=Paper=
{{Paper
|id=Vol-1664/w8
|storemode=property
|title=Quantified Group Responsibility in Multi-Agent Systems
|pdfUrl=https://ceur-ws.org/Vol-1664/w8.pdf
|volume=Vol-1664
|authors=Vahid Yazdanpanah,Mehdi Dastani
|dblpUrl=https://dblp.org/rec/conf/woa/YazdanpanahD16
}}
==Quantified Group Responsibility in Multi-Agent Systems==
https://ceur-ws.org/Vol-1664/w8.pdf
Quantified Group Responsibility in Multi-Agent
Systems
Vahid Yazdanpanah Mehdi Dastani
Department of Industrial Engineering Department of Information
and Business Information Systems and Computing Sciences
University of Twente Utrecht University
Enschede, The Netherlands Utrecht, The Netherlands
Email: v.yazdanpanah@utwente.nl Email: m.m.dastani@uu.nl
Abstract—This paper1 builds on an existing notion of group towards a state of affairs in strategic settings, e.g., collective
responsibility and proposes two ways to define the degree of group decision making scenarios. In this paper, we build on a
responsibility: structural and functional degrees of responsibility. forward-looking approach to group responsibility and define
These notions measure potential responsibilities of agent groups
for avoiding a state of affairs. According to these notions, a degree two notions of responsibility degrees. The first concept is
of responsibility for a state of affairs can be assigned to a group based on the partial or complete power of an agent group to
of agents if, and to the extent that, the group of the agents have preclude a state of affairs while the second concept is based
potential to preclude the state of affairs. These notions will be on the potentiality of an agent group to reach a state where
formally specified and their properties will be analyzed. the agent group possesses the complete power to preclude the
I. I NTRODUCTION state of affairs. This results in a distinction between what we
The concept of responsibility has been extensively inves- will call the “structural responsibility” versus the “functional
tigated in philosophy and computer science. Each proposal responsibility” of an agent group. In our proposal, an agent
focuses on specific aspects of responsibility. For example, group has the full responsibility, if it has an action profile
[2] focuses on the causal aspect of responsibility and de- to preclude the state of affairs. All other agent groups that
fines a notion of graded responsibility, [3] focuses on the do not have full responsibility, but may have contribution to
organizational aspect of responsibility, [4] argues that group responsible agent groups, will be assigned a partial degree of
responsibility should be distributed to individual responsibility, responsibility.
[5] focuses on the interaction aspect of responsibility and II. G ROUP R ESPONSIBILITY: A P OWER - BASED A NALYSIS
defines an agent’s responsibility in terms of the agent’s causal
contribution, and [6] focuses on the strategic aspect of group In order to illustrate our conception of group responsibility
responsibility and defines various notions of group responsibil- and the nuances in degrees of responsibility, we follow [2] and
ity. In some of these proposals, the concept of responsibility use a voting scenario to explain the degree of responsibility
is defined with respect to a realized event “in past” while of agents’ groups for voting outcomes. The voting scenario
in other approaches it is defined as the responsibility for considers a small congress with ten members consisting of five
the realization of some event “in future”. This introduces a Democrats (D), three Republicans (R), and two Greens (G).
major dimension of responsibility, namely backward-looking We assume that there is a voting in progress on a specific bill
and forward-looking responsibility [7]. Backward-looking ap- (B). Without losing generality and to reduce the combinatorial
proaches reason about level of causality or contribution of complexity of the setting, we assume that all members of a
agents in the occurrence of an already realized outcome party vote either in favour of or against the bill B. Table I
while forward-looking notions are focused on the capacities illustrates the eight possible voting outcomes. Note that in this
of agents towards a state of affairs. scenario, six positive votes are sufficient for the approval of
Although some of the existing approaches are designed to B. For example, row 4 shows the case where R and D vote
measure the degree of responsibility, they either constitute against B and the bill is disapproved. For this case we say that
a backward-looking (instead of forward-looking) notion of the group RD votes against B. It should also be noted that our
responsibility [2], provide qualitative (instead of quantitative) assumption reduces parties to individual agents with specific
levels of responsibility [8], or focus on individual (instead weights such that the question raises as why we use this party
of group) responsibility [5]. To our knowledge, there is no setting instead of a simple voting of three agents whose votes
forward-looking approach that could measure the degree of have different weights. The motivation is that this setting is
group responsibility quantitatively. Such notion would enable realistic and makes the weighted votes of each agent (party)
reasoning on the potential responsibility of an agent group more intuitive.
Following [6] we believe that it is reasonable to assign the
1 The original version of this work appears in [1]. responsibility for a specific state of affairs to a group of agents
44
TABLE I group that shares members with responsible groups, should be
VOTING RESULTS assigned a degree of responsibility that reflects its proportional
G(2) R(3) D(5) Result contribution to the groups with preclusive power. For example,
0 − − − × group R with three members, has larger share in GR than the
1 − − + × group G has. Therefore, we believe that the relative size of a
2 − + − ×
3 − + + X
group and its share in the groups with the preclusive power
4 + − − × are substantial parameters in formulation of the notion of
5 + − + X responsibility degree. In this case, the larger share of R in GR
6 + + − ×
7 + + + X
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
TABLE II
WAR INCIDENCE this concept of responsibility degree is supported by the fact
that lobby groups do proportionally support political parties
Congress President War
that can play a role in some key decisions. In a sense, the
0 − − ×
1 − + ×
lobby groups consider political parties responsible for some
2 + − × decision and therefore they are willing to support the parties.
3 + + X 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
if they jointly have the power to avoid the state of affairs2 .
about declaration of the congress to the President (P ) which
According to [9], the preclusive power is the ability of a
enables P to start a war (Table II). Roughly speaking, P
group to preclude a given state of affairs which entails that
will be in charge only after the approval of the congress.
a group with preclusive power, has the potential but might
When we are reasoning at the moment when the voting is
not practice the preclusion of a given state of affairs. For our
in progress in the congress, it is reasonable to assume that
voting scenario, this suggests to assign responsibility to the
groups GR and D are responsible as they have preclusive
group GR consisting of parties G and R since they can jointly
power to avoid the war. Moreover, after the approval of B,
disapprove B. Note that the state of affair to be avoided can
the President P is the only group with preclusive power to
also be the state of affairs where B is disapproved. In this
avoid the war. Hence, we believe that although P alone would
case, the group can be assigned the responsibility to avoid
not have the preclusive power before the approval of B in the
disproving B. Similarly, groups D, GD, RD, and GRD have
congress, it is rationally justifiable for an anti-war campaign
preclusive power with respect to the approval of B as they
to invest resources on P , even before the approval voting of
have sufficient members (weights) to avoid the approval of B.
the congress, simply because there exists possibilities where P
Note that none of the other two groups, i.e., G and R, could
will have the preclusive power to avoid the war. Accordingly,
preclude the approval of B independently. However, based on
a reasonable differentiation could be made between the groups
[6], the agent groups that consist of a smaller sub-group with
which do have the chance of acquiring the preclusive power
preclusive power, must be excluded from the set of responsible
and those they do not have any chance of power acquisition.
groups. Hence, we consider GR and D as being responsible
This functional notion of responsibility degree addresses the
groups for the approval of B. The intuition for this concept of
eventuality of a state in which an agent group possesses the
responsibility is supported by the fact that the lobby groups are
preclusive power regarding a given state of affairs.
willing (i.e., it is economically rational) to invest resources in
parties that have the power to avoid a specific state of affairs. III. M ODELS AND P RELIMINARY N OTIONS
We build on the ideas in [6] and propose two orthogonal
approaches to capture our conception of degree of group The behaviour of a multi-agent system is often modelled
responsibility towards a state of affairs. Our intuition suggests by concurrent game structures (CGS) [10]. Such structures
that the degree of responsibility of a group of agents towards specify possible state of the system, agents’ abilities at each
a state of affairs should reflect the extent they structurally or state, and the outcome of concurrent actions at each state.
functionally can contribute to the groups that have preclusive Definition 1 (Concurrent game structures [10]). A concurrent
power with respect to the state of affairs. game structure is a tuple M = (N, Q, Act, d, o), where N =
Our conception of structural responsibility degree is based {1, ..., k} is a nonempty finite set of agents, Q is a nonempty
on the following observation in the voting scenario. We deem set of system states, Act is a nonempty and finite set of atomic
that regarding the approval of B, although the groups G and actions, d : N × Q → P(Act) is a function that identifies
R have no preclusive power independently, they nevertheless the set of available actions for each agent i ∈ N at each
have a share in the composition of GR with preclusive state q ∈ Q, and o is a deterministic and partial transition
power regarding the approval of B. Hence, we say that any function that assigns a state q 0 = o(q, α1 , ..., αk ) to a state
2 See [6] for a detailed discussion on why to focus on avoiding instead q and an action profile (α1 , ..., αk ) such that all k agents in
of enforcing a state of affairs. N choose actions in the action profile respectively. An action
45
profile ᾱ = (α1 , ..., αk ) is a sequence that consists of actions the second notion C can gain preclusive power for S in some
αi ∈ d(i, q) for all players in N . In case o(q, α1 , ..., αk ) is potentially reachable state.
undefined then o(q, α10 , ..., αk0 ) is undefined for each action Let M be a multi-agent system, S a state of affairs in M , C
profile (α10 , ..., αk0 ). For the sake of notation
Q simplicity, d(i, q) an arbitrary group, and Ĉ be a (weakly) q-responsible group
will be written as di (q) and dC (q) := i∈C di (q). for S in M .
A state of affairs refers to a set S ⊆ Q, S̄ denotes the set Definition 3 (Power measures). We say that the structural
Q \ S, and (αC , αN \C ) denotes the action profile, where αC power difference of C and Ĉ in q ∈ Q with respect to
is the actions of the agents in group C and αN \C denotes the S, denoted by ΘS,M q (Ĉ, C), is equal to cardinality of Ĉ\C.
actions of the rest of the agents. Following the setting of [6], Moreover, we say that C has a power acquisition sequence
we recall the definitions of q-enforce, q-avoid, q-responsible hα¯1 , ..., α¯n i in q 0 ∈ Q for S in M iff for qi ∈ Q, o(qi , ᾱi ) =
and weakly q-responsible (See [6] for details and properties of qi+1 for 1 ≤ i ≤ n such that q 0 = q1 and qn+1 = q 00 and C
these notions). is (weakly) q 00 -responsible for S in M .
Definition 2 (Agent groups: strategic abilities and responsi- Consider the war approval declaration of the congress to
bility [6]). Let M = (N, Q, Act, d, o) be a CGS, q ∈ Q the president (P ) in Section II. Here, we can see that the
be a specific state, and S a state of affairs. We have the structural power difference of the group G and the weakly qs -
following concepts: 1) C ⊆ N can q-enforce S in M iff there responsible group GR is equal to 3. Moreover, the singleton
is a joint action αC ∈ dC (q) such that for all joint actions group P that is not responsible in qs has the opportunity of
αN \C ∈ dN \C (q), o(q, (αC , αN \C )) ∈ S; 2) C ⊆ N can q- being responsible for the war in states other than qs . Note that
avoid S in M iff for all αN \C ∈ dN \C (q) there is αC ∈ dC (q) power acquisition sequence does not necessarily need to be
such that o(q, (αC , αN \C )) ∈ S̄; 3) C ⊆ N is q-responsible unique. If the group C is not (weakly) responsible in a state q,
for S in M iff C can q-enforce S̄ and for all other C 0 ⊆ N the existence of any power acquisition sequence with a length
that can q-enforce S̄, we have that C ⊆ C 0 ; 3) C ⊆ N is higher than zero implies that the group could potentially reach
weakly q-responsible for S in M 3 iff C is a minimal group a state q 0 (from the current state of q) where C is (weakly) q 0 -
that can q-enforce S̄. responsible for S. This notion also covers the cases where C
is already in a (weakly) responsible state where the minimum
Considering the voting scenario from Section II, groups length of power acquisition sequence is taken to be zero.
GD, RD and GRD can qs -enforce the approval of B while In this case, the group is already (weakly) q-responsible for
groups D, GR, GD, RD, and GRD can qs -avoid the approval S. For example, in the voting scenario, group D is weakly
of B. In this scenario, qs denotes the starting moment of the responsible for the state of affairs and therefore, the minimum
voting progress. Note that the notions of q-enforce and q-avoid length of a power acquisition sequence is zero. When we are
correlate with the notions of, respectively, α-effectivity and β- reasoning in a source state q, the notion of power acquisition
effectivity in [11]. In this scenario, we have no qs -responsible sequence, enables us to differentiate between the non (weakly)
group for approval of B and two groups D and GR are weakly q-responsible groups that do have the opportunity of becoming
qs -responsible for the approval of B. Note that the groups GD, (weakly) q 0 -responsible for a given state of affairs (q 6= q 0 ) and
RD, and GRD are not weakly qs -responsible for the approval those they do not. Moreover, we emphasize that the availability
of B as they are not minimal. of a power acquisition sequence for an arbitrary group C
The concept of (weakly) q-responsibility merely assigns from a source state q to a state q 0 in which C is (weakly)
responsibility to groups with preclusive power and considers q-responsible for the state of affairs, does not necessitate the
all other groups as not being responsible. As we have argued existence of an independent strategy for C to reach q 0 from q.
in section II, we believe that responsibility can be assigned
to all groups, even those without preclusive power, though IV. S TRUCTURAL D EGREE OF R ESPONSIBILITY
to a certain degree including zero degree. In order to define Structural degree of responsibility addresses the preclusive
our notions of responsibility degree, we first introduce two power of a group for a given state of affairs by means of
notions of structural power difference and power acquisition the maximum contribution that the group has in a (weakly)
sequence. Given an arbitrary group C, a state q, and a state responsible group for the state of affairs. To illustrate the
of affair S, the first notion concerns the number of missing intuition behind this notion, consider again the voting scenario
elements in C that when added to C makes it a (weakly) q- in the section II. If an anti-war campaign wants to invest its
responsible groups for a S, and the second notion concerns a limited resources to prevent the bill start a war, we deem that
sequence of action profiles from given state q that leads to a it is reasonable to invest more on R than G, if the resources
state q 0 where C is (weakly) q 0 -responsible for S. According admit such a choice. Although neither R nor G could prevent
to the first notion, group C can gain preclusive power for S the war individually, larger contribution of R in groups with
if supported by some additional members, and according to preclusive power, i.e. GR and D, entitles R to be assigned
with larger degree of responsibility than G. This intuition
3 In further references, “in M ” might be omitted wherever it is clear from will be reflected in the formulation of structural degree of
the context. responsibility.
46
Definition 4 (Structural degree of responsibility). Let WS,M
q
denote the set of all (weakly) q-responsible groups for state
of affairs S in multi-agent system M , and C ⊆ N be an q7 q5 q3
S
arbitrary group. In case WS,M
q = ∅, the structural degree of (1
, 1)
(1, 0, 1)
q-responsibility of any C for S in M is undefined; otherwise, 1,
1 ,1
,
) (0
the structural degree of q-responsibility of C for S in M
denoted SDRS,M q (C), is defined as follows: qs
ΘS,M (Ĉ,C) ᾱ0
SDRS,M
q (C) = max ({i | i = 1 − q
}) 0)
(0 (0,
, 0) ,0
(0, 1, 0)
|Ĉ| , 0, 0
Ĉ∈WS,M 1 0 ,1
q
(1, 1, )
)
(
Proposition 1 (Full structural responsibility). The structural
degree of q-responsibility of group C for S is equal to 1 iff C S̄ q6 q4 q2 q1 q0
is either a (weakly) q-responsible group for S or C ⊇ Ĉ such
that Ĉ is (weakly) q-responsible for S.
Proof. Follows directly from Definition 4 and definition of
(weak) responsibility in [6]. Fig. 1. Voting scenario
Example 1. Consider again the voting scenario from Section
II (Figure 1). In this scenario, we have an initial state qs culpability, our focus as a forward-looking approach will be
in which all voters can use their votes in favour or against on maximum expected preclusive power of a group regarding
the approval of the bill B (no abstention or null vote is a given state of affairs.
allowed). The majority of six votes (or more) in favour of B The following lemma introduces a responsibility paradox
will be considered as the state of affairs consisting of states case in which our presented notion of structural degree of
q7 , q5 and q3 . This multi-agent system can be modelled as responsibility is not applicable as a notion for reasoning about
CGS M = (N, Q, Act, d, o), where N = {1, ..., 10}, Q = responsibility of groups of agents.
{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 Lemma 1 (Applicability constraint: responsibility paradox).
situated in three parties such that G = {1, 2}, R = {3, 4, 5} The empty group is (unique) q-responsible for S iff the
and D = {6, 7, 8, 9, 10}. For notation convenience, actions structural degree of q-responsibility of all possible groups C
of party members will be written collectively in the action for S is equal to 1.
profiles, e.g., we write (0, 1, 0) to denote the action profile Proof. See [1] for the proof.
(0, 0, 1, 1, 1, 0, 0, 0, 0, 0). The outcome function is as illus-
trated in Figure 1 (e.g., o(qs , (0, 0, 1)) = q1 is illustrated by
The common avoidability of S implies that the occurrence
the arrow from qs to q1 ). Moreover, the simplifying assumption
of S is impossible by means of any action profile in q. In
that all party members vote collectively is implemented by
other words, given the specification of a CGS model M , a
o(qs , ᾱ0 ) = qs for all possible action profiles ᾱ0 in which party
state of of affairs S and a source state q in M , no action
members act differently. We observe that the set of weakly
profile ᾱ leads to a state qs ∈ S. Common avoidability of a
qs -responsible groups in this example is {GR, D}. Using
state of affairs, correlates with the impossibility notion ¬♦S
Definition 4, the structural degree of qs -responsibility of G will
in modal logic [12]. An impossible state of affairs S in q,
be equal to max({2/5, 0/5}) = 2/5 and SDRSqs (R) = 3/5.
entitles all the possible groups to be “fully responsible”. The
A similar calculation leads to the conclusion that the structural
impossibility of S neutralizes the space of groups with respect
degree of qs -responsibility for all (weakly) qs -responsible
to their structural degree of q-responsibility for S. Therefore,
groups, i.e., GR and D, and their super-sets is equal to 1.
we believe that in cases where the empty group is responsible
The structural degree of qs -responsibility of empty group (∅)
for a given state of affairs, as S is impossible, full degree
is equal to 0 as the structural power difference of the empty
of structural responsibility of a group is not an apt measure,
group with all (weakly) qs -responsible groups Ĉ is equal to
does not imply the preclusive power of any group, and hence,
the cardinality of Ĉ.
not an applicable reasoning notion for one who is willing
A group C might share members with various (weakly) q- to invest resources in the groups of agents that have the
responsible groups, therefore the largest structural share of C preclusive power over S. Note that in case the empty set is
in (weakly) q-responsible groups for S, will be considered not responsible for S, its structural degree of responsibility
to form the SDRSq (C). We would like to stress that our is equal to 0 because its structural power difference with all
notions for responsibility degrees are formulated based on the (weakly) responsible groups Ĉ is equal to the cardinality of
maximum expected power of a group to preclude a state of Ĉ.
affairs. While we believe that in legal theory, and with respect The next theorem illustrates a case in which a singleton
to its backward-looking approach, the minimum preclusive group possesses the preclusive power over a state of affairs.
power of a group need be taken into account for assessing The existence of such a dictator agent in a state q, polarizes
47
the space of all possible groups with respect to their structural the group which has the shorter path has a higher potential
degree of q-responsibility for the state of affairs. preclusive power and thus gets the larger functional degree
of responsibility. Accordingly, a group which is already in a
Theorem 1 (Polarizing dictatorship). Let Ĉ be a singleton
responsible state, has full potential to avoid a state of affairs.
group, q an arbitrary state and S a possible state of affairs
Hence, it will be assigned with maximum functional degree
(in sense of Lemma 1). Then, Ĉ is a (unique) q-responsible
of responsibility equal to one.
group for S iff for any arbitrary group C, SDRSq (C) ∈ {0, 1},
where SDRSq (C ∈ I) = 1 and SDRSq (C ∈ O) = 0 for Definition 5 (Functional degree of responsibility). Let
I = {C|C ⊇ Ĉ} and O = {C|C + Ĉ}. PS,M
q (C) denote the set of all power acquisition sequences
of group C ⊆ N in q for S in M . Let also ` = min ({i |
Proof. See [1] for the proof. k∈PS,M
q (C)
i = length(k)}) be the length of a shortest power acquisition
As our concept of group responsibility is based on the sequence. The functional degree of q-responsibility of C for S
preclusive power of a group over a given state of affairs, the in M , denoted by FDRS,M (C), is defined as follows:
q
following monotonicity property shows that increasing the size
of a group by adding new elements, does not have a negative 0 if PS,M
q (C) = ∅
FDRS,M (C) =
effect on the preclusive power. This property, as formulated q 1
(`+1) otherwise
below, correlates with the monotonicity of power and power Proposition 3 (Full functionality implies full responsibility).
indices [13]. Let Ĉ be a group, q an arbitrary state and S a given state
Proposition 2 (Structural monotonicity). Let C and C 0 be of affairs. If FDRSq (Ĉ) = 1, then the structural degree of
two arbitrary groups such that C ⊆ C 0 . If WS,M
q 6= ∅ then q-responsibility of Ĉ for S is equal to 1.
S S 0
SDRq (C) ≤ SDRq (C ). Proof. See [1] for the proof.
Proof. See [1] for the proof. Example 2 (War powers resolution). Consider again the
The following theorem shows that in case of existence of voting scenario in the congress, as explained in Section II;
a unique nonempty q-responsible group for a state of affairs, but now extended with a new president agent P . The decision
the structural degree of q-responsibility of any group could of starting a war W should first be approved by a majority of
be calculated cumulatively based on the degrees of disjoint the congress members (six votes or more in favour of W )
subsets. In this case, for any two arbitrary groups C1 and C2 , after which the president makes the final decision. Hence,
the summation of their structural degree of q-responsibility P has the preclusive power which is conditioned on the
will be equal to the degree of the unified group. approval of the congress members. Moreover, we have a sim-
plifying assumption that no party member acts independently
Theorem 2 (Conditional cumulativity). If there exists a and thus assume that all members of a party vote either
nonempty (unique) q-responsible group for S, then for any in favor of or against the W . In this scenario, which is
arbitrary
Pn group C and partition P = {C1 , ..., Cn } of C, we illustrated in Figure 2, we have an initial state qs in which
have i=1 SDRSq (Ci ) = SDRSq (C). all the congress members could use their votes in favour
Proof. See [1] for the proof. or against the approval of W (no abstention or null vote
is allowed). In this example, W will be considered as the
V. F UNCTIONAL D EGREE OF R ESPONSIBILITY state of affairs consisting of states q11 , q12 , and q13 . This
Functional degree of responsibility addresses the dynamics multi-agent scenario can be modelled by the CGS M =
of preclusive power of a specific group with respect to a given (N, Q, Act, d, o), where N = {1, ..., 11} (the first ten agents
state of affairs. We remind the example from Section II where are the voters in the congress followed by the president),
the president will be in charge, regarding the war decision, Q = {qs , q0 , ..., q13 }, Act = {0, 1, wait}, di (qs ) = {0, 1}
only after the approval of the congress. It is our understanding for all i ∈ {1, ..., 10}, d11 (qs ) = {wait}, di (q) = {wait} for
that the existence of a sequence of action profiles that leads to all i ∈ {1, ..., 10} and q ∈ {q0 , ..., q13 }, d11 (r) = {wait} for
a state where the president becomes responsible for the war r ∈ ({q0 , q1 , q2 , q4 , q6 } ∪ {q8 , ..., q13 }), and d11 (t) = {0, 1}
decision rationalizes the investment of an anti-war campaign for t ∈ {q3 , q5 , q7 }. The outcome function o is illustrated in
on the president, even before the approval of the congress. Figure 2 where for example o(qs , (1, 0, 0, ?)) = q4 in which
The functional degree of responsibility of a group C in the war W will not take place because of the disapproval
a state q will be calculated based on the notion of power of the congress (? represents any available action). For
acquisition sequence by tracing the number of necessary state notation convenience, actions of party members will be written
transitions from q, in order to reach a state q 0 in which the collectively in the action profiles, e.g., we write (0, 1, 0, ?)
group C is (weakly) q 0 -responsible for S. The length of a to denote the action profile (0, 0, 1, 1, 1, 0, 0, 0, 0, 0, ?). More-
shortest power acquisition sequence form q to q 0 , illustrates over, the simplifying assumption that all party members vote
the potentiality of preclusive power of the group C. If two collectively is implemented by o(qs , ᾱ0 ) = qs for all possible
groups have the capacity of reaching a state in which they have action profiles ᾱ0 in which at least one party member acts
the preclusive power over the state of affairs S, we say that independently.
48
The set of all weakly qs -responsible groups WW qs 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 q13 q12 q11
S
campaign wants to negotiate and invest its limited resources in
order to avoid the war W , convincing any of groups in WW qs ,
can avoid the war. However, it is observable that convincing
(?, ?, ?, 1)
(?, ?, ?, 1)
(?, ?, ?, 1)
the president is also adequate. Although the president has no S̄ q10 q9 q8
preclusive power in qs over W , there exist some accessible
(?
(?
(?
,?
,?
, ?,
,?
,?
states from qs (i.e., q3 , q5 , and q7 ), in which P is responsible
?,
,0
,0
0
)
)
)
for the state of affairs. This potential capacity of P , will be q7 q5 q3
addressed by means of the introduced notion of functional de- (1 )
(1, 0, 1, ?)
gree of responsibility. Two weakly qs -responsible groups GR ,1
,1 ,?
,? 1,1
and D, have the functional degree of qs -responsibility of 1 for ) (0
,
W because they already have sufficient power to avoid W in
source state qs . Groups ∅, G, R, D, GD, RD, and GRD are qs ᾱ0
not (weakly) qs -responsible for W and no power acquisition ) (0 (0,
(0, 1, 0, ?)
?) ,? ,0
sequence exists for these groups. Accordingly, their functional 1, 0
,
,0 ,1 0, 0
, ?)
(1, , 0 ,?
degree of qs -responsibility for W is 0. Groups P G, P R, P D, (1 )
P GR, P GD, P RD and P GRD, have the potentiality of
possessing the preclusive power in other states, i.e., q3 , q5 , S̄ q6 q4 q2 q1 q0
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 Fig. 2. War powers resolution
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 R EFERENCES
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Proof. See [1] for the proof.
49