Published in Theory and Decision 32 (1992), 165-202
(Note: This is a manuscript version. Please quote only the published version.)
Raimo Tuomela
ON THE STRUCTURAL ASPECTS OF COLLECTIVE ACTION AND FREE-RIDING
I WHAT IS THE PROBLEM OF COLLECTIVE ACTION?
1. One of the main aims of this paper is to study the
possibilities for free-riding type of behavior in various kinds of
many-person interaction situations. In particular it will be of
interest to see what kinds of game-theoretic structures, defined in
terms of the participants' outcome-preferences, can be involved in
cases of free-riding. I shall also be interested in the related
problem or dilemma of collective action in a somewhat broader sense.
By the dilemma of collective action I mean, generally speaking, the
conflict between individual and collective rationality and the
conflict between corresponding actions, in the sense it has been
discussed in recent literature. Typically (although not invariably)
collective action problems and free-rider problems coexist.
Let me start my discussion by considering what Elster (1985) has to
say about the subject. First, the notion of collective action itself
should be characterized. Elster defines it as follows (p. 137): "By
collective action I mean the choice by all or most individuals of the
course of action that, when chosen by all or most individuals, leads
to the collectively best outcome." While this characterization is
informative in the present context, I think that it is not
appropriate as a general characterization. It may provide a
sufficient condition, but it fails as a necessary condition. One
reason for this is that there may not be a single collectively best
outcome at all. Instead, I suggest we follow common sense and take
collective action simply to be action by a collection or group of
people, where these people (or at least many of them) act with the
aim of achieving a common end or goal (this notion understood very
broadly so as to include e.g. following norms, practices, and
customs). We also require of a situation of collective action that
the participants have several (or at least two) possible courses of
action open to them.
Elster's above definition of collective action goes in terms of the
collectively best outcome or goal. As a general account this seems
too strong (compare with my above definition), but in the context of
the problem of collective action and the free-rider problem we need a
notion like this. I prefer to call Elster's above notion collectively
or socially best collective action. Suppose now that we can make
sense of the notion of individually best action. The problem of
collective action can then be taken in a preliminary way to be a
dilemma or conflict between collectively and individually best
action, where the action required for achieving the collectively best
outcome or goal is different from (and in conflict with) the action
required for achieving the individually best outcome - provided the
notions of collectively and individually best outcome are applicable
to the context in question. Or, as we may also put it, means-end
rational action realizing what is collectively best (or, more
broadly, what is collectively good) is in conflict with means-end
rational action realizing individual best (or good). Given this
general preliminary characterization let me now consider our problem
in some more detail and in part by reference to how the problem has
been analyzed in the literature. Before that I would like to point
out that the problem of collective action is closely related to,
although not equivalent with, the free-rider problem, which will be
the main concern of the present paper. My approach to the free-rider
problem will be presented in Section II. The rest of the paper is
devoted to a discussion of structurally different many-person games,
mainly three-person games, with special reference to the free-rider
problem.
2. Elster (1985), p. 139, following Schelling (1978), gives two
different analyses of the notion of a collective action problem: the
strong sense and the weak sense of the notion. Consider first the
strong sense, which is defined by the following two clauses for a
standard two-choice situation, with cooperation or contribution, C,
in the production of the public good (or whatever job is at stake)
and defection, D, as the choice-alternatives: 1) Each individual
derives greater benefits under conditions of universal cooperation
than he does under conditions of universal noncooperation; 2) each
derives more benefits if he abstains from cooperation, regardless of
what others do. Prisoner's dilemma is of course a typical case of a
situation which should satisfy an analysis of the problem of
collective action. Concentrating on the two person case here, we
recall that in a Prisoner's dilemma (PD) the row player has the
following preference ordering of the outcomes: DC, CC, DD, CD; and
the column player has symmetric preferences. (Note that I shall in
this paper formulate preference orderings in the above fashion,
letting the comma - or in some cases semicolon - represent the
better-than-or-equal-to -relation.) Now (assuming that the players
indeed choose over outcomes rather than e.g. disjunctive bundles of
them) each player gains more from cooperation than from cooperation
whether or not the other player cooperates.
However, as Elster himself notes, this definition is too strong in
many cases. For instance, it excludes pivotal cases of voting - cases
in which an individual is in a decisive position (in which his vote
will decide an issue) and in which it accordingly is rational for him
to vote (see Elster, 1985, p. 139). There are also many other cases
of the collective action problem which fail to satisfy Elster's
strong definition. Thus certain situations which have the structure
of a Chicken game, Assurance game, or the Battle of the Sexes, for
instance, qualify, as also Elster notices and as we shall soon
see.
Elster's weak sense of a collective action problem is defined by
means of two clauses the first of which coincides with clause 1) of
the definition of the strong sense. Clause 2) says that cooperation
is individually unstable and individually inaccessible. Individual
instability means that each individual has an incentive to defect
from a situation of universal cooperation. Individual inaccessibility
means that an individual has no incentive to take the first step away
from a situation of universal noncooperation. Now Elster points out
that there are, however, cases of collective action in which
cooperation is either individually unstable or individually
inaccessible but not both - for example, Chicken and Assurance games
- but which nevertheless present collective action problems (though
less serious ones in the case of Assurance games). An example of an
individually inaccessible but stable case is given by Elster's (1985)
example about erosion occurring in a fictional village - I shall call
this the erosion case of the first kind. It goes as follows. On each
plot being cultivated at a riverbank village the erosion can be
stopped if and only if trees are planted on it and on both adjoining
plots. (Planting qualifies as cooperation or contribution here.) This
gives a game of Assurance.1 An example of an accessible but
individually unstable case is given by Chicken - illustrated by the
second type of erosion example. In it the assumption is made that
erosion will occur if and only if trees are cut down on the
individual plot and both the adjoining plots. (Here non-cutting
represents contribution.)
Also Taylor (1987) discusses the characterization of collective
action dilemmas. According to his preliminary characterization the
defining characteristic of a collective action problem is roughly
that rational egoists are unlikely to succeed in cooperating to
promote their common interests (Taylor, 1987, p. 3). But, as I shall
argue, the egoism-altruism distinction as well as other similar
psychological characteristics are actually irrelevant to the problem
of collective action. In his discussion Taylor correctly points out
that the category of collective action problems includes many (but
not all) public good problems, where we take a good or service to be
a public good (if society-wide) or collective good (if relative to a
specific collective, not necessarily a society) if it is in some
degree indivisible and nonexcludable in the collective in question.
Indivisibility (or jointness of supply) relative to a collective
involves that the good, once produced, is available to any member of
the collective in principle. Nonexcludability means the impossibility
(or at least prohibitive costliness) of preventing individual
consumption of the good, once produced. (But note that a collective
action problem and a free-rider problem can arise when there is
nonexcludability but not indivisibility - cf. user-sensitive
recreation areas open to everybody.)
Taylor thinks that the following, final definition covers all cases
of collective action problems: A collective action problem exists
where rational individual action can lead to a strictly
Pareto-inferior outcome, i.e., an outcome which is strictly less
preferred by every individual than at least one other outcome (see,
Taylor, 1987, p. 19). This definition, while on the right track,
relies on an unclear notion of rational action. As Taylor points out,
almost anything can be rationalized in this context, and this makes
his analysis problematic.
I shall not here discuss rationalization generally but only some
examples. In the case of the Prisoner's dilemma (PD) obviously the
dominance of D over C (irrespective of what the others are believed
to be doing) serves to rationalize it. Next consider the more
interesting case of Chicken (CG) and let the two choice alternatives
again be C (typically: cooperation) and D (typically: defection).
What is being rationalized in the first place is (conditional)
actions - or strategies - but not individual joint outcomes. For
instance, even defection and the all-defection outcome in CG can be
rationalized. Defection (D) can be rationalized as follows. If a
player believes (with objectively good or bad reasons) that the other
player (or, in the many-person case at least K-1 others - see below)
will do C, then it is rational (in a utility-maximizing sense) for
him to do D. This rationalizes D and derivatively the outcomes DC and
DD. Of course, if DD results, that is not what our agent expected.
But it is still a consequence of a rational action, even if an
unexpected one. Taylor speaks of rationalizing outcomes, and this
must be understood in the just mentioned derivative sense, I think.
(It would of course not be rational in any sense that all the players
intentionally jointly did D and arrived at the mutual defection
outcome. This is not the way to rationalize the mutual defection
outcome.) Note also the following: it is enough for
outcome-rationalization that rational individual action in the case
of some (e.g. a few or, perhaps, a great minority, as the case may
be) individuals leads to a collective outcome different from the
socially optimal one. This possibility arises in the case of
collective action (or collective action situation) where the
participants do not have symmetric payoffs.
Note that Taylor's present account does not speak about egoism (or
altruism). It also had better not to do so, because we can have
collective action problems also in a world of altruists. For
instance, complete altruists could clearly face a Prisoner's dilemma.
Accordingly we should not always take the choice alternatives C and D
to represent cooperation and defection (these notions understood in
their normal, full-blooded sense). For taken in this full sense it
seems that we build in altruism and egoism to some extent, at least
in a behavioral sense. But that is more than we want here.
Accordingly, I want to emphasize here that we can have collective
action dilemmas to which the altruism-egoism dichotomy (as well as
the cooperation-defection interpretation of C and D) is not directly
relevant (see below).
As noted, the basic problem with Taylor's analysis is that, while it
works well in many contexts, there are situations in which it is not
clear what individually rational action amounts to. For instance,
Taylor excludes Assurance games (cf. Section II below) from the class
of collective action problems, because of its several equilibrium
outcomes the Pareto-preferred one is strictly preferred by everyone.
He accordingly claims that individually rational action does not lead
to a Pareto-inferior outcome. But I would like to claim that in some
contexts also other actions in an Assurance game (AG) can be regarded
as rational - possibly depending on the common history and the
resulting beliefs of the participants. It should be kept in mind that
even if we pertain to standard game-theoretic characterizations of
rationality, individual rationality can be characterized in many
different ways: in terms of dominance, the maximin criterion,
maximization of expected utility, and so on. (In Section II I will
present a two-person AG in which defection is the individually
rational alternative in the sense of the maximin criterion.) It is
worth noticing that in some game situations different background
beliefs may affect the subjective probabilities in such a way that
the expected utility criterion may give a different result than
acting according to plain preference or maximin. Indeed, accepting
the expected utility criterion as a possible criterion of rationality
not only takes us out from purely structural considerations (viz.
interactive preference-dependencies) but leads to a more
subjectivistic notion of collective action problems (at least as long
as subjective probabilities are involved). A purely subjective notion
of a collective action problem would make it wholly dependent on what
the players regard as individually and collectively rational in the
situation in question. (I shall here resist the temptation to go far
in the subjective direction.)2
In accordance with what was said above it seems reasonable to
characterize collective action problems or dilemmas as conflicts
between collectively and individually rational action (my phrasing is
geared to player-symmetric situations):
(CAD ) Let S be an interpersonal outcome structure
(viz. a game-theoretic structure) or a token of such a structure.
Then S involves a collective action problem (or
dilemma ) if and only if S involves a conflict between
collectively rational and individually rational action, viz. if and
only if there are actions or strategies C and D such that C is a
collectively rational action (e.g. in the sense of leading to a
Pareto-preferred outcome when chosen by all or most) and D is an
individually rational action (e.g. in the sense of individual
expected utility-maximization) and C and D are incompatible actions
or strategies.
Structures are taken to be types here. Not all tokens of such a type,
e.g. a CG-structure, need involve a collective action dilemma - this
may depend on the participants' beliefs, as seen. Therefore S must be
allowed to be only a token. It is worth noting that in interpersonal
structures in which the participants have symmetric preference
orderings - this is the case to which the phrasing in
(CAD ) alludes to in the first place - D is assumed
to be an individually rational action in the case of all
participants. In other, asymmetric cases it suffices that D be
individually rational in the case of at least some (and preferably
most) participants; D must be better than C at least for those
individuals. (The criterion of Pareto-preference can normally be used
for defining collective rationality. It involves the assumption that
the outcomes be comparable and that the strategies or actions
deterministically lead to the outcomes in question; the strategies
are to applied without knowledge of other players' choices. It is not
necessary for our philosophical purposes to discuss these technical
qualifications.)
When applying (CAD ) to particular cases it may be
necessary occasionally (e.g. in the context of some Assurance games)
to rely in part on the players' conceptions of what is individually
and collectively rational and evaluate the applicability of the
analysans of (CAD ) on that basis. But with that
qualification (and small concession to "subjectivism" and understood
in view of our preceding discussion, (CAD ) will
work as expected at least in the context this paper, as we will
mostly operate with specific numeric examples which do not pose
problems. Notice that we may have conflicts between two individually
rational actions which are rational in different sense, and we may
have rational collective actions which are analogously in conflict
(viz. there is in this case no unique collectively rational action
alternative but two or more collectively rational actions which are
in conflict; cf. simple coordination problems). Such cases are not
(at least directly) at stake when we are discussing collective action
problems.
II FREE-RIDING DEFINED
1. In this section I will present an analysis of the notion of
free-riding related to a collective action X. X can be a joint action
performed together by some members of the collective or it can be an
action performed by a collective. (An action performed by a
collective involves a joint action by the members or representatives
of that collective bringing about that action of the collective; see
Tuomela, 1989b, and 1991b. Even if free-riding is perhaps most
naturally discussed in the context of actions assumed to be
performable together, we need not assume quite that much here. What
we need is that the action X is in the present context assumed to
produce a nonexcludable public good in joint supply when at
least K agents (out of N) participate. We shall also need to assume
that there some sort of obligation for the agents to participate in
the production of the good. (If X is a joint action that we
jointly plan or "we-intend" to perform together the required kind of
obligation, indeed a joint obligation, will exist; cf. Tuomela,
1991a, 1991b, Chapter 3.3) When the public good has been produced, we
say that X has been performed successfully. We take the action
alternatives to be C (doing one's part of X) and D (defecting from
doing one's part of X).
There is an extensive literature on free-riding. I shall not survey
it here (see e.g. Hardin, 1982). My aims will be twofold. First, I
will present an analysis of what is basically involved in the
free-riding problem. Secondly, I will discuss several kinds of
many-person situations (actually three-person ones) in which
free-riding can take place, and I shall characterize these situations
in terms of the various kinds of interpersonal or social control
involved (cf. Tuomela, 1985). This section will be devoted to a
characterization of the key features of free-riding, and my account
will emphasize the role of the potential free-rider's intention to
free-ride (and I shall below draw from Tuomela and Miller, 1991).
Before presenting my analysis of free-riding it is appropriate to say
a few words about the production function involved in the provision
of the public good in question. Our analysis is meant to be
compatible with any kind of production function - as long as it
allows the good to be produced for at least some argument values. The
production function will be of the general form x=f(y,z,...) where x
represents the public good in question and y the number of
cooperative action tokens. The other arguments (z,...) can deal with
the qualitative features of the situation, taking care that the right
kind of variety of actions is present, if the production of the
public good is based on a division of tasks and labor. For
simplicity's sake, we shall below suppress these other variables and
deal with a simple function of the kind x=f(y), assuming the variable
y to be of the right kind so that it measures the number of actions,
of the right kind, needed for the production of any amount of the
good x in the first place and, in the second place, for the
production of an increment in the variable x. The only assumption
about the form we shall really need in this paper is that there is a
number, say K, which is an allowed value of the variable y such that
f(K)=x(0), where x(0) represents a token of the minimally existing
quantity of the public good X. f may be a monotonously growing
(discrete or continuous) function of the number of cooperative action
tokens or it may be S-shaped so that f(K+N)>f(K) for some N but
yet f(K+N+P)<f(K+N), for some positive number P; and so on; the
possibilities are many and various, as has been documented in the
literature (cf. Hardin, 1982, Hampton, 1987).
Fortunately, we do not here have to rely much on any particular
assumptions about the form of the production function. It is possible
to relax even the idealized assumption that K participants are needed
for the production of a minimal amount of public good. We can do with
the vaguer condition that there will be sufficiently many
contributors to provide at least some amount of the public good.
Let me now propose an analysis of an agent's intention to free-ride.
This analysis will of course also clarify the notion of free-riding,
for a person's intention to free-ride will under favorable conditions
(including that his relevant beliefs mentioned in the analysans, are
true and that he does not change his mind) lead to his free-riding
intentionally. Intentional free-riding may also come about due to a
person's conditional intention to free-ride, provided he
appropriately deconditionalizes the intention. (I shall also comment
on such conditional intentions below.) Arguably, these are the only
ways intentional free-riding can come about (and in saying this I
include also so-called action-intentions or "endeavorings" among the
intentions just mentioned). Following Tuomela and Miller (1991), my
analysis of the notion of an intention to free-ride in the case of
full-fledged and adequately informed members of G can be rendered as
follows (I phrase it to fit a good X which exists to a maximal degree
once it has been produced):
(FR ) A member A of a collective G intends to
free-ride relative to a public good produced by a joint action X
if and only if
1) A intends to defect (viz. not to contribute or do his part of
X).
2) A has a belief to the effect that the joint action opportunities
for the performance of X will obtain, especially that at least K
members (or a sufficient number of members required for the provision
of the public good produced by the performance of X) contribute (or
do their parts).
3) A believes that he ought to participate in the production of X and
that there is (or will be) a mutual belief among the full-fledged and
adequately informed members of G to the effect that the joint action
opportunities for the performance of X will obtain, and to the effect
that each full-fledged and adequately informed member ought to
contribute.
4) A believes that he will gain more from defection (doing D) than
from contribution (doing C) if at least K agents contribute, viz. if
at least K agents out of N do C, K being the minimal number of agents
capable of jointly performing X.
5) A believes that the outcome resulting from all the agents
contributing (doing C), is better than the outcome when all defect
(do D).
6) A believes that his defection involves a cost (possibly nil) to
the contributing members of G.
Of our clauses 1) gives the basic intention in question here. An
intention to defect is an intention to omit doing one's share in a
situation where the provision of a public (or, as I shall
synonymously say, collective) good is at stake. It is, however, a
weaker intention than the intention to free-ride. Clauses 2) and 3)
are conceptual presuppositions of an intention to defect, while 4),
5), and 6) are conceptual presuppositions of an intention to
free-ride but not of an intention to defect. As I have in part
stipulated the difference between an intention to defect and an
intention to free-ride to be this, the previous claim does not seem
to need further defense. That something like clauses 2) and 3) are
indeed needed as presuppositions should be clear from the idea that
we are concerned with defection in the case of a situation of the
provision of a public good. Clause 3) requires that A believes that
he is expected to, or that he should, contribute to X. This is an
obvious idea based on a group obligation stemming from his group's,
G's, commitment (intention) to produce the public good related to X.
Clause 3) also says that he believes that there is at least some
degree of consensus among the members of G that the relevant members
of G ought to contribute. (Note that the intentions to defect and the
intention to free-ride must be distinguished from the intention to
choose the alternative D. The last mentioned intention does not have
to satisfy any of the clauses 2)-6).)
4) states a central idea involved in free-riding: Once the
nonexcludable good has been provided a free-rider can enjoy it
without participating in its production costs. (Recall that we are
speaking of gains as net gains, viz. gross gains less costs,
in this paper.) As the first clause formulates a quite obvious
requirement, widely accepted in the literature on the topic, it does
not seem to need extra defense here. Let me note that when I later in
the paper speak of a free-rider effect I mean just the
condition here formulated by 4).
Clause 5) formulates the idea that - to put it somewhat misleadingly
- joint action is preferable to acting alone, and serves to define a
sense of collective rationality. This clause can be explicated in
different ways. The standard way, to be accepted in this paper at
least for typical cases, is the distributive sense called Pareto
optimality. Considering a two-choice situation with C and D as the
alternatives, Pareto optimality means that the outcome resulting from
everyone's doing C is preferred by everyone to the outcome resulting
from everyone's doing D. But I would like to emphasize that this
formulation is somewhat idealized. This is because the production
function might be so shaped that for some value of K it might be the
case that f(K) > f(K+R), where R is any positive number. In this
kind of case of "crowding" we should replace 5) by
5') A believes that the outcome resulting from all those agents'
contribution (doing C) who are ought (or are normatively expected) to
contribute in G is better than the outcome when they all defect (do
D).
The notion of someone's being normatively expected to contribute will
depend on the believed shape of the production function (and will
typically be connected to the accepted division of labor in G). In
spite of 5') being more realistic than 5) I will below continue to
use the latter in our discussion, to keep matters simple.
Clause 6) introduces the idea that free-riding may be costly to
others. It need not be so, but on the other hand it can even be so
costly as to bring some of the cooperators below the universal
defection line (cf. Pettit's (1986) distinction between free-riders
and foul dealers and see also Tuomela (1988)). We can in fact speak
of free-riders in three senses. First there are the free-riders in
the literal sense. They are ones who do not (at least ideally) impose
any costs to the contributing members of G. Secondly we have
free-riders who do impose some cost on them; but the cost is
"tolerable" in the sense of leaving some gain to the contributors as
compared with the case where no good or no amount of good is
produced. This second kind of free-riders might be called parasites.
And thirdly we have foul-dealers.
Note that (FR ) involves only a subjective notion of
the free-riding situation in that clauses 2)-6) only concern the
agent's beliefs, which are not required to be true. But if they are
true then (FR ) of course serves to characterize the
intention to free-ride in an objective social free-riding situation,
and an action satisfying this intention accordingly will be a
free-riding action.4
We may also discuss conditional intentions to free-ride. In such
cases the intention in the first clause is some contingent condition,
e.g. that at least K other members contribute. Let us now consider
the mentioned situation: the condition that at least K other members
contribute may be i) sufficient, ii) necessary and sufficient, iii)
necessary, or iv) it, together with some other conditions - left
implicit in clause 4) - is necessary and sufficient. Consider i), the
case where A has the intention to defect if at least K others
contribute. This allows for the possibility that A contributes if
fewer than K others participate (and in particular in the critical
case where K-1 others contribute), and A may then also contribute on
the basis of his we-intention to do X. (At least it is rational for A
to form the we-intention to cooperate, for in this case cooperation
simply is more valuable to him than defection.) Case i) represents a
typical case of free-riding. With fewer than K other participants,
the agent may often decide on the basis of the costs of participation
and the expected gains of participation whether he will participate,
and the present possibility leaves open the results of such
deliberation.
The case ii) where A intends to defect if and only if at least K
other members participate is a strong one. It requires A to
participate when fewer than K others participate. If intentional
participation in a collective or joint action indeed requires the
we-intention to perform that joint action, as we have required, we
have here the somewhat special combination of we-intention to
cooperate (when fewer than K other persons participate) and defect
(when at least K other persons participate). In the third case iii)
no commitment to defection follows from the fulfillment of the
condition, viz. that A defect only if at least K others contribute.
This case involves that A is committed to participation if fewer than
K other persons participate and he is needed (this is the case if K-1
others participate - and in our present setup only then). If at least
K others participate he can decide whether to cooperate or to defect;
in the case of a rational agent that clearly depends on the
production function of the public good in question and of course on
which particular values of its argument the agent will at that moment
have to consider. In any case, this kind of case seems typical in
real life. Also case iv) seems rather typical - in this case A still
requires that some further conditions hold before forming the
intention to defect.
Do the above cases i)-iv) all qualify for representing free-riding as
we normally understand this notion? The answer is positive in the
case of i) and ii), for in their case the agent satisfying our
analysans of (FR ) will indeed under normal
circumstances intentionally free-ride (precisely with the
free-rider's reason of gaining something due to what the others do)
if the collective good is provided by the others. But also iii) and
iv) seem acceptable, keeping in mind that we are dealing with a
disposition to free-ride. In fact iv) is not problematic here at all.
Case iii) is problematic and indeed unacceptable only if A believes
that there cannot be any further conditions, which, when added to the
present one would make him free-ride. For then we presumably would
not say that A is disposed to free-ride - and it is just this
disposition we are trying to analyze here.
2. Free-riding situations can be illustrated in part in terms of
game-theoretic structures. The "classical" account of free-riding, to
my knowledge originating from Olson (1965) and Hardin (1971),
connects free-riding exclusively to Prisoner's dilemmas. But, as
various authors (Taylor and Ward, 1982, Taylor, 1987, Hampton, 1987,
and Tuomela, 1988) have shown, there are actually several
game-theoretic structures that are relevant for studies of
free-riding. In this paper I shall continue this line of thought and
argue that depending on the situation at hand, elements of conflict
(with exchange as its special case) and coordination will both be
present to varying degrees. I shall start by discussing some simple
game-theoretic situations in which free-riding can be exemplified. In
this section my numeric examples will concern only two-person
two-choice situations. Later in the paper I will take up numeric
examples also of three-person games as well as consider some
additional questions related to many-person situations.
The aim of this section is to show that conditional intentions can be
connected to various games in the context of free-riding. The game of
Chicken (CG) is of special interest here, and I shall consider a
numeric example of two-person Chicken. Many-person games are trickier
to analyze and I shall discuss them later in the paper. I shall here
start with the single-shot case and only later comment on the dynamic
case. For the time being the reader can think of a single-shot game
as representing the structure of a situation related to a particular
argument value of the production function f.
Let us now consider the following numeric example of Chicken, where
the ranking of the outcomes is given for our reference point player 1
(the row player):
|
|
C |
D |
|
|
C |
3, 3 |
2, 4 |
|
|
D |
4, 2 |
1, 1 |
|
|
|
|
|
(CG): DC, CC, CD, DD |
|
|
C |
D |
|
|
C |
3, 3 |
1, 4 |
|
|
D |
4, 1 |
2, 2 |
|
|
|
|
|
(PD): DC, CC, DD, CD |
Clause 1) of (FR ) can here be taken to be
satisfied in the case of rational players in the case of one-shot
games, for defection is 1's sole equilibrium strategy, and thus it is
rational for player 1 to form not only the conditional intention to
defect but in this case even the absolute intention to defect. The
presuppositional clauses obviously can be satisfied in PD. Conditions
4) and 5) of (FR ) are clearly satisfied (taking
K=2). Also clause 6) is obviously satisfied, as player 2 will lose 2
units by player 1's (viz. row-player's) defection. PD obviously then
can represent a case where a person intends to free-ride, and it
clearly involves a collective action dilemma. It worth noting,
however, that in the case of PD and CG one cannot - except in their
limiting cases giving the second player the same payoff in the
outcomes CC and DC - be the kind of free-rider that imposes no costs
on the other.
Next follow Assurance game (AG) and Imitation game (IG), which also
have some relevance to the issues discussed:
|
|
C |
D |
|
|
C |
4, 4 |
1, 3 |
|
|
D |
3, 1 |
2, 2 |
|
|
|
|
|
(AG): CC, DC, DD, CD |
|
|
C |
D |
|
|
C |
4, 4 |
1, 2 |
|
|
D |
2, 1 |
3, 3 |
|
|
|
|
|
(IG): CC, DD, DC, CD |
In AG D is a maximin strategy. Furthermore, D can maximize a
player's expected utility. Let player 1's subjective probability for
2's choice of C be relatively low, for instance 0.3. Then the
expected utility of C for 1 will be 1.9 and that of D 2.3. Thus D is
a rational choice also on this ground. (Changing the utility of DC
for 1 from 3 to 4 and that of CD for 2 similarly would make this new
AG a borderline game and give it a zero rather than negative
free-rider effect in the defined sense. Then, if 'more' in clause 4)
of (FR ) is taken to allow for zero increase,
(FR ) would be satisfied.) So we see that while the
above two-person AG does not strictly speaking involve the free-rider
problem, it still involves a collective action dilemma. The same
general conclusion holds for IG. D is a maximin strategy and under
some conditions it analogously with AG has D has a strategy
maximizing expected utility. (As we shall see, in the many-person
case both AG and IG can involve a free-rider effect, depending on how
exactly the generalization to the many-person situation is made.)
Consider still the following two-person Threat Game (TG):
|
|
C |
D |
|
|
C |
3, 3 |
2, 3 |
|
|
D |
2, 1 |
2, 2 |
|
|
|
|
|
(TG): CC, DC, CD, DD (player 1) |
|
|
|
|
CD, CC, DD, DC (player 2) |
In this asymmetric game there is no free-rider effect, but the
game still poses a collective action dilemma (recall my comments
related to (CAD )). The last mentioned claim is true
because in the case of player 2 D is the dominant strategy. Thus
rational utility maximizing action on the part of both players leads
to the CD outcome rather than the Pareto optimal cooperative outcome
CC. In our example CC is collectively somewhat better than CD. Thus
even if we changed our criterion (CAD ) to allow
that different strategies be allowed for individually rational or
best ones in the case of differen agents, still the cooperative
outcome would be better in some sense. (Note that if we made such a
change, then we would have to account for the fact that both CC and
CD are Pareto optimal - thus CC would have to be regarded as
collectively better in another sense.)
Next we shall consider coordination games (or games with a high
coordination component). Here we must interpret the situation
differently. Above we have considered conjunctive collective
actions. In them each participant or (or at least K participants)
must contribute before the collective action comes about. Now we must
interpret the collective action disjunctively and give up
the interpretation of C and D as contribution and defection,
respectively. To see what this amounts to consider the following
simple Turn-taking game:
|
|
C |
D |
|
|
C |
0, 0 |
0, 2 |
|
|
D |
2, 0 |
0, 0 |
|
|
|
|
|
(TT): DC, CD, CC, DD (player 1) |
|
|
|
|
CD, DC, DD, DD (player 2) |
Here the outcomes DC and CD can be taken to make the collective
action come about. We can say that the result of the collective
action consists of the disjunction DCvCD. As each player only can
make one choice at a time this disjunction will be an exclusive one.
The turn-taking aspect then comes into picture if both players want
to have some gain from their action, if e.g. an even distribution is
wanted. Then the players obviously have to alternate between DC and
CD and in this sense coordinate their actions. Note that we do have a
collective action dilemma in the defined sense here, for D is the
dominant strategy in the case of both players, and using it would
result in the outcome DD and thus to no collective action in the
meant sense. In the two-person case there is no free-rider effect,
but in the many-person case even that becomes a possibility if only a
subset of the players is needed for the collective action to come
about.
Finally consider the following Battle of the Sexes game, which is
primarily a coordination type of game (preference-ranking given for
player 1):
|
|
C |
D |
|
|
C |
2, 2 |
3, 4 |
|
|
D |
4, 3 |
1, 1 |
|
|
|
|
|
(BS): DC, CD, CC, DD |
We may interpret this game in two ways. First, the above
disjunctive interpretation can be used by defining DCvCD to be the
disjunctive result of the collective action in question. As in the
previous game CD and DC are both equilibrium points, and there is no
other way to come to a solution than by some kind of agreement to
select one of them (and stay with it) or then to alternate. Note that
there is a collective action dilemma here. If, for instance, player 1
believes that 2 will (at least with a high probability) choose C he
will choose D, and vice versa, and DD will result.
(There is also another relevant interpretation of BS available here.
Suppose only one of the players can represent (or volunteer for) the
dyad. How is the representative to be chosen? The players must
obviously coordinate either on CD or DC, if D means representing (or,
more broadly, actively contributing) and C not doing it. We shall
later comment on this aspect of collective action dilemmas when
discussing many-person games in detail.)
As discussed in Tuomela (1984), pp. 141-144, there are many other
kinds of joint and collective actions besides conjunctive and
disjunctive ones. Serial collective actions, where the
participants must do their shares in a certain order for the
collective action to come about, represent one such type. They seem
to require the kind of multi-play treatment (e.g. in terms of an
iterated Prisoner's dilemma game) to be commented on below.
Let us still return to our (FR ). We may relax it by
liberalizing the first condition so that not every participant but
perhaps instead most of them satisfy its requirement 1). Then various
nonsymmetric games also qualify for characterizing the structure of
the free-riding situation. One such game is the following: Agent A1
plays Chicken with the utilities mentioned in our earlier Chicken
game, and A2 has the following preference ranking: CC, DC, DD,
CD.
To conclude, let me finally comment briefly on the dynamic case. We
have been assuming up to now that each participant acts only once,
viz. produces one appropriate action token reflected in the value of
the argument y in the production function; or at least we have
assumed that he behaves as if he acted only once (even if in fact he
may have produced several action tokens at different times).
Depending on what the production function is like at a given argument
value, it may be represented by different games for different such
values (see Hampton, 1987, for a discussion). One typical dynamic
situation is to have a long succession of Prisoner's dilemmas - all
values of y may even correspond to such cases. Or we may have Chicken
games for some values of y and Prisoner's dilemmas for other values
and no dilemmas for some values (e.g. when there are exactly K-1
other contributors in the case of a particular value of y, viz. when
the value of y is K-1). It all depends on the production function in
question what kind of situations must be faced.
The best hope for finding rational cooperative solutions to social
action dilemmas of course relates to the case of iterated games. For
then various psychological "continuity" factors, such as reputation
and trust, can be argued to affect the strategic situation. Even
without such factors solutions have been found - see e.g. Taylor
(1987) (also cf. Axelrod, 1984). For instance, Taylor (1987) shows
that under certain specific (and, indeed, quite strict) conditions it
is rational (in the utility maximization sense) for agents (even
egoistic agents) constantly to cooperate. (In fact, he speaks for the
importance of conditional cooperation strategies, which indeed seem
closely related to conditional we-intentions.) Tit-for-tat is one
such strategy, which under certains conditions will result in an
equilibrium, indeed a coordination equilibrium (one in which no one
benefits from unilaterally changing strategy). But in real life the
conditions required may be hard to satisfy: the solutions are in many
ways problematic. In fact here the situation is often plagued by the
problem that there are too many cooperative or partly cooperative
equilibrium-solutions and the problem that too much problem-solving
capacity and rationality is required from the agents.
While I shall not in this paper seriously discuss how to solve the
free-rider problem(s) or the collective action dilemma, let me here
point out that the feedback based interaction possibilities between
the agents should be utilized whenever feasible. At least in small
groups this kind of interaction seems central. Nevertheless studies
such as Taylor's otherwise first-rate investigations fail to do that.
The players are treated by him as independent except that the
discount parameter defined over the trials reflects some of the
agents' expectations about each other's behavior (but that is not
feedback-based interaction.) A proper discussion of the dynamic case
will be left for another context.
III COORDINATION VERSUS CONFLICT IN SELECTION SITUATIONS
When a collective performs an action, say X, this involves that there
are some operative members of the collective who jointly perform
something which brings about (the result event of) X (see Tuomela,
1989b). But how are the operative members to be selected in a
rational way? And how to allocate shares or parts to them? How to get
them to perform their parts? These are three interesting problems
worth investigating.
In the case of formal collectives such as organizations there are
typically rules which make it clear who can act for the collective.
But in the case of many informal groups, at least, it may be a
problem who can and will act on behalf of the collective. To be sure,
many collectives (e.g. crowds) act in virtue of all their members
suitably acting, but here we will not be interested in those cases.
Rather we will be concerned with cases like this: 1) Our informal
running team will receive a trophy after having won a race. But only
one of us is allowed to represent the team. Everyone of us would like
to be the person to receive the trophy on behalf of the team. Who
will be the one to do it? 2) Our army patrol needs one or two men to
check the wires when crossing the enemy line. We all dislike the job.
How shall we choose the man to do the job, supposing that none of us
has the power to order anyone for the task?
Another example (related to case 2)) is Hume's famous meadow draining
case:
"Two neighbours may agree to drain a meadow, which they possess in
common; because 'tis easy for them to know each other's mind; and
each must perceive, that the immediate consequence of his failing in
his part, is, the abandoning of the whole project. But tis' very
difficult, and indeed impossible, that a thousand persons shou'd
agree in any such action; it being difficult for them to concert so
complicated a design, and still more difficult for them to execute
it; while each seeks a pretext to free himself of the trouble and
expense, and wou'd lay the whole burden on others." (Hume, 1965, p.
538)
Here the meadow has got to be drained, and at least two of us, let us
suppose, are needed for the job. Who will do it? Let us now discuss
this problem (and related ones) involving both a coordination and a
conflict (or exchange) aspect. In the case of wanted jobs the members
of the collective face a coordination problem. Often this problem
seems to be of the Battle of the Sexes (BS) type, occasionally it may
have the structure of an Imitation game (IG), an Assurance game (AG),
or a Turn-taking game (TT) - and this list could be extended to cover
other games involving a coordination problem. But in the case of
unwanted jobs there is conflict involved, and the game of Chicken is
often appropriate for representing the structure of the situation.
Also PD seems possible in the case of unwanted jobs. In the case of
CG our agents regard it as crucial that the job will be done, while
in the case of PD that is less important than to avoid being a sucker
(the sole contributor - or, more generally, one of few contributors).
(Conflict here involves that the players want the others to act
against their - the latters' - preferences.) Considering first, for
simplicity, the two-person case, let me present the preference
orderings for the Battle of the Sexes (BS), Imitation game (IG),
Assurance game (AG), Chicken (CG) as well as the Prisoner's Dilemma
(PD) - in the case of player 1 (the games are player-symmetric):
IG: CC, DD, DC, CD
AG: CC, DC, DD, CD
BS: DC, CD, CC, DD
TT: DC, CD, CC, DD
CG: DC, CC, CD, DD
PD: DC, CC, DD, CD
There is a central difference between Chicken, which is a game of
conflict, and BS, which is a game of coordination with some conflict
of interest. Chicken games have only Nash equilibria, while BS games
have coordination equilibria.5 IG seems to be pertinent when the
players consider doing something together to be important, while what
is being done together is not so important. AG can come about when an
agent agrees to participate as long as the other one will - but
dislikes doing the job alone. TT is often at stake in the case of
disjunctive collective tasks.
Consider now the meadow draining situation (and analogous cases). In
the two-person case we get the result that we are dealing with CG,
supposing - at least seemingly contra Hume - that even one player
alone can drain the meadow. Draining the meadow is a cost for someone
and thus CC represents a cheaper alternative for player 1 than CD.
Consider next the general many-person case, illustrated, for
simplicity, on the basis of the three-person case. Suppose thus that
there are three participants and that, for a change, at least two
persons are needed for the job. Here we get for player 1:DCC, CCC,
CCD, CDC, DCD, DDC, CDD, DDD
(Strictly speaking, we only have DCC, CCC, CCD=CDC, ...., DDD,
allowing for the other triplets to come in any order.) This is a CG
(and so is the case where one alone is taken to be able to drain the
meadow, as the reader can easily check). However, note that if I
count my being the only one to contribute as a very costly thing
(assuming that two persons are needed for the job), DDD goes before
CDD and we get not quite a Prisoner's Dilemma but what has been
called a PD-structured situation (Ullmann-Margalit, 1977, p. 23).
(This situation is not a full-blown PD because each player still
prefers contributing to not doing so if exactly one of the other two
participants contributes.)6
The selection of operative members accordingly is sometimes a CG (or
even PD-resembling) problem and sometimes a coordination problem such
as BS. When specialization and differentiation of tasks are centrally
involved we typically have a coordination type of problem, e.g. BS or
perhaps turn-taking. Consider playing a violin sonata with piano
accompaniment (in the case of a specific violinist and an
accompanying pianist). This can be treated as a BS problem. But if
there are several violinists and pianists we must first choose one of
each, and this often (to be precise, when the job is wanted) involves
another coordination type of problem.
An interesting feature about selection situations is that there is a
different kind of problem when the job is wanted as compared with a
case where it is hated. Let us consider a case where the job is
wanted and where there is a surplus of candidates for it. We consider
the problem of selecting one representative (or more representatives
if you like, but here we shall for simplicity only consider the case
of selecting one). We let there be three players who want to elect a
king or leader or simply a person to represent the collective (or
whatever similar task you can think of). Supposing that everyone
wants the job, we face a coordination problem which is of the BS
type. For the preference ordering can be taken to be (in the case of
player 1): CDD, DCD, DDC, ... (the other alternatives are all bad).
This game is a BS game, but let me emphasize that C now is
interpreted as the "good" action relative to the other players' doing
D. In this sense we have now reversed the roles of C and D here, but
this does not affect the basic nature of the game. The three outcomes
mentioned above will be equilibrium points (in the sense that it pays
for no one to switch to another alternative given that the other two
stick to their present ones).
Let us now change the situation so that if the players agree on a
specific triplet, they may form a triumvirate, viz. select CCC. Now
CCC will be fourth in their preference ranking and it will also be an
equilibrium (for a change by one player would give a triplet with two
C's which is worse). Put more exactly we now have the following
rankings for the players:
1: CDD, DCD, DDC, CCC, DCC, CCD, CDC, DDD
2: DCD, CDD, DDC, CCC, CDC, CCD, DCC, DDD
3: DDC, CDD, DCD, CCC, CCD, CDC, DCC, DDD
(The ranking obviously goes from left to right, the leftmost outcome being the most preferred.) Of course, the order of the fifth, sixth, and seventh triplet in these rankings could be different, too. To get a numeric example, we assign the following numbers to our triplets: CDD=<4,3,3>, DCD=<3,4,3>, DDC=<3,3,4,>, CCC=<2,2,2>, DCC=<1,1,1>, CCD=<1,1,1>, CDC=<1,1,1>, DDD=<1,1,1>. A still more realistic possibility would be to think that the triumvirate-alternative CCC is the second best so that e.g. player 1's ranking becomes: CDD (4,2,2), CCC (3,3,3), DCD (2,4,2), DDC (2,2,4), DCC (1,1,1), CCD (1,1,1), CDC (1,1,1), DDD (1,1,1).
However, if nobody wants to be the king (or the representative or
whatever operative member) the players face a CG or a PD or something
like that. For then we have the same situation as with meadow
draining. Accordingly, we have found something informative to say
about the selection of representatives or - more generally -
operative members both when the job is wanted and when it is not
wanted.
As I have above discussed the selection problem in the rather general
case of three-person games, it is easy to do it for the general
N-person case (N>3). I shall not bore the reader with that nor
with the equally obvious generalization of the number of operative
members exceed one agent. But let me here emphasize one feature of
N-person games. In case of step goods which require at least K
persons to participate there will be a selection problem involved for
an agent when N is larger than K. If N=K, our reference point agent
believes that he must participate, given that the K-1 others
participate. Or more generally, whenever our agent thinks that he is
needed to produce the good, there will be no selection problem (or
surplus problem).
Let me conclude this section by a couple of general remarks. We have
discussed the selection of producers of a good and noted that it may
involve not only a coordination problem but conflict (e.g. in the
sense of CG), too. This is contrary to what Hampton (1987) at least
seems to think. She conceptualizes the problem of the provision of a
public good as follows: first there is the coordination problem of
selecting the producers of the good, and after that there is the
problem of how to get the selected producers to act - and here CG may
be appropriate. But why should we analyze it so? In view of what was
said above, it seems that we should be prepared to face all kinds of
conflict/coordination combinations in selecting the producers; and
sometimes the selection of the producers may motivationally involve
or determine that they indeed will produce, so that we only need to
solve one game. On the other hand, as said in the beginning of this
section, we may also have the three problems of 1) selecting the
operative members (using our earlier terminology), 2) allocating
tasks to them and 3) getting them to perform. In solving problems 1)
and 2) we may deal with situation structures exemplified by e.g. BS,
AG, CG, PD - or there may be a simple unproblematic situation
involving no problem of strategic interaction at all. (Hampton, 1987,
only speaks of BS, AG, and CG but not PD in this connection.) In the
case of problem 3) more conflict may be involved, but all of the
mentioned games still represent possibilities also in this case. All
kinds of combinations of coordination and conflict seem possible in
principle, and, it seems, we cannot a priori exclude any interesting
types of structures.
IV THREE-PERSON GAMES WITH A COLLECTIVE ACTION PROBLEM
1. Two-person games can be used to give guidelines as to how to
characterize many-person games. Thus, in the case of the Prisoner's
dilemma we go about as follows. As is usual, we accept from the
two-person case that dominance of defection and Pareto-optimality of
the joint cooperative outcome are properties to be preserved (cf.
Pettit, 1986). We thus get in the case of n agents with the two
choice-alternatives C and D the following partial ranking for agent
i: C1,...,Di,...,CN; ... C1,...,Ci,...,CN; ... D1,...,Di,...,DN; ...
D1,...,Ci,...,DN. Altogether we have M=2N choice-alternatives, and
here we fix the order of four of them and make the first one of the
aforementioned alternatives the first also in the total ranking and
the last one of them the last in the total ranking. The rest of the
2N-4 alternatives (for instance, C1,D2,...,Di,...,Cn) go somewhere in
between the first and the last alternative. In the case of Chicken we
similarly can get on the basis of the two-person game the following
partial ranking for i: C1,...,Di,...,CN; ... C1,...,Ci,...CN; ...
D1,...,Ci,...,DN; ... D1,...,Di,...,DN.
The three-person case can of course be taken to represent the general
N-person case in many contexts (except that three persons do not
amount to more than a small group, not e.g. a mass or crowd) and as
the number of rankings grows rapidly with growing N I will here
concentrate on three-person structures. (As far as I know there is no
previous systematic work available on the three-person case - and
note, too, that the technique to be described of course applies to
any finite number N.) I will present a methodology or technique one
may use in the study of three-person structures. I will be less
concerned with the strategic and other features of the resulting
structures. For lack of space, my general remarks on three-person
games concentrate on CG (and PD); but other relevant games (such as
AG and BS) can be discussed quite analogously. The reader may be
reminded here that by their very characterization CG and PD - except
for some borderline cases of these games - involve a free-rider
effect of some kind and a collective action dilemma.
Let me now proceed to a discussion of three-person Chicken. (What I
will say applies by rather direct analogy to PD, too.) We must here
obviously deal with 23 =8 possible choice-combinations here. We know
that DDD ranks lowest and CCC somewhere between the highest and the
second lowest alternatives. For instance, DCC will be preferred by
player 1 to CCC; and he will also prefer CCC to CDD, and CDD to DDD.
But there are a great many total rankings compatible with this. In
fact, given certain plausible alternative assumptions about the
provision of public goods, there are 20 possible CG-structures in the
context of the provision of public goods. (Altogether, if we - as is
natural - glue CDD and DDD together and let nothing go in between
there will be 86 possible CG-structures satisfying merely the
constraints on many-person Chicken games stated above.) Let me now
present the details.
We must assume in all Chicken games that two persons be able to
provide the good in question - or at least to provide a substantial
amount of it which makes the third person's free-riding possible (so
that it pays for him to defect rather than cooperate). I shall divide
the Chicken games into three categories depending in part on how much
1 person and how much 2 persons can contribute toward the provision
of the good. To characterize the first class, suppose first that one
contributor can provide enough of the good for it to become
free-ridable. In this category of Chicken games the first person
ranks highest the alternatives DDC, DCD, and DCC, in some order.
These give us 3! (=6) permutations. Thus above CCC we have those
alternatives in which the first person defects and one or two of the
other players contribute. After CCC we have those combinations in
which the first person's cooperation is combined with one other
person's contribution, viz. the alternatives CDC and CCD, or with the
other two persons' defection, viz. CDD. We get 3! permutations, and
altogether there are then 36 Chicken games of this kind. Let us in
addition make the natural assumption that CDC and CCD give more of
the good than what the first person alone can provide. We should then
put these two combinations above CDD. Given this, we have in our
first category 12 Chicken games satisfying the assumptions made.
In the second category we have the possibilities in which, again, one
contributor can provide a sufficient, free-ridable amount of the
good. In addition we assume in this class of games that a person does
not much mind if one other person defects but does mind being himself
the sole contributor. Here we put below CCC only CDD and DDD. Thus
DDC, DCD, DCC, CDC, and CCD go above CCC in the ranking. There are 24
possibilities here. Of them 4 are structures in which the first
person values free-riding more than contribution, and we will include
only those.
Finally we have as our third category the class of cases where two
persons' contribution is required for a free-ridable amount of the
good. Therefore only DCC goes above CCC. Furthermore, we assume that
two persons will provide more of the good than one person will (so
that CCD and CDC rank above DDC and DCD); and CDD comes last except
for DDD. There are four structures of this kind. Thus, we have in our
three categories altogether 20 structures of CG which are compatible
with the assumptions made. One can of course make many other kinds of
typologies as well, depending on what assumes of the situation. I
shall not here go into a deeper discussion.
Three-person PD can be handled completely analogously. The only
essential difference of course is that DDD always comes after CCC so
that we only have to consider the combinations which go above CCC and
those which go below DDD. In partial contrast to CG we may want to
consider cases in which some alternatives go in between DDD and CDD
(the lowest ranking alternative). Over and above the possibilities we
discussed in the case of CG we should then include these additional
preference structures.
2. I shall below consider some numerical examples of three-person
Imitation game (IG), Assurance game (AG), Chicken game (CG), Battle
of the Sexes (BS), and the Prisoner's dilemma (PD). (These examples
are from Tuomela, 1989a, from a somewhat different context.) No kind
of exhaustiveness for my selections will be assumed or claimed, but
the point is in any case to indicate how these game-theoretic
structures can exemplify collective action dilemmas and
free-riderism.
In the case of IG the preferences in the two-person case with two
alternatives C and D are as follows. Player 1: CC, DD, DC, CD; player
2: CC, DD, CD, DC. In the three-person case we can get the following
ordering (for player 1, and symmetrically for the others) illustrated
by two numeric examples within parentheses: CCC (4,4,4; 4,4,4), DDD
(3,3,3; 3,3,3), DCC (3,1,1; 1,1,1) CDD (1,1,1; 1,1,1), DCD (1,1,1;
1,1,1), DDC (1,1,1; 1,1,1), CDC (1,3,1; 1,1,1), CCD (1,1,3; 1,1,1). I
shall call the first of these examples IG1 and the second IG2. In IG1
it is assumed that the sole defector gains and that there thus is a
free-rider effect. In IG2 there is no free-rider effect. (Actually
IG2 is a very special case of an Imitation game because it is really
a coordination game: the three agents must all try to do the same
thing (C or D), preferably C; otherwise they get only 1 utile.)
In AG the preferences in the two-person case are: CC, DC, DD, CD for
player 1 and symmetrically for player 2, viz. CC, CD, DD, DC. The
three-person case becomes the following, illustrated by two numeric
examples : CCC (5,5,5; 5,5,5), DCC (4,3,3; 3,4,4), CDC (3,4,3;
4,3,4), CCD (3,3,4; 4,4,3), DDC (2,2,0; 2,2,0), DCD (2,0,2; 2,0,2),
DDD (1,1,1; 1,1,1), CDD (0,2,2; 0,2,2). Here we are assuming that two
cooperators can provide the good (although not as fully as three can)
and give a better result for everyone than does one sole cooperator,
who still can provide some of the good (although it yields a negative
net benefit to him). In AG1 there is a free-rider effect, but there
is none in AG2. There is one in AG1 because each person prefers the
outcome in which the other two contribute but he does not to the two
outcomes in which he and exactly one of the other two participants
contribute. However, all three participants prefer the outcome where
they all three contribute to any of the other possible outcomes,
including those in which they do not - but the other two do -
contribute; and, furthermore, each prefers to contribute if exactly
one of the other two contributes. (One might say that free-riding is
not "individually accessible" in AG1.)
Consider now CG. We recall that in the two-person case we get for
both players the ranking: DC, CC, CD, DD (player 1), and CD, CC, DC,
DD in the case of player 2. With specific reference to the meadow
draining example, we can consider, in the three-person-case, the
following for player 1 (and symmetrically for the others), using now
three slightly different numeric illustrations: DCC (4,2,2; 4,2,2;
4,2,2),DCD (3,1,3; 4,1,4; 3,1,3), DDC (3,3,1; 4,4,1; 3,3,1), CCC
(3,3,3; 3,3,3; 2,2,2;), CCD (2,2,4; 2,2,4; 2,2,4), CDC (2,4,2; 2,4,2;
2,4,2), CDD (1,3,3; 1,4,4; 1,3,3), DDD (0,0,0; 0,0,0; 0,0,0).
considering the meadow draining example, in the first illustration I
have assumed that the sole free-rider gets 4 utiles. Doing one third
of the job gives 3 utiles, one half 2 utiles and doing it alone gives
1 utile. In the first numeric case I have assumed that a sole agent
is not able alone to do the job to the same degree or as well as
jointly with one or two other agents, viz. that two agents are needed
for doing the job properly. In the second case, in contrast, one
agent is assumed to be able to do the job fully satisfactorily. I
have also presented as my third case a case of CG. In it the third
participant brings about a slight crowding effect (cf.
CCC=(2,2,2)).
Next consider the Battle of the Sexes. In the two-person case the
ranking for player 1 is: DC, CD, CC, DD and for player 2 CD, DC, CC,
DD. The idea in the three-person case is to think of the selection of
a leader or something like that. I have assigned the numbers in two
different ways to get for player 1 (and symmetrically for the others)
the following: CDD (4,3,3; 4,2,2), DCD (3,4,3; 2,4,2), DDC (3,3,4;
2,2,4), CCC (2,2,2; 3,3,3), DCC (1,1,1; 1,1,1), CCD (1,1,1; 1,1,1),
CDC (1,1,1; 1,1,1), DDD (1,1,1; 1,1,1). My numeric examples, BS1 and
BS2, correspond to the kind of situations discussed in the previous
section in the case of selecting a leader or representative.
We recall that in the case of the two-person Prisoner's dilemma we
have the ranking DC,CC,DD,CD, in the case of player 1 and CD, CC, DD,
DC in the case of player 2. In the three-person case I give as my
illustration the following ranking, with three different numeric
assignments: DCC (4,2,2; 4,2,2; 5,2,2), CCC (3,3,3; 3,3,3; 4,4,4),
DCD (2,1,2; 3,1,3; 3,1,3), DDC (2,2,1; 3,3,1; 3,3,1), CCD (2,2,4;
2,2,4; 3,3,5), CDC (2,4,2; 2,4,2; 3,5,3), DDD (2,2,2; 2,2,2; 2,2,2),
CDD (1,2,2; 1,3,3; 1,3,3).
Next I present an example of a collective action dilemma which is
not, however, a free-rider dilemma. This is a borderline game between
PD and AG, and I will call it A/P (or AP). In the two-person case the
following illustration can be given:
|
|
C |
D |
|
C |
3, 3 |
0, 3 |
|
C |
3, 0 |
2, 2 (A/P) |