1. INTRODUCTION
This paper is about the measurement of specific freedom – the freedom of an agent to undertake some particular action. In this regard its general subject matter is not new. In a recent paper, Dowding and van Hees (Reference Dowding and van Hees2003) discuss, for example, the need for, and general form of, a “freedom function” that assigns a value between 0 and 1 to a freedom or right and that describes the expectation that a person may have about being in a position to exercise (“being free to perform”) that freedom or right. The usefulness of such a function is that in principle it could be used to define threshold values for indicating whether or not a person has a particular freedom or right in a material (or effective) sense – as against a mere formal or legal sense – and therefore for making non-welfaristic judgments about social states or to design the assignment of rights related to government policy, public regulation, or legal rules.Footnote 1 In a nutshell, such a function would allow us to make comparative judgments about the extent to which a person can expect, to some degree or another, to exercise a freedom or have that freedom respected.Footnote 2
Much light, however, still needs to be shed on the nature of such a function. In their contribution, Dowding and van Hees leave the matter more or less open, claiming only that the extent to which a person is free to perform a particular type of action or right depends only on the probabilities with which each of the relevant instances of the action or right will not be prevented. A straightforward example is that of determining our “freedom of expression.” According to Dowding and van Hees this is given by the probability that shouting “Down with the Government” at Whitehall at a given time and date and doing the same thing at Piccadilly Circus, etc. will go unprevented.
Dowding and van Hees refer to the recent and burgeoning literature on measuring freedom for a hint as to how such a function could variously be defined (Arrow Reference Arrow, Basu, Pattanaik and Suzumura1995; Carter Reference Carter1999; Dowding Reference Dowding1992; Pattanaik and Xu Reference Pattanaik and Xu1990; Pattanaik and Xu Reference Pattanaik and Xu1998; Sugden Reference Sugden1998; Rosenbaum Reference Rosenbaum2000). A perusal of this literature indicates, however, that as yet there is no agreed upon framework for defining this function as the “probability of being unprevented.”Footnote 3 The papers by Arrow, Dowding, Pattanaik and Xu, Rosenbaum and Sugden are all concerned with “freedom of choice” rather than with the “freedom to do something” per se (freedom simpliciter).Footnote 4 Even Carter's (1999) extensive analysis of measuring overall freedom as an aggregation of the probability of being unprevented to do something does not suggest an explicit model for determining the “input probabilities” into a freedom function. Instead, they enter into his measure as an exogenous variable.Footnote 5 There is, therefore, still an open question about (i) the source of the input probabilities and by implication (ii) how to aggregate these probabilities into a value as suggested by Dowding and van Hees. This paper provides a partial and tentative framework for answering both questions.
Starting from a purely negative conception of freedom, I will construct a function that describes i's freedom to perform an action as the “conditional probability of success.” This model makes an agent's freedom to do something a function of the propensities of other agents to choose certain actions (strategies) and the “decision rule,” which is a function that maps those actions into a unique outcome in which the agent can or cannot perform the specified action. The basic idea is that an agent is guaranteed to be unprevented to perform a specific action if, and only if, she belongs to a subset of agents (a coalition) that can guarantee the performance of that action. Roughly speaking, “negative freedom is membership of powerful coalitions,” and a measure, in the sense of ascertaining the expectation that an agent can perform a specified action, is the probability of being a member of a such a coalition.Footnote 6
In the process of constructing a freedom function I make two other contributions of general theoretical importance. First, I address the issue of how to measure freedom in a strategic rather than the parametric setting of social choice theory that developed since Sen's (Reference Sen1970) seminal contribution. Although a number of writers have, for some time, considered this to be a necessary step (Nozick Reference Nozick1974; Gärdenfors Reference Gärdenfors1981; Sugden Reference Sugden1985; Gaertner et al. Reference Gaertner, Pattanaik and Suzumura1992; Pattanaik and Suzumura Reference Pattanaik and Suzumura1996; van Hees Reference van Hees2000), it is still a largely underdeveloped area (Deb Reference Deb2004).
Second, I add to the nascent literature that attempts to develop formal models of freedom in an explicit philosophical framework (Steiner Reference Steiner and Griffiths1983; Carter Reference Carter1999; Dowding and van Hees Reference Dowding and van Hees2003; Bavetta Reference Bavetta2004), rather than deriving such a model from a set of intuitive desiderata. In other words, I do not take for granted a particular notion of freedom, but instead base my measure on a philosophically grounded generic concept and syntax of freedom.
2. THE CONCEPTION AND SYNTAX OF SPECIFIC FREEDOM
When constructing a measure of freedom, it is essential to be clear from the outset about the conception or type of freedom that we want to deal with. Primarily, this requires us to distinguish between an “opportunity” and an “exercise” concept of freedom.Footnote 7 As Carter (Reference Carter2004), who employs this distinction in his dissection of Pattanaik and Xu's (1990) axioms of freedom of choice, puts it, “where freedom is treated as an opportunity concept, it means the possibility for an agent of performing some action or actions” (Carter Reference Carter2004: 64), where “possibility” is understood as meaning a lack of constraints of various kinds. Taken in this sense, freedom is concerned with actions that might be performed, given the absence of constraints, at some moment subsequent (or identical) to that at which the agent possesses the freedom in question. On this view, freedom is a matter “of what we can do, of what it is open to us to do, whether or not we do anything to exercise these options” (Taylor Reference Taylor and Ryan1979: 177).
In contrast, freedom as an “exercise concept” concerns the performance by an agent of some action or actions; it is “to do certain things or to achieve certain outcomes in a certain way” (Carter Reference Carter2004: 64). On this view, freedom usually involves exercising control over one's life, so that one is free to the “extent that one has effectively determined oneself and the shape of one's life” (Taylor Reference Taylor and Ryan1979: 177). Clearly the Hobbes–Bentham notion of negative freedom as simply the “absence of external physical or legal obstacles” (Taylor Reference Taylor and Ryan1979: 176) is an opportunity concept, while the Rousseau–Marx notion of positive freedom as “self-realization” or “collective self-government” is an exercise concept.
The aim in this paper is to define a freedom function that is applicable to an opportunity concept of freedom. I make no apology for this restriction for the simple reason that I want such a function to also be applicable to the language of rights, and rights generally concern the “opportunity” to do things (voting, protesting, reading) and not “exercising”: if I have a right to read a certain book, then I have that right whether or not I ever read it; or whether or not I read it as a Marxian “species being” (and the same is true if it is merely a freedom to read that book, i.e. irrespective of the right to do so). Furthermore, in this connection I will basically assume a more or less empirical conception of freedom in that I will not explicitly concern myself with the thorny issues of the deontic permissibility of the freedoms in question (the moral freedom to do something, such as entering your house, irrespective of the actual or empirical freedom to do so). The normative conception of negative freedom will take a back seat in this analysis, although, as it will become evident, it is not excluded from the freedom function that is derived. For similar reasons I will also exclude the issue of the eligibility of the freedoms in terms of the reasonableness or rationality of a particular option (e.g. Sen's (Reference Sen1991:24) famous example, “being beheaded at dawn”).
Now, in order to specify a freedom function in a systematic manner, the conception of freedom must be further fleshed out with a syntax of freedom. MacCallum's (Reference MacCallum1967) now canonical analysis is the obvious candidate. MacCallum formulates a freedom ascription as a triadic relation between agents, constraints (preventing conditions), and possible actions: “of something (an agent or agents), from something, to do, not do, become, or not become something . . .” (p. 314). This relation can be summarized in the format of:
“i is (is not) free from j to do (not do, become, not become) ϕ,”
where i ranges over agents, j ranges over “preventing conditions” such as constraints, restrictions, interferences, and barriers, and ϕ ranges over actions (“doings”) or conditions of character (Marxian “self-fulfillment” or “realization of one's true nature”) or circumstance (“becoming angry”).
To define a freedom function that can be used to analyze the extent of my material freedoms and rights along the lines that Dowding and van Hees (Reference Dowding and van Hees2003) suggest, MacCallum's rather opened-ended syntax must be honed down. In particular, the range variables have to be restricted to capture only those opportunities that are social:
• i covers only natural or juridical persons (or groups);
• j covers only preventing conditions that are inflicted by the actions of other such agents or groups of agents;Footnote 8
• ϕ covers only possible actions or ‘doings.'Footnote 9
3. THE COALITIONAL NATURE OF SPECIFIC FREEDOM ASCRIPTIONS
Having set out a conception and syntax of specific freedom, the next natural step is to adopt a formal framework from which we can derive a freedom function. Prior to doing this, however, we have to first sort out a logical conundrum that afflicts canonical definitions of negative specific freedom. The solution to this conundrum is a mainstay of the contribution of this paper.
The usual method, which Carter (1999: 27), Kramer (Reference Kramer2003b: 3), and Steiner (1994: 8) employ, is to define prevention in terms of the action(s) of natural or juridical individual agents. Indeed, a review of the literature on specific freedom indicates a preoccupation with dyadic relations: j preventing or not preventing i from ϕ-ing. This, it is easy to show, is not true in general: prevention arises not from the action of an individual agent, but from the combined actions of a non-empty set of individual agents. Rather, prevention is a function of individual agents who “belong together” by dint of a common action (or omission), coordinated or otherwise, that is inimical to (“is against”), but not necessarily an action that can alone “prevent,” some other agent performing ϕ. In n-person game theory, these sets are referred to as coalitions (we can regard a coalition (a group) as an index for a certain collection of actions by individuals that can be mapped into the same possible outcome).Footnote 10
Establishing this thesis is straightforward. Suppose, for instance, a society consists of four members, denoted by the set N = {a, b, c, d}. Suppose further that there is some action ϕ that for a to perform it, at least two others must not perform some action inimical to a performing ϕ. Now, if we apply an individual agent-based definition of specific freedom and unfreedom we find that there is a configuration of agents who together prevent a from performing ϕ but none of these agents can be said to be doing the “preventing” as such. This is the case when all other agents, i.e. the subset {b, c, d}, perform actions inimical to a-ing ϕ.
To prove this proposition, let us, without loss of generality, take Carter's (1999: 27) definitions as our point of departure. (I do not take Steiner's because he only formally defines unfreedom and I do not take Kramer's because he includes a condition for an agent's personal ability to do something which in this context only complicates the issue without adding anything.) Carter says that an agent is free to ϕ,
“if every other agent refrains from preventing her doing it”;
and she is unfree to ϕ,
“if some other agent prevents her doing it.”
Take the configuration {b, c, d}, the members of which have chosen, either jointly or severally, to perform, consciously or otherwise, some action that is inimical to a performing ϕ. Obviously a cannot be free to ϕ because, assuming that by “refrains from preventing” Carter means the same as “not preventing,” it is not the case that “every other agent does not prevent a from ϕ-ing”; they clearly are.Footnote 11 But by Carter's account, a cannot be said to be unfree to ϕ either because this would require that “some other agent” (at least one) is preventing a from ϕ-ing, which is not the case. This can be shown as follows. If some other agent is preventing a from ϕ-ing, it means that there is at least one agent who, if, ceteris paribus, she were to decide otherwise, she would see to it that a was free to ϕ. Otherwise this agent could not be said to be doing the preventing. As is usual when assessing counterfactuals we have to take some state of the world as given. In this case we hold the decisions of c and d constant and ask whether a would be free to ϕ if b were not to oppose her. The answer is “no,” because at least two agents must do so. So it cannot be said that b is doing the preventing. Applying the same reasoning to c and d we find that in both cases the answer is also “no.” Hence no member of {b, c, d} is doing the preventing as such. Thus, while the non-fulfillment of Carter's conditions for a specific unfreedom (it is not the case that some agent j, k, . . ., n is preventing i) logically entails the fulfillment of his conditions for a specific freedom (all other agents j, k, . . ., n are not preventing i) this entailment does not necessarily imply that an agent will possess a specific freedom, as our example unambiguously demonstrates.Footnote 12
There are two important points to observe here. The first is that there can be no dispute that the logical relationship between Carter's two definitions is correct: his use of the universal quantifier for freedom and existential quantifier for unfreedom assures this. What is disputed is the acceptability of both his definitions together because they fail to pick up cases where there is absence of “agental prevention.”
Secondly, there is an important conceptual issue that is thrown up. The source of the difficulty with the individual agent-based definition of freedom is that it is based on a very strong assumption about the nature of power relations that govern prevention: that they are individualistic. The assumption is that every social state can be forced by some (at least one) individual agent (either i can see to it that she performs ϕ or there is some j, k, . . ., n who can see to it that she does not perform ϕ). This is neither logically nor empirically true. To belabor the point, if “agental prevention” exists this is simply the special case of the singleton set. In the case of {b, c, d} preventing a from ϕ-ing there is no such “agental prevention,” because neither {b} nor {c} nor {d} can see to it that a is free to perform ϕ; but there is if {c, d} is the preventing coalition, because either {c} or {d} can see to it that she is free to do so. In a social context, then, power is a property to be ascribed to coalitions and not to individuals. Tempting as it may be, i is not to be confused with {i} (Holler and Widgrén Reference Holler and Widgrén1999). To confuse the two is to commit what is best called the “individualistic fallacy.” Once we drop the requirement of agental prevention a solution to the problem is obvious: preventing conditions must be formulated in terms of sets of agents (coalitions) because a “preventing set” can always be identified.
4. FORMAL FRAMEWORK
We can now formalize a generic definitional framework for ascribing a specific freedom or specific unfreedom:
Definition 4.1 (Specific freedom) “i is free to ϕ”: if i were to attempt to ϕ, then no non-empty set of agents prevent i from ϕ-ing.
Definition 4.2 (Specific unfreedom) “i is unfree to ϕ”: if i were to attempt to ϕ, then some non-empty set of agents prevent i from ϕ-ing.
There are three observations to be made here. Firstly, the reference to hypothetical actions in the form of what an agent attempts to do means that in an important sense the definitions concern an “impure” conception of negative freedom. In line with Gray (Reference Gray1980) and Flathman (Reference Flathman198), what counts for freedom is not merely unimpeded or impeded movement or “behavior” but action.
The restriction to hypothetical actions instead of merely hypothetical behavior (or “movement”) is not innocent. In its absence, the definition of a specific freedom would merely say that i is free to ϕ “if no non-empty set of agents prevent i from ϕ-ing.” Without any further specification of the meaning of ϕ-ing,Footnote 13 an implication of the simpler formulation is that it ascribes i the freedom to ϕ even if i were compelled by the physical force (not simply the coercion or threats) of others to do so (which i could not resist). In the now proverbial case of a’s freedom to ϕ, this would be if {b, c, d} or any of its two player subsets could not only see to it that a performs ϕ if a were to attempt to do so, but also see to it that a do so even if a were not to make such an attempt. The “pathology” – if one can call it that – of the simpler formulation is that it ascribes a the freedom to ϕ even if {a} cannot prevent the outcome in which a performs ϕ. More concretely, the simpler formulation counts as an instance of my freedom to smoke, the case where I do not attempt to smoke but two of my friends hold me down, put a cigarette in my mouth, hold my mouth closed, and block my nose so that by mere reflex I inhale the smoke as I breath. In contrast, Definitions 4.1 and 4.2 require that I would perform some action that could be denoted as “attempting to smoke” before it can be said that I am free or unfree to smoke. In the case of my friends holding me down and blocking my nose, etc. while I am clearly “smoking,” I am not really “doing” anything. It is a mere undertaking.
The definitions do not, however, exclude as an instance of my freedom to smoke a scenario where I attempt to smoke and then my friends force me to do so in the above manner the moment I reach for the cigarette packet, because what counts is that my attempt goes unprevented. My friends have merely made me unfree to not smoke. Technically, the restriction to hypothetical actions says that the potential coalitions that we inspect to determine a‘s freedom must include a, i.e. {a, b, c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {a, b}, {a, c}, {a, d}, {a}. This is the basis of the slogan introduced at the beginning: “freedom is membership of powerful coalitions” – because in the game theoretic jargon we will use later, the coalitions that are necessary and sufficient for an agent to perform a particular action are denoted as “powerful.”
The second observation is that Definitions 4.1 and 4.2 do not require that the freedom to ϕ logically entails the freedom to not-ϕ. Although linguistic intuitions might suggest this tack, such a requirement is too strong as there are many things that I am free to do but unfree to not do. My attempt to smoke may go unprevented as my friends stand behind me making sure that I smoke (they do not physically force me to do so, however) and therefore I am free to do so; but at the same time I may be unfree to not smoke (they would hold me down, block my nose, etc. if I attempted not to smoke).
Thirdly, the definitions make no direct reference to i’s power to ϕ or the power of j,k,. . .,n with respect to i ϕ-ing. All that is required under this conception of freedom is that potential constraints are inoperative; not that they do not exist. As above, a is free to ϕ in {a, b, c, d} although any two member subset of {b, c, d} could, but does not, prevent a from ϕ-ing. That is, the definitional framework ascribes i the freedom to ϕ even if constraints could have been operative had i attempted to ϕ, but would not have been, because the set of agents that could have made the constraints operative would not have done so (because, for example, the members of such a set had no common intention to do so). An implication of this account of freedom is that we ascribe i the freedom to ϕ even if it is at the grace and favour of some set of agents who could, at will and with impunity, make i unfree to ϕ. Freedom here is not absence of potential “domination.”
Although this hiatus between power and freedom would be troublesome for some theorists such as Pettit (1996, Reference Pettit1997), it is not really something that we need to worry about here given that we have assumed a negative conception of freedom. Isaiah Berlin (Reference Berlin1969: 130) famously remarked, “Liberty is principally concerned with the area of control, not with its source”; and, he continued, “The answer to the question ‘Who governs me?’ is logically distinct from the question ‘How far does government interfere with me?’.”
5. A GAME THEORETIC MEASURE
5.1 Types and tokens
Having specified (i) a conception of specific freedom (unfreedom) and (ii) the conditions for ascribing a specific freedom (unfreedom), we can now turn our attention to the problem of constructing a function that describes i’s expectation that she is free (unfree) to ϕ, that is, a measure for making judgments about the degree to which an individual can enjoy the freedom to ϕ. Following Dowding and van Hees’ (2003) proposal, we want this expectation to reflect the different instantiations r 1,. . .,r n, called act-tokens, of performing a particular type of action R, called an act-type, given by:
where p(ri) is the probability that an act-token ri will not be prevented (the agent is free to perform an instance of the act-type or right, R).Footnote 14 The basic idea is that while the formal existence of a class of acts (an act-type), R, is given by the possibility that at least one of its instantiations, the tokens r 1,. . .,r n, are possible, we want to determine how probable each of these tokens or instantiations are and from this derive a probabilistic judgment about the extent to which R can be said to materially (or effectively) and not just formally exist given the probability of each of these instantiations.
In the language that I have been using, an action ϕ can be taken as either an act-type, R, or an act-token, ri, because a “specific freedom” can be more or less “specific” (Carter Reference Carter1999; Steiner Reference Steiner1994; van Hees Reference van Hees2000). To use the example of “freedom of expression” again, this is an act-type R that can be instantiated in the different ways we have said: r 1 is shouting “Down with the government” at Whitehall at a particular time and date, r 2 is doing so at Piccadilly Circus, and so on. Each of these tokens can be specified further as act-types themselves: shouting “Down with the government” at Whitehall alone or doing so with others, etc. An action to which there is a unique corresponding particular event is an act-token; it is an action in which all spatiotemporal and physical components are specified. Thus in the example of a performing ϕ, each of the coalitions {a, b, c, d}, {a, b, c}, {a, b, d}, {a, c, d} are the instantiations (tokens) r 1,. . .,r n of a performing the act-type ϕ. Hence, given our definitional framework, we arrive at the central idea of this paper: the natural way to define Γ (·) is on the domain of possible coalitions.
5.2 Game forms
To define Γ(·) on the domain of coalitions in a systematic manner, we have to skip through some game theoretic preliminaries. The basic concept that we need is that of a game form (all of which has been implicit in our example of a‘s freedom to ϕ). A game form is a specification of a finite set of outcomes X, a finite set of individuals (or players) N = {1,. . .,n}, a finite set of feasible actions or strategies Ai for each i∈ N,Footnote 15 and an outcome function π (or decision rule) that yields some single outcome x∈ X for any given n-tuple [ai] of strategies, one strategy a i ∈ A i for each i, i.e. g = (N,{A i}_i∈ N,π). A game form can be said, therefore, to specify the “rules of the game.”
For our purposes, we are interested in a particular game form in which the outcome set, X, has two elements, either i can perform ϕ (ϕi) or cannot perform ϕ (¬ϕi), i.e. X = {φi,¬ φi} and in which each player (including i) has two possible strategies: to either “agree” (do not act in a manner inimical to i performing ϕ) that i should be free to ϕ or not (attempt to “resist”), which we designate as Ai={yes, no}. To be clear, by “strategy” is not necessarily meant a particular action as such, but rather a “bundle of actions”; they should be seen as courses of action. Depending upon the context of the specific freedom, the act of agreeing to or hindering i's freedom to ϕ may involve different things. It could be as minor as a nod or a wink or providing a signature; or it could involve moving a heavy object; or it could even be an “omission” in the sense of not doing something that is required, either consciously or unconsciously. In any case, what are involved are many actions (to provide a signature I must pick up a pen, put the pen to paper, hand over the signed form, etc.), each of which I must be free to perform.Footnote 16 Note, then, that under this construction a specific freedom or unfreedom presupposes other prior specific freedoms and that the specific freedom or unfreedom in question is the outcome of a combination of such bundles of actions as determined by a “decision rule,” π.
Now, in accord with our definitional framework, π defines the subsets of agents, 
, called coalitions, that can force an outcome in X. That is, we are looking at a game form with a very sharp distribution of power: a coalition S, which is a collection of members of N who have made the same strategy choice, has either full power (is “winning”) or zero power (is “losing”). Thus, as before, a has the support of b and c in {a, b, c} and this coalition has the power to see to it that a can ϕ, while its complement, {d}, is powerless (cannot prevent a from ϕ-ing); while a only has the support of b in {a, b}, which because it is not enough is therefore powerless to see to it that a can ϕ, while its complement {c, d} is powerful (can prevent a from ϕ-ing).
Such a game form is also called a simple game and can be represented by a non-empty set consisting of the winning (“powerful”) coalitions. We assume, as is usual, that W satisfies three basic conditions: (i) , otherwise all coalitions would be winning and no player could prevent anything; (ii) , i.e. the grand coalition is powerful; and (iii) if and , then , i.e. if a coalition is winning then additional support will not alter the outcome. Note that the first condition, , guarantees the freedom game to be non-trivial because if the empty set is winning then every set would be winning because every subset includes the empty set, which would imply that both ϕi and ¬ϕi would be the outcome. Note also that the non-emptiness of W implies that the specific freedom formally exists because it means that at least one act-token, given by a coalition, is possible.
5.3 The conditional probability of success
Once we have identified the set of winning coalitions for a given specific freedom ϕ, it can be said that we have a freedom game form, . We need make no other assumption as regards the decision rule π (in particular, we need not specify which or how many players constitute a winning coalition); it can simply be assumed to be “natural” in the sense that no social law or convention need be contained in it.Footnote 17
With this apparatus at hand, Definitions 4.1 and 4.2 reincarnate as:
Definition 5.1 (Specific freedom) ”i is free to ϕ” if, for some S, ( does not prevent i from performing ϕ, i.e. it is not a blocking coalition).
Definition 5.2 (Specific unfreedom) ”i is unfree to ϕ” if, for some S, i∈ S ∉ W(ϕ) ( prevent i from performing ϕ, i.e. it is a blocking coalition).
Note: (i) In accord with definitions 4.1 and 4.2, implies that no set of agent prevents i from ϕ-ing because is not preventing and neither is because it is powerless; and if then at least one set of agents is preventing because has the power to do so. (ii) The requirement that we restrict our attention to attempted actions in the definition of specific freedom and unfreedom is captured by the conditions and . The two conditions also capture the subjunctive element in Definitions 4.1 and 4.2 because i's necessarily being a member of a winning coalition guarantees that i’s ϕ-ing is preceded by i’s attempting to ϕ. Without these two restrictions we would have the much weaker and respectively and would yield a different measure. (iii) To avoid indeterminacy of the freedom ascription it is assumed that (a) S and form and (b) if then i is free to ϕ, i.e. for each we exclude the possibility that any decisive subset of S will not in fact permit i to ϕ;Footnote 18 if this would be the case then these members by definition belong to (it is also assumed that those members of N not in S are in ). (iv) As discussed above, there is no reference to i‘s power; this would require a further restriction such that but .
Thus to speak of i's freedom to ϕ in a freedom game form is to speak of membership of a powerful coalition. Following the idea that is to be an aggregation of the probabilities of act-tokens ri of an act-type R as in (5.1), then is precisely the probability of such an .Footnote 19
To define in a more precise fashion we need some additional structure and notation because calculating the probability of an requires a probability model for S. This means incorporating a minimal, but necessary, amount of behavioral information. That is, for any coalition S that may arise we may either know, be able to estimate, or be able to make a reasonable a priori judgment as to the probability p(S) that the players in N will choose an element of their strategy set such that S occurs.Footnote 20 In other words, is made up of two components, the 2N elementary events denoted by each and a probability distribution that associates each S with its probability of occurrence p(S). That is, p(S) gives the probability that players in S consent to i performing ϕ (by choosing “yes” from their strategy set Ai) and those in will not (by choosing “no” from their strategy set Ai). (As is usual, for any , and .) Our freedom function Γ(R) is, then, specified by the pair .Footnote 21
With this basic set-up, the natural specification of is given by a conditional probability that describes the expectation that i can perfom ϕ, based on all hypothetical instances of ϕ-ing. For a given W(ϕ) and p:
To summarize, a measure of the expectation that an agent has a specific freedom is none other than a conditional variant of the notion of “success” used in the voting power literature that was independently introduced by Penrose (Reference Penrose1946) and Rae (Reference Rae1969) and more fully discussed by Barry (Reference Barry1980a, Reference Barry1980b).
6. CONCLUDING REMARKS
Although the formalization of a measure of specific freedom as has, except in very special circumstances, limited practical applicability,Footnote 22 it does have a fair amount of conceptual and programmatic value.
For a start, is a partial and tentative answer to an open question in the literature. This is an important question because if we are serious about making comparative judgments of the extent to which a person will be in a position to perform some freedom or have a freedom respected, rather than just saying that a person is free or unfree merely on the grounds that an instance of that freedom (act-token) exists or does not exist, then we need a measure that permits us to distinguish between different grades of freedom.
Next, the fine-grained process of constructing from a conception and syntax of specific freedom has churned up a significant conceptual finding: an individualistic definition of specific freedom taken in a broadly negative sense appears to be logically unsustainable. An agent derives her freedom from membership of collectivities (coalitions); or following Dowding and van Hees’ analysis (2003), coalitions are a necessary component in the construction of freedom and rights. This clearly has conceptual implications that extend well beyond the narrow scope of this paper. One such implication is that my material freedom to perform some action is increased by the creation of institutions that “simulate” the coalitions necessary for me to perform that action; or by giving me additional means that reduce my reliance on others and therefore increase the number of possible winning coalitions of which I am a member. If we have a concern for expanding a person's specific freedom, we must look at the game form (the natural or formal procedural rules) that governs this activity.
Finally, the game theoretic formalization allows us to characterize different conceptions of freedom in a single framework. Consider, for instance, Pettit's (Reference Pettit1997) “republican” conception of freedom that was briefly touched upon above.Footnote 23 As stated, this account of freedom is based on a criterion of “non-domination” or what Pettit (Reference Pettit1996) also calls “antipower.” In precise terms, Pettit would ascribe i the freedom to ϕ if, and only if, i cannot be prevented from ϕ-ing by any set of agents, viz. is immune from any interference. This idea says that a person can only be ascribed a freedom if they are not only free in the actual world where there is no effective resistance (where no set of agents can prevent i from ϕ-ing), but also no such resistance in all nearby possible worlds (List Reference List2004). In the model introduced here, the set of possible worlds is the set of act-tokens which is denoted by the set of possible coalitions. Non-domination merely says that the set of preventing coalitions must be empty, which means that none of the act-tokens can be denied to the agent. It may be trivial, but this is just another way of saying that with respect to ϕ-ing, i must be dictator, i.e. {i} is a winning coalition and any coalition without i is losing. To summarize, we can discriminate accounts of freedom on the domain of coalitions because the key properties of power and robustness will single out which coalitions are considered as providing us with the freedom to do something. The measure we have defined informs us of the likelihood of these coalitions.
