Axiomatic characterization of belief merging by negotiation

Abstract

Belief merging has been an active research field with many important applications. The major approaches for the belief merging problems, considered as arbitration processes, are based on the construction of the total pre-orders of alternatives using distance functions and aggregation functions. However, these approaches require that all belief bases are provided explicitly and the role of agents, who provide the belief bases, are not adequately considered. Therefore, the results are merely ideal and difficult to apply in the multi-agent systems. In this paper, we approach the merging problems from other point of view. Namely, we treat a belief merging problem as a game, in which rational agents participate in a negotiation process to find out a jointly consistent consensus trying to preserve as many important original beliefs as possible. To this end, a model of negotiation for belief merging is presented, a set of rational and intuitive postulates to characterize the belief merging operators are proposed, and a representation theorem is presented.

Appendix—Proofs

Proposition 1

Δ^N satisfies for IC0, IC1, IC2, IC3, IC7, IC8; it does not satisfy properties IC4, IC5, and IC6.

Proof

For the sake of proof, we recall about the axiomatic approach of Social Contraction function as follows:

Given a belief vector $\overrightarrow{S} \in \mathcal{B}^n$ and a function $f: \mathcal{B}^n \rightarrow \mathcal{B}^n$. We have:

Theorem 2

[7] Function f is a Social Contraction function, i.e. f = f ^N , if and only if f satisfies the following postulates:

• SC1.   $\overrightarrow{S} \subseteq f(\overrightarrow{S})$ ;

• SC2.   $f(\overrightarrow{S})$ is consistent;

• SC3.    if $\overrightarrow{S}$ is consistent, then $\overrightarrow{S} = f(\overrightarrow{S})$ ;

• SC4.   $f_n(\overrightarrow{S}) = S_n$ .

Now, we refer to these postulates to prove Proposition 1 as follows:

• IC0 is satisfied, it means that $\Delta^N(\overrightarrow{S}) \subseteq S_n$. Indeed, by SC4 we have $f_n(\overrightarrow{S}) = S_n$. Therefore, by (1) we have:

  $$ \Delta^N(\overrightarrow{S}) =\bigcap\nolimits_{i=1}^{n}{f_i(\overrightarrow{S})} = \bigcap\nolimits_{i=1}^{n-1}{f_i(\overrightarrow{S})} \bigcap f_n(\overrightarrow{S}) = \bigcap\nolimits_{i=1}^{n-1}{f_i(\overrightarrow{S})} \bigcap S_n. $$

  It implies that $\Delta^N(\overrightarrow{S}) \subseteq S_n$.

• By SC2 we have $f(\overrightarrow{S})$ is consistent, it means that $\bigcap_{i=1}^{n}{f_i(\overrightarrow{S})} \neq \emptyset$. Thus, by (1) we have $\Delta^N(\overrightarrow{S}) \neq \emptyset$, it implies that $\Delta^N(\overrightarrow{S})$ is consistent, so IC1 is satisfied.

• By SC3, if $\overrightarrow{S}$ is consistent then $f(\overrightarrow{S}) = \overrightarrow{S}$, therefore by (1), we have $\Delta^N(\overrightarrow{S}) =\bigcap_{i=1}^{n}{f_i(\overrightarrow{S})} = \bigcap_{i=1}^{n}{S_i}$. Thus, IC2 is satisfied.

• IC3 is obvious.

• IC7 and IC8 are satisfied. Truly, we consider $S_{\mu}\in \mathcal{B}$ and $\overrightarrow{S'}$ from $\overrightarrow{S}$ by replacing S _N by S′ _N  = S _N  ∩ S _μ . IC7 and IC8 are understood equivalently as follows:

  • IC7′.   $\Delta^N(\overrightarrow{S}) \bigcap S_{\mu} \subseteq \Delta^N(\overrightarrow{S'})$.

  • IC8′. if $\Delta^N(\overrightarrow{S}) \bigcap S_{\mu} \neq \emptyset$, then $\Delta^N(\overrightarrow{S'}) \subseteq \Delta^N(\overrightarrow{S}) \bigcap S_{\mu}$.

  Firstly, if $\Delta^N(\overrightarrow{S}) \bigcap S_{\mu} = \emptyset$ then IC7′ is trivially satisfied. Then, we consider the case $\Delta^N(\overrightarrow{S}) \bigcap S_{\mu} \neq \emptyset$, we need to prove that both IC7′ and IC8′ are satisfied, it is equivalent to $\Delta^N(\overrightarrow{S'}) = \Delta^N(\overrightarrow{S}) \bigcap S_{\mu}$. Indeed, by (1), we have:

  $$ \begin{array}{llll} \Delta^N(\overrightarrow{S'}) & = & \bigcap_{i=1}^{n}{f_i(\overrightarrow{S'})} &\\ & = & \bigcap_{i=1}^{n-1}{f_i(\overrightarrow{S'})} \bigcap S'_{N} &\textrm{(by $SC4$, we have } f_n(\overrightarrow{S'})= S'_{N})\\ &=& \bigcap_{i=1}^{n-1}{f_i(\overrightarrow{S'})} \bigcap (S_{N} \bigcap S_{\mu})& \textrm{(because of } S'_{N} = S_{N} \bigcap S_{\mu}) \\ &=& \bigcap_{i=1}^{n-1}{f_i(\overrightarrow{S})} \bigcap (S_{N} \bigcap S_{\mu})&(because~S_i= S'_i, \forall i \in [1..n-1]) \\ &=& \bigcap_{i=1}^{n}{f_i(\overrightarrow{S})} \bigcap S_{\mu} &\textrm{(by $SC4$, we have } S_{N} = f_n(\overrightarrow{S}))\\ &=& \Delta^N(\overrightarrow{S}) \bigcap S_{\mu}& \mbox{(by (\ref{DN:eq}))} \end{array} $$

□

Proposition 2

If F is a Simultaneous Concession Solution, then Δ ^G satisfies properties IC1, IC2, IC5, IC7, and IC8; it does not satisfy properties IC0, IC3, IC4, and IC6.

Proof

In the original work, the author does not mention about the integrity constraint, but in the spirit of the work, we can understand the integrity constraint as a formula μ added in the highest priority level of the demand set, which has the biggest height of the hierarchy. The merging operators are modified as $\Delta^G(G) = \bigcup_{a_i\in\mathcal{A}} f_i(G)\bigcup\{\mu$}. For the convenience of proof, we remind the following result from original work:

Definition 16

[30] Given a belief profile $G=(\{(X_i, \succcurlyeq_i)|{a_i\in \mathcal{A}}\},\mu) \in g^{\mathcal{A},\mathcal{L}}$, belief profile $G'=(\{(X'_i, \succcurlyeq'_i)|{a_i\in \mathcal{A}}\},\mu)\in g^{\mathcal{A},\mathcal{L}}$ is a max proper sub-game of G, denoted $G' \sqsubset_{\max} G$ if for all $a_i\in \mathcal{A}$, we have:

• 1. X′ _i  ⊂ X _i and $\not\exists X$ such that X′ _i  ⊂ X ⊂ X _i ;

• 2. $\phi \in X'_i, \psi \in X_i, \psi \succcurlyeq_i \phi$ implies ψ ∈ X′ _i ;

• 3. $\succcurlyeq'_i = \succcurlyeq_i \bigcap (X'_i \times X'_i)$.

Theorem 3

[30] Given belief profile $G=(\{(X_i, \succcurlyeq_i)|{a_i\in \mathcal{A}}\},\mu) \in g^{\mathcal{A},\mathcal{L}}$ , function $f:g^{\mathcal{A},\mathcal{L}}\rightarrow \mathcal{O}(G)$ is a Simultaneous Concession Solution, i.e f = F, if and only if it satisfies the following axioms:

• Axiom1.    $\bigcup_{a_i\in \mathcal{A}}f_i(G)$ is consistent.

• Axiom2.    $\forall a_i\in \mathcal{A}, \phi \in f_i(G), \psi \in X_i$ and $\psi \succcurlyeq_i \phi$ implies ψ ∈ f _i (G).

• Axiom3.     if $\bigcup_{a_i\in \mathcal{A}} X_i$ is consistent, then $\forall a_i\in \mathcal{A}(f_i(G) = X_i)$ .

• Axiom4.     if $\exists a_i\in \mathcal{A}(f_i(G) = \emptyset)$ , then $\forall a_i\in \mathcal{A}(f_i(G) = \emptyset)$ .

• Axiom5.     if $G' \sqsubset_{\max} G$ and $\bigcup_{a_i\in \mathcal{A}}X_i$ is inconsistent, then f(G) = f(G′).

Now, we refer to the these axioms to prove Proposition 2 as follows:

• IC0 is not satisfied. Indeed, suppose that for some $a_i\in \mathcal{A}$, f _i (G) = ∅, then by Axiom4 f _i (G) = ∅ for all $a_i\in \mathcal{A}$, thus by (2) we have Δ^G(G) = ∅, it implies that $\Delta^G(G) \nvdash \mu$.

• Axiom1 requires that f(G) is consistent, it means that Δ^G(G) is also consistent by (2), thus IC1 is satisfied.

• ByAxiom3, if $\bigcup_{a_i\in \mathcal{A}} X_i$ is consistent, then $\forall a_i\in \mathcal{A}(f_i(G) = X_i)$ if G is non-conflictive. Therefore, $\Delta^G(G) =\bigcup_{i=1}^n f_i(G) = \bigcup_{i=1}^nX_i \bigcup \{\mu\}$, it implies that IC2 is satisfied.

• IC4 is fail. Truly, let us consider bargaining game

  $$ G = (\{(X_1,\succcurlyeq_1),(X_2,\succcurlyeq_2)\},\top), $$

  in which X ₁ = {α, α→β} and $\alpha\succcurlyeq_1\alpha \rightarrow \beta$, $X_2 = \{\lnot\beta, \alpha \rightarrow \beta\}$ and $\alpha \rightarrow \beta \succcurlyeq_2 \lnot\beta$. We have $\Delta^G(G) = \{\alpha, \alpha \rightarrow \beta\} = X_1$, so $\Delta^G(G)\vdash X_1$ and $\Delta^G(G)\nvdash X_2$. It violates IC4.

• By means of Axiom5, we have

  $$\label{DZ:pr} \bigcup f(G_1\bigcup G_2)\subseteq(\bigcup{f(G_1)}) \bigcup (\bigcup{f(G_2)}) $$

  (4)

  and f(G ₁),f(G ₂),  and f(G ₁ ∪ G ₂) are consistent (by Axiom1), thus IC5 is satisfied.

• However, even if ( ∪ f(G ₁)) ∪ ( ∪ f(G ₂)) is consistent, but because of (4), IC6 is not satisfied.

• Now, we prove that Δ^G satisfies both IC7 and IC8. First, we consider a bargaining game G and two formulas μ and ψ as the integrity constraints. We have G′ and G′′ by respectively adding μ and $\mu \land \psi$ into the highest priority level of the demand set, which has the biggest height of the hierarchy of G. We have the counters of IC7 and IC8 as follows:

  • IC7′′.   $\Delta^G(G') \land \psi \vdash \Delta^G(G'')$.

  • IC8′′.   if $\Delta^G(G') \land \psi$ is consistent then $\Delta^G(G'') \vdash \Delta^G(G') \land \psi$.

  If $\Delta^G(G') \land \psi$ is inconsistent, IC7′′ is trivially satisfied. We consider the case when $\Delta^G(G') \land \psi$ is consistent, then to prove IC7′′ and IC8′′ we need to prove $\Delta^G(G') \land \psi = \Delta^G(G'')$. Indeed, we have Δ^G(G′) = ( ∪ f(G)) ∪ {μ}, thus ψ is consistent with μ and ψ is consistent with ( ∪ f(G)) and obviously, ( ∪ f(G)) is consistent with μ. Therefore, ∪ f(G) is consistent with $(\mu\land \psi)$, so $\Delta^G(G') \land \psi = \Delta^G(G) \land (\mu \land \psi) =\Delta^G(G'')$. It is clear that both IC7 and IC8 are satisfied.□

Proposition 3

If Δ ^M(G) is a Negotiation Belief Merging operator, it satisfies for IC0, IC1, IC2, IC7, IC8 and, it does not satisfy properties IC3, IC4, IC5, IC6

Proof

• Because of (3), it is easy to see that IC0 is satisfied.

• Suppose that μ is consistent, we also have $\neg \mu$ consistent. Thus, CO implies that $\bigcup_{a_i\in\mathcal{A}}f_i(G)$ is consistent and by (3) we also have Δ^M(G) consistent. It means that IC1 is satisfied.

• We assume that $\bigcup_{a_i\in\mathcal{A}}X_i \bigcup \{\mu\}$ is consistent. We have

  $$ MAXCONS(\bigcup\nolimits_{a_i\in\mathcal{A}}X_i,\mu) =\{\bigcup\nolimits_{a_i\in\mathcal{A}}X_i\bigcup \{\mu\}\}. $$

  Therefore, by CP we have

  $$ \bigcup\nolimits_{a_i\in\mathcal{A}}X_i\bigcup \{\mu\} \subseteq \bigcup\nolimits_{a_i\in\mathcal{A}}f_i^M(G)\bigcup \{\mu\}. $$

  (*)

  On the other hand, by IR we have

  $$ \bigcup\nolimits_{a_i\in\mathcal{A}}f_i^M(G)\bigcup \{\mu\} \subseteq \bigcup\nolimits_{a_i\in\mathcal{A}}X_i\bigcup \{\mu\}. $$

  (**)

  Thus, from (*) and (**) we have

  $$ \bigcup\nolimits_{a_i\in\mathcal{A}}f_i^M(G)\bigcup \{\mu\} = \bigcup\nolimits_{a_i\in\mathcal{A}}X_i\bigcup \{\mu\} $$

  (***)

  Lastly, from (3) and (***), we have

  $$ \Delta^M(G) = \bigcup\nolimits_{a_i\in\mathcal{A}}X_i \bigcup \{\mu\}. $$

  It means that IC2 is satisfied.

• We consider a belief profile $G=(\{(X_i, \succcurlyeq_i)|{a_i\in \mathcal{A}}\},\mu)$ and a formula μ ₁. If $\Delta^M(G)\land \mu_1$ is inconsistent, IC7 is obviously satisfied. We consider the other case when $\Delta^M(G)\land \mu_1$ is consistent. Let $G'=(\{(X_i, \succcurlyeq_i)|{a_i\in \mathcal{A}}\},\mu\land\mu_1)$. In order to prove that both IC7 and IC8 are satisfied, we need to show that $\Delta^M(G)\land \mu_1 \equiv \Delta^M(G')$. Indeed, because $\Delta^M(G)\land \mu_1$ is consistent, μ and μ ₁ are consistent, and f _i (G) and μ ₁ are also consistent. Therefore, $f_i(G') \equiv f_i(G)\land \mu_1$, and together (3) we have:

  $$\begin{array}{rll} \bigcup\nolimits_{a_i\in \mathcal{A}} f_i(G')&\equiv & \bigcup\nolimits_{a_i \in \mathcal{A}} (f_i(G)\land \mu_1)\\ &\equiv& \bigcup\nolimits_{a_i\in \mathcal{A}} f_i(G)\land \mu_1. \end{array}$$

  Consequently, we have:

  $$\begin{array}{rll} \Delta^M(G') &=& \bigcup\nolimits_{a_i\in \mathcal{A}} f_i(G') \land (\mu \land \mu_1)\\ &\equiv & \left(\bigcup\nolimits_{a_i \in \mathcal{A}}f_i(G)\land \mu\right) \land \mu_1\\ & = & \Delta^M(G) \land \mu_1. \end{array}$$

  Thus, we have $\Delta^M(G)\land \mu_1 \equiv \Delta^M(G')$ if $\Delta^M(G)\land \mu_1$ is consistent.

  Generally, IC7 and IC8 are satisfied.

• IC4 is fail. Truly, let us consider bargaining game $G = ((X_1,\succcurlyeq_1),(X_2,\succcurlyeq_2),\top)$, in which X ₁ = {α, α→β} and $\alpha\succcurlyeq_1\alpha \rightarrow \beta$, $X_2 = \{\lnot\beta, \alpha \rightarrow \beta\}$ and $\alpha \rightarrow \beta \succcurlyeq_2 \lnot\beta$. We have $f^M(G) = \{\alpha, \alpha \rightarrow \beta\} = X_1$, so $f^M(G)\vdash X_1$ and $f^M(G)\nvdash X_2$. It violates IC4.□

Theorem 1

Negotiation Belief Merging solution f ^M satisfies properties IR , CO , CP , PO , and SY .

Proof

We assume to have a Negotiation Belief Merging solution f ^M for any belief merging negotiation model $\mathcal{M}$. We need to show that f ^M satisfies properties IR, CO, CP, PO, and SY. With a initial negotiation S = (G,O), where $G=(\{(X_i, \succcurlyeq_i)|{a_i\in \mathcal{A}}\},\mu)\in g^{\mathcal{A},\mathcal{L}}$ and $O = \{\emptyset|a_i\in \mathcal{A}\}$, we have $s^M(S) =\lambda = (S_0,\ldots,S_k)$.

• For IR, we need to show that if $S_k = (G, O^*)$ and f(G) = O ^* then $f_i(G)\subseteq X_i$. Indeed, because of choice function c, an agent is chosen if she still has some belief to submit and by update function w, she only allows to submit the belief in her belief set. Therefore, an agent obtains the set of beliefs which does not exceed her initial belief set i.e. $f_i(G)\subseteq X_i$.

• Because the integrity constraint μ is put into the initial constructing common belief set and in each round of negotiation the chosen beliefs do not allow to jointly cause conflict with the common belief set, thus the solution f(G) is always jointly consistent with μ. Consequently, CO is satisfied.

• In each round of negotiation, one of maximal consistent subsets of the current common belief set and submitting belief with the current common belief set as integrity constraint is chosen. Thus, both CP and PO are satisfied.

• Because the possible outcome is gradually constructed by based on the beliefs and the preferences of agents on these beliefs without the identifier of agents, therefore SY is satisfied.□

Proposition 4

if Δ ^M(G) is a Negotiation Belief Merging operator,

$$ \Delta^M(G) \vdash \Delta^G(G). $$

Proof

Because Δ^G(G) has to satisfy Axiom1 and Axiom2, it is easy to see that Δ^M(G) ⊢ Δ^G(G) by CO and CP. □