Is Promoting Beliefs Useful to Make Them Accepted in Networks of Agents? (extended version including proofs of propositions)

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Is Promoting Beliefs Useful to Make Them Accepted in Networks of Agents?

(extended version including proofs of propositions)

Nicolas Schwind

National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

Katsumi Inoue

National Institute of Informatics / SOKENDAI, Tokyo Institute of Technology, Tokyo, Japan

Gauvain Bourgne CNRS & Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7606,

LIP6, F-75005, Paris, France [email protected]

S´ebastien Konieczny CRIL - CNRS Universit´e d’Artois

Lens, France [email protected]

Pierre Marquis CRIL - CNRS Universit´e d’Artois

Lens, France [email protected] Abstract

We consider the problem of belief propagation in a network of communicating agents, modeled in the recently introduced Belief Revision Game (BRG) framework. In this setting, each agent expresses her belief through a propositional formula and revises her own belief at each step by considering the beliefs of her acquaintances, using belief change tools. In this paper, we investigate the extent to which BRGs satisfy some monotonicity properties, i.e., whether promoting some desired piece of belief to a given set of agents is actually always useful for making it accepted by all of them. We formally capture such a concept of promotion by a new family of belief change operators. We show that some basic monotonicity properties are not satisfied by BRGs in general, even when the agents merging-based revision policies are fully rational (in the AGM sense). We also identify some classes where they hold.

1 Introduction

We are interested in the issue of monotonicity in a multi- agent system, represented as a Belief Revision Game (BRG) [Schwind et al., 2015]. The BRG setting is a framework for modeling the belief dynamics of a group of agents V, for instance agents involved in a social network. A BRG is a dynamical system where agents have their own belief bases (representing their belief states), and communicate syn- chronously with their acquaintances. The acquaintance relationship is given through a binary relationAoverV, i.e., (V, A)is a graph. Let us introduce a motivating example:

Example 1. Consider a group of friends Alex, Beth and Chris who are discussing on whether they should trust the quality of the food served in a given restaurant. Alex and Beth know each other, Alex and Chris as well, but Beth and Chris do not know each other. Two meals are considered by them. At

the beginning, Alex believes that either the first meal or the second one is healthy, but not both of them; Beth believes that none of the two meals is healthy; whereas Chris believes that at least one of the two meals is healthy.

At each communication step, each agent revises her beliefs by considering the beliefs of her acquaintances. Several merging-based revision policies have been defined, each of them reflecting how much an agent is ready to question her current beliefs in front of the beliefs of her acquaintances.

Then a piece of beliefϕis accepted by an agentiofV when there exists a step of the game from whichϕholds in the belief bases ofiat each subsequent step; andϕis unanimously accepted when it is accepted by every agent ofV.

Now, the question is, given a piece of beliefϕ, is adding

“moreϕ” in a BRG always beneficial toϕ? More precisely, whenever a piece of beliefϕis unanimously accepted in a BRG, is it always harmless to replace at the beginning some bases byϕ, or more generally, by a base that is “closer” to ϕ, i.e., by a “promotion” of ϕ? This monotonicity condition is essential when one wants to investigate the potential manipulation of such sytems, in particular the control issue:

consider a set of agents from a predefined subsetCofV and an additional agentM who can “control” the agents fromC, i.e.,M can modify the initial beliefs of agents fromC, then is it possible for M to make a piece of belief unanimously accepted? Such a control issue is significant for a number of multi-agent problems, including brand crisis management [Dawar and Pillutla, 2000]; in such applications, it is useful to know what information agents fromCshould convey to their acquaintances in order to avoid the propagation of negative perceptions.

In the next section we provide some preliminaries on BRGs. We then define and investigate in Section 3 the notion of promoting a formulaϕin a belief base, and show that many rational change operators (including revision and contraction operators) can be used for the promotion purpose.

We then formalize in Section 4 the monotonicity property based on promotion operators and show that this condition is not always satisfied by BRGs. As a consequence, replacing

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some agents’ initial beliefs byϕis not always the best way to makeϕunanimously accepted. In Section 5 we show that the monotonicity property holds for BRGs based on the complete graph topology, focusing on revision policies based on a specific merging operator. We discuss some related work in Section 6 before concluding.

The proofs of propositions are given in an appendix.

2 Belief Revision Games

LetL_P be a propositional language built up from a finite set of propositional variables P and the usual connectives, including⊕, the xor connective. ⊥(resp. >) is the Boolean constant always false (resp. true). Formulae are interpreted in a standard way. M od(ϕ)denotes the set of models of the formulaϕ,|=denotes logical entailment and≡logical equivalence, i.e., ϕ |= ψ iffM od(ϕ) ⊆ M od(ψ) andϕ ≡ ψ iffM od(ϕ) =M od(ψ). Abelief baseBdenotes the set of beliefs of an agent, it is a finite set of propositional formulae interpreted conjunctively, so thatBis identified with the conjunction of its elements. AprofileC=hB1, . . . , B_niis a finite vector of belief bases. A Belief Revision Game (BRG for short) is formalized as follows [Schwindet al., 2015]:

Definition 1 (Belief Revision Game). A Belief Revision Game (BRG)is a tupleG= (V, A,L_P,B,R)where

• V ={1, . . . , n}is a finite set of agents;

• A⊆V ×V is an irreflexive binary relation onV which represents theacquaintancesbetween the agents;

• LP is a finite propositional language;

• Bis a mapping fromV toLP where for eachi ∈ V, B(i)(notedBi) is the initial belief base of agenti;

• R={R1, . . . , Rn}, where eachRiis the revision policy of agenti, i.e., a mapping fromLP× LPin(i)

toLP with in(i) =|{j|(j, i)∈A}|the in-degree ofi, such that if in(i) = 0, thenR_iis the identity function.

LetG= (V, A,LP,B,R)be a BRG and let us denoteCi

thecontextofi, defined as the profileCi=hBi₁, . . . , Bi_in(i)i where{i1, . . . , i_in(i)}={ij |(ij, i)∈A}. ThenRi(Bi,Ci) is the belief base of agentionce revised by taking into account her own current beliefsBiand her current contextCi.¹ In a BRG, the beliefs of each agent evolve at each time step according to her revision policy. This induces for eachi∈V abelief sequence(B^s_i)_s∈NwhereB_i^sdenotes the belief base of agent i after ssteps, defined as B_i⁰ = B_i and for each s ≥0,B^s+1_i =Ri(B^s_i,C_i^s), whereC_i^sis the context ofiat steps. Schwindet al. [2015] showed that in any BRG, the belief sequence of each agentiiscyclic, i.e., in(B^s_i)s∈Nthere exists a finite subsequence B_i^b, . . . , B_i^e such that for every j > e, we haveB_j^j≡Bb+((j−b)mod(e−b+1))

i ; thebelief cycle

Cyc(i)of an agenticorresponds to the series of this subsequence of belief basesCyc(i) =B^b_i, B_i^b+1, . . . , B_i^e. As we are interested in determining the pieces of beliefs resulting

1Abusing notations, Ri(Bi,Ci) stands for Ri(Bi, Bi₁, . . . , Bi_in(i))whereCi=hBi₁, . . . , Bi_in(i)i.

from the interaction of the agents, we focus on theoutcome of each agentiinG, denotedAccG(i)and defined as:

Acc_G(i) =_

{B^s_i |B_i^s∈Cyc(i)}.

We say that a formulaϕisacceptedbyiinGifAccG(i)|= ϕ, ϕis unanimously accepted inGif ϕis accepted by all i ∈ V inG, andGconverges at step sif for eachi ∈ V, B_i^s+1=B_i^s.

The formalization of a BRG allows each agent ito consider any revision policyRi∈ R. However, one can take ad- vantage of theoretical tools from Belief Change Theory (see e.g. [Alchourr´onet al., 1985]), more precisely, belief revision and merging operators for defining revision policies. A merging operator∆ associates any formulaµ(the integrity constraint) and any profileCwith a new formula∆µ(C)(the merged base). A merging operator ∆ aims at defining the merged base as the beliefs of a group of agents represented by the profile, under some integrity constraint. Standard properties (denoted(IC0)–(IC8)) are expected for merging operators, and such operators are called IC merging operators (see [Konieczny and Pino P´erez, 2002] for details on these properties).

IC merging operators include some distance-basedope- rators, i.e., operators that are based on the selection of some models of the integrity constraint, the “closest” ones to the given profile. These operators are characterized by a distance d between interpretations and an aggregation function f [Konieczny et al., 2004]. They associate with every formula µ and every profile C a belief base

∆^d,f_µ (C)satisfyingM od(∆^d,f_µ (C)) = min(M od(µ),≤^d,f_C ), where≤^d,f_C is the total preorder over interpretations induced by C defined by ω ≤^d,f_C ω⁰ if and only if d^f(ω,C) ≤ d^f(ω⁰,C), whered^f(ω,C) =f_B∈C{d(ω, B)}andd(ω, B) = min_ω0|=Bd(ω, ω⁰). Usual distances ared_D, the drastic distance (d_D(ω, ω⁰) = 0ifω = ω⁰ and1otherwise), andd_H the Hamming distance (dH(ω, ω⁰) = n if ω and ω⁰ dif- fer on n variables). Using aggregation functions such as Σand GMax lead to IC merging operators. For instance, GMax operators consider for each profile C the total preorder over interpretations≤^d,GMax_C defined byω ≤^d,GMax_C ω⁰ if and only if d^GMax(ω,C)≤^lexd^GMax(ω⁰,C) (where≤^lexis the lexicographic ordering induced by the natural order) and d^GMax(ω,C) is the vector of numbersd1, . . . , dnobtained by sorting in a descending order the vectorhd(ω, B_i)|B_i∈ Ci.

Lastly,belief revision operatorscan be seen as belief merging operators applied to singleton profiles: indeed, if∆is an IC merging operator then the revision operator ◦∆ induced by

∆defined for all basesB1, B2 asB1◦∆B2 = ∆B₂(hB1i) satisfies the standard KM revision postulates [Katsuno and Mendelzon, 1992].

Let us go back to BRGs. Six classes of revision policies have been proposed in [Schwindet al., 2015]. Each of them, denotedR^k_∆(k∈ {1, . . . ,6}) is parameterized by an IC merging operator∆. Each class is defined as follows,² at each stepsand for any agentisuch thatCi6=∅:

2When using a merging operator without integrity constraints we just note∆(C)instead of∆>(C)for improving readibility.

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stepss B^s₁ B^s₂ B₃^s 0 p1⊕p2 ¬p1∧ ¬p2 p1∨p2

2k+ 1 ¬p1∨ ¬p2 p1⊕p2 p1⊕p2

2k+ 2 p1⊕p2 ¬p1∨ ¬p2 ¬p1∨ ¬p2

Table 1: The belief sequences of Alex, Beth and Chris inG_∗.

• R¹_∆(B_i^s,C^s_i) = ∆(C_i^s);

• R²_∆(B_i^s,C^s_i) = ∆_∆(C^s

i)(hB^s_ii) [=B_i^s◦∆∆(C_i^s)];

• R³_∆(B_i^s,C^s_i) = ∆(hB_i^s,C_i^si);

• R⁴_∆(B_i^s,C^s_i) = ∆(hB_i^s,∆(C_i^s)i);

• R⁵_∆(B_i^s,C^s_i) = ∆B_i^s(∆(C_i^s)) [= ∆(C_i^s)◦∆B_i^s];

• R⁶_∆(B_i^s,C^s_i) = ∆_B^s

i(C_i^s).

Example 1 (continued). We consider the BRG G_∗ = (V_∗, A_∗,L_P_∗,B_∗,R_∗) defined as follows. V_∗ = {1,2,3}

where 1 corresponds to Alex, 2 to Beth, and 3 to Chris.

A_∗={(1,2),(2,1),(1,3),(3,1)}expresses that Alex knows Beth and vice-versa, and Alex knows Chris and vice-versa, but Beth and Chris are not connected. L_P_∗ is the propositional language defined from the variablesP∗ = {p1, p₂}, wherep1stands for “the first meal is healthy” andp2means

“the second meal is healthy.” The initial beliefs of the group members are B1 = p1⊕p2,B2 = ¬p1∧ ¬p2and B3 = p₁∨p₂. Assume that all agents use the same revision policy,Ri =R¹_∆_{dH ,}GMaxfor eachi ∈V∗. The belief sequences associated with the three agents are given in Table 1: the belief cycle of agent 1 (resp. 2, 3) is given by (B₁⁰, B₁¹) (resp. (B¹₂, B₂²), (B¹₃, B₃²)). We have for each i ∈ V_∗, Cyc(i) =p₁⊕p₂,¬p1∨ ¬p2andAcc_G(i) =¬p1∨ ¬p2.

[Schwindet al., 2015] studied the extent to which BRGs satisfy some basic logical properties depending on the class of revision policies used by the agents, and focused on the case when all agents use the same revision policy ranging overR^k_∆,k∈ {1, . . . ,6}. It turned out that when the revision policy is induced from the merging operator∆^d^D^,Σ, i.e., the merging operator based on the drastic distance and the summation function, the underlying BRGs satisfy a number of expected properties [Schwindet al., 2015]. The authors also developed a software allowing one to play BRGs, available online athttp://www.cril.univ-artois.fr/software/brg.html.

3 On the Notion of Promotion

Belief control in a multi-agent system can take various forms, depending on the meaning given to “control.” Here, we are specifically interested in control strategies that consist in promoting a certain beliefϕin the beliefs of the agents. A key issue to be addressed is then to determine what “promoting”

precisely means in this context. A simple view is to consider that promotingϕin the belief baseBof an agent consists in replacingB by a base equivalent to ϕ. While such a drastic way of promotingϕmakes sense, it is not the only one.

Thus, for instance, revisingBbyϕis another approach to do the job. Considering the whole spectrum of promotion tech- niques is interesting because in some scenarios it could be the case that the agent under consideration can be ready to

promoteϕby revising her own beliefsB with it, while she would be reluctant in questioning her whole baseB and replacing it byϕ. In a bribery context, she could for instance ask much more money to accept to change her baseB toϕ than to change it to the revision ofBbyϕ.

We now characterize the notion of “promotion” of ϕ through a preorder ϕ over belief bases which reflects the closeness relationship toϕ; thus,B⁰ ϕ Bstates thatB⁰ is at least as close to ϕas B. On this ground, promoting B consists in replacingBby anyB⁰satisfyingB⁰ϕB:

Definition 2(ϕ-promotion). For every formulaϕ∈ LP, the ϕ-promotion relation is the binary relationϕonL_P× L_P defined for all belief basesB, B⁰ asB⁰ ϕ B if and only if B∧ϕ|=B⁰|=B∨ϕ.

Obviously, one can check that for anyϕ∈ L_P, the binary relationϕis reflexive and transitive. More precisely:

Proposition 1. For every formulaϕ ∈ L_P,(L_P,ϕ)is a Boolean lattice(L_P,u_ϕ,t_ϕ,¬, ϕ,˙ ¬ϕ), where:

• uϕ is the meet operation defined for eachB, B⁰ ∈ LP

asBuϕB⁰= (B∧B⁰)∨((B∨B⁰)∧ϕ),

• t_ϕis the join operation defined for eachB, B⁰ ∈ L_P as BtϕB⁰= (B∧B⁰)∨((B∨B⁰)∧ ¬ϕ),

• ¬˙ is the complement corresponding to the standard logical negation¬,

• ϕis the least element, i.e., for eachB∈ LP,ϕϕB,

• ¬ϕis the top element, i.e., for eachB∈ L_P,Bϕ¬ϕ.

where formulae ofL_Pare considered up to equivalence.

Every baseB⁰promotingϕinBsatisfiesϕ_ϕB⁰ _ϕB.

ThusB is (up to logical equivalence) the greatest formula w.r.t. ϕpromotingϕinB, andϕis (up to logical equivalence) the least formula w.r.t. ϕpromotingϕinB. Stated otherwise, the least demanding promotion ofϕw.r.t. Bcon- sists in lettingBunchanged, while the promotion ofϕw.r.t.

Bleading to a formula as close as possible to ϕconsists in replacingBbyϕ.

The following model-theoretic characterization of the notion of promotion can be derived easily.Ndenotes the symmetric difference between sets:

Proposition 2. Given a formula ϕ, let B and B⁰ be two belief bases such that B⁰ promotes ϕ w.r.t. B. Then

∃S⊆Mod(B)NMod(ϕ)such thatMod(B⁰) = (Mod(B)∩ Mod(ϕ))∪S.

This proposition also illustrates the fact thatBandϕplay symmetric roles in the notion of promotion. Indeed, making Bcloser toϕis precisely the same as makingϕcloser toB: Proposition 3. B⁰ϕBif and only ifB⁰Bϕ.

When a promotion ofϕinBis achieved, the set of interpretations assigning different truth values toBand toϕmay only diminish. Formally:

Proposition 4. Given a formula ϕ, let B and B⁰ be two belief bases such that B⁰ promotes ϕ in B. Then Mod(B⁰)NMod(ϕ)⊆Mod(B)NMod(ϕ).

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We define a promotion operator as a mapping from L_P × L_P toL_P such thatψϕϕ ψ. Interestingly, some belief change operators of the literature are promotion operators. We recall below some of the postulates for rational revision operators◦, and contraction operators −[Katsuno and Mendelzon, 1992, Caridroitet al., 2015], as well as some postulates for arbitration (alias symmetric revision) operators [Liberatore and Schaerf, 1998]:

(R1) ϕ◦µ|=µ.

(R2) Ifϕ∧µis consistent, thenϕ◦µ≡ϕ∧µ.

(C1) ϕ|=ϕ−µ.

(C2) Ifϕ6|=µ, thenϕ−µ|=ϕ.

(C4) Ifϕ|=µ, then(ϕ−µ)∧µ|=ϕ.

(LS2) ϕ∧µ|=ϕµ.

(LS7) ϕµ|=ϕ∨µ.

Proposition 5. Letϕbe a formula and letBbe a belief base.

Let◦be any revision operator satisfying (R1)and(R2), let

−be any KM contraction operator satisfying(C1),(C2)and (C4), and letbe any arbitration operator satisfying(LS2) and(LS7). We have:

1. (i)B◦ϕ_ϕB∧ϕ_ϕB, (ii)B∨ϕ_ϕB−¬ϕ_ϕB, and (iii)BϕϕB;

2. B◦ϕ,B∧ϕ,B∨ϕ,B−¬ϕandBϕare incomparable w.r.t.ϕin the general case.³

We can now lift the relation of formula promotion to BRGs defined on the same set of variablesV, acquaintance relation A, propositional languageL_P and revision policiesR. Given two BRGsG= (V, A,LP,B,R)andG⁰ = (V, A,LP,B⁰, R)and a formulaϕ, we noteG⁰ ϕGif and only if for each agenti∈V,B_i⁰ϕB_i. Finally, we noteGϕany BRGG⁰ such thatG⁰ϕG. Observe that such a promotion operation ofϕinGcan be non-uniform, i.e., it is not necessarily the case that the promotion ofϕin distinct bases ofG⁰is achieved thanks to the same promotion operator. For instance, it can be the case thatB_i⁰ =Bifor agenti∈ V, whileB_j⁰ =Bj◦ϕ for agentj∈V, andB⁰_k=ϕfor agentk∈V.

4 On Monotonicity in BRGs

In this section, we focus on the issue of monotonicity for BRGs instantiated with revision policies from the six classes defined in Section 2. Given a merging operator∆andE ⊆ {1, . . . ,6}RÊ_∆ denotes the set{R_∆^k | k ∈ E}. We use the simpler notationR^k_∆instead ofRÊ_∆whenE={k}. Given a classGof BRGs andE⊆ {1, . . . ,6},G(RÊ_∆)is the subclass of all BRGs(V, A,L_P,B,R)fromGwhere for eachR_i∈ R, Ri ∈RÊ_∆. Additionally, a set of revision policiesRÊ_∆is said to satisfy a given propertyP on a given classG of BRGs if all BRGs fromG(RÊ_∆)satisfyP.

Given a BRGG = (V, A,L_P,B, R), a subsetC ofV of so-called “controllable agents” whose initial beliefs can be modified, and a formulaϕ, one is interested in determining

3This is even the case for◦and−when they correspond one another thanks to Levi/Harper identities.

stepss B₁^s B^s₂ B₃^s

0 p1⊕p2 ¬p1∧ ¬p2 p1∧p2

s≥1 p1⊕p2 p1⊕p2 p1⊕p2

Table 2: An example of control strategy forG_∗whereϕ= p₁∨p₂is unanimously accepted.

how to modify the belief bases of agents inCin order to make ϕunanimously accepted in the resulting game (when possible). The objective is thus to determine a successful “control strategy” to be implemented in order to reach the goal when it can be reached. A control strategy forGgivenCtakes here the form of a mappingσfromCtoL_P, stating for eachi∈C thatB_imust be replaced byσ(i); it is successful forϕifϕis unanimously accepted in the BRG obtained by applyingσto G. Typically, one wants to minimize the number of agents in Cto be controlled, but the optimization problem under consideration can also be much more complex (for instance, it may take into account the cost of controlling each agent inC which may not be uniform).

Among the potential control strategies is thebasic strategy σ_ϕ for GgivenC defined for any i ∈ C byσ(i) ≡ ϕ: it simply amounts to promotingϕas much as possible in the belief base of each agent fromC by just replacing it byϕ.

We have the following surprising result:

Proposition 6. Given a BRGG = (V, A,L_P,B,R), a set C ⊆ V of controllable agents, and a formulaϕ, it can be the case that the basic strategyσϕforGgivenCis not successful forϕ, while a control strategy forGgivenCwhich is successful forϕexists.

Example 1(continued). Assume that the goal of the restaurant manager is to convince all protagonists that at least one of the meals is healthy, i.e.,ϕ=p₁∨p₂. Note that at the beginning, Chris’ beliefs coincide with this goal (sinceB3=ϕ) and thatϕis not unanimously accepted. So if Chris is the only controllable agent, the basic strategy σϕ is not successful.

However, when we replace Chris’ beliefs byp1∧p2instead, we get thatAcc_G_∗(i) =p₁⊕p₂for eachi∈V, sop₁∨p₂ is unanimously accepted (see Table 2). This shows that control is possible here givenC ={3}as the only controllable agent, but not with the basic strategy.

This example illustrates the complexity of the controllabi- lity issue for BRGs in the general case. This explains why it is important to identify some conditions on BRGs for which focusing on the basic strategy would be enough to decide whether a positive answer can be given or not to the control- lability question. In the following, we show that some monotonicity properties can be used as such conditions. A first property of monotonicity has been introduced in [Schwindet al., 2015]. Given a BRGG= (V, A,L_P,B,R), a formulaα and an agenti∈V, let us denoteG_i→αthe BRG(V,A,L_P, B⁰,R)whereB⁰mapsV to{B⁰₁, . . . , B_n⁰}, withB_i⁰=Bi∧α and for everyj∈V,j6=i,B_j⁰ =Bj.

Definition 3(Monotonicity (Mon)). A BRGG= (V, A,L_P, B,R)satisfies(Mon)if wheneverϕis unanimously accepted inG,ϕis unanimously accepted inGi→ϕfor everyi∈V.

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B_i^s∧ϕmax(B^s)|=⊥(case (i)) ϕmax(B^s)|=B^s_i (case (ii)) Otherwise (case (iii)) R¹_∆_{dD ,}_Σ ϕ_max(B^s) ϕ_max(B^s)∨(ϕ_max₋₁(B^s)∧ ¬B_i^s) ϕ_max(B^s)∧ ¬B_i^s R²_∆_{dD ,}Σ ϕmax(B^s) ϕmax(B^s) ϕmax(B^s)∧ ¬B_i^s R³_∆_{dD ,}Σ ϕmax(B^s) ϕmax(B^s) ϕmax(B^s) R⁴_∆_{dD ,}Σ ϕ_max(B^s)∨B_i^s ϕ_max(B^s) ϕ_max(B^s)∨B_i^s

R⁵_∆_{dD ,}Σ B_i^s ϕmax(B^s) B^s_i

R⁶_∆_{dD ,}_Σ B_i^s ϕ_max(B^s) ϕ_max(B^s)∧B_i^s Table 3:B^s+1_i depending on the revision policy applied by agentiin any BRG fromGcom(R^{1,...,6}_∆_{dD ,}Σ ).

(Mon) is similar to the monotonicity criterion in Social Choice Theory. In the BRG context, a formulaϕthat is unanimously accepted should still be so if some agent’s initial beliefs were “strengthened” byϕ. Note that the strengthe- ning of a belief base by a formulaϕinvolved here consists in just expanding the base byϕ. However, one could consider weaker versions than expansion, i.e., the promotion operators we formalized in the previous section. So we introduce now a stronger version of the monotonicity property (cf. De- finition 3) on BRGs based on the promotion relation:

Definition 4 (Strong Monotonicity(SMon)). A BRG G = (V, A,L_P,B,R)satisfies(SMon)if for eachi∈V, ifϕis unanimously accepted inG, thenϕis unanimously accepted in any BRGGϕ.

It is easy to see that (SMon)implies(Mon)and that the converse does not hold in general. BRGs satisfying(SMon) are interesting in terms of strategy-proofness. Indeed, Propo- sition 1 tells us thatϕis the least element of(L_P,_ϕ). As a consequence, for such BRGs, determining whether it is possible to convince all the agents involved in the BRG to accept some piece of beliefϕsimply amounts to determining whetherϕis unanimously accepted in the BRG obtained by the promotion ofϕin everyBiassociated with a controllable agent so thatBiϕ=ϕ. Stated otherwise:

Proposition 7. Let G = (V, A, L_P, B, R) satisfying (SMon), a setC ⊆V of controllable agents, and a formula ϕ. If the basic strategyσϕforGgivenCis not successful for ϕthen no control strategy forGgivenCis successful forϕ.

An interesting issue now is to know whether there are BRGs satisfying(SMon). We provide a positive answer to this question in the next section.

5 The Case of Complete Graphs

We now study the extent to which (SMon) is satisfied by BRGs whose acquaintance graph is acompletegraph. This simple topology is adequate to the cases when all the agents ofV know each other (for instance, this is the case in mee- tings where all agents are sitting around a table). We simply call the corresponding class of BRGs the complete BRGs:

Definition 5(Complete BRG). A BRGG = (V, A,L_P,B, R)iscompleteif(V, A)is a complete graph, i.e., for alli, j∈ V,i6=j,(i, j)∈A. Given a classGof BRGs,G_comdenotes the subclass of complete BRGs fromG.

In the general case,(SMon)is not satisfied by BRGs from Gcom(R^k_∆)withk ∈ {1, . . . ,6}, even when∆is “fully” rational in the sense that it satisfies all IC postulates. Indeed:

Proposition 8. For∆ ∈ {∆^d^H^,Σ,∆^d^H^,GMax}, for anyk ∈ {1, . . . ,6},R_∆^k does not satisfy(SMon)onGcom.

In the following, we investigate the monotonicity issue for complete BRGs when the merging operator used for defining the revision policies is the distance-based merging operator based on thedrastic distance(and the summation function) ∆^d^D^,Σ. Computing∆^d_µ^D^,Σ(C)consists in selecting in the models of the integrity constraint µthose satisfying as many bases of the profileCas possible. Several works have proved this specific operator to satisfy a number of expected properties, e.g., some (language) independence conditions [Konieczny et al., 2011, Marquis and Schwind, 2014]. In particular,∆^d^D^,Σis robust from the point of view of strategy- proofness [Everaere et al., 2007], this is why this operator appears as a good candidate for the monotonicity issue.

The following notations are used in the rest of this section.

LetCbe a profile andωbe an interpretation.VC=V{B_i | Bi ∈ C}. #E denotes the number of elements from a finite setE.#max(C)is the maximal number of bases inCthat are consistent when interpreted conjunctively, i.e., #max(C) = max(#{C⁰ | C⁰ ⊆ C,VC⁰ 6|= ⊥}). ϕmax(C) = ∆^d^D^,Σ(C).

Andϕ_max₋₁(C)is any formulaϕsuch thatM od(ϕ) ={ω|

#{Bi∈ C |ω|=Bi}= #max(C)−1}.

Let us first show that a stronger version of the agreement preservation property(AP)(see [Schwindet al., 2015]) is satisfied by complete BRGs:

Proposition 9. LetG = (V, A,L_P,B,R)be a BRG from Gcom(R^{1,...,6}_∆_{dD ,}_Σ ). If there is a stepswhereV{B_i^s | i ∈V} is consistent, then for eachi∈V,AccG(Bi) = V

{Bi |i∈ V}. Moreover,Gconverges at steps+2at most. It converges at steps+ 1at most ifG∈ Gcom(R^{2,...,6}_∆_{dD ,}_Σ ).

We now intend to characterize the outcome of a BRG from its initial state. Beforehand, let us introduce an intermediary result characterizing the belief base of any agent in a BRG fromGcom(R^k_∆_{dD ,}_Σ)wherek∈ {1, . . . ,6}, at some steps+1 given the belief bases of all agents at steps:

Lemma 1. Let E ⊆ {1, . . . ,6} and G be a BRG from G_com(R^E_∆_{dD ,}_Σ). Then for each agenti ∈ V and each step s, her baseB_i^s+1can be characterized as shown in Table 3.

Obviously enough, a consequence of Table 3 is that all six revision policies fromR^k_∆_{dD ,}_Σ,k ∈ {1, . . . ,6}, remain distinct. Moreover, in the case when all agents have the same

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revision policy amongR^k_∆_{dD ,}Σ,k∈ {1, . . . ,4}, the outcome of complete BRGs can be fully characterized:

Proposition 10. Letk ∈ {1, . . . ,4}and G = (V, A,L_P, B,R)be a BRG fromGcom(R^k_∆_{dD ,}Σ). Then for eachi∈V, AccG(i) = ∆^d^D^,Σ(hB1, . . . , Bni).

The cases when all agents have the same revision policy amongR^k_∆_{dD ,}Σ,k∈ {5,6}is simpler to characterize:

Proposition 11. Letk ∈ {5,6}andG= (V, A,L_P,B,R) be a BRG fromGcom(R^k_∆_{dD ,}Σ).

• If k = 5, then for all i ∈ V, AccG(i) = Bi

if ∆^d^D^,Σ(hB1, . . . , Bni) 6|= Bi, otherwise AccG(i) =

∆^d^D^,Σ(hB1, . . . , B_ni).

• If k = 6, then for all i ∈ V, Acc_G(i) = B_i if

∆^d^D^,Σ(hB1, . . . , Bni)∧ Bi |= ⊥, otherwise AccG(i) =

∆^d^D^,Σ(hB1, . . . , Bni)∧Bi.

These results allow us now to derive monotonicity results for the class of complete BRGs we are dealing with:

Proposition 12. Letk∈ {1, . . . ,6}. Then all BRGs from the classG_com(R^k_∆_{dD ,}_Σ)satisfy(SMon).

A consequence of Propositions 7 and 12 is that considering the basic strategy is enough for BRGs fromGcom(R^k_∆_{dD ,}Σ), i.e., to decide whether a control strategy exists for making a formulaϕaccepted by all agents in such a BRG, it is enough to focus on the basic strategy.

Let us illustrate how these monotonicity results apply to the following slightly modified version of our running example:

Example 1 (continued). Consider again the BRG G_∗ = (V_∗, A_∗,LP∗,B_∗,R_∗)from our running example, and letG⁰_∗ be the BRG(V_∗, A⁰_∗,L_P_∗,B⁰_∗,R_∗)where:

•A⁰ is the total relation overV_∗, i.e., Beth and Chris know each other as well;

• B_∗⁰ mapsV_∗ to{B₁⁰, B⁰₂, B₃⁰}, withB₁⁰ = B₁,B⁰₂ = B₂ andB⁰₃=¬p1∧ ¬p2, i.e., the beliefs of Alex and Beth inG⁰_∗ remain the same as inG_∗, but Chris now initially believes in G⁰_∗that none of the two meals is healthy.

From Proposition 10, we get thatAcc_G⁰_∗(i) = ∆^d^D^,Σ(hB⁰₁, B₂⁰, B⁰₃i)≡ ¬p1∧¬p2for eachi∈V_∗. Note thatp₁∨p₂is not accepted by any agent. By applying the basic strategyσϕ

whereϕ =p₁∨p₂and when Chris is the only controllable agent, one gets that the outcome of each agent in the resulting BRG becomesp1⊕p2. We have nowϕbeing unanimously accepted, which means thatσϕis successful.

6 Related Work

As far as we know, the control problem for multi-agent systems has not been investigated to date in settings similar to BRGs. Though there are many works on opinion dynamics where some control or influence issues have been studied, the opinion of an agent takes typically the form of a real number, as in Hegselmann-Krause’s model [Hegselmann and Krause, 2005]. Let us mention among others [Tsang and Lar- son, 2014, Bloembergenet al., 2014, Ranjbar-Sahraeiet al., 2014, Blondel et al., 2009]. The very nature of the agents opinions (real number vs. propositional formula) makes these studies and our own approach quite unrelated. Indeed, unlike

real numbers that can be simply averaged, merging propositional formulae is a much more complex process (this is why many works on the topic in the knowledge representation area have been done for two decades). The quite paradoxical result presented in the paper (roughly, promoting some belief is not guaranteed to be the safest way to make it unanimously accepted) echoes paradoxes pointed out in other settings for multi-agent systems, in particular social network games [Apt et al., 2013]. However, the resemblance is only at a shal- low level since the BRG model departs a lot from social network games. Especially, social network games are “true”

games, unlike BRGs (in the BRG model, agents do not have strategies and there is no payoff based on the adequacy of an agent’s beliefs with the beliefs of her acquaintances).

On the other hand, some control issues (under various forms: manipulation, bribery, etc.) have received much atten- tion recently in voting theory, especially from the viewpoint of computational complexity. See among many others [Ama- natidiset al., 2015, Brederecket al., 2015, Faliszewskiet al., 2015, Loreggiaet al., 2015, Faliszewski et al., 2014, Bou- veret and Lang, 2014, Christianet al., 2007]. However, the results obtained in voting theory cannot be exploited directly for BRGs since in BRGs the agents beliefs may evolve after each communication step (when voting is considered, the users preferences are typically considered as fixed and agents do not influence each other). An exception is [Hassanzadehet al., 2013] which shows how to reach a consensus on rankings (encoding the agents preferences) by iterative voting. How- ever, the corresponding model is quite different from the BRG one since it deals with preferences, and not with beliefs, and the system dynamics is ruled by distinct mechanisms (voting vs. revision). Furthermore, belief control has not been investigated in this model (the focus is on convergence issues).

Finally, other approaches have been considered for modeling the belief dynamics of a group of agents (see dynamic epistemic logic [Harelet al., 2000, Hendricks and Symons, 2006], or in propositional logic [Gauwinet al., 2007, Del- grandeet al., 2007, Grandiet al., 2015]), but the belief control issue for such settings has not been considered as well.

7 Conclusion

We pointed out a quite paradoxical result in belief diffusion:

in a network of agents, replacing some agents’ belief bases by a piece of beliefϕmay not be the “optimal” way to makeϕ unanimously accepted. However, we have identified a class of BRGs satisfying a property of strong monotonicity, and thus, avoiding this paradox, making for these BRGs the belief control issue easier to manage.

As perspectives for further research, we plan to identify additional classes of BRGs for which the strong monotonicity property holds, e.g., by considering other topologies as the line, single loop and more generally regular graph topologies. For all these BRGs, the next step will be to search for control strategies by focusing now on the “who” issue, i.e., which agents from a predefined setC should be considered.

Another interesting perspective is to study further the notion of “promotion,” as an attempt to provide a uniform axiomati- zation of a large class of standard belief change operators.

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Appendix: Proofs of Propositions

Proposition 1

For every formula ϕ ∈ L_P, (L_P,ϕ)is a Boolean lattice (LP,uϕ,tϕ,¬, ϕ,˙ ¬ϕ), where:

• uϕis the meet operation defined for eachB, B⁰ ∈ L_P asBu_ϕB⁰ = (B∧B⁰)∨((B∨B⁰)∧ϕ),

• tϕis the join operation defined for eachB, B⁰∈ LP as BtϕB⁰ = (B∧B⁰)∨((B∨B⁰)∧ ¬ϕ),

• ¬˙ is the complement corresponding to the standard logical negation¬,

• ϕis the least element, i.e., for eachB∈ LP,ϕϕB,

• ¬ϕis the top element, i.e., for eachB ∈ L_P,B ϕ¬ϕ.

where formulae ofL_P are considered up to equivalence.

Proof. Letϕ ∈ L_P. Since the standard inference relation

|=onL_P induces a Boolean lattice where formulae fromL_P are considered up to equivalence, it is enough to build an iso- morphismfϕof Boolean lattices from(L_P,ϕ)to(L_P,|=).

Let us define fϕ as follows, for eachB ∈ LP, fϕ(B) = (B∨ϕ)∧(¬B∨ ¬ϕ). LetB, B⁰ ∈ L_P. We have to show thatB⁰ ϕ Bif and only iff_ϕ(B⁰)|=f_ϕ(B). Yet we have B⁰ϕBif and only if (i)B⁰ |=B∨ϕand (ii)B∧ϕ|=B⁰. On the one hand, (i) is equivalent toB⁰∧ ¬ϕ|=B∧ ¬ϕ. On the other hand, (ii) is equivalent to¬B⁰ |=¬B∨ ¬ϕ, which is equivalent¬B⁰∧ϕ|=¬B∧ϕ. ThusB⁰_ϕBif and only ifB⁰∧ ¬ϕ|=B∧ ¬ϕand¬B⁰∧ϕ|=¬B∧ϕif and only if (B⁰∧ ¬ϕ)∨(¬B⁰∧ϕ)|= (B∧ ¬ϕ)∨(¬B∧ϕ)if and only iff_ϕ(B⁰)|=f_ϕ(B). Then it is easy to verify that:

• fϕ(BuϕB⁰) =fϕ(B)∧fϕ(B⁰),

• f_ϕ(BtϕB⁰) =f_ϕ(B)∨f_ϕ(B⁰),

• fϕ(¬B) = ˙¬fϕ(B),

• f_ϕ(ϕ) =⊥,

• fϕ(¬ϕ) =>.

Proposition 2

Given a formulaϕ, letBandB⁰be two belief bases such that B⁰promotesϕw.r.t.B. Then∃S⊆Mod(B)NMod(ϕ)such thatMod(B⁰) = (Mod(B)∩Mod(ϕ))∪S.

Proof. Letϕ be a formula, B and B⁰ be two belief bases and assume thatB⁰ promotesϕw.r.t. B. Then we have (a) B∧ϕ |=B⁰ and (b)B⁰ |=B ∨ϕ. LetαS be the formula defined asα_S =B⁰∧(¬B∨ ¬ϕ). From (b) we get thatα_S |= (B∨ϕ)∧(¬B∨ ¬ϕ), soM od(αS)⊆Mod(B)∆Mod(ϕ).

YetB⁰can be written as the formula(B⁰∧(B∧ϕ))∨α_S. so from (a)B⁰is equivalent to the formula(B∧ϕ)∨αS. Hence, Mod(B⁰) = (Mod(B)∩Mod(ϕ))∪Mod(φ_S).

Proposition 3

B⁰ϕBif and only ifB⁰Bϕ.

Proof. Direct by definition.

Proposition 4

Given a formulaϕ, letB andB⁰ be two belief bases such that B⁰ promotes ϕ in B. Then Mod(B⁰)NMod(ϕ) ⊆ Mod(B)NMod(ϕ).

Proof. IfB⁰promotesϕinB, then we haveB∧ϕ|=B⁰ |= B∨ϕ. By contraposition, we have¬B ∧ ¬ϕ |= ¬B⁰ |=

¬B∨ ¬ϕ, i.e.,¬B⁰ _¬ϕ¬B. Letωbe a model of¬B⁰∧ϕ.

Then since¬B⁰ |=¬B ∨ ¬ϕholds, we also have thatω is a model of(¬B∨ ¬ϕ)∧ϕ, i.e., a model of¬B∧ϕ. Letω be a model ofB⁰∧ ¬ϕ. Then sinceB⁰ |=B∨ϕholds, we also have thatωis a model of(B∨ϕ)∧ ¬ϕ, i.e., a model of B∧ ¬ϕ. This concludes the proof.

Proposition 5

Letϕbe a formula and letB be a belief base. Let◦be any revision operator satisfying(R1)and(R2), let−be any KM contraction operator satisfying(C1),(C2)and(C4), and let be any arbitration operator satisfying(LS2)and(LS7). We have:

1. (i)B◦ϕϕB∧ϕϕB, (ii)B∨ϕϕB−¬ϕϕB, and (iii)BϕϕB;

2. B◦ϕ,B∧ϕ,B∨ϕ,B−¬ϕandBϕare incomparable w.r.t.ϕin the general case.

Proof.

• B∧ϕϕBholds sinceB∧ϕ|=B∧ϕ|=B∨ϕ.

• B◦ϕϕ B∧ϕϕ B sinceB∧ϕ∧ϕ |=B◦ϕis equivalent to state thatB∧ϕ|=B◦ϕ(which holds as soon as◦satisfies(R2)) andB◦ϕ|= (B∧ϕ)∨ϕholds as soon as◦satisfies(R1)).

• B∨ϕϕB− ¬ϕholds since(B− ¬ϕ)∧ϕ|=B∨ϕ holds, andB∨ϕ |= (B− ¬ϕ)∨ϕholds sinceB |= B− ¬ϕholds, as soon as−satisfies(C1).

• B− ¬ϕϕ Bholds sinceB∧ϕ|=B− ¬ϕbecause B∧ϕ|=B holds andB |=B− ¬ϕholds, as soon as

• Bϕϕ B trivially holds whensatisfies(LS2)and (LS7).

Incomparabilities are easy to show as well using counter- examples.

Proposition 6

Given a BRGG= (V, A,L_P,B,R), a setC ⊆V of controllable agents, and a formulaϕ, it can be the case that the basic strategyσϕforGgivenCis not successful forϕ, while

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a control strategy forGgivenC which is successful forϕ exists.

Proof. The example directly following the proposition serves as a proof.

Proposition 7

LetG= (V, A,L_P,B,R)satisfying(SMon), a setC ⊆V of controllable agents, and a formulaϕ. If the basic strategy σϕforGgivenCis not successful forϕthen no control strategy forGgivenCis successful forϕ.

Proof. First, let us point out that for every formulaϕ, sinceϕ is the least element of(L_P,ϕ)for any formulaϕ, then by definition of a promotion operator we get thatϕ_ϕBϕfor every baseB and every promotion operator. This means that for every BRGG= (V, A,L_P,B,R), for any setC⊆V and any formulaϕ, we have that

Gσ_ϕ ϕGσϕ, (1) whereσϕis the basic strategy forGgivenC,σis any strategy forGgivenC, andGσ_ϕ(resp. Gσ) is the BRG obtained by applyingσ_ϕ(resp.σ) toG.

Now, let G = (V, A, L_P, B, R) be a BRG satisfying (SMon), C ⊆ V be a set of controllable agents, andϕbe a formula. Let us show the contraposite of the statement to be proved. Assume that there is a control strategyσfor G givenCwhich is successful forϕ, that is, there exists a map- pingσfromC toL_P stating for eachi ∈ C thatB_i is replaced byσ(i), such that ϕis unanimously accepted in the BRGG_σ obtained by applying σ toG. But since Gsatis- fies(SMon),ϕmust be also be unanimously accepted in any BRGGsigmaϕ. In particular, from Equation 1 aboveϕis unanimously accepted in the BRGBσϕobtained by applying the basic strategy σϕ toG. So the basic strategyσϕ for G givenCis successful forϕ. This concludes the proof.

Proposition 8

For∆ ∈ {∆^d^H^,Σ,∆^d^H^,GMax}, for anyk ∈ {1, . . . ,6},R^k_∆ does not satisfy(SMon)onGcom.

Proof. Let ∆ ∈ {∆^d^H^,^P,∆^d^H^,GMax,∆^d^H^,GMin} and k ∈ {1, . . . ,6}. LetG = (V, A, L_P,B,R)be the BRG from Gcom(R^k_∆)defined as V = {1,2,3,4}, LP is the propositional language defined fromP ={p, q, r},B1=p∧q∧r, B2 = p∧ ¬q∧r,B3 = ¬p∧ ¬q∧randB4 = >. Let ϕ ≡ (p⊕q)∧r. Then one can verify that B4 is stable and consists of the single belief basep∧ ¬q∧r, i.e., ϕ is accepted by agent4 inG. On the other hand, consider the BRGG_1→q∧r = (V, A,L_P, B⁰,R)and note that we have q∧rϕ p∧q∧r, i.e.,B₁⁰ promotesϕw.r.t. B1, however B₄⁰ is stable and consists of the single belief base¬q∧r, i.e., ϕis no longer accepted by agent4inG1→q∧r.

Proposition 9

LetG= (V, A,LP,B,R)be a BRG fromGcom(R^{1,...,6}_∆_{dD ,}Σ ).

If there is a stepswhereV

{B_i^s| i∈V}is consistent, then for eachi∈V,AccG(Bi) = V{Bi |i∈V}. Moreover,G converges at steps+ 2at most. It converges at steps+ 1at most ifG∈ Gcom(R^{2,...,6}_∆_{dD ,}Σ ).

Proof. Let G = (V, A, L_P, B, R) be a BRG from Gcom(R^{2,...,6}_∆_{dD ,}_Σ ),s ≥ 0and assumeϕ = V{B^s_i | i ∈ V} is consistent. Leti ∈ V. It is easy to see that B_i^s can be expressed as a formulaϕ∨α^s_i and∆^d^D^,Σ(hC_i^si)as a formula ϕ∨β_i^s, where both formulaeϕ∧α^s_i,ϕ∧β_i^sandα^s_i ∧β_i^sare inconsistent. We use these notations in the rest of the proof.

We fall into the following cases for eachi∈V:

• R_i = R²_∆_{dD ,}_Σ. ThenB_i^s+1 = B_i◦^d^D ∆^d^D^,Σ(hC_i^si) = (ϕ∨α^s_i)◦^d^D(ϕ∨β^s_i) =ϕ.

•Ri=R³_∆_{dD ,}Σ. ThenB_i^s+1= ∆^d^D^,Σ(hB_i^s,C_i^si) =ϕ.

• Ri = R⁴_∆_{dD ,}Σ. Then B^s+1_i =

∆^d^D^,Σ(hBi,∆^d^D^,Σ(hC_i^si)) = ∆^d^D^,Σ(hϕ∨α^s_i, ϕ∨β_i^si) =ϕ.

• R_i ∈ {R⁵_∆_{dD ,}_Σ, R⁶_∆_{dD ,}_Σ}. Then B_i^s+1 =

∆^d^D^,Σ(hC_i^si)◦^d^DB_i = (ϕ∨β_i^s)◦^d^D(ϕ∨α^s_i) =ϕ.

We just showed that ifG∈ G_com(R^{2,...,6}_∆_{dD ,}_Σ ), then for each i∈V,B_i^s+1=ϕandGconverges at steps+ 1at most.

Now, if G ∈ G_com(R^{1,...,6}_∆_{dD ,}_Σ ), then we may fall into the additional case where for an agenti ∈ V,Ri = R¹_∆_{dD ,}Σ; thenB_i^s+1 = ∆^d^D^,Σ(hC^s_ii) = ϕ∨β_i^s |= ϕ. Yet one can remark thatV

{β_i^s|i∈V}is inconsistent. Indeed, towards a contradiction if there were an interpretationω|=V

{β_i^s|i∈ V}, this would mean thatω|=ϕmax(C_i^s+1)for everyi∈V, i.e., ω |= ϕ_max(B), that is, ω |= ∆^d^D^,Σ(hB^s_i,C_i^si) |= ϕ, which contradicts the fact thatϕ∧β_i^sis inconsistent for each i ∈ V. This shows that at steps+ 2,B_i^s+2 = ϕ. We just showed that ifG ∈ Gcom(R^{1,...,6}

∆^{dD ,}^Σ ), then for each i ∈ V, B_i^s+2=ϕandGconverges at steps+ 2at most.

Proposition 10

Letk ∈ {1, . . . ,4}andG = (V, A,LP,B,R)be a BRG from Gcom(R^k_∆_{dD ,}Σ). Then for each i ∈ V, AccG(i) =

∆^d^D^,Σ(hB1, . . . , Bni).

Proof.

•Letk= 3, then the result is direct. Then for eachi∈V, we haveB_i¹ = ϕmax(B⁰) = ∆^d^D^,Σ(hBi). So for eachi ∈ V,AccG(i) = ∆^d^D^,Σ(hBi).

• Letk = 2. First, Table 3 shows that for eachi ∈ V, B_i¹|=ϕmax(B⁰), so by(UP)we get that for each steps≥1 and for eachi∈V,

B_i^s|=ϕmax(B⁰). (2) Then, let us show that for each steps ≥ 0, ϕ_max(B^s) |= ϕmax(B^s+1) by induction ons. In the case where there is