefs of each agent evolve each time a revision step is performed. This calls for a notion of belief sequence , which makes precise the dynamics of the game: Definition 2 (Belief Sequence) Given a BRG G = (V, A, LP, B, R) and an agent i V , the belief sequence of i, denoted (Bs i )s N, states how the beliefs of agent i evolve while moves take place. (Bs i )s N is inductively defined as follows:

B0 i = Bi; Bs+1 i = Ri(Bs i , Cs i ) for every s N, where Cs i is the context of i at step s.

Bs i denotes the belief state of agent i after s moves. Since LP is a finite propositional language, there exists only finitely many formulae up to equivalence, hence only finitely many belief states can be reached. To make it formal, we need the concept of belief cycle:

Definition 3 (Belief Cycle) A sequence (Ks)s N of formulae from LP is cyclic if there exists a finite subsequence Kb, . . . , Ke such that for every j > e, we have Kj Kb+((j b)mod(e b+1)). In this case, the (characteristic) belief cycle of (Ks)s N is defined by the subsequence Kb, . . . , Ke for which b and e are minimal.

By the above argument, it is easy to prove that:

Proposition 1 For every BRG G = (V, A, LP, B, R) and every agent i V , the belief sequence of i is cyclic.

As a consequence, each agent i is associated with a belief cycle which we simply denote Cyc(Bi): the belief sequence of every agent i (which is an infinite sequence) can always be finitely described, since it is entirely characterized by its initial segment B0 i , B1 i , . . . , Bb 1 i and its belief cycle Cyc(Bi) = Bb i , Bb+1 i , . . . , Be i , which will be repeated (up to equivalence) ad infinitum in the sequence. In the following, we are interested in determining the pieces of beliefs which result from the interaction of the agents in a BRG, focusing on the agents belief cycles. A formula ϕ is considered accepted by an agent when it holds in every state of its belief cycle, which means that from some step s, ϕ will always hold. Then we define the notion of acceptability at the agent level and at the group level:

Definition 4 (Acceptability) Let G = (V, A, LP, B, R) be a BRG and ϕ LP. ϕ is accepted by i V if and only if for every Bs i Cyc(Bi), we have Bs i |= ϕ. ϕ is unanimously accepted in G if and only if ϕ is accepted by all i V .

A case of interest is when |Cyc(Bi)| = 1, i.e., the belief cycle of agent i has length 1. In such a case, the beliefs of agent i stabilize once the belief cycle is reached. A specific case is achieved by stable BRGs:

Definition 5 (Stability) Let G = (V, A, LP, B, R) be a BRG. A belief state Bi B is said to be stable in G if |Cyc(Bi)| = 1. The BRG G is said to be stable iff each Bi B is stable in G.

Stability of a game is an interesting property, since it says in a sense that we reach some equilibrium point, where no agent further changes her belief. These two concepts will take part of some further properties on BRGs which we will introduce and investigate in the following.

Merging-Based Revision Policies While all kinds of possible revision policies are allowed for BRGs, we now focus on revision policies R that are rationalized by theoretical tools from Belief Change Theory (see e.g. (Alchourr on, G ardenfors, and Makinson 1985)), in

particular belief merging and belief revision operators. Before introducing specific classes of revision policies of interest, let us introduce some necessary background on belief merging and belief revision. Formally, given a propositional language LP a merging operator is a mapping from LP LP n to LP. It associates any formula µ (the integrity constraints) and any profile K = K1, . . . , Kn of belief states with a new formula µ(K) (the merged state). A merging operator aims at defining the merged state as the beliefs of a group of agents represented by the profile, under some integrity constraints. A set of nine standard properties denoted (IC0) (IC8) are expected for merging operators (Konieczny and Pino P erez 2002). Such operators are called IC merging operators. For space reasons, we just recall those used in the rest of the paper:

(IC0) µ(K) |= µ;

(IC1) If µ |= , then µ(K) |= ;

(IC2) If V K K K µ |= , then µ(K) V K K K µ;

(IC3) If K1 K2 and µ1 µ2, then µ1(K1) µ2(K2);

(IC4) If K1 |= µ, K2 |= µ and µ( K1, K2 ) K1 |= , then µ( K1, K2 ) K2 |= .

A couple of additional postulates have been investigated in the literature, which are appropriate for some merging scenarios. We recall below one of them, Disjunction (Everaere, Konieczny, and Marquis 2010):

(Disj) If W K µ is consistent, then µ(K) |= W K.

(Disj) is not satisfied by all IC merging operators but is expected in the case when it is assumed that (at least) one of the agent is right (her beliefs hold in the actual world), but we do not know which one. Distance-based merging operators d,f are characterized by a pseudo-distance d (i.e., triangular inequality is not mandatory) between interpretations and an (aggregation) function f from R+ R+ to R+ (some basic conditions are required on f, including symmetry and nondecreasingness conditions, see (Konieczny, Lang, and Marquis 2004) for more details). They associate with every formula µ and every profile K a belief state d,f µ (K) which satisfies Mod( d,f µ (K)) = min(Mod(µ), d,f K ), where d,f K is the total preorder over interpretations induced by K defined by ω d,f K ω if and only if df(ω, K) df(ω , K), where df(ω, K) = f K K{d(ω, K)} and d(ω, K) = minω |=K d(ω, ω ). Usual distances are d D, the drastic distance (d D(ω, ω ) = 0 if ω = ω and 1 otherwise), and d H the Hamming distance (d H(ω, ω ) = n if ω and ω differ on n variables). IC merging operators include some distance-based ones. We mention here two subclasses of them: the summation operators d,Σ (i.e., the aggregation function is the sum Σ) and the GMin operators d,GMin. GMin operators1 associate with every formula µ and every profile K

1Here we give an alternative definition of d,GMin by means of lists of numbers. However using Ordered Weighted Averages, one

ω K1 K2 K3 dΣ H(ω, K) d GMin H (ω, K) 11 0 2 2 4 (0, 2, 2) 10 1 1 1 3 (1, 1, 1) 01 1 1 1 3 (1, 1, 1)

Table 1: The merging operators d H,Σ and d H,GMin.

a belief state d,f µ (K) which satisfies Mod( d,f µ (K)) = min(Mod(µ), K), where d,GMin K is the total preorder over interpretations induced by K defined by ω d,GMin K ω if and only if d GMin(ω, K) lex d GMin(ω , K) (where lex is the lexicographic ordering induced by the natural order) and d GMin(ω, K) is the vector of numbers d1, . . . , dn obtained by sorting in a non-decreasing order the multiset d(ω, Ki) | Ki K .

Example 2 Let P = {a, b}, K = K1, K2, K3 where K1 = a b, K2 = K3 = a b, and µ = a b. We consider both summation and GMin operators based on the Hamming distance. Table 1 shows for each interpretation ω Mod(µ) the distances d H(ω, Ki) for i {1, 2, 3}, and the distances dΣ H(ω, K) and d GMin H (ω, K) (interpretations ω are denoted as binary sequences following the ordering a < b). We get that d H,Σ µ (K) (a b) ( a b) and d H,GMin µ (K) a b.

Noteworthy, summation operators and GMin operators satisfy all (IC0) (IC8) postulates (whatever the pseudodistance under consideration), and additionally, GMin operators satisfy (Disj), as well as the operator d D,Σ = d D,GMin ((Disj) is not satisfied by d H,Σ). Belief revision operators can be viewed as belief merging operators restricted to singleton profiles: the revision K1 K2 of a belief state K1 by another belief state K2 consists in merging the singleton profile K1 under the integrity constraints K2. Accordingly, if is an IC merging operator then the revision operator induced by defined for all states K1, K2 as K1 K2 = K2( K1 ) satisfies the standard AGM revision postulates (Alchourr on, G ardenfors, and Makinson 1985; Katsuno and Mendelzon 1992).

We are now ready to introduce several classes of revision policies Ri which are parameterized by an IC merging operator and for some of them, by the corresponding revision operator .2 Let G = (V, A, LP, B, R) be a BRG. In the following, we assume for the sake of simplicity that all agents i V apply the same revision policy, i.e., given an IC merging operator , for all Ri R, Ri = R . Then let us consider the following revision policies, defined at each step s for any agent i who has a non-empty context Ci:

Definition 6 (Merging-Based Revision Policies)

R1 (Bs i , Cs i ) = ( Cs i );

could fit the definition of a distance-based operator (Konieczny, Lang, and Marquis 2004). 2When using a merging operator without integrity constraints we just note (K) instead of (K) for improving readibility.

step i Bi 1 Bi 2 Bi 3 0 s b s (b q) s 1 b q s b q b q 2 s b q s b q s b q

Table 2: The belief sequences of Alice, Bob and Charles.

R2 (Bs i , Cs i ) = ( Cs i )( Bs i ) [= Bs i ( Cs i )];

R3 (Bs i , Cs i ) = ( Bs i , Cs i ); R4 (Bs i , Cs i ) = ( Bs i , ( Cs i ) ); R5 (Bs i , Cs i ) = Bs i ( ( Cs i )) [= ( Cs i ) Bs i ];

R6 (Bs i , Cs i ) = Bs i ( Cs i ).

First of all, please note that since (IC3) requires to be syntax-independent (i.e., profiles and integrity constraints are considered up to equivalence), these revision policies are all consistent with the conditions given in Definition 1. Intuitively, these strategies are ranked according to the relative importance given to each agent s beliefs compared to her neighbors opinion. For R1 , only the aggregated opinion of the neighbors is relevant. For R2 , the current opinion of the agent is revised by the aggregated opinion of the neighbors; doing so, an agent is ready to adopt the part of the merged beliefs of her neighbors which are as close as possible to her own current beliefs. For R3 the agent considers that her opinion is as important as each one of her neighbors. For R4 the agent considers that her opinion is as important as the aggregated opinion of her neighbors. For R5 and R6 , the agent does not give up her current beliefs and just accepts additional information compatible with them. Noteworthy, R5 and R6 are not equivalent: for R5 the agent first aggregates her neighbors opinion, and then revise the merged result by her own opinion; for R6 the agent proceeds with her neighbors opinion and her own one in a single step.3

Example 1 (continued) We formalize the example presented in the introduction as the BRG G = (V, A, LP, B, R) defined as follows. Let V = {1, 2, 3} where 1 corresponds to Alice, 2 to Bob, and 3 to Charles. A = {(1, 2), (2, 1), (2, 3), (3, 2)} expresses that Alice and Bob are connected, and that Bob and Charles are connected. LP is built up from the set of propositional variables P = {s, b, q}, where s stands for Binary Search , b for Bubble Sort and q for Quick Sort . The initial beliefs of agents are expressed as B1 = s b, B2 = s (b q) and B3 = s. Since in the case of conflicting beliefs, each agent considers to merge her friends opinions and her own one together, revision policies R3 are appropriate candidates for each agent. Let us consider the summation operator based on the Hamming distance. We have R1 = R2 = R3 = R3 d H ,Σ. The belief sequences associated with the three agents are given in Table 2: the belief cycle of agent 1 (resp. 2, 3) is given by (B2 1) (resp. (B1 2), (B2 3)). G is a stable game. Note that s b q is unanimously accepted in G (as well as all formulae entailed by it).

3Consider for instance Ci = p q, p, p q and Bi = p. Then R5 d D,Σ(Bi, Ci) p q whereas R6 d D,Σ(Bi, Ci) p q.

Logical Properties for Belief Revision Games We introduce now some expected logical properties for BRGs, and investigate which BRGs satisfy them depending on the chosen revision policy. While the properties hereafter are relevant to all BRGs, we focus on BRGs which are instantiated with revision policies from the six classes defined in the previous section, and assume that the same revision policy is applied for each agent. Given a revision policy Rk , G(Rk ) is the set of all BRGs (V, C, LP, C, R) where for each Ri R, Ri = Rk . Additionally, Rk is said to satisfy a given property P on BRGs if all BRGs from G(Rk ) satisfy P. We start with a set of preservation properties which are counterparts of some postulates on belief merging operators (cf. previous section). These properties express the idea that the interaction between agents should not lead them to degrade their belief states.

Definition 7 (Consistency Preservation (CP)) A BRG G = (V, A, LP, B, R) satisfies (CP) if for each Bi B, if Bi is consistent then all beliefs from (Bs i )s N are consistent.

(CP) requires that agents with consistent initial beliefs never become self-conflicting in their belief sequence. It is the direct counterpart of (IC1) for merging operators:

Proposition 2 For every k {1, . . . , 6}, Rk satisfies (CP) if satisfies (IC1).

Definition 8 (Agreement Preservation (AP)) A BRG G = (V, A, LP, B, R) satisfies (AP) if given any consistent formula ϕ LP, if for each Bi B, ϕ |= Bi then for each Bi B and at every step s 0, ϕ |= Bs i .

(AP) requires that if all agents initially agree on some alternatives, then they will not change their mind about them. It corresponds to (IC2) for merging operators:

Proposition 3 For every k {1, . . . , 6}, Rk satisfies (AP) if satisfies (IC2).

Definition 9 (Unanimity Preservation (UP)) A BRG G = (V, A, LP, B, R) satisfies (UP) if given any formula ϕ LP, if for each Bi B, Bi |= ϕ then for each Bi B and at every step s 0, Bs i |= ϕ.

(UP) states that every formula which is a logical consequence of the initial agents beliefs should remain so in their belief sequence; note that in such a case, the formula is unanimously accepted in the BRG under consideration (cf. Definition 4). It is interesting to note that the statements of (AP) and (UP) have quite a similar structure. However, (AP) expresses a unanimity on models whereas (UP) is concerned with unanimity on formulae. The corresponding properties for merging operators have been presented in (Everaere, Konieczny, and Marquis 2010), where the authors also showed that the corresponding postulate of unanimity on formulae for merging operators is equivalent to (Disj) (cf. previous section).

Proposition 4 For every k {1, . . . , 6}, Rk satisfies (UP) if satisfies: (IC0) when k {5, 6};

(Disj) when k {1, 3, 4}; (IC0) and (Disj) when k = 2.

In the general case, revision policies Rk with k {1, 2, 3, 4} do not satisfy (UP) for merging operators which do not satisfy (Disj). This is because such merging operators may produce new beliefs absent from the states of the profile under consideration: some interpretations that do not satisfy any of the input belief states can be models of the merged state. However, for R5 and R6 , is not required to satisfy (Disj) since in the presence of (IC0) alone these policies are the most change-reluctant ones: each agent who accepts ϕ at some step will keep accepting ϕ at the next step since she will only refine her own beliefs. We address precisely the behavior of all merging-based revision policies in terms of agents responsiveness to their neighbors:

Definition 10 (Responsiveness (Resp)) A BRG G = (V, A, LP, B, R) satisfies (Resp) if for each Bi B such that Ci is not empty, for every step s 0, if (i) for every Bs ij Cs i , Bs ij Bs i |= , and (ii) V Bs ij Cs i Bs ij |= , then

Bs+1 i |= Bs i .

Informally, (Resp) demands that an agent should take into consideration the beliefs of her neighbors whenever (i) her beliefs are inconsistent with the beliefs of each one of her neighbors, and (ii) her neighbors agree on some alternatives. Accordingly, (Resp) is not satisfied by R5 and R6 :

Proposition 5 If satisfies (IC0), then R5 and R6 do not satisfy (Resp).

But (Resp) is satisfied by most of the remaining revision policies Rk under some basic conditions on :

Proposition 6 For every k {1, 2, 4}, Rk satisfies (Resp) if satisfies:

(IC2) when k = 1; (IC0) and (IC2) when k = 2; (IC2) and (IC4) when k = 4.

Intuitively, R3 seems to be less change-reluctant than R4 , since for R3 the agent considers her beliefs as being as important as each one of her neighbors whereas for R4 , she considers her beliefs as being as important as the aggregated beliefs of her neighbors. However, surprisingly R3 does not satisfy (Resp) even when some fully rational IC merging operators are used:

Proposition 7 R3 d H ,Σ does not satisfy (Resp).

Recall that the merging operator d H,Σ satisfies all the standard IC postulates (IC0) (IC8). Thus, the fact that satisfies those postulates is not enough for R3 to satisfy (Resp). However, we show below that these postulates are consistent with (Resp), in the sense that there exists a merging operator satisfying (IC0) (IC8) (and (Disj)) which makes R3 a responsive policy:

Proposition 8 For any aggregation function f, R3 d D,f satisfies (Resp).

In particular, the revision policy R3 d D,Σ = R3 d D,GMin satisfies (Resp). Given a BRG G = (V, A, LP, B, R), a formula ϕ and an agent i V , let us denote Gi ϕ the BRG (V , A, LP, B , R) defined as B i = B i ϕ and for every j V , j = i, B j = Bj.

Definition 11 (Monotonicity (Mon)) A BRG G = (V, A, LP, B, R) satisfies (Mon) if whenever ϕ is unanimously accepted in G, ϕ is also unanimously accepted in Gi ϕ for every i V . (Mon) is similar to the monotonicity criterion in Social Choice Theory. It is expressed in (Woodall 1997) as the condition where a candidate should not be harmed if she is raised on some ballots without changing the orders of the other candidates. In the BRG context, a formula ϕ which is unanimously accepted should still be unanimously accepted if some agent s initial beliefs were strengthened by ϕ. For each revision policy Rk , k {1, . . . , 6}, (Mon) is not guaranteed even when the merging operator under consideration satisfies the postulates (IC0) (IC8): Proposition 9 For every k {1, . . . , 6}, Rk d H ,Σ does not satisfy (Mon). The existence of revision policies Rk which satisfy (Mon) remains an open issue. However, one conjectures that for every k {1, . . . , 6}, Rk d D,Σ satisfies (Mon). This claim is supported by some empirical evidence. We have conducted a number of tests when four propositional symbols are considered in the language LP, for various graph topologies up to 10 agents and for k {1, . . . , 6}. All the tested instances supported the claim. The last property we provide concerns the stability issue: Definition 12 (Convergence) A BRG satisfies (Conv) if it is stable. Proposition 10 The revision policies R5 and R6 satisfy (Conv) if satisfies (IC0). None of the remaining revision policies Rk , k {1, 2, 3, 4} satisfy (Conv) in the general case. In fact, for these policies the stability of BRGs cannot be guaranteed as soon as the merging operator under consideration satisfies some basic IC postulates. Proposition 11 For every k {1, 2, 3, 4}, Rk does not satisfy (Conv) if satisfies: (IC2) when k = 1; (IC0) and (IC2) when k = 2; (IC1), (IC2) and (IC4) when k {3, 4}.

All the results are summarized in Table 3. For each class Rk of revision policies and each property on revision policies, for some (set of) postulate(s) (P) on merging operators or directly for some merging operators, (P) (resp. (P)) means that Rk satisfies (resp. does not satisfy) the corresponding property when satisfies (P) or is one of the merging operators which are specified. One can observe that under some basic conditions on , for k {1, 2, 4} the revision policies Rk are well-behaved in terms of responsiveness but do not guarantee the stability of all BRGs, while the converse holds for the revision policies R5 and R6 .

(CP) (AP) (UP) (Resp) (Mon) (Conv) R1

(IC2) ( d H ,Σ) (IC2) R2

(IC0) & (Disj)

(IC0) & (IC2) ( d H ,Σ) (IC0) & (IC2) R3

( d D,f ) / ( d H ,Σ) ( d H ,Σ) (IC1) & (IC2) & (IC4) R4

(IC2) & (IC4) ( d H ,Σ) (IC1) & (IC2) & (IC4) R5

(IC0) (IC0) ( d H ,Σ)

(IC0) (IC0) ( d H ,Σ)

Table 3: Properties satisfied by the revision policies Rk for k {1, . . . , 6}.

Before closing the section, we go further in the investigation of the convergence property by considering a subclass of so-called directed acyclic BRGs (V, A, LP, B, R) which require the underlying graph (V, A) not to contain any cycle:

Proposition 12 For k {1, 2, 3, 4}, all directed acyclic BRGs from G(Rk ) satisfy (Conv) when k = 1 or if: when k = 2, satisfies (IC0) and (IC2); when k = 3, is a distance-based merging operator; when k = 4, satisfies (IC2), (IC4) and (Disj), or is a distance-based merging operator.

Related Work Belief revision games are somehow related to many settings where some interacting agents are considered, including cellular automata (Wolfram 1983), Boolean networks (Kauffman 1969; 1993; Aldana 2003), opinion dynamics (Hegselmann and Krause 2005; Riegler and Douven 2009; Tsang and Larson 2014), and many complex systems (Latane and Nowak 1997; Kacpersky and Holyst 2000; Olshevsky and Tsitsiklis 2009; Bloembergen et al. 2014; Ranjbar-Sahraei et al. 2014). We focus here on related work strongly connected to Belief Revision Games. In (Delgrande, Lang, and Schaub 2007), the authors introduce a general framework for minimizing disagreements among beliefs associated with points connected through a graph. They define a completion operator which consists in revising the belief state of each point with respect to the belief states of its neighbors . This operator outputs a new graph where each belief state is strengthened and restricted to the models which are the closest ones to the neighbor states. Suitable applications include the case when points in the graph are interpreted as regions in space (W urbel, Jeansoulin, and Papini 2000). Though the idea of embedding belief states into a graph structure is similar to our approach, it differs from BRGs on several aspects. First, only undirected graphs are considered. Second, their completion operator is idempotent so it cannot be used iteratively. Third, belief states are strengthened by the operation of completion, whereas in BRGs agents can give up beliefs (e.g., when considering responsive policies such as R1, R2 and R4). In (Gauwin, Konieczny, and Marquis 2007), the authors introduce and study families of so-called iterated merging conciliation operators. Such operators are considered to rule the dynamics of the profile K of belief states associated with a group of agents. At each step the state Bi of agent i is modified, by revising the merged state (K) by Bi

(skeptical approach), or by revising Bi by the merged state (K) (credulous approach). Such merge-then-revise change functions are closely related to our merging-based revision policies R2 (for the credulous one) and R5 (for the skeptical one). They do not coincide with them nevertheless since in our approach Bi does not belong to its context Ci; clearly enough, this amounts to giving more importance to Bi when majoritarian merging operators are considered, and as a consequence the states obtained after the revision of Bi may differ. Notwithstanding the merging-based revision policies used, such conciliation processes correspond to specific BRGs where the topology is the clique one. One of the main issues considered in (Gauwin, Konieczny, and Marquis 2007) is the stationarity of the process (i.e., the convergence of the policies), which is proved in the skeptical approach; however, preservation issues, as well as responsiveness and monotonicity are not studied.

Conclusion In this paper, we formalized the concept of belief revision game (BRG) for modeling the dynamics of the beliefs of a group of agents. We pointed out a set of properties for BRGs which address several preservation issues, as well as responsiveness, monotonicity and convergence. As a first attempt to investigate the behavior of BRGs with respect these properties, we introduced several classes of revision policies which are based on belief merging operators. We considered the case where all agents use the same revision policy and investigated the extent to which the BRGs concerned with these policies satisfy the properties. Additionally, we developed a software available online at http://www.cril.fr/brg/brg.jar. It consists of graphical interface which allows one to play BRGs considering any of the 18 revision policies from {Rk | k {1, . . . , 6}, { d D,Σ, d H,Σ, d H,Gmin}}. Some instances of BRGs are provided together with the software, including the BRG from our motivating example (Example 1) and the counterexamples used in the proofs of some propositions. Practical applications of the BRG model are numerous. For instance, in brand crisis management, negative content regarding a brand could disseminate rapidly over social media and generate negative perceptions (Dawar and Pillutla 2000). In such a case, identifying how information is propagated within a social network and which are the influential agents (the opinion leaders) is a hot research topic. As a consequence, our general framework leaves the way open to many extensions and additional theoretical studies. Pers-

pectives include a further investigation of the robustness of BRGs in terms of belief manipulation.

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