Dictatorship, Group Agency, and Freedom: Comments on van

Hees' Impossibility Theorem

November 30, 2014

Abstract

1 Introduction: The Impossibility Theorem

In two papers, van Hees (2010) and van Hees and Braham (2014), Martin van Hees develops a formalargument showing that sharing individual responsibility in an equal manner, while avoiding gaps inindividual responsibility for group acts, can only be done when there is a dictator forcing all of theoutcomes. Van Hees says that a decision procedure for a group is uniform when everyone who has someresponsibility is responsible for the same aspects of an outcome. He says that there is a responsibilityvoid when there is an outcome, but no individual is responsible for it. A decision procedure is complete

when there are no responsibility voids for any outcome. The formal argument then shows that

Theorem 1.1 (van Hees' Impossibility Theorem). If a group decision procedure is Uniform and Com-plete, then it must be Dictatorial.

where a decision procedure is dictatorial when there is one individual who can force any outcome,regardless of what others decide to do. This result is a generalization of the results in van Hees (2010),but is derived directly in van Hees and Braham (2014). In the rest of this section I will give somebackground to their work, and outline the formal framework of the impossibility theorem. Then, in thefollowing sections, turn to generalizing the theorem to see the full scope of this result.

As van Hees and Braham see it, they are dealing with Dennis Thomson's �Problem of Many Hands".They quote ` �[B]ecause many di�erent o�cials contribute in many ways to decisions and policies ofgovernment, it is di�cult even in principle to identify who is morally responsible for political outcomes�(Thompson 1980: 905).'(van Hees and Braham, 2014, p. 2) The issue for van Hees and Braham iswhether there is a way to identify individuals who are responsible for outcomes even though the outcomeis a result of a large set of individual actions. What their result shows is that, since dictatorship isto be avoided, one must either be comfortable with outcomes not being the responsibility of anyoneinvolved (a void), or not everyone being responsible for the same aspects of the outcome (non-uniformityor fragmentation). They argue that neither fragmentation nor voids are totally benign; the institutionsthat humans design tend not to have voids, and fragmentation can lead to individuals being responsiblefor things that they do not value, morally speaking. I will discuss the intuitive justi�cations of theseconditions in section 2.

The impossibility result uses a formal, game theoretic gloss of a general account of moral responsibility,and shows that assuming uniformity and completeness requires a decision procedure, represented by astrategic game form, to be dictatorial. I will provide that formal framework to give context for laterdiscussion. A strategic game form is de�ned as follows.

De�nition 1.1. A Strategic Game Form (usually called a game form)G is a tuple (N, {Σa : a ∈ N } , S, o)where

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• N is a �nite set of agents/players,

• Each Σa is a (�nite) set of strategies for each agent/player a,

• S is a set of outcomes for the plays of the game, and

• o :∏a∈N Σa → S is the outcome function, assigning to each strategy pro�le (σa)a∈N (referred to

as σN ), an outcome o(σN ) ∈ S.

A game form is surjective i� o is surjective, and a strategic game is a game form augmented withpreference pro�les over S for each player.

In the de�nition of a game form there is no speci�cation of the preference pro�les or utilities of theplayers. When those are added it is a game rather than a game form. For further discussion somenotation is needed.

Notation 1. Let G be a game form.

• Given a strategy pro�le σN and a non-empty A ⊆ N , σA = (σa ∈ σN : a ∈ A).

• Alternatively, starting with a partial strategy pro�le σA, that is a strategy for each a ∈ A, (σA, σ′A

)is a strategy pro�le where σ′

A∈∏b∈N\A Σb.

• Similarly, σa ∈∏b∈N\{ a } Σb.

• The set of possible outcomes given a partial strategy pro�le σA is de�ned as

o(σA) =

o(σA, σA) ∈ S : σA ∈∏

b∈N\A

Σb

.

Against this formal background, van Hees sets out to capture the notion of moral responsibilityas consisting of three things: a condition of autonomy1, a condition of causality, and a condition ofavoidance. For an agent to be autonomous they need to be able to make plans, know right from wrong,in short: be a moral agent. The autonomy condition doesn't play a role in the formal model; each agentis simply assumed to satisfy it.

The avoidance condition is simply that one could have done something, followed some other strategy,that made the occurrence of the outcome less likely. That de�nition requires the game form to beaugmented with a set of probability assignments: one for each agent which assigns a probability toeach combination of strategies of the other agents. Thus, each pa is a probability assignment over∏b∈N\{ a }Σb. The augmented game form is called a responsibility game. An avoidance potential for

each strategy of an agent σa for an outcome X, denoted ρa(σa, X), is de�ned by looking at the sum overthe likelihood that X occurs given that a plays σa. The set

ha(σa, X) = {σ′a : o(σa, σ′a) ∈ X }

denotes the set of strategies the other players could play which would result in X, given that a plays σa.Then ρ(σa, X) is calculated by the sum

ρ(σa, X) =∑

σ′a∈h(σa,X)

pa(σ′a).

For an agent's strategy σa to meet the avoidance condition relative to X, there must be another σ′a suchthat ρ(σ′a, X) < ρ(σa, X).

The causal relationship is termed `causal e�cacy' and is de�ned as follows:

1van Hees and Braham call this an agency condition, but I will avoid that so that the terminology doesn't get mixedup.

2

De�nition 1.2 (Causally E�ective). Let G be a surjective game form (all of the game forms in vanHees' work are surjective). a ∈ N is causally e�ective for X relative to σN i� there is A ⊆ N such that

i) a ∈ A (so σa ∈ σA ⊆ σN ),

ii) o(σA) ⊆ X, and

iii) o(σA\a) * X.

An agent is causally e�ective i� there is some Y for which it is causally e�ective.

The essence of van Hees' de�nition is that a's contribution of σa ∈ σN to the outcome being in Xis crucial for A; without it, A can not guarantee X. But built into condition iii is that there is someaction/strategy that a can perform/choose that might not result in X; whether it does or not will dependon what others do. It also seems that underlying condition iii a is no longer participating in the jointstrategy of σA. The notion of causal e�cacy is supposed to give a condition for a's action which canpass a NESS-test for causation. I.e., relative to σN , σa is a Necessary Element of a Su�cient Set ofconditions (σA) for X. Van Hees and Braham give it an additional gloss, �c is a cause of e if e dependson c under some contingency that was present on the occasion� (van Hees and Braham, 2014, p. 6).That interpretation seems to suggest that A must be N . If o(σN\a) ⊆ X, then σa ∈ σN would hardly benecessary for σA to be su�cient for X on that contingent occasion of σN . But the real contingency thatvan Hees and Braham intend is σA, rather than σN . Just to summarize, individual moral responsibilityof an agent for a group outcome X ( S, i.e., where o(σN ) ∈ X, is then de�ned as 1) a's contribution σais causally e�ective for X, and 2) there is σ′a such that ρ(σ′a, X) < ρ(σa, X).

The question van Hees and Braham engage with is whether group decision procedures can be mademore transparent so that the problem of many hands can be overcome. Put another way, can assumptionsbe made about how to distribute individual responsibility so that we are not left asking �who's to blame�for collective outcomes? The conditions of completeness and uniformity are supposed to play thatsimplifying role. Consider uniformity.

De�nition 1.3 (Uniformity). An individual a ∈ N is responsibility bearing i� there is some non-emptyX ( S such that a is responsible for X. A responsibility game is uniform i� for any σN , a, b ∈ N whoare responsibility bearing, and X ( S such that X 6= ∅, a is responsible for X i� b is responsible for X.

When a group decision procedure is uniform, there isn't much of an issue in �guring out whatoutcomes, or aspects of the outcome, each member is responsible for: if they are responsible for anything,they are responsible for every aspect. The issue, then, is whether anybody at all can be held responsible.That is what the completeness condition takes care of:

De�nition 1.4 (Completeness). A responsibility game Gp displays a void i� there is X ( S such thatX 6= ∅, and σN with o(σN ) ∈ X, but no a ∈ N is responsible for X. A game form is complete i� itdoesn't display a void for any X ( S such that X 6= ∅, and σN with o(σN ) ∈ X.

The �nal condition of being a dictator is that there is a ∈ N that can bring about each state,regardless of what the others do. Formally,

De�nition 1.5. A game form is dictatorial i� there is a ∈ N such that for any s ∈ S, there is σa ∈ Σafor which o(σa) = { s }.

Another way of simplifying a group decision procedure is explored in van Hees (2010) which rendersthe avoidance condition inert. Thus when responsibility is completely a matter of a causal relationshipbetween the agent and the outcome, in this formal sense, something very much like uniformity alsoleads to dictatorship. The proofs of both of these results rely heavily on the framework introduced forthe avoidance condition. Indeed, they should since uniformity, completeness, and non-dictatorship arecompatible when reformulated just using the causal e�ectiveness condition. For example, in the followinggame form I will call G1

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G σa σ′a σ′′aσb 1 3 2σ′b 3 2 1σ′′b 2 1 3

both players a and b are causally e�ective for any non-empty X ( { 1, 2, 3 } when o(σN ) ∈ X. Thus, itis uniform and complete. But it is not dictatorial. The probability assignments make all the di�erencein this sort of case. What van Hees' theorem tells us is that when it is made into a responsibilitygame, either uniformity or completeness (using the avoidance condition) must fail. But we should noticesomething about this game: none of the individuals has any unilateral power. That would naturally leadone to ask whether that is a general fact.

In what follows I will explore the extent to which van Hees' result can be generalized where the focusis on causal agency rather than responsibility, and what that generalization says about the possibilityof group agency. In section 2 I will discuss the literature on group agency to show that assumptionslike uniformity are required for certain kinds of group activity. I then o�er a generalized frameworkfor discussing group decisions/action which separates the causal power groups have from how they mayexercise that power. Finally I show that dictatorship arises when group members are allowed to exercisetheir power in any manner they wish. I take this to be an incompatibility between group agency andindividual freedom.

2 Group Agency

Van Hees' theorem concerns the relationship between the results of group action, and individual respon-sibility as part of such a group. The rationale in van Hees (2010) and van Hees and Braham (2014) forrequiring that a group decision procedure be uniform�or something like it�is to simplify the distribu-tion of responsibility for outcomes of group action. But in the literature on group agency certain kindsof group agency, I will argue, require uniformity, perhaps in all circumstances.

For the moment, I will stick to using the terminology of an agent or group being causally e�ectivefor an outcome. In the sequel I will switch to agents being agentive for an outcome. That will serve todistinguish the framework in the work of Belnap et al. (2001), Horty (2001), and Broersen (2014), fromthe framework of game theory that van Hees' uses.

Consider the uniformity condition restricted to just the causal component: a decision procedure, i.e.,a game form, is uniform i� for all σN , non-empty X ( S, and causally e�ective a, b ∈ N , a is causallye�ective for X relative to σN i� b is. What that formulation says is that causal agency must be shared forany result of a group act among those that are causally e�ective for anything. The question is whetherthat kind of condition is sensible to impose on group agency? I will argue that it is, at least in somecases.

In the literature, e.g., Bratman (1992), Tuomela (1993), and List and Pettit (2011), authors lookat what it is for a group to be acting as a group rather than being simply involved in some accidentalcoaction. What follows of course is that there are di�erent kinds of group agents, di�erent ways of actingas a group. I will specify the kind of group agent I think uniformity applies to by looking at some of thecore aspects of each of these views.

The notion of shared cooperative activity in Bratman's work, and fully cooperative joint action inTuomela's share a common core which basically is that the members of groups that are involved in thesekinds of cooperative action are committed to helping each other. I will describe the basic componentsof joint activity�mixing the terminology of the two authors�as follows. First, there is some goal tothe joint activity, what Tuomela would call a `we-intention content'. This can be a state of a�airs, someaggregation of states of a�airs, or some joint action type. Second, the agents must have a we-intentiontoward that content; they must we-intend to do, or being about, that content. We-intending is thatspecial form of intention needed for group agency in general which I will not defend here. Tuomelaanalyzes we-intentions as requiring commitments of the following form:

(W)(i) We will do X

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(ii) X cannot be performed by us unless we perform Z

(iii) we will do Z

(iv) Unless I perform Y we cannot perform Z

(v) I will do Y (as my contribution to Z). (Tuomela, 1993, pp. 89�90)

What this amounts to is an intention for each group member to do their part of the joint goal. Thatcommitment of each agent to do their part, as Bratman describes shared cooperative activity (SCA),must consist of three things, all of which are common knowledge among the members of the group.2

(i) Mutual responsiveness: In SCA each participating agent attempts to be responsive tothe intentions and actions of the other, knowing that the other is attempting to besimilarly responsive.

(ii) Commitment to the joint activity : In SCA the participants each have an appropriatecommitment (though perhaps for di�erent reasons) to the joint activity, and their mutualresponsiveness is in the pursuit of this commitment.

(iii) Commitment to mutual support : In SCA each agent is committed to supporting thee�orts of the other to play her role in the joint activity. . . . These commitments tosupport each other put us in a position to perform the joint activity successfully even ifwe each need help in certain ways. (Bratman, 1992, p. 328)

As Tuomela describes his version of these commitments:

Basically, we are jointly committed to perform extra actions required for our joint action X,where the extra actions (e.g., actions helping to bring about the other participants' part-performances) are actions not foreseen in our joint plan to perform X and hence were notincluded in our preassigned part-actions. (Tuomela, 1993, p. 90)

All Tuomela is saying which adds to Bratman's account is that the helping may not be subsumable intothe initial parts that a group member agrees to do in order to realize the joint goal.3 To be properlyengaged in this kind of cooperative group activity the actions of the individuals must also be un-coercedon both views.4 How does all of this result in uniformity?

First, the formal model is an abstract representation of things. Thus a great deal can be �t into howstrategies are interpreted. When two people are carrying a sofa up some stairs in a cooperative manner,their actions are interdependent in the sense that they are aware of each other and trying not to makethings more di�cult for each other. A strategy in such a case would have to consist of all the particularmovements made by a player.5 How the outcomes are described in such an example should also re�ectthe particular granularity of the strategies. Thus, where the sofa ends up at the top of the stairs, howtired both agents are, etc., might all be included in the descriptions of states in S. Such �ne-graineddescriptions make it clearer how the states can depend causally on an agent's choice of strategy.

I will summarize this situation in terms of outcomes to make it relevant for discussion to come. Anyaspect of a joint outcome X (proper non-empty subset of S) is a superset of X which is a proper subset ofS. Any agent may have a particular aspect of X, Y , that is their particular part as in (W) above. Everyaspect of X is within the causal e�ectiveness of some agent, which follows from the �niteness of N .6

2I.e., I know that you know, you know that I know that you know, ad in�nitum.3For Bratman, he would say that one agent's initial subplan did not include helping the other agents, but there are

subplans for all the agents involved which mesh (are consistent) and include that helping.4In the case of Tuomela, there are additional requirements that make the di�erence between being engaged in cooperative

vs. un-cooperative activity. Particularly, in cooperative activity the bene�t of acting as part of the group and helping theother agents perform their parts in that group is more bene�cial for the individuals than acting alone towards the sameend.

5That would result in an in�nite, perhaps uncountably in�nite, set of strategies. Thus van Hees' formulation wouldn'tsu�ce since he assume S, N and Σa all to be �nite. In the sequel we will correct that issue, no assumptions are made onthe sizes of S, and reference to particular strategies is abstracted away.

6In any game form G, non-empty X ( S, and σN , if o(σN ) ∈ X, there is some a ∈ N such that a is causally e�ectivefor X relative to σN since there must be some non-empty set A of agents such that o(σA) ⊆ X because o(σ∅) = S.

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But the commitment to mutual support would require that each aspect may involve some contributionof each agent that is causally in�uential for something.

The reader may notice that although cooperative joint activity requires commitment, the agents maynot be required to exercise that commitment. It may be that each agent only has to do their part, nomore, and thus needn't be casually involved in every aspect of the outcome of a joint act. Put anotherway, the requirement is a counterfactual one; agents only need to help, if help were needed, but help maynot be needed. But consider the following from Bratman:

Suppose (a) you and I [we-intend to J], and are embarked on our J-ing; (b) a problem arisesfor you: you continue to have the relevant intentions but you need help from me to act in waysnecessary for our J-ing successfully; (c) I could successfully help you without undermining myown contribution to our J-ing; (d) there are no new reasons for me to help you in your role inour J-ing (you do not, for example, o�er me some new incentive to help you); (e) this is allcommon knowledge. Let us say that circumstances satisfying (a)-(e) are cooperatively relevantto our J-ing. For our J-ing to be a SCA there must be at least some cooperatively relevantcircumstance in which I would be prepared to provide the necessary help. And similarly foryou. (Bratman, 1992, p. 337)

For Bratman, then, cooperatively relevant circumstances must be possible. That is to say that, at thevery least, there must be cases where each agent is involved in every aspect of the outcome. Althoughit may not be the case that every circumstance is one in which each agent is causally involved in everyaspect, there must be such a circumstance which were are capable of planing for. Without that level ofcommitment one can question what that commitment would amount to.

What uniformity amounts to in the case of cooperative joint activity, then, is that there must beactions/strategies which would correspond to the actuality of cooperatively relevant circumstances. Thereare a few notes of clari�cation to be made about this requirement after the formal framework is exposed.

3 Separating Decisions from Power

The mathematical theory of coalitional power, or group ability, is the theory of e�ectivity functions.E�ectivity functions come out of social choice/game theory. But here I want to be a bit more explicitabout what they represent. Generally, e�ectivity functions represent the power distribution amongst thegroups, or coalitions as they are called, of a (�nite) society/group of individuals N . Individual agents willbe referred to as a, b, c; A,B,C will be used to denote sets of agents. The way that power is representedby an e�ectivity function is by assigning each group of agents A, a collection of subsets X,Y . . . from aset of social states S. The subsets X of S are the sets of social states A is e�ective for; i.e., the group Aby acting together can ensure that the next social state is among those in X, or those in Y , et cetera.In game theory, e�ectivity functions represent the power a group of players has to determine outcomeswhen they decide to form a coalition and act according to a joint strategy�which is made up of theindividual strategies of the players. As such, e�ectivity functions are kinds of so-called coalitional gameforms since they are a generalized representation of coalitional ability/power for game forms. In this �rstsection we will review e�ectivity functions, and then review their relationships with games. So althoughthere is a strong historical connection to games, which I will explain in section 3.2, e�ectivity functionswill not be assumed to derive only from games.

3.1 E�ectivity Functions

How e�ectivity functions are de�ned di�er somewhat in the literature, but the di�erences are minor, cf.Abdou and Keiding (1991), Peleg (1998), Pauly (2001). Here we will de�ne an e�ectivity function ina more general manner than Peleg (1998), but less general than Abdou and Keiding (1991) or Pauly(2001).

De�nition 3.1. Let N be a set of individuals, S a set of social states. An e�ectivity function E, is afunction E : P(N)→ P(P(S)), such that

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(E1) ∅ 6∈ E(A) for all A ⊆ N , (Safe)

(E2) S ∈ E(A) for all A ⊆ N (Live) and

(E3) If X ∈ E(A) and X ⊆ Y ⊆ S, then Y ∈ E(A) (outcome monotonic).

When X ∈ E(A), then A is capable to ensure that the outcome will be amongst those in X: Acan eliminate or veto all of the possibilities outside X. The conditions E1�3 de�ne a fairly weak set offunctions, but they all match the interpretation of E(A). Not much can be said about the abilities ofagents for the basic e�ectivity functions, but the intuitive interpretations are as follows. E1 says thatno mater what the agents in A do, they never break the system. As long as they (A) are e�ective forsomething, the society will persist in some social state no matter what choices we make. It could beconstrued as a `No Apocalypse' condition, hence calling such functions safe. E2 says that each groupis at least as e�ective as the empty coalition, and also guarantees that every group will be e�ective forsomething, even if it is not able to in�uence the ultimate outcome alone. Finally, E3 says that if A canguarantee that the outcome will be inside X, then A can guarantee that it will be inside any supersetof X. That is an eminently reasonable condition given the intuitive interpretation of E(A). Of courseother conditions can be added to e�ectivity functions resulting in more interesting functions that enableus to draw more relationships between the e�ectiveness of various coalitions. A few of the crucial onesare as follows:

De�nition 3.2. Let E be an e�ectivity function.

• E(∅) = {S } (Ine�ectivity of ∅),

• E(N) = P(S) \ {∅ } (Citizen Sovereignty),

• E is Regular when for all A ⊆ N , and X,Y ⊆ S:

[X ∈ E(A) & Y ∈ E(N \A)]⇒ X ∩ Y 6= ∅

• E is Coalition Monotonic (OutMon) when for all X ⊆ S:

[X ∈ E(A) & A ⊆ B ⊆ N ]⇒ X ∈ E(B)

• E is Superadditive (SupAdd) when all A,B ⊆ N , and X,Y ⊆ S:

[X ∈ E(A), Y ∈ E(B) & A ∩B = ∅]⇒ X ∩ Y ∈ E(A ∪B)

It is easily seen that when E is superadditive, it is regular and coalition monotonic; but it the conversemay not hold. Ine�ectivity says that the empty coalition cannot veto anything, but the whole group ofagents�the whole society�can guarantee any social state whatsoever, i.e., Citizen Soverignty. Indeed,{ s } ∈ E(N) for each s ∈ S. For the moment, these conditions will su�ce for the discussion, but therewill be other conditions later. When E is coalition monotonic, it follows that E(a) ⊆ E(A) for all a ∈ A.That is, a group is at least as e�ective as its members. But, even for supperaditive Es, it does notfollow that A is no more e�ective than its members; a coalition may be more powerful than its members.When an E does not allow a group to be more powerful than its members it will be called strongly

individualistic, i.e., E(A) = ∪a∈AE(a).But there is another notion, which is weaker than strong individualism which I call weak individualism.

To de�ne this notion precisely, an operation needs to be de�ned:

Notation 2. Let E be an e�ectivity function, A,B ⊆ N .

E(A) f E(B) = {X ∩ Y : X ∈ E(A) & Y ∈ E(B) }

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De�nition 3.3. Let E be an e�ectivity function. E is weakly individualistic i� for every non-emptyA ⊆ N ,

E(A) =

{X ⊆ S : ∃Y ∈

k

a∈AE(a) & Y ⊆ X

}That means the e�ectiveness of A is determined completely by the e�ectiveness of its members, but

isn't simply the collection of the members' e�ectiveness. Intuitively, one would suspect that groups canbe more powerful than the sum of their parts, i.e., there can be synergy. Indeed, Pauly has shown (Pauly,2001, p. 31) that, for a certain class of e�ectivity functions to be de�ned in the next section, if E isstrongly individualistic, then there is a ∈ N such that for all X ⊆ S, X ∈ E(a): there is a dictator�Iwill discuss dictators more in section 5.1. But it is important to note that there are non-dictatoriale�ectivity functions which are weakly individualistic. For example: Let N = { a, b } and S = { 1, 2, 3 },and E be de�ned by:

• E(∅) = { { 1, 2, 3 } }

• E({ a }) = { { 1, 2 } , { 1, 3 } , { 1, 2, 3 } }

• E({ b }) = { { 2, 3 } , { 1, 2 } , { 1, 2, 3 } }

• E({ a, b }) = P({ 1, 2, 3 }) \ {∅ }

As the reader may check this function is superadditive and weakly individualistic. Indeed, every weaklyindividualistic e�ectivity function is superadditive. But it is not strongly individualistic: none of theindividuals are e�ective for the unit sets. One of the most studied subjects for e�ectivity functions isthe relationship between them and game forms, which is the topic of the next section.

3.2 Game Forms and E�ectivity Functions

As was mentioned above, e�ectivity functions can be a concise representation of the power of groupswithin strategic games. And one of the most studied aspects of e�ectivity functions is their relationship tostrategic games. Recall de�nition 1.1 above. There are two questions that arise: what kind of e�ectivityfunction correspond to game forms? and what kinds of solution concepts can be represented by e�ectivityfunctions? Here I will discuss the �rst, and only brie�y since that is all that will be need in the sequel.

The easy connection between game forms and e�ectivity functions are the notions of α and β functions.

De�nition 3.4. Let G be a game form.

• The α function for G is de�ned as: for any A ⊆ N ,

EαG(A) = {X ∈ P(S) : ∃σA∀σA, s.t. o(σA, σA) ∈ X }

• The β function for G is de�ned as: for any A ⊆ N ,

EβG(A) = {X ∈ P(S) : ∀σA,∃σA s.t. o(σA, σA) ∈ X }

EαG(A) consists of the sets of outcomes X such that A can guarantee, regardless of what the agentsoutside of A do, that the outcome will be in X. Notice that X ∈ EαG(A) i� there is σA such that

o(σA) ⊆ X. EβG(A) consists of those sets of outcomes X such that whatever the agents outside of Ado, A can �nd some way to guarantee that the outcome will be in X. It is easy to see that EαG(∅)

is the principle �lter generated by the set:{o(σN ) : σN ∈

∏b∈N\A Σb

}; for a surjective game form

EαG(∅) = {S }. The conditions under which a function E : P(N)→ P(P(S)) is an α function of a gameform G (demonstrated ultimately in Goranko et al. (2013) building on the crucial work in Pauly (2001))are those which constitute what is called a `truly playable' function.

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De�nition 3.5. Let E be a function from P(N) to P(P(S)). Call it `truly playable' i�

• Outcome Monotonic: If X ∈ E(A) and X ⊆ Y , then Y ∈ E(A)

• N-maximal: Xc 6∈ E(∅), then X ∈ E(N)

• Live: ∅ 6∈ E(A)

• Safety: S ∈ E(A)

• Super Additive: If A ∩B = ∅, X ∈ E(A) and Y ∈ E(B), then X ∩ Y ∈ E(A ∪B).

• Crown: If X ∈ E(N), then there is x ∈ X such that {x } ∈ E(N).

To put the result another way: For any function E from P(N) to P(P(S)), E is truly playable i�there is a game form G such that E = EαG. Note that a truly playable function E, such that E(∅) = {S }will also obey citizen sovereignty (by N-maximality).

What is interesting about this relationship between e�ectivity functions and game forms is thatgame forms are a representation of action which is fundamentally individualistic: the outcome of agame in decided by a selection of individual strategies for each agent. However, when we look atthe α e�ectivity function for a game form, it may not be interactive. Indeed, consider the functionE : P({ a, b })→ P(P({ 1, 2, 3 })) de�ned by

• E(∅) = { { 1, 2, 3 } }

• E({ a }) = { { 1, 2 } , { 1, 2, 3 } }

• E({ b }) = { { 1, 2 } , { 1, 2, 3 } }

• E({ a, b }) = P({ 1, 2, 3 }) \ {∅ }

As the reader may check, E is a truly playable e�ectivity function. Thus, it corresponds to the αfunction of some game. But it is not interactive. There is power that { a, b } has that is not simply acombination of the powers of the individuals separately. One �nal subject to introduce which relatese�ectivity functions to game forms.

One might notice that in the case of game forms no player ever gets to choose S, even though eachis e�ective for it. One of the things that is common to the e�ectivity functions derived from game formsshare the characteristic such that EαG(A) has a basis, i.e., all X ∈ EαG(A) can be determined by some setin{o(σA) : σA ⊆ σN ∈

∏i∈N Σi

}. This set will generally be called the non-monotonic core of E:

De�nition 3.6. The non-monotonic core of E(A) is de�ned as

Enc(A) = {X ∈ E(A) : ∀Y ∈ E(A)[Y ⊆ X only if Y = X] } .

Not all e�ectivity functions have non-monotonic cores. But when, for each A, Enc(A) is non-empty,I say that the e�ectivity function is grounded. Enc(A) acts as the ground for the total e�ectiveness of Arepresented by E(A). These notions give a representation of power, but not of decision which I will turnto next.

3.3 Decisions

E�ectivty functions provide a representation of power or ability, but they don't provide a representationof choice. In game forms, each strategy pro�le�each play�represents a particular course of action for theindividual players: one action for each player. E�ectivity functions hide this direct connection betweenindividual action and outcome. In the impossibility result the decision mechanism is represented by agame form; the game form is supposed to represent a social institution and how outcomes are determinedwithin it. In this framework, the e�ectivity function represents the social institution. Indeed, Peleg(1998) o�ers a way to make that connection explicit by deriving e�ectivity functions from structures

9

which assign rights to individuals in an abstract manner; what he calls a constitution. Although we havea correspondence between truly playable e�ectivity functions and game forms, the general setting that Iintend to work in doesn't share that direct connection to game forms. E�ectivity functions need to beaugmented to include some representation of choice, i.e., some way of indicating that certain abilitieshave been exercised, some decisions made. To that end I introduce decision pro�les.

De�nition 3.7. Let E be an e�ectivity function as per de�nition 3.1. A Decision Pro�le over E is apartial function D from P(N) to P(S) s.t.

(1) D(∅) is always de�ned

(2) ∪{A ⊆ N : D(A) is de�ned } = N

(3) D(A) ∈ E(A).

Notation 3. Let D be a decision pro�le.

• The image of D is a multiset, that is it can have repetitions of members while order still doesn'tmatter. The image of D will also simply be written as D, and D(A) is written DA, when de�ned.

• Since D is a multiset, it may have repetitions of sets. Nonetheless, certain operations still makesense in this case. ∩D = { s ∈ S : s ∈ X∀X ∈ D } as per usual, but set di�erence is slightlydi�erent: D ÷ {X } is D with only one instance of X removed. What will be important lateris D ÷ {DA } which will be D without one instance of the set DA. It will usually be written asD ÷DA.

• Since the image of a decision pro�le is represented as a multiset rather than a set, 〈x : ϕ(x)〉 willbe used to denote a multiset of objects that satisfy ϕ(x). The usual braces will be used for sets,and parentheses will be used for order tuples as above.

Since decision pro�les are partial functions, DA may not be de�ned. But what is key is that everyonemakes a decision; each member of N is included in some decision as part of some coalition. Decisionpro�les, however, may be de�ned such that multiple coalitions that involve a single agent exercise theirpower. Of course that is what people often do, they act as part of multiple groups, as well as perhaps inmore self serving ways. Nonetheless, Da is always a single value; I am only permitting a to exercise oneof its unilateral powers per moment of decision. Also, a concept that will be useful later is the idea of onedecision pro�le being an alternative to another. This can be made precise with the following de�nition.

De�nition 3.8. Let D and D′ be decision pro�les over E for which A ⊆ N is de�ned. D′ is anA-alternative to D (D′ ∼A D) i� For all B ⊆ N such that DB and D′B are de�ned, if B ∩ A = ∅,D′B = DB .

A-alternatives represent other ways that A could have chosen. How A chooses is determined, to acertain extent, by how its members have chosen. But as long as a coalition doesn't overlap with A, theirchoices need not be a�ected.

Introducing decision pro�les o�ers the chance to tease apart a decision procedure from the powerstructure. In a game form they are both combined together. Thus, the rules of the institution can besplit into two parts: those that de�ne the powers of groups and individuals, and those that constrain howthose powers are exercised. Thus, the group decision method itself can have certain properties. Here area few:

De�nition 3.9. Let E be an e�ectivity function.

• A decision pro�le D is total i� D is a total function.

• A decision pro�le is thin when for all a ∈ N , there is unique A ⊆ N such that DA is de�ned anda ∈ A.

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• A decision pro�le D is individualistic over E i� D is a total, and DA = ∩ 〈Da : Da ∈ D & a ∈ A〉for all non-empty A ⊆ N .

• A decision pro�le D is monotonic over E i� for all A ⊆ B ⊆ N , if DA and DB are de�ned,DB ⊆ DA.

There are a few worries concerning how decision pro�les might interact with e�ectivity functions. Fora general e�ectivity function, given two individuals a, b, their choices could block each other. That is:X ∈ E(a) and Y ∈ E(b) such that X ∩Y = ∅. What that means is that no social state could result fromthat interaction of a and b's decisions. Call a decision pro�le D acceptable i� ∩D 6= ∅. If E is a regulare�ectivity function, and D a thin or individualistic e�ectivity function, then it is acceptable. However, fortotal decision pro�les over even superadditive e�ectivity functions, the worry of an unacceptable pro�lemay still be troubling when, instead of individuals a and b, one considers overlapping sets of agents Aand B.

Decision pro�les can be re�ned in such a way that they only take decisions from the non-monotoniccores of e�ectivity functions. A decision pro�le D is proper when DA ∈ Enc(A). It is individually properwhen Da ∈ Enc(a). This re�ects better the idea that decisions are the proper decisions within the rangeof e�ectiveness.

To sum up, roughly, a decision procedure can be represented as a set of decision pro�les over ane�ectivity function. When all the decision pro�les in a decision procedure D have a property ϕ I will saythat D has the property ϕ. So the question becomes: what conditions on a decision procedure de�nedon an e�ectivity function guarantee which properties of the e�ectivity function, and vice versa?

For example, if a proper and total decision procedure is de�nable on the e�ectivity function E, eachE(A) must have a non-empty non-monotonic core. Also, if a decision procedure is individualistic andtotal over E, then E must be weakly individualistic. I will look at more connections of this sort in thenext section.

4 Freedom and Responsibility

In this framework freedom may take on di�erent forms. Since there is a separation between the powerstructure and the decision structure, there can be freedom within the power structures and freedomwithin how that power is exercised. I will call the �rst kinds of freedom structural freedoms and thelatter procedural freedoms. There are often complaints about the disparity between the rights that weare promised and those we may exercise. Sen's condition of minimal liberalism from (1970) can betranslated7 into e�ectivity functions as follows:

De�nition 4.1. Let E be an e�ectivity function. E is minimally liberal i� there are distinct a, b ∈ Nsuch that there are X,Y ∈ P(S) \ {S } (not necessarily distinct), such that X ∈ E(a) and Y ∈ E(b).

In minimal liberalism there are at least two individuals that can exercise some power: they caneach exclude at least one social state, although it may be the same state. Liberty often is directed atindividuals, it seems to be most important that institutions do not completely suppress individual liberty.Consider the following notion which is another form structural freedom.

De�nition 4.2. Say that an e�ectivity function E is individually interesting i� |N | ≥ 2 and there isa ∈ N which has at least two distinct X,Y ∈ Enc(a) with X ∩ Y = ∅.

What this condition requires is that the non-monotonic core of at least one individual's e�ectivenessconsists of disjoint choices. It is individually interesting because it gives that individual completelydistinct choices. One might even say that that individual is faced with a choice. Indeed, it is only undersuch circumstances that Belnap et al. (2001) would say that a genuine choice could be made. Next I willintroduce some notions of procedural freedom.

7This translation comes from Peleg (1998).

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Here I switch from saying that E is liberal to saying that a decision procedure D is liberal. Thenotions above were framed in terms of individual liberty, but there could also be liberty for a group.And as one might expect, liberty obtains when that group may exercise its powers.

De�nition 4.3 (Liberal for A). Let A ⊆ N . A decision procedure D is liberal for A i� there is D ∈ Dsuch that DA is de�ned, and for all D′ ∼A D, D′ ∈ D. A decision procedure is liberal when it is liberalfor all non-empty A ⊆ N .

A decision procedure is liberal for A when all of A's powers can be exercised within the decisionprocedure. When D′ ∼A D that means D′A is de�ned, but since the condition says that all A-variantsof D are to be considered, that means every element of E(A) will make an appearance in some D′. Theissue is that there is no restriction on how that may happen. Thus, some ways A could exercise itspower may interfere with how other groups exercise their power resulting in an inconsistent outcome,i.e., ∩D′ = ∅. Thus, certain constraints on the properties of the decision procedure, e.g., acceptability,will also limit the amount of procedural freedom that is available. But such restrictions can work theother way as well. If a certain kind of decision procedure is de�nable, then the e�ectivity function musthave certain properties. For example,

Proposition 4.1. Let E be an e�ectivity function. Suppose that a total, monotonic and liberal decision

procedure is de�nable on E, then E is coalition monotonic.

Proof. Suppose that A ⊆ B and X ∈ E(A). By de�nition, since D is liberal, there must be D ∈ D withDA = X. Since D is total and monotonic, DB is de�ned and DB ⊆ DA = X. Since DB ∈ E(B) byde�nition of a decision pro�le, X ∈ E(B) by outcome monotonicity.

Another concept which will be useful later is a libertarian decision procedure. This is to say that itenables each individual to make any unilateral decision they wish.

De�nition 4.4 (Libertarian Decision Procedure). Let E be an e�ectivity function. D is a libertarian

decision procedure over E i� for each a ∈ N , D is liberal for a.

But there are complications with libertarian decision procedures. When a decision pro�le is proper orindividually proper, it can't be libertarian in the general sense since there may be sets an individual maybe e�ective for which are not included in some decision pro�le. To correct the concept of a libertariandecision procedure I introduce a revised version.

De�nition 4.5. A decision procedure D is properly libertarian when for each a ∈ N , Y ∈ Enc(a), andD ∈ D there is D′ ∈ D such that D′ ∼a D and D′a = Y .

This notion extends the notion of libertarian decision procedure since it not only says that the decisionprocedure allows each power of each individual to be exercised, it allows each individual to exercise anypower in any decision situation. The decisions of the individuals are independent. If one considershow strategy pro�les are composed, proper libertarianism mimics that. Whatever individual strategiesplayers other than a decide to play, a may play whatever strategy it wishes. The outcome may change,but all options are available to her.

As I have already mentioned the kinds of decision procedures which are de�nable imply conditionson the e�ectivity functions, and many of the properties of decision procedures are inspired by propertiesof game forms. So it is natural to ask how close decision procedures can get to game forms. Or evenhow far away complex decision procedures can be from game forms. Consider the following result:

Lemma 4.1. Let E be an e�ectivity function. A properly libertarian, individually proper, individualistic

and acceptable decision procedure D is de�nable on E i� for any A ⊆ N and b ∈ N ,ca∈AE(a) ⊆ E(A)

& Enc(b) 6= ∅.

Proof. (only if) To see Enc(b) 6= ∅, D is individualistic, so it is total. Thus Db is de�ned for eachb ∈ N , and since it is individually proper Db ∈ Enc(b). For

ca∈AE(a) ⊆ E(A), suppose Xa ∈ E(a)

for a ∈ A. Then there is X ′a ∈ Enc(a) with X ′a ⊆ Xa for each a ∈ A, and there will be D ∈ D

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with Da = X ′a because D is properly libertarian. Since by de�nition of decision pro�le, DA ∈ E(A),hence ∩a∈ADa ∈ E(A) because D is individualistic. But DA = ∩a∈AX ′a ⊆ ∩a∈AXa, so by outcomemonotonicity, ∩a∈AXa ∈ E(A).

(if) Conversely, let D be de�ned from the set of functions f ∈ ∪a∈NEnc(a)N such that f(a) ∈ Enc(a)where for each f de�ned a decision pro�le Df by:

DfA =

⋂a∈A

f(a), if A 6= ∅

S otherwise

So de�ned, each Df is clearly individualistic. It is also individually proper since Dfa = f(a) ∈ Enc(a) for

a ∈ N . It is also clearly properly libertarian because there is a Df de�ned for every f . To see that it isacceptable, note that

ca∈N E(a) ⊆ E(N). If ∩Df = ∅, then ∩a∈NDf

a = ∅, but ∩a∈NDfa = ∩a∈Nf(a) ∈c

a∈N E(a) by de�nition. Thus ∅ ∈ E(N) contrary to safety. Therefore ∩Df 6= ∅.

The kind of e�ectivity function involved in this lemma can be a long way from that of a game form.It needn't be a crown, nor superadditive for example. But in the other direction, every game form cangive rise to an equally nice collection of decision procedures. For example:

Lemma 4.2. Let EαG be the α-e�ectivity function of a game form G. The D de�ned by:{〈o(σA);σA ⊆ σN 〉 : σN ∈

∏i∈N

Σi

},

will be properly libertarian, proper, monotonic, total, and acceptable over EαG.

The proof is left as an exercise. Just note that Eα−ncG (A) ={o(σA) : σA ∈

∏a∈A Σa

}, and o(σ∅) = S.

Of course what one might notice is that none of the formulations of e�ectivity functions and decisionprocedures assumes �niteness of S or E(A). It is only the set of agents that is �nite. But it is notassumed either that ∩D is a unit set in any of the cases either. Thus this formulation extends thatof game forms, technically speaking. In the interests of brevity, I will move on to a brief discussion ofresponsibility, or rather, agency in this new framework.

The obvious approach would be to translate van Hees' notion of causal e�ectiveness in de�nition 1.2.But there are some issues with this approach. First is that each D would have to be de�ned for someset A with a ∈ A for each a ∈ N . That is �ne, but DA\a also has to be de�ned. Thus arbitrary decisionpro�les won't cut it. When a decision procedure is total, however, there will be no problem.

De�nition 4.6 (Cooperatively Agentive). Let E be an e�ectivity function, and D be a total decisionpro�le over E. a is Cooperatively agentive for X relative to E and D i� there is A ⊆ N with a ∈ A suchthat

i) DA ⊆ X, and

ii) DA\a * X.

In van Hees' work this is his notion of individual causal e�ectiveness. However, in other theoriesof agency, this is a notion of causal agency that would apply more to groups than to individuals. Anindividual is individually agentive when it is the individual who is, individually, the causal nexus. Butwhen one is part of a group, one can stand to shed some of that individuality in favour of the group.Of course van Hees and Braham make a case for saying that this notion is perfectly �ne as a notion ofindividual causal e�ectiveness, but to be deferential to the work outside of theirs, I give it another name.Having said that, when A = { a } this notion is one of individual causal agency, properly so-called.

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5 Uniformity and Dictatorship

As van Hees' result suggests there should be some connection between uniformity and dictatorship. Inthis section I will explore the notion of dictatorship in this new framework, then turn to expressing theproblematic conditions and what results they lead to.

5.1 Dictators

The intuitive idea of a dictator is someone who can impose their will for an overall outcome. In socialchoice theory, à la Arrow (1951), a dictator is an individual whose preferences determine the societalpreferences under aggregation. In this framework of e�ectivity functions and decision pro�les, the resultsare a bit di�erent. Van Hees' de�nition of a dictator a is when for any s ∈ S there is a σa ∈ Σa, suchthat o(σa) = s.

With the introduction of decision pro�les, although the power structure may not be dictatorial, aparticular decision procedure, i.e., set of decision pro�les D, may result in there being a dictator overhow the power is exercised. This provides two tiers of dictators: those relative to the e�ectivity functionand those relative to a decision procedure over an e�ectivity function. I will call the �rst kind of dictatorstructural dictators and the later, procedural dictators, mirroring the notions of freedom above. Althoughthere are di�erent kinds dictators that can be speci�ed within each of these categories, I will postponein-depth investigations of those di�erences.

De�nition 5.1 (Structural Dictators). Let E be an e�ectivity function. a ∈ N is a T(otal)-dictatorrelative to E i� E(a) = E(N).

The idea behind the structural dictator is that there is one individual who has all of the e�ectivenessof the whole group. This generalizes van Hees' version since the e�ectiveness of the whole group maynot be to such a degree that it can guarantee any social state whatsoever, i.e., no citizen sovereignty. Ihave called it a total dictator although it is the only structural dictator I mention, I want to indicatethat there are others. Total dictators are the only version I will be concerned with for the moment.

For procedural dictators, there needs to be substantial revisions. The major barrier is that dictatorsare individuals, and it may not be the case that Da is de�ned for every decision pro�le. However, I willskirt that issue here in the interests of simplicity. The general form of a procedural dictator I want toemphasize here will be one individual whose decision determines the overall decision.

De�nition 5.2 (Procedural Dictator). Let D be a decision procedure over an e�ectivity function E. ais a D(ecision)-dictator relative to D i� for all D ∈ D, ∩D = Da.

Even if a power structure is totally dictatorial, one could constrain how that power is exercised sothat that all powerful individual couldn't dictate everything. Although, there would be other versionsof procedural dictatorship which would be satis�ed in such cases.

The next question to ask is whether there are relations between the procedural dictators or betweenthe procedural and structural dictators. Whether there are such relationships depends primarily on whatthe decision procedure is like. In this limited case, when the decision procedure can be used to arriveat any X ∈ E(N), a D-dictator must be a T-dictator. What is interesting is how dictatorship can arisefrom the burden of too much responsibility:

Lemma 5.1. Suppose that E is a coalition monotonic e�ectivity function, and D is a total decision

procedure such that D∅ = S for all D ∈ D. If for all X ∈ E(N), there is D ∈ D such that ∩D ⊆ X, and

there is a unique agency bearing individual for all D ∈ D, E is T-dictatorial.

Proof. Suppose for all X ∈ E(N), there is D ∈ D such that ∩D ⊆ X, and there is a unique agencybearing individual for all D ∈ D. Let the unique agency bearing individual be a. Since a is the only agentthat bears any agency in each D, and there is always someone agentive for any result (see lemma 5.2below), a is agentive for ∩D, in each D. Thus, there is B such that a ∈ B, DB ⊆ ∩D, and DB\{ a } * ∩D.But that means Da ⊆ ∩D. For suppose that a ∈ B, b ∈ B, and b 6= a, then DB\{ b } ⊆ ∩D otherwise b

14

would be agency bearing. (Note that means ∩D = Da and so a is a D-dictator.) By the de�nition ofdecision pro�le, Da ∈ E(a), so by outcome monotonicity X ∈ E(a). Hence E(N) ⊆ E(a). By coalitionmonotonicity, E(a) = E(N), i.e., E is totally dictatorial.

5.2 Uniformity and Completeness

Van Hees' theorem, as mentioned above, shows that requiring a group decision/action mechanism todistribute responsibility similarly across the members of the group, and avoid any failures of assigningresponsibility for outcomes leads to a mechanism that is dictatorial (in the outcome sense of dictatorial).The two conditions of uniformity and completeness are phrased in terms of responsibility, but for themost part their formal rendering simply leaves a place holder for `responsibility'. A decision mechanismis complete when every outcome has some, individual, agent. Generally, completeness is phrased as theabsence of voids, and voids are when there is an outcome, but no one is responsible for it. Using de�nition4.6 completeness admits of a two stage de�nition:

De�nition 5.3 (Completeness). Let E be an e�ectivity function, and D a decision procedure.

• D ∈ D displays a void for non-empty X ( S relative to E i� there is no individual a ∈ N suchthat a is cooperatively agentive for X, although ∩D ⊆ X.

• D is complete i� for every D ∈ D and non-empty X ( S, D does not display a void for X relativeto E.

I am simply translating van Hees' de�nitions into my framework as I did with de�nition 4.6. I willpause to note something about completeness in this new framework:

Lemma 5.2. Let E : P(N) → P(P(S)) be an e�ectivity function for which ∅ is ine�ective. Further,

suppose an acceptable, monotonic, and total decision procedure D is de�nable over E. If for each D ∈ D,∩D 6= S, D is complete.

Proof. Suppose ∩D = X ( S and X ⊆ Y 6= S. From the ine�ectiveness of ∅, D∅ = S. Since D ismonotonic, DN = ∩D ⊆ Y 6= S. From the �niteness of N , there must be a minimal non-empty set Asuch that DA ⊆ Y , and so for any a ∈ A, DA\{ a } * Y . So there is some a ∈ N that is cooperativelyagentive for Y .

Thus, completeness is not really a substantial condition to impose on, su�ciently complex decisionprocedures. Uniformity, however, is.

The idea behind uniformity is that if an agent is responsible for X, then every responsibility bearingagent is also responsible for X. I recast it as:

De�nition 5.4 (Uniformity). Let E be an e�ectivity function and D a decision procedure over E.

• a ∈ N is agency bearing relative to D i� there is non-empty X ( S such that a is cooperativelyagentive for X relative to D.

• A decision pro�le D is uniform (over E) i� for every non-empty X ( S and all agency bearingindividuals a, b ∈ N : a is cooperatively agentive for X i� b is.

• A decision procedure D is uniform i� every decision pro�le D ∈ D is uniform (over E).

From here the initial result isn't that problematic.

Theorem 5.1. Let E be an individually interesting e�ectivity function for a ∈ N . Suppose a properly

libertarian, proper, total, monotonic, and acceptable decision procedure D is de�nable on E such that for

all D ∈ D, D∅ = S. If there is D ∈ D which is uniform with Da = X and there is some Y ∈ Enc(a)such that Y ∩X = ∅, then there is a unique agency bearing individual in D.

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Proof. Suppose that E is individually interesting for a. So |Enc(a)| ≥ 2, and there is b ∈ N such thata 6= b. It follows by assumptions on D that D is complete, and for each D ∈ D, ∩D 6= S. Let D ∈ Dbe uniform, and Da = X ( S. Thus ∩D ⊆ X. Since D∅ = S, a is cooperatively agentive for X as perde�nition 4.6.

Suppose for reductio that there is another agency bearing individual in N , say b, at D. By uniformitythere is B with b ∈ B such thatDB ⊆ X, andDB\{ b } * X. If a ∈ B, thenDB\{ b } ⊆ Da by monotonicityof D, so DB\{ b } ⊆ X. Hence a 6∈ B. There must be D′ ∼a D in D such that D′a = Y , while D′B = DB

because D is properly libertarian. But then D′B ∩ D′a = DB ∩ D′a = ∅ since Y ∩ X = ∅, D′a 6= Da,and the assumption of individually interesting. Thus ∩D′ = ∅, contrary to the assumption that D isacceptable. Therefore, a is the unique agency bearing individual for D.

Under the substantial assumptions, all that is required is that there be a unique agency bearingindividual. And that is only required when that agent can exercise its power in a way where the otheragents may also have to be agentive for all aspects of the overall outcome. The result starts to becomeproblematic when uniformity is required more ubiquitously.

Corollary 5.1. Let E be an e�ectivity function. Suppose a proper, total, monotonic, and acceptable

decision procedure D is de�nable on E such that for all D ∈ D, D∅ = S. If E is individually interesting

while D is properly libertarian and uniform, then there is a unique agency bearing individual for each

D ∈ D.

In this result, all of the decision pro�les in D are assumed to be uniform. Since it is also properlylibertarian, a may choose any other proper choice it has, but that will con�ict with what any otheragency bearing individual might be able or have to do. But as we have seen above in lemma 5.1, toomuch agency for one individual results in dictatorship. Thus, there is a trade o� between very minimalstructural freedom, individual procedural freedom and uniformity.

6 Conclusion: Uniformity and Group Agency

To put the result in corollary 5.1 in perspective, we need to see how uniformity relates to the discussionin section 2. At the end of section 2, I concluded that the commitment to mutual responsiveness,which incidentally survives in to Bratman's later work (2013), has to result in the possibility that eachagency bearing individual be agentive for the same outcomes. Thus, it seems that such a restriction isn'tparticularly problematic for group agency, generally speaking.

The situation would bear more resemblance to that in theorem 5.1 than in corollary 5.1. The caseswhere the agency bearing individuals share causal links to each aspect of the outcome are remote, andso needn't interfere with the structural or procedural freedoms of the individuals. But that depends onhow the power structures and decision procedures are interpreted.

The mathematical models are highly abstract, thus they can admit of many interpretations. Butwhat results is that when the model requires ubiquity of uniformity, there will be a necessary trade o�between structural freedom and procedural freedom. Therefore, when individuals have power they arenot able to exercise it, and when they are able to exercise power completely, they can't have any.

Having said that, perhaps that is the result one should expect for complete cooperation. For truecooperation to obtain in a decision procedure, each individual must relinquish their access to theirindividual, unilateral power so that they may act as part of a group. Along those lines, one might lookat uniformity as obtaining because, whatever interventions each agent might make in the group, there isa persistent intervention that they must make in order for the group structure to obtain.

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