Quantified Group Responsibility in Multi-Agent Systems

Abstract

-This paper 1 builds on an existing notion of group responsibility and proposes two ways to define the degree of group responsibility: structural and functional degrees of responsibility. These notions measure potential responsibilities of agent groups for avoiding a state of affairs. According to these notions, a degree of responsibility for a state of affairs can be assigned to a group of agents if, and to the extent that, the group of the agents have potential to preclude the state of affairs. These notions will be formally specified and their properties will be analyzed.

I. INTRODUCTION

The concept of responsibility has been extensively investigated in philosophy and computer science. Each proposal focuses on specific aspects of responsibility. For example, [ 2 ] focuses on the causal aspect of responsibility and defines a notion of graded responsibility, [ 3 ] focuses on the organizational aspect of responsibility, [ 4 ] argues that group responsibility should be distributed to individual responsibility, [ 5 ] focuses on the interaction aspect of responsibility and defines an agent’s responsibility in terms of the agent’s causal contribution, and [ 6 ] focuses on the strategic aspect of group responsibility and defines various notions of group responsibility. In some of these proposals, the concept of responsibility is defined with respect to a realized event “in past” while in other approaches it is defined as the responsibility for the realization of some event “in future”. This introduces a major dimension of responsibility, namely backward-looking and forward-looking responsibility [ 7 ]. Backward-looking approaches reason about level of causality or contribution of agents in the occurrence of an already realized outcome while forward-looking notions are focused on the capacities of agents towards a state of affairs.

Although some of the existing approaches are designed to measure the degree of responsibility, they either constitute a backward-looking (instead of forward-looking) notion of responsibility [ 2 ], provide qualitative (instead of quantitative) levels of responsibility [ 8 ], or focus on individual (instead of group) responsibility [ 5 ]. To our knowledge, there is no forward-looking approach that could measure the degree of group responsibility quantitatively. Such notion would enable reasoning on the potential responsibility of an agent group 1The original version of this work appears in [ 1 ]. towards a state of affairs in strategic settings, e.g., collective decision making scenarios. In this paper, we build on a forward-looking approach to group responsibility and define two notions of responsibility degrees. The first concept is based on the partial or complete power of an agent group to preclude a state of affairs while the second concept is based on the potentiality of an agent group to reach a state where the agent group possesses the complete power to preclude the state of affairs. This results in a distinction between what we will call the “structural responsibility” versus the “functional responsibility” of an agent group. In our proposal, an agent group has the full responsibility, if it has an action profile to preclude the state of affairs. All other agent groups that do not have full responsibility, but may have contribution to responsible agent groups, will be assigned a partial degree of responsibility.

II. GROUP RESPONSIBILITY: A POWER-BASED ANALYSIS

In order to illustrate our conception of group responsibility and the nuances in degrees of responsibility, we follow [ 2 ] and use a voting scenario to explain the degree of responsibility of agents’ groups for voting outcomes. The voting scenario considers a small congress with ten members consisting of vfie Democrats (D), three Republicans (R), and two Greens (G). We assume that there is a voting in progress on a specific bill (B). Without losing generality and to reduce the combinatorial complexity of the setting, we assume that all members of a party vote either in favour of or against the bill B. Table I illustrates the eight possible voting outcomes. Note that in this scenario, six positive votes are sufficient for the approval of B. For example, row 4 shows the case where R and D vote against B and the bill is disapproved. For this case we say that the group RD votes against B. It should also be noted that our assumption reduces parties to individual agents with specific weights such that the question raises as why we use this party setting instead of a simple voting of three agents whose votes have different weights. The motivation is that this setting is realistic and makes the weighted votes of each agent (party) more intuitive.

Following [ 6 ] we believe that it is reasonable to assign the responsibility for a specific state of affairs to a group of agents

TABLE I

VOTING RESULTS G(2)

R(3)

D(5)

Result if they jointly have the power to avoid the state of affairs2. According to [ 9 ], the preclusive power is the ability of a group to preclude a given state of affairs which entails that a group with preclusive power, has the potential but might not practice the preclusion of a given state of affairs. For our voting scenario, this suggests to assign responsibility to the group GR consisting of parties G and R since they can jointly disapprove B. Note that the state of affair to be avoided can also be the state of affairs where B is disapproved. In this case, the group can be assigned the responsibility to avoid disproving B. Similarly, groups D, GD, RD, and GRD have preclusive power with respect to the approval of B as they have sufficient members (weights) to avoid the approval of B. Note that none of the other two groups, i.e., G and R, could preclude the approval of B independently. However, based on [ 6 ], the agent groups that consist of a smaller sub-group with preclusive power, must be excluded from the set of responsible groups. Hence, we consider GR and D as being responsible groups for the approval of B. The intuition for this concept of responsibility is supported by the fact that the lobby groups are willing (i.e., it is economically rational) to invest resources in parties that have the power to avoid a specific state of affairs.

We build on the ideas in [ 6 ] and propose two orthogonal approaches to capture our conception of degree of group responsibility towards a state of affairs. Our intuition suggests that the degree of responsibility of a group of agents towards a state of affairs should reflect the extent they structurally or functionally can contribute to the groups that have preclusive power with respect to the state of affairs.

Our conception of structural responsibility degree is based on the following observation in the voting scenario. We deem that regarding the approval of B, although the groups G and R have no preclusive power independently, they nevertheless have a share in the composition of GR with preclusive power regarding the approval of B. Hence, we say that any 2See [ 6 ] for a detailed discussion on why to focus on avoiding instead of enforcing a state of affairs. group that shares members with responsible groups, should be assigned a degree of responsibility that reflects its proportional contribution to the groups with preclusive power. For example, group R with three members, has larger share in GR than the group G has. Therefore, we believe that the relative size of a group and its share in the groups with the preclusive power are substantial parameters in formulation of the notion of responsibility degree. In this case, the larger share of R in GR in comparison with the share of G in GR will be positively reflected in R’s responsibility degree. These parameters will be explained in details later. We would like to emphasize that this concept of responsibility degree is supported by the fact that lobby groups do proportionally support political parties that can play a role in some key decisions. In a sense, the lobby groups consider political parties responsible for some decision and therefore they are willing to support the parties.

The second approach in capturing the notion of functional responsibility degree addresses the dynamics of preclusive power of a specific group. Suppose that the bill B was about declaration of the congress to the President (P ) which enables P to start a war (Table II). Roughly speaking, P will be in charge only after the approval of the congress. When we are reasoning at the moment when the voting is in progress in the congress, it is reasonable to assume that groups GR and D are responsible as they have preclusive power to avoid the war. Moreover, after the approval of B, the President P is the only group with preclusive power to avoid the war. Hence, we believe that although P alone would not have the preclusive power before the approval of B in the congress, it is rationally justifiable for an anti-war campaign to invest resources on P , even before the approval voting of the congress, simply because there exists possibilities where P will have the preclusive power to avoid the war. Accordingly, a reasonable differentiation could be made between the groups which do have the chance of acquiring the preclusive power and those they do not have any chance of power acquisition. This functional notion of responsibility degree addresses the eventuality of a state in which an agent group possesses the preclusive power regarding a given state of affairs.

III. MODELS AND PRELIMINARY NOTIONS

The behaviour of a multi-agent system is often modelled by concurrent game structures (CGS) [ 10 ]. Such structures specify possible state of the system, agents’ abilities at each state, and the outcome of concurrent actions at each state. Definition 1 (Concurrent game structures [ 10 ]). A concurrent game structure is a tuple M = (N, Q, Act, d, o), where N = {1, ..., k} is a nonempty finite set of agents, Q is a nonempty set of system states, Act is a nonempty and finite set of atomic actions, d : N × Q → P(Act) is a function that identifies the set of available actions for each agent i ∈ N at each state q ∈ Q, and o is a deterministic and partial transition function that assigns a state q0 = o(q, α1, ..., αk) to a state q and an action profile (α1, ..., αk) such that all k agents in N choose actions in the action profile respectively. An action profile α¯ = (α1, ..., αk) is a sequence that consists of actions αi ∈ d(i, q) for all players in N . In case o(q, α1, ..., αk) is undefined then o(q, α10, ..., αk0) is undefined for each action profile (α10, ..., αk0). For the sake of notation simplicity, d(i, q) will be written as di(q) and dC (q) := Qi∈C di(q).

A state of affairs refers to a set S ⊆ Q, S¯ denotes the set Q \ S, and (αC , αN\C ) denotes the action profile, where αC is the actions of the agents in group C and αN\C denotes the actions of the rest of the agents. Following the setting of [ 6 ], we recall the definitions of q-enforce, q-avoid, q-responsible and weakly q-responsible (See [ 6 ] for details and properties of these notions).

Definition 2 (Agent groups: strategic abilities and responsibility [ 6 ]). Let M = (N, Q, Act, d, o) be a CGS, q ∈ Q be a specific state, and S a state of affairs. We have the following concepts: 1) C ⊆ N can q-enforce S in M iff there is a joint action αC ∈ dC (q) such that for all joint actions αN\C ∈ dN\C (q), o(q, (αC , αN\C )) ∈ S; 2) C ⊆ N can qavoid S in M iff for all αN\C ∈ dN\C (q) there is αC ∈ dC (q) such that o(q, (αC , αN\C )) ∈ S¯; 3) C ⊆ N is q-responsible for S in M iff C can q-enforce S¯ and for all other C0 ⊆ N that can q-enforce S¯, we have that C ⊆ C0; 3) C ⊆ N is weakly q-responsible for S in M 3 iff C is a minimal group that can q-enforce S¯.

Considering the voting scenario from Section II, groups GD, RD and GRD can qs-enforce the approval of B while groups D, GR, GD, RD, and GRD can qs-avoid the approval of B. In this scenario, qs denotes the starting moment of the voting progress. Note that the notions of q-enforce and q-avoid correlate with the notions of, respectively, α-effectivity and βeffectivity in [ 11 ]. In this scenario, we have no qs-responsible group for approval of B and two groups D and GR are weakly qs-responsible for the approval of B. Note that the groups GD, RD, and GRD are not weakly qs-responsible for the approval of B as they are not minimal.

The concept of (weakly) q-responsibility merely assigns responsibility to groups with preclusive power and considers all other groups as not being responsible. As we have argued in section II, we believe that responsibility can be assigned to all groups, even those without preclusive power, though to a certain degree including zero degree. In order to define our notions of responsibility degree, we first introduce two notions of structural power difference and power acquisition sequence. Given an arbitrary group C, a state q, and a state of affair S, the first notion concerns the number of missing elements in C that when added to C makes it a (weakly) qresponsible groups for a S, and the second notion concerns a sequence of action profiles from given state q that leads to a state q0 where C is (weakly) q0-responsible for S. According to the first notion, group C can gain preclusive power for S if supported by some additional members, and according to 3In further references, “in M ” might be omitted wherever it is clear from the context. the second notion C can gain preclusive power for S in some potentially reachable state.

Let M be a multi-agent system, S a state of affairs in M , C an arbitrary group, and Cˆ be a (weakly) q-responsible group for S in M .

Definition 3 (Power measures). We say that the structural power difference of C and Cˆ in q ∈ Q with respect to S, denoted by ΘqS,M (Cˆ, C), is equal to cardinality of Cˆ\C. Moreover, we say that C has a power acquisition sequence hα¯1, ..., α¯ni in q0 ∈ Q for S in M iff for qi ∈ Q, o(qi, α¯i) = qi+1 for 1 ≤ i ≤ n such that q0 = q1 and qn+1 = q00 and C is (weakly) q00-responsible for S in M .

Consider the war approval declaration of the congress to the president (P ) in Section II. Here, we can see that the structural power difference of the group G and the weakly qsresponsible group GR is equal to 3. Moreover, the singleton group P that is not responsible in qs has the opportunity of being responsible for the war in states other than qs. Note that power acquisition sequence does not necessarily need to be unique. If the group C is not (weakly) responsible in a state q, the existence of any power acquisition sequence with a length higher than zero implies that the group could potentially reach a state q0 (from the current state of q) where C is (weakly) q0responsible for S. This notion also covers the cases where C is already in a (weakly) responsible state where the minimum length of power acquisition sequence is taken to be zero. In this case, the group is already (weakly) q-responsible for S. For example, in the voting scenario, group D is weakly responsible for the state of affairs and therefore, the minimum length of a power acquisition sequence is zero. When we are reasoning in a source state q, the notion of power acquisition sequence, enables us to differentiate between the non (weakly) q-responsible groups that do have the opportunity of becoming (weakly) q0-responsible for a given state of affairs (q 6= q0) and those they do not. Moreover, we emphasize that the availability of a power acquisition sequence for an arbitrary group C from a source state q to a state q0 in which C is (weakly) q-responsible for the state of affairs, does not necessitate the existence of an independent strategy for C to reach q0 from q.

IV. STRUCTURAL DEGREE OF RESPONSIBILITY

Structural degree of responsibility addresses the preclusive power of a group for a given state of affairs by means of the maximum contribution that the group has in a (weakly) responsible group for the state of affairs. To illustrate the intuition behind this notion, consider again the voting scenario in the section II. If an anti-war campaign wants to invest its limited resources to prevent the bill start a war, we deem that it is reasonable to invest more on R than G, if the resources admit such a choice. Although neither R nor G could prevent the war individually, larger contribution of R in groups with preclusive power, i.e. GR and D, entitles R to be assigned with larger degree of responsibility than G. This intuition will be reflected in the formulation of structural degree of responsibility. Definition 4 (Structural degree of responsibility). Let WqS,M denote the set of all (weakly) q-responsible groups for state of affairs S in multi-agent system M , and C ⊆ N be an arbitrary group. In case WqS,M = ∅, the structural degree of q-responsibility of any C for S in M is undefined; otherwise, the structural degree of q-responsibility of C for S in M denoted SDRqS,M (C), is defined as follows:

SDRqS,M (C) = Cˆ∈mWaqSx,M ({i | i = 1 − ΘqS,M (Cˆ,C) ) |Cˆ| } Proposition 1 (Full structural responsibility). The structural degree of q-responsibility of group C for S is equal to 1 iff C is either a (weakly) q-responsible group for S or C ⊇ Cˆ such that Cˆ is (weakly) q-responsible for S.

Proof. Follows directly from Definition 4 and definition of (weak) responsibility in [ 6 ].

Example 1. Consider again the voting scenario from Section II (Figure 1). In this scenario, we have an initial state qs in which all voters can use their votes in favour or against the approval of the bill B (no abstention or null vote is allowed). The majority of six votes (or more) in favour of B will be considered as the state of affairs consisting of states q7, q5 and q3. This multi-agent system can be modelled as CGS M = (N, Q, Act, d, o), where N = {1, ..., 10}, Q = {qs, q0, ..., q7}, Act = {0, 1, wait}, di(qs) = {0, 1} and di(q) = {wait} for all i ∈ N and q ∈ Q \ {qs}. Voters are situated in three parties such that G = {1, 2}, R = {3, 4, 5} and D = {6, 7, 8, 9, 10}. For notation convenience, actions of party members will be written collectively in the action profiles, e.g., we write (0, 1, 0) to denote the action profile (0, 0, 1, 1, 1, 0, 0, 0, 0, 0). The outcome function is as illustrated in Figure 1 (e.g., o(qs, (0, 0, 1)) = q1 is illustrated by the arrow from qs to q1). Moreover, the simplifying assumption that all party members vote collectively is implemented by o(qs, α¯0) = qs for all possible action profiles α¯0 in which party members act differently. We observe that the set of weakly qs-responsible groups in this example is {GR, D}. Using Definition 4, the structural degree of qs-responsibility of G will be equal to max({2/5, 0/5}) = 2/5 and SDRqSs (R) = 3/5. A similar calculation leads to the conclusion that the structural degree of qs-responsibility for all (weakly) qs-responsible groups, i.e., GR and D, and their super-sets is equal to 1. The structural degree of qs-responsibility of empty group (∅) is equal to 0 as the structural power difference of the empty group with all (weakly) qs-responsible groups Cˆ is equal to the cardinality of Cˆ.

A group C might share members with various (weakly) qresponsible groups, therefore the largest structural share of C in (weakly) q-responsible groups for S, will be considered to form the SDRqS (C). We would like to stress that our notions for responsibility degrees are formulated based on the maximum expected power of a group to preclude a state of affairs. While we believe that in legal theory, and with respect to its backward-looking approach, the minimum preclusive power of a group need be taken into account for assessing (0,1,1)

α¯0 (0,0,1) (0, 0, 0) ¯ S q6 q1 q0 culpability, our focus as a forward-looking approach will be on maximum expected preclusive power of a group regarding a given state of affairs.

The following lemma introduces a responsibility paradox case in which our presented notion of structural degree of responsibility is not applicable as a notion for reasoning about responsibility of groups of agents.

Lemma 1 (Applicability constraint: responsibility paradox). The empty group is (unique) q-responsible for S iff the structural degree of q-responsibility of all possible groups C for S is equal to 1.

Proof. See [1] for the proof.

The common avoidability of S implies that the occurrence of S is impossible by means of any action profile in q. In other words, given the specification of a CGS model M , a state of of affairs S and a source state q in M , no action profile α¯ leads to a state qs ∈ S. Common avoidability of a state of affairs, correlates with the impossibility notion ¬♦S in modal logic [ 12 ]. An impossible state of affairs S in q, entitles all the possible groups to be “fully responsible”. The impossibility of S neutralizes the space of groups with respect to their structural degree of q-responsibility for S. Therefore, we believe that in cases where the empty group is responsible for a given state of affairs, as S is impossible, full degree of structural responsibility of a group is not an apt measure, does not imply the preclusive power of any group, and hence, not an applicable reasoning notion for one who is willing to invest resources in the groups of agents that have the preclusive power over S. Note that in case the empty set is not responsible for S, its structural degree of responsibility is equal to 0 because its structural power difference with all (weakly) responsible groups Cˆ is equal to the cardinality of Cˆ.

The next theorem illustrates a case in which a singleton group possesses the preclusive power over a state of affairs.

The existence of such a dictator agent in a state q, polarizes Theorem 1 (Polarizing dictatorship). Let Cˆ be a singleton group, q an arbitrary state and S a possible state of affairs (in sense of Lemma 1). Then, Cˆ is a (unique) q-responsible group for S iff for any arbitrary group C, SDRqS (C) ∈ {0, 1}, where SDRqS (C ∈ I) = 1 and SDRqS (C ∈ O) = 0 for I = {C|C ⊇ C} and O = {C|C + Cˆ}.

ˆ

As our concept of group responsibility is based on the preclusive power of a group over a given state of affairs, the following monotonicity property shows that increasing the size of a group by adding new elements, does not have a negative effect on the preclusive power. This property, as formulated below, correlates with the monotonicity of power and power indices [ 13 ].

SDRqS (C) ≤ SDRqS (C0).

Proposition 2 (Structural monotonicity). Let C and C0 be two arbitrary groups such that C ⊆ C0. If WqS,M 6= ∅ then

The following theorem shows that in case of existence of a unique nonempty q-responsible group for a state of affairs, the structural degree of q-responsibility of any group could be calculated cumulatively based on the degrees of disjoint subsets. In this case, for any two arbitrary groups C1 and C2, the summation of their structural degree of q-responsibility will be equal to the degree of the unified group.

Theorem 2 (Conditional cumulativity). If there exists a nonempty (unique) q-responsible group for S, then for any arbitrary group C and partition P = {C1, ..., Cn} of C, we have Pn

i=1 SDRqS (Ci) = SDRqS (C).

Functional degree of responsibility addresses the dynamics of preclusive power of a specific group with respect to a given state of affairs. We remind the example from Section II where the president will be in charge, regarding the war decision, only after the approval of the congress. It is our understanding that the existence of a sequence of action profiles that leads to a state where the president becomes responsible for the war decision rationalizes the investment of an anti-war campaign on the president, even before the approval of the congress.

The functional degree of responsibility of a group C in a state q will be calculated based on the notion of power acquisition sequence by tracing the number of necessary state transitions from q, in order to reach a state q0 in which the group C is (weakly) q0-responsible for S. The length of a shortest power acquisition sequence form q to q0, illustrates the potentiality of preclusive power of the group C. If two groups have the capacity of reaching a state in which they have the preclusive power over the state of affairs S, we say that Definition 5 (Functional degree of responsibility). Let PqS,M (C) denote the set of all power acquisition sequences of group C ⊆ N in q for S in M . Let also ` = min ({i | k∈PqS,M (C) i = length(k)}) be the length of a shortest power acquisition sequence. The functional degree of q-responsibility of C for S in M , denoted by F DRqS,M (C), is defined as follows: 0 F DRqS,M (C) =

1 (`+1) Proposition 3 (Full functionality implies full responsibility). Let Cˆ be a group, q an arbitrary state and S a given state of affairs. If F DRqS (Cˆ) = 1, then the structural degree of q-responsibility of Cˆ for S is equal to 1. if PqS,M (C) = ∅ otherwise

Proof. See [1] for the proof.

Example 2 (War powers resolution). Consider again the voting scenario in the congress, as explained in Section II; but now extended with a new president agent P . The decision of starting a war W should first be approved by a majority of the congress members (six votes or more in favour of W ) after which the president makes the final decision. Hence, P has the preclusive power which is conditioned on the approval of the congress members. Moreover, we have a simplifying assumption that no party member acts independently and thus assume that all members of a party vote either in favor of or against the W . In this scenario, which is illustrated in Figure 2, we have an initial state qs in which all the congress members could use their votes in favour or against the approval of W (no abstention or null vote is allowed). In this example, W will be considered as the state of affairs consisting of states q11, q12, and q13. This multi-agent scenario can be modelled by the CGS M = (N, Q, Act, d, o), where N = {1, ..., 11} (the first ten agents are the voters in the congress followed by the president), Q = {qs, q0, ..., q13}, Act = {0, 1, wait}, di(qs) = {0, 1} for all i ∈ {1, ..., 10}, d11(qs) = {wait}, di(q) = {wait} for all i ∈ {1, ..., 10} and q ∈ {q0, ..., q13}, d11(r) = {wait} for r ∈ ({q0, q1, q2, q4, q6} ∪ {q8, ..., q13}), and d11(t) = {0, 1} for t ∈ {q3, q5, q7}. The outcome function o is illustrated in Figure 2 where for example o(qs, (1, 0, 0, ?)) = q4 in which the war W will not take place because of the disapproval of the congress (? represents any available action). For notation convenience, actions of party members will be written collectively in the action profiles, e.g., we write (0, 1, 0, ?) to denote the action profile (0, 0, 1, 1, 1, 0, 0, 0, 0, 0, ?). Moreover, the simplifying assumption that all party members vote collectively is implemented by o(qs, α¯0) = qs for all possible action profiles α¯0 in which at least one party member acts independently.

The set of all weakly qs-responsible groups WqWs consists of two groups of GR and D. These two are the minimal groups with the preclusive power over W in qs. If an anti-war campaign wants to negotiate and invest its limited resources in order to avoid the war W , convincing any of groups in WqWs , can avoid the war. However, it is observable that convincing the president is also adequate. Although the president has no preclusive power in qs over W , there exist some accessible states from qs (i.e., q3, q5, and q7), in which P is responsible for the state of affairs. This potential capacity of P , will be addressed by means of the introduced notion of functional degree of responsibility. Two weakly qs-responsible groups GR and D, have the functional degree of qs-responsibility of 1 for W because they already have sufficient power to avoid W in source state qs. Groups ∅, G, R, D, GD, RD, and GRD are not (weakly) qs-responsible for W and no power acquisition sequence exists for these groups. Accordingly, their functional degree of qs-responsibility for W is 0. Groups P G, P R, P D, P GR, P GD, P RD and P GRD, have the potentiality of possessing the preclusive power in other states, i.e., q3, q5, and q7, but none of them will be minimal group with preclusive power over W . Note that minimality is a requirement for being a (weakly) responsible group [ 6 ]. Hence, the functional degree of qs-responsibility for all these groups will be 0. The group which has a chance of becoming a (weakly) responsible group in states other than qs (i.e., q3, q5, and q7) is P . In fact, the President is the (unique) responsible group for W in states q3, q5, and q7. As the minimum length of power acquisition sequence for P is 1, the functional degree of qs-responsibility of P for W is 1/2. Although, P has no independent action profile to avoid W in qs, there exists a power acquisition sequence for P through which P acquires the preclusive power over W .

The next proposition illustrates that through a shortest power acquisition sequence, the potentiality that the group is responsible for the state of affairs, increases strictly. This potential reaches its highest possible value where the group “really” has the preclusive power over the state of affairs as a (weakly) responsible group. Note that there is a one-to-one correspondence between any power acquisition sequence P = hα¯1, ..., α¯ni in q for a group C for S and the sequence of states hq1 = q, ..., qn+1i due to the deterministic nature of the action profiles α¯i for 1 ≤ i ≤ n, i.e., o(qi, α¯i) = qi+1 and q = q1 and q0 = qn+1 and C is (weakly) q0-responsible for S. Hence, in the following, we write P = hq1, ..., qn+1i and interchangeably use it instead of P = hα¯1, ..., α¯ni. Therefore, we simply refer to any state qi as a state “in” the power acquisition sequence P .

Proposition 4 (Strictly increasing functionality). Let P = hq1, ..., qn+1i (n ≥ 1) be a power acquisition sequence in q = q1 for a group C for S. Then, for any tuple of states (qi, qi+1), 1 ≤ i ≤ n, F DRqSi (C) < F DRqi+1 (C) iff P is a S shortest power acquisition sequence in q for C for S. ) 1 , q8 (?,?,?,0) ,,(???

(1,1,1,?) (1,0,0,?) (0,1,1,?)

α¯0 (0,0,1,?)(0, 0, 0, ?) q3

References

[1] V. Yazdanpanah and M. Dastani , “Quantifiesd degrees of group responsibility,” in Coordination, Organizations, Institutions, and Norms in Agent Systems XI. Springer, 2016 , p. in press.

[2] H. Chockler and J. Y. Halpern , “ Responsibility and blame: A structural-model approach,” J. Artif . Intell. Res. (JAIR) , vol. 22 , pp. 93 - 115 , 2004 . [Online]. Available: http://dx.doi.org/10.1613/jair.1391

[3] D. Grossi , L. M. M. Royakkers , and F. Dignum , “ Organizational structure and responsibility , ” Artif. Intell. Law , vol. 15 , no. 3 , pp. 223 - 249 , 2007 . [Online]. Available: http://dx.doi.org/10.1007/s10506- 007-9054-0

[4] S. Miller , “ Collective moral responsibility: An individualist account,” Midwest studies in philosophy , vol. 30 , no. 1 , pp. 176 - 193 , 2006 .

[5] M. Braham and M. Van Hees , A“ n anatomy of moral responsibility , ” Mind , vol. 121 , no. 483 , pp. 601 - 634 , 2012 .

[6] N. Bulling and M. Dastani , “ Coalitional responsibility in strategic settings,” in Computational Logic in Multi-Agent Systems - 14th International Workshop, CLIMA XIV, Corunna, Spain, September 16-18 , 2013 . Proceedings, 2013 , pp. 172 - 189 .

[7] I. van de Poel, “ The relation between forward-looking and backwardlooking responsibility,” in Moral Responsibility, ser . Library of Ethics and Applied Philosophy , N. A. Vincent , I. van de Poel, and J. van den Hoven, Eds. Springer Netherlands, 2011 , vol. 27 , pp. 37 - 52 .

[8] K. Shaver , The attribution of blame: Causality, responsibility, and blameworthiness . Springer Science & Business Media , 2012 .

[9] N. R. Miller , “Power in game forms,” in Power, voting, and voting power. Springer, 1982 , pp. 33 - 51 .

[10] R. Alur , T. A. Henzinger , and O. Kupferman , A“lternating-time temporal logic,” J. ACM , vol. 49 , no. 5 , pp. 672 - 713 , 2002 . [Online]. Available: http://doi.acm. org/10 .1145/585265.585270

[11] M. Pauly , Logic for social software . Universiteit van Amsterdam , 2001 .

[12] S. A. Kripke , “ Semantical analysis of modal logic i normal modal propositional calculi , ” Mathematical Logic Quarterly , vol. 9 , no. 5-6 , pp. 67 - 96 , 1963 .

[13] M. J. Holler and S. Napel , “ Monotonicity of power and power measures , ” Theory and Decision , vol. 56 , no. 1-2 , pp. 93 - 111 , 2004 .