A collaborative group is commonly defined as a set of agents, which share information and coordinate activities while working towards a common goal. How do groups decide which are their common goals and what to intend? According to the much cited theory of Cohen and Levesque, an agent intends g if it has chosen to pursue goal g and it has committed itself to making g happen. Following the same line of reasoning, a group intention should be a collectively chosen goal with commitment. The literature often considers a collective goal to be one of those individual goals that are shared by all members. This approach presumes that a group goal is also an individual one and that the agents can act as a group if they share the beliefs relevant to this goal. This is not necessarily the case. We construct an abstract framework for groups in which common goals are determined by social choice. Our framework uses judgment aggregation to choose a group goal and a multi-modal multi-agent logic to define commitment and revision strategies for the group intentions.
1 Introduction
An agent acts according to its beliefs and intentions. According to what does a group
act? We would expect that in order for groups to act, jointly or by coordinating their
activities, they need to establish what to believe i.e., form epistemic attitudes, and what
to aim for, i.e., to form motivational attitudes. In existing frameworks for collaborative
activities [
Consider a group of robots in charge of office building maintenance. One candidate group goal for them is to clean the meeting room. The decision rule to adopt the goal is that the room needs to be cleaned when the following conditions (reasons) are met: the floors are dirty or the garbage bin is full, and people do not occupy the room. To decide whether to pursue this goal, the robots need to decide if the reasons to adopt the goal are true. The robots cannot check whether the room is occupied or what state it is in. Hence, to estimate the state of affairs, the robots need to rely on their individual beliefs, which may diverge.
Assume there are three robots in the group. One robot believes that the room is
occupied and thus, according to it, the group should not adopt the goal. According to the
other two robots, the group should adopt the goal. One robot believes that the garbage
bin is full and the floors clean and the other that the floors are dirty. The question is how
to should one aggregate the beliefs of the robots in this case? The majority of the robots
would estimate that the goal should be adopted. However the group is not univocal
and as a result, the goal would not be chosen for a group goal when the robots reason
according to [
Consider now that the group adopts the goal, but before pursuing it, the robots learn that there is a seminar scheduled in the meeting room. The group has to de-commit from their intention to clean the room. However, to be able to do so, the group has to have a commitment strategy that allows de-committing upon change in the reasons for the goal adoption. Furthermore, the group needs to be able to reconsider its reasons for goals, and goals, after de-committing. If they simply drop the goal, without “remembering” why, they would not be able to re-deliberate and commit again once the seminar is over.
An intelligent agent reacts to the changes in its environment and a group should be
capable of doing the same. When new information becomes available the group faces
a choice: to remain committed to its group attitudes or to reconsider them. The most
sensitive commitment strategy proposed by Rao and Georgeff [
The research question we address in this paper is:
A good methodology for collectively choosing and reconsidering goals is one that
can be used by any group of agents regardless of the homogeneity of the individual
beliefs of its members and their persuasion abilities. The relation between individual goals
and beliefs can be specified and analyzed in modal agent logics like BDILT L [
Our research question thus breaks down into the following sub-questions: 1. How to aggregate individual (epistemic and motivational) attitudes? 2. What are the desirable properties for this aggregation?
A good methodology for collectively choosing and reconsidering goals is also a methodology that is dynamic enough to allow for the group to change its epistemic and motivational attitudes in light of new information. Having such a methodology increases the autonomy and reactivity of groups. Hence we need to answer the following sub-question as well: 3. Which commitment and reconsideration strategies should be available for groups?
We thus focus on finding the following solutions: Formal framework. We extend a multi-agent modal language to be able to represent judgment aggregation in it.
Choice of aggregation. Judgment aggregation is an abstract framework that allows for various desirable properties to be specified for the aggregation procedure. The task is to determine which aggregation properties are necessary and desirable for aggregating individual beliefs and goals.
Commitment and reconsideration strategies. Within our formal framework, we define when to de-commit from intentions and how to change them.
Multiple goals. Since a group can have more than one goal, we need to model the effect that the commitment to (and reconsideration of) one goal has over the commitment to (and reconsideration of) other goals.
We make the following assumptions. The group has a fixed membership – agents do not join or depart from the group. The group has a set of candidate group goals and an order of priory over these goals. The decision rules for each candidate goal are available to the group. How the decision rules are learned is outside of the scope of this paper. Here we do not consider how plans are generated, executed and revised once the group goals are selected or reconsidered, nor we consider whether the group goals are executed via individual or joint actions.
The layout of the paper is as follows. In Section 2 we introduce judgment aggregation. In Section 3 we extend a multi-modal agent formalism to capture the aggregation of individual attitudes. Sections 4 and 5 respectively study the commitment and reconsideration strategies. Related work, conclusions and outlines for future work are in Section 6. 2
From individual attitudes to group goals Let us consider again the example of the robot cleaning crew from Section 1. Example 1. Let C w 1, w2, w3 be a crew of cleaning ro bots. We denote the group goal to clean the meeting room with g1, and the reasons to adopt this goal with: there are no people in the room (p1), the room is dirty (p2), the garbage bin in it is full (p3). The individual beliefs of the robots on whether g1 should be the group goal are justified by individual beliefs on p1, p2, p3 using the decision rule p 1 p 2 p 3 g 1.
A group of agents could collectively decide to adopt or reject a goal by voting.
However, the goals of the agents are not independent from their beliefs [
Consider a logic L with entailment operator . In a judgment aggregati on framework, an agenda A L is the pre-defined set of issues, on which every agent casts her judgments. E.g., the agenda of Example 1 is A p 1, p2, p3, g1. Consider a valuation function v such that a judgment “yes” on issue a is a valuation va 1, while a judgment “no” on the same issue is a valuation va 0. A profile is the set of a ll judgments assigned, on the agenda issues, by the decision-making agents. Definition 1 (Profile). Let N 1, 2, . . . be a set of agent names, A L an agenda and A A a a A. A profile π is the set of judgments fo r the agenda items, submitted by the agents in the group: π N A. We define two operators: The judgment set for agent i is π i a i, a π.
The set of all the agents who judged “yes” to a A is π a i i, a π.
Definition 1 extends the definition of a profile given [
In judgment aggregation, the judgments over A, both individual and aggregates, are constrained by decision rules R L. The set R contains only formulas r such that all the non-logical symbols of A occur in r R. For instance, in Exampl e 1, the decision rule is: p 1 p 2 p 3 g 1 and all agents respect it. In general, each agent could follow a different decision rule and yet another decision rule can be imposed for the group. In judgment aggregation, the agents are allowed to submit only those judgment sets which are consistent with r R. Often, the agents are als o required to cast judgments on all the agenda issues. We construct Definition 2 to formalize what it means for a judgment set to be admissible.
Definition 2 (Admissible profiles). A judgment set π i is complete if and only if for every a A, either i, a π or i, a π. A judgment set π i is consistent with r R if and only if r π i . A judgment set π i is admissible if it is consistent with the given r R and complete for A. The set of all admissible judgment sets for a given A and r is denoted by W.
A profile π is admissible if π i is admissible for all i N . The set of all admissibl e profiles for agents N is denoted by Π.
We denote conr, ϕ 1 if ϕ is consistent with r and conr, ϕ 0 otherwise. For the profile in Figure 1, conp 1 p 2 p 3 g 1, π i 1 for every i C.
We now present the definition of a judgment aggregation function, as given in [
We refer to f π as the collective judgmen t set for N . We denote the result of f π with when the JA function pr oduces an inadmissible judgment set. If f π then we call f π a decision and denote it b y Dπ. Figure 1 illustrates an example of a judgment aggregation function and profile, for which the collective judgment set is because conp 1 p 2 p 3 g 1, p 1, p2, p3, g 1 0.
The JA function is a useful abstraction, because many properties of judgment aggregation can be defined in terms of it. It then can be studied which properties can be accepted together (avoiding impossibility results). Given a JA function f , we describe the most common properties in the JA literature.
Universal domain. f satisfies universal domain if and only if its domain is Π. Anonymity. Given a profile π Π, letπ π 1, . . . , π n, be the multiset of all the individual judgment sets in π. Two profiles π, π Π are permutations of each other if and only ifπ π . f satisfies anonymity if and only if f π f π for all permutation π and π .
Unanimity on a A. f satisfies unanimity on a A if and only if for every profile π Π it holds: if for all i N , i, a π, then a f π.
Collective rationality. f satisfies collective rationality if and only if for all π Π, conr, f π 1 for a given r R, and either a f π or a f π for every a A. Constant. f is constant when there exists ϕ 2 A such that for every π Π, f π ϕ. Independence. Let Φ π a a A for every π Π. Let f 1, . . . , fm be functions defined as fj : A Φ 0, 1. The JA function f satisfies independence if and only if for all π Π, there exist functions f i such that for all ϕ f π it holds that ϕ a a A, f j a, π a 1 a a A, f j a, π a 0.
The best known example of fj : A Φ 0, 1 is the simple majority v oting fm which counts how many agents expressed judgment “yes” on agenda item a. The function fm returns a if that number of agents is greater than n2 and a otherwise.
fmaj ϕ f mπ a a A if ϕ is complete and consisten otherwise t
The JA function fmaj satisfies universal domain, anonymity, unanimity,
completeness, and independence but it does not satisfy collective rationality, as it can be seen on
Figure 1. All judgement aggregation functions which satisfy universal domain, anonymity,
independence and collective rationality are constant [
Two aggregation procedures that violate independence but guarantee universal
domain, anonymity and collective rationality have been proposed in the literature:
premisebased and conclusion-based procedures. A distinguishing feature of judgment
aggregation with respect to social choice theory is the distinction between premises and
conclusions. The agenda is a union of two disjoint sets: the premise set (Ap), and the
acnodncAlucsiong set2(.AWc)e. AgiveAa gpenAeracl,dAefipnAitiocn o.nInwEhxeanmaplJeA1,fuAnctionp isp pre1m,pis2e,-p3or
conclusion-based in Definition 4. The literature on judgment aggregation [
An example of premise and conclusion-based aggregations is given in Figure 1. Intuitively, a JA function is premise-based when the collective judgment set on the premises is obtained as a result of aggregating only the premise profile πp. The decisions f π are those ϕ W for which f pπ p ϕ (or alternatively f cπ c ϕ for conclusion-based aggregations). However, it is possible that, depending on the decision rule, there are more than one ϕ W that satisfy the conditio n f pπ p ϕ (or alternatively f cπ c ϕ). This is what happens fo r the conclusion-based procedure we illustrate in Figure 1. 2.2
In this section first we set the problem of finding collective decisions for group goals in the context of judgment aggregation. We then argue that the aggregation function used for this problem should satisfy: collective rationality, universal domain, unanimity and select a unique ϕ W. For a democratic group , in which all agents have equal say on what the group attitudes should be, anonymity should be satisfied. For a group in which the agents have different levels of expertise, anonymity can be omitted.
We use Gg to denote that “g is a group goal”. A set of the candidate group goals is the set G Gg g L which contains all the g oals which the group considers to adopt. For each goal Gg G, the group has at its dispo sal decision rules Rg R and an agenda Ag composed of the goal g (conclusion), and all the reasons (premises) relevant for g, which were given by the decision rule. Each agent submits her judgments on the agenda thus generating a profile πg, such that conR g, πg i 1 for all i N .
The decision rules contain all the constraints which the individual and collective judgment sets should satisfy. These constraints contain three types of information: just), rules rules describing how the goal depends on the reasons (justification rules Rg describing the constraints of the world inhabited by the agents (domain knowledge RgDK ) and rules that describe how g interacts with other candidate goals of the group coord). Hence, the decision rule for a group goal g is Rg (coordination rules Rg Rjgust R gDK R cgoord. just Example 2 (Example 1 revisited). Consider the cleaning crew from Example 1. Rg1 is p 1 p 2 p 3 Gg 1 and Ag1 p 1, p2, p3, Gg1. Suppose that the crew h as the following candidate group goals as well: place the furniture in its designated location (g2) and collect recyclables from garbage bin (g3). The agendas are Ag2 p 4, p5, p6, p7, Gg2, A g3 p 3, p8, p9, Gg3. The justification rules are Rjgu2st p 4 p 5 p 6 p 7 Gg 2 and Rjgu3st p 8 p 9 p 3 Gg 3. The formulas p4 p 9 are: the furniture is out of place (p4), the designated location for the furniture is empty (p5), the furniture has wheels (p6) , the furniture has handles (p7), the agents can get revenue for recyclables (p8), there is a container for the recyclables (p9). An example of a domain knowledge could be RgD2K p 4 p 5, since it can not happen that the designated location for the furniture is empty while the furniture is not out of place. Group goal Gg3 can be pursued at the same time as Gg1, however, Gg2 can only be pursued alone. Thus the coordination rule for all three goals is Rcgo1ord R cgo2ord R cgo3ord Gg 2 Gg 1 Gg 3 Gg 2.
We want the justifications for a goal to explain when a goal should be adopted/refuted. Having collective justifications for a goal enables the agents to, when the world changes, consider adopting a goal that has been rejected previously. To this end, we take into consideration justification rules which are of form Gg Γ , where Γ L such that all the non-logical symbols of Γ occur in Agp and Gg A cg.
We now continue to discuss the desirable properties for the aggregation of individual beliefs and goals. We need a JA function that satisfies universal domain to be able to aggregate all admissible profiles. We can use only JA functions that satisfy collective rationality. If the collective set is not complete, for example, if it contains only a collective judgment on the goal, then we do not know why the goal was (not) adopted and consequently we would not know when to revise it. For example, the cleaning crew decides for the goal g3 (to collect recyclables), without having the reasons like p9 (a container where to put them). If the world changes and p 9 holds, the robots will continue to collect recyclables.
The aggregation of all admissible profiles should produce a set consistent with the decision rule because otherwise we would not be generating the group goals and justifications for them. For the same reason we need an aggregation method that selects a unique ϕ W.
To guarantee that conR g, f π g 1, we need to choose betw een conclusionbased and premise-based procedures. At first glance, the premise-based procedure seems an obvious choice since it will produce complete collective judgment sets under our decision rules. However, upon closer inspection, this procedure has notable drawbacks.
As it is observable from the profile in Figure 1; a premise-based procedure may lead the group to adopt a conclusion that the majority (or even the unanimity) of the group does not endorse. In our case, the conclusion is the goal and a premise-based aggregation may establish a group goal which none of the agents is interested in pursuing. To avoid this we need to aggregate using a conclusion-based procedure. In particular we want a conclusion-based procedure that has the property of unanimity on Gg.
Given that our decision rules are of the form g Γ , there exist profiles for w hich
a conclusion-based procedure will not produce complete collective set of judgments.
However, the conclusion-based aggregation can be supplemented with an additional
procedure that completes the produced set of judgments when necessary. Such
aggregation procedure is the complete conclusion-based procedure (CCBP) developed in [
The CCBP from [
Formal framework for modeling group attitudes In this section we introduce the language of modal multi-agent logic in which we represent individual and collective mental attitudes. We then combine the methodology of judgment aggregation with this representation language and show how collective attitudes are generated. To model commitment to a group goal and reconsideration of a group goal we use temporal logic. 3.1
Just like modal agent logic is concerned mainly with the relation between the individual goals and beliefs over time, modal multi-agent logic is concerned with the relation between group goals and beliefs over time. We use modal multi-agent logic to represent: the agenda, individual judgments, collective judgments and new information that may prompt goal revision. In line with judgment aggregation proper, we do not use the formal language to represent the judgment aggregation function, but only the arguments of this function and the results from it. We assume that there is a service, available to the agents, that elicits the individual judgments, performs the aggregation and makes the aggregation results available, to the members and for plan-generation.
Agenda issues in judgment aggregation are usually represented by propositional formulas. This is not a viable option in our case. First, we want to represent the difference between a goal and the supporting reasons by means of representing the distinction between conclusions and premises explicitly in the logic. Second, the logic should represent the distinction between individual and collective judgments. We distinguish conclusions from premises by using a single K modal operator Gg for representing that “g is a group goal”.
The obvious choice for modeling the judgment “true” on agenda issue a, of an agent
i, is the modal operator belief Bia (correspondingly Bia for a judgment “false” o n
a). However, we find that beliefs are ill suited for modeling collective judgments of
agents. While a belief Bia is an exclusively private mental state, judgments are public
and contributed for the decision-making purposes of the group. A judgment is thus
closer to a public commitment than to a private belief. Hence, we model judgments
by using the modal operator of acceptance AS a [
We represent the new information that becomes available to the agents with a normal modal K operator E. Ea denotes: “it is observed that a is true”.
Lastly, to model how the group attitudes evolve with reconsideration we need a temporal logic. By using LT L we do not need to distinguish between path formulas and state formulas, but we are able to quantify over traces. Just as in BDILT L, where Ba denotes that a is believed to be necess ary, we use Ea to denote that a is observed to be impossible.
We now give the syntax of our modal multi-agent logic AGELT L.
Definition 5 (Syntax). Let Agt be a non-empty set of agents, with S Agt, and L P be a set of atomic propositions. The admissible formulae of AGELT Lare formulae ψ0, ψ1 and ψ2 of languages Lprop, LG and LAELT L correspondingly: ψ0 :: p ψ 0 ψ 0 ψ 0 ψ1 :: ψ 0 Gψ 0 ψ2 :: ψ 0 A S ψ1 Eψ 2 Xψ2 ψ 2Uψ2 where p ranges over LP and S over 2Agt. Moreover, ♦φ Uφ, φ ♦φ, and φRφ φUφ .
We can now adjust the definition for a judgment aggregation function. We represent individual judgments with Ai a with intuitive reading “agent i judges a as true” and Ai a with reading “agent i judges a as false”.
Definition 6 (JA in AGELT L ). Let N 1, 2, . . . be a set of agent name s and G L GL prop a set of candidate goals.
An agenda Ag L G for goal Gg G is a set of formulas such that Ag Agp Acg, with Agp A gp a a A gp, A gp L prop and Acg Gg, Gg.
A profile of judgments is the set π A i a i N , a A g. π i a A i a π is the judgment set of age nt i. π a i A i a π is the set of all the agent s that accepted a.
Given a set of decision rules Rg L prop, a profile is acceptable if and only if, for all i N and all a A, conR g, π i 1 and either A i a or Ai a. The set of all acceptable profiles for N and Ag, given Rg is Π.
The decision for a profile π, Dπ A a a f a π, f a : Π 2
AGELT Lhas Kripke semantics. As in Schild [
Temporal formulas are validated in the standard manner [
The mental state of the group is determined by the mental states of the members and the choice of judgment aggregation function. We represent the mental state of the group by a set χ of AGELT L formulas. The set χ contains: the set of all candidate goals for the group G L GL prop and, for each Gg G, the corresponding decis ion rules Rg, as well as the individual and collective acceptances made in the group regarding agenda Ag. The set χ is common knowledge for the group members. An agent uses χ when it acts as a group member and its own beliefs and goals when it acts as an individual.
To deal with multiple, possibly mutually inconsistent goals, the group has a priority order x over the group goals G χ. To avoid overburdening the language with a x operator, we incorporate the priority order within the decision rules Rjguist Γ i Gg i. We want the decision rules to capture that if Ggi is not consistent (according to the coordination rules) with some higher priority goals Gg1, . . . , Ggm, then the group can accept Ggi if and only if none of Gg1, . . . , Ggm is accepted. Hence, we replace the jGusgtjifiGca,tGiogn rjulexRGjguigsitandχ GwgitihGRg gpjijuRst cgΓoiordi . jmA N Gg j Gg i, where Example 3. Consider the goals and rules of the robot crew C from Example 2. Assume the crew has been given the priority order Gg1 χ Gg2 χ Gg3. χ contains: G Gg 1, Gg2, Gg3, one background knowle dge rule, one coordination rule, three justification rules, out of which two are new priority modified rules: G, p 4 p 5, Gg 2 Gg 1 Gg 3 Gg 2, Gg1 p 1 p 2 p 3, Gg2 p 4 p 5 p 6 p 7 A C Gg 1, Gg 3 p 8 p 9 p 3 A C Gg 2.
The agents give their judgments on one agenda after another starting with the agenda for the highest priority candidate goal. Once the profile π and the decision Dπ for a goal g are obtained, they are added to χ. To avoid the situation in which the group casts judgments on an issue that has already been decided, we need to remove decided issues from Ag before eliciting the profile for this agenda.
The group goals are generated by executing GenerateGoals(χ, N ). function GenerateGoals(χ, S): for {eaBch: GagAi G s.tN.[aGgχ aj AG: (Gg j Gag χi)A(A N Ggig;j χ or A C:χGg j χ)] comment // B is the set of already cNollectively accpted issues from Agi elicit requests the agents to submit complete judgment sets for πgi χ. We require that elicit is such that for all returned π it holds conχ, π i 1 for all i N and that conχ, f a π 1. When a higher priority goal Ggi is accepted by the group, a lower priority incompatible goal Ggj cannot be adopted regardless of the judgments on the issues in Agj . Nevertheless, Although elicit will provide individual judgments for the agenda Agj . If the acceptance of Ggi is reconsidered, we can obtain a new decision on Ggj because the profile for Ggj is available.
Ag1 , Ag1 p 8, p9, Gg3.
Example 4. Consider the χ for robots given in Example 3. The following calls to elicit are made in the given order. First, πg1 elicitN , A g1 , χ with the GenerateGoalsχ χ χ πg1 f a πg1 . Second, πg2 elicitN , A g2 , χ , with GenerateGoalsχ χ χ πg2 f a πg 2. Last, πg3 elicitN , A g3 , χ , with GenerateGoalsχ χ χ πg3 f a πg3 . Since there is no overla pping between agendas Ag2 and
A g1 and Ag2 A g2 . However, since Ag2 A g3 p 3, then Ag3 Proposition 1. GenerateGoals terminates if and only if elicit terminates and does not violate the candidate goal preference order.
4
Commitment strategies
The group can choose to reconsider the decision (acceptance or rejection) on a group
goal in presence of new information. Whether the group chooses to reconsider depends
on how committed it is to bring the goal about. According to Cohen and Levesque
[
N N acceptance as resulting from judgment aggregation, which is a social choice method. Thus, in our framework, group intention is social choice with commitment. The level of commitment of a group to a collective acceptance depends on the choice of commitment strategy.
These are the three main commitment strategies (introduced by Rao and Georgeff [
These commitment strategies only consider the relation between beliefs regarding g and Gg. Instead, a commitment to a goal can now be reconsidered upon new information on either one of the agenda issues in Ag and also upon new information on a higher priority goal.
The strength of our framework is exhibited in its ability to describe the groups’ commitment not only to its decision to adopt a goal, but also to its decision to reject aangdoaAl. Ngamje.lyC, oifmtmheitamgeennttstoderecjiedcetdgI NallgoiwasnfdoArgNtgo bje arreecocnosmidmeritetdedantod beovtehntIuNalglyi adopteNd if the state of the world changes.
Let N be a set of agents with a set of candidate goals G. Let Ggi, Ggj G have agendas Agi , Agj . We use p A gpi and qi A cgi , qj A cgj . The profiles and decisions are πgi and f a πgi ; Gg j Gg i, and Ggj cannot be pursued at the same time as Ggi.
We use the formulas α 1 α 7 to refine the blind, single- minded and open-minded commitment. Instead of the “until”, we use the temporal operator release: ψ R φ ψ U φ, meaning that φ has to be true until and in cluding the point where ψ first becomes true; if ψ never becomes true, φ must remain true forever. Unlike the until operator, the release operator does not guarantee that right hand-side formula will ever become true, which in our case translates to the fact that an agent could be forever committed to a goal. (α1) Egi R I g (α2) R A NGgi i (α3) Eg (α4) A (α5) A
Ni Eg i R A N qi
NN qp EjpRRAAN qi N qi Blind commitment: α1 α 2.
Only the observation that the goal is achieved (Egi) can release the intention to achieve the goal I gi. If the goal is never achieved, the group will always be committed to it.
N If a goal is not accepted, then the agents will not reconsider accepting it.
Single-minded commitment: α3.
Only new information on the goal (either that the goal is achieved or had become impossible) can release the decision of the group to adopt /reject the goal. Hence, new information is only regarded if it concerns the conclusion, while information on the remaining agenda items is ignored.
Extended single-minded commitment: α3 α 4.
Not only new information on gi, but also the collective acceptance to adopt a more important incompatible goal gj can release the intention of the group to achieve gi. Similarly, if gi is not accepted, the non-acceptance can be revised, not only if gj is observed to be impossible or achieved, but also when the commitment to pursue gj is dropped (for whatever reason). Open-minded commitment: α3 α 5.
A group will maintain its collective acceptances to adopt/reject a goal as long as the new information regarding all collectively accepted agenda items is consistent with f a πgi . Extended open-minded commitment: α3 α 4 α 5.
Extending on the single-minded commitment, a change in intention to pursue a higher priority goal Ggj can also release the acceptance of the group on Ggi.
Once an intention is dropped, a group may need to reconsider its collective acceptances. This may cause for the dropped goal to be re-affirmed, but a reconsideration process will be invoked nevertheless. 5
Reconsideration strategies In Section 3.2 we defined the mental state of the group χ. We can now define what it means for a group to be coherent.
N
N Definition 7 (Group coherence). Given a Kripke structure M and situations s W , a group of N agents is coherent if the following conditions are met: (ρ1): M A S a A S a for any S N and any a A g. (ρ2): If M, s χ then χ . (ρ3): M, s G g for all Gg G. (ρ4): Let Gg G and G GGg, then M G Eg XGg. (ρ5): Let p A gp and q Gg, Gg. Ep Ep R A q XA p The first condition ensures that no contradictory judgments are given. The second condition ensures that the mental state of the group is logically consistent in all situations. The third and fourth conditions ensure that impossible goals cannot be part of the set of candidate goals and if g is observed to be impossible in situation s, then it will be removed from G in the next situation. ρ5 enforces the acceptance of the new information on the group level, when the commitment strategy so allows – after a is observed and that lead the group to de-commit from g, the group necessarily accepts a.
A coherent group accepts the observed new information on a premise. This may cause the collective acceptances to be inconsistent with the justification rules. Consequently, the decisions and/or the profiles in χ need to be changed in to ensure that ρ1 and ρ2 are satisfied. If, however g or g is observed, the group r econsiders χ by removing Gg from G. In this case, the decisions and profiles are not changed.
For simplicity, at present we work with a world in which the agents’ knowledge can only increase, namely the observed information is not a fluent. A few more conditions need to be added to the definition of group coherence, for our model to be able to be applicable to fluents. E.g., we need to define which observation is accepted when two subsequent contradictory observations happen.
For the group to be coherent at all situations, the acceptances regarding the group goals need to be reconsidered after de-commitment. Let Dg χ contain the group acceptances for a goal g, while πg χ contain the profile for g. There are two basic ways in which a collective judgment set can be reconsidered. The first way is to elicit a new profile for g and apply judgment aggregation to it to obtain the reconsidered Dg . The second is to reconsider only Dg without re-eliciting individual judgments. The first approach requires communication among agents. The second approach can be done by each agent reconsidering χ by herself. We identify three reconsideration strategies available to the agents. The strategies are ordered from the least to the most demanding in terms of agent communication.
Decision reconsideration (D-r). Assume that Ep, p A gp, q Gg, Gg and the group de-commited from A q. The reconsidered decision Dg is such that p is acrcuelpetse,dn, aim.ee.,lyAcNonpR Dgpjusgt,, DanNgd tNhe e1n.tiIrfethdeecDis-irosnpeiscicfioenssaisnteunntiqwuiethDthe jugst,ififocrataionyn observed information and any Dg, then χ can be reconsidered without any communication among the agents. Given the form of Rgpjust (see Section 3.2), this will always be the case.
However D-r is not always an option when the de-commitment occurred due to a change in collective acceptance of a higher priority goal g . Let q Gg , Gg . Let the new acceptance be AN q . D-r is possible if and only if Dg D g and caoccneRptagpnjcuesto,fDqg oAr q Nisq neve1r.inRethcealdl etchiastioAn for Nπgq. was not in Ag and as such the Partial reconsideration of the profile (Partial π-r). Assume that Ea, a A g, Gg G. Not only the group, but also the individual agents need to accept a. The Partial π-r asks for new individual judgments be elicited. This is done to ensure the logical consistency of the individual judgment sets with the observations. New judgments are only elicited from the agents i which Ai a.
Let W N be the subset of agents i s.t. A i a χ. Agents i are s.t. A i a χ when the observation is Ea. Let πgW πg be the set of all acceptances made by agents in W . We construct χ χ πgW . The new profile and decision are obtained by executing GenerateGoals (χ , W ).
Example 5. Consider Example 4. For πg1 , πg2 and πg3 of the robot crew C, the decisions Dg1 A C p1, AC p 2, AC p3, AC Gg1, D g2 A C p4, AC p5, AC p6, AC p7, AC Gg 2 and Dg3 A C p8, AC p9, AC Gg3 are made. Assume the gro up de-commits on Gg1because of Ep 2. If the group is committed to Gg3, the commitment on Gg3 will not allow for A p3 to be modified when reconsidering Gg1. Since A p3 exists in χ , p3 will be
N N excluded from the (new) agenda for g1, although it was originally in it. elicit calls only on the agents in W to complete πg1 χ with their judgment sets.
Full profile reconsideration (π-r). The full profile reconsideration is the same with the partial reconsiderations in all respects except one – now W N . Namely, within the full profile revision strategy, each agent is asked to revise his judgment set by accepting the new information, regardless whether he had already accepted the information or not.
Unlike the Rao and Georgeff commitment strategies [
Example 6. Consider the profiles and decisions in Example 5 . Assume that initially the group chose open-minded commitment for IC g1 and blind commitment for IC g3, with goal open-minded commitment for AC Gg 2. If Eg1 and thus IC g1 is dropped, then the extended open-minded commitment would allow AC Gg 2 to be reconsidered and eventually IC g2 established. However, since the group is blindly committed to IC g3, this change will not cause reconsideration and as a result both IC g2 and IC g3 will be in χ thus making χ incoherent.
Problems arise when subR gpijust subR gpjjust , where subR gpjust denotes the set of atomic sub-formulas of g (Ggi, Ggj G). Proposition 2 summari zes under which conditions these problems are avoided.
Proposition 2. Let α and α be the commitment strategies selected for gi and gj correspondingly. χ α α (in all situations): a) if φ subR gpijust subR gpjjust and p A gi A gj , then α5 is either in both α and α or in none; b) if Ggi is more important than Ggj and Gj and Gi cannot be accepted at the same time, then α4 α .
Proof. The proof is straightforward. Namely, if the change on acceptance of Ggi causes the decision on Ggj to induce group incoherence, we are able to de-comit from Ggj . If we were not able to de-comit on Ggj group coherence is blocked. If the change on collective acceptance on Ggi is caused by an observation on a premise p A gi A gj then condition a) ensures that the commitment to the collective acceptance regarding Ggj does not block group coherence. If the change on collective acceptance on Ggj is caused by a change in commitment to a higher priority goal the condition b) ensures that a commitment regarding Ggj does not block group coherence. Condition b) allows only “goal sensitive” commitments to be selected for lower level goals. 6
Conclusions
We constructed a group decision-making framework by combining judgment
aggregation and multi-agent modal logic. We identified the desirable judgment
aggregation properties for aggregation in collaborative groups. Our multi-agent modal logic
AGELT L is an extension of BDILT L with modal operators for representing individual
and collective acceptances and observations of new information. We extend the
commitment strategies of Rao and Georgeff [
Our framework is intended for groups that engage in joint activity. Our framework
is applicable when it is not possible to assume that the agents persuade each other on
a single position and goal, but it is necessary anyway that the group presents itself as
a single whole from the point of view of beliefs and goals, and above all as a rational
entity that has goals justified by the beliefs it holds, and it is able to revise these goals
under the light of new information. This requirement was held by Tuomela [
Work on collaborative group activities [
Icard et al. [
In our framework, the new information is simultaneously available to the entire
group. In the future we intend to explore the case when only some members of the group
observe the new information. The only assumptions we make regarding the connectivity
of the members is that they are able to communicate their acceptances and receive the
aggregation result. The problem of elicitation and communication complexity in voting
is a nontrivial one [
In the work we presented, we do not consider how individual acceptances are formed.
We can take that Biφ A i φ, but this need not be the case. We can define dishonest
agents as those for which Biφ A i φ does not hold. In this case, the agent might
declare Ai φ while it does not believe φ. The question is whether there are scenarios in
which incentives for doing so arise. Furthermore, given that the some reconsideration
strategies call for re-elicitation of judgments, can an agent have the incentive to
deliberately give judgments that would lead to sooner re-elicitation? We intend to devote more
attention to answering these questions as well as studying the manipulability properties
of our decision-making framework.
Appendix – Relations between AGELT L and acceptance logic and
axiomatization of AGELT L
Here we elaborate in more detail on the fusion logic AGELT L we use. The modal
operator AS φ we use is equivalent to the modal operator AS:xφ of the acceptance logic
of [
The operator AS:xφ uses x ranging over a set of labels to describe the context under which the acceptance is made. In our case the context is that of the group and since we deal with only one group, we have no use of these labels. The context labels play no role in the semantics of the acceptance logic formulas.
On the semantic level, the axioms for AS φ are all the axioms of AS:xφ except two: the axiom inclusion (Inc.) and the axiom unanimity (U n.). Dropping (U n.) and (Inc.) does not affect the decidability of the logic of acceptance. (U n.) (not to be confused with unanimity in judgment aggregation) states that if A N , A i:x φ. In our case, it is the aggregation of individual acceptances Nth:axtφd,ettheremniines the collective acceptance. Since we use the acceptances to model judgments, we do not want an axiom that states that the individual judgments mirror the collective judgments. The agents use the collective acceptance when functioning as a group and their private beliefs when acting as individuals. In our framework we do not model the private mental states, but only individual acceptances which are “declared” to all the agents in the group.
(Inc.) states that if a the group C accepts ϕ, so will any subgroup B C. In our case, the judgment aggregation over the profile containing only the judgment sets of B can produce a different collective judgment set than the profile containing all the judgment sets of C.
The axiomatization of the AGELT Llogic is thus: (ProTau) All principles of propositional calculus (LTLTau) All axioms and derivation rules of LTL (K-G) Gφ ψ Gφ Gψ (K-E) Eφ ψ Eφ Eψ (K-A) AS φ ψ A S φ A S ψ (PAccess) AS φ A M AS φ if M S (NAccess) A S φ A M A S φ if M S (Mon) A S A M if M S (MP) From φ and φ ψ infer ψ (Nec-A) From φ infer A S φ (Nec-G) From φ infer Gφ (Nec-E) From φ infer Eφ
Given M W, R , G , E , A, L and s W , the truth conditions for t AGELT L(in a situation s) are: he formulas of
N – M, s ; – M, s p if and only if p Lp; – M, s φ if and only if M, s φ; – M, s φ ψ if and only if M, s φ and M, s ψ; – M, s A φ if and only if M, s φ for all s, s
A;
A formula φ is true in an AGELT Lmodel M if and only if M, s φ for every situation s W . The formula φ is valid (noted AGELT L ) if and only if φ is true in all AGELT Lmodels. The formula φ is AGELT L-satisfiable if and only if the formula ϕ is not AGELT Lvalid.