Abstract
Most definitions of cooperation provide sufficient but not necessary conditions. This paper describes a form of minimal cooperation, corresponding to mass actions implying many agents, such as demonstrations. It characterizes its intentional, epistemic, strategic, and teleological aspects, mostly obtained from weakening classical concepts. The rationality of minimal cooperation turns out to be part of its definition, whereas it is usually considered as an optional though desirable feature. Game-theoretic concepts thus play an important role in its definition. The paper concludes by answering concrete questions about what should and should not be called cooperation.
1. Introduction: Standard Definitions of Cooperation
When is a set of individual actions a genuine case of cooperation? For decades, philosophers have strived to solve this problem of the right definition of cooperation. A first obvious fact is that the cooperative nature of a set of individual actions is underdetermined by observable behavior. We talk of a collective action as soon as a salient global effect is produced—for instance, in the case of a crowd movement. Almost any juxtaposition of individual actions that has a somewhat noteworthy effect could be called a collective action. However, for an action to be cooperative requires additional ingredients that guarantee a certain link between agents. To cooperate is to act together, in unison, in agreement, which suggests some coherence between the agents’ mental states. In a nutshell, cooperation = collective action + X, where X states properties of and relations between the agents’ mental states.
In other words, although human collective behaviors often look like cases of cooperation, whether they actually are cases of cooperation depends on further properties of a group of agents; these properties in turn depend on human cognitive abilities. Moreover, from basic conversation and carrying of objects to highly sophisticated forms of coordination, human cooperation is widespread and highly successful on a daily basis. Intuitively, such activities have something in common; definitions of cooperation aim to elucidate the nature of these nonobservable links, both for the sake of conceptual inquiry and to account for the ubiquity of cooperation in our lives.
So what kind of mental states does cooperation involve? In one of the first analyses of cooperation, Regan (1980), although he did not intend to define it, suggested the following ideas. In a situation where cooperative and noncooperative actions are predefined, cooperation can fail when an agent chooses the wrong action, when she chooses the right one while thinking it is not cooperative, or when she chooses the right one while mistakenly thinking that the other agent chose the wrong one. So for there to be cooperation, agents must intend to do the right action, for adequate reasons to act and from a correct understanding of the situation. Moreover, they must believe that others have those too, believe that others believe it, and so on. 1 The set of mental states that lead to cooperation is called a collective intention.
For instance, one could say that a collective intention is composed of agents’ individual intentions to do their part and of their being common knowledge. Most existing definitions actually enrich this extremely basic structure by further specifying how individual 2 intentions and beliefs are intertwined. Bratman’s account provides a good example of such definitions:
We intend to J if and only if
(a) I intend that we J and (b) you intend that we J;
(a) I intend that we J in accordance with and because of (1)(a), (1)(b), and meshing subplans of (1)(a) and (1)(b); (b) you intend that we J in accordance with and because of (1)(a), (1)(b), and meshing subplans of (1)(a) and (1)(b);
(1) and (2) are common knowledge between us. (1999, 121)
A definition of cooperation thus typically has the following basic structure: a set of individual intentions that have certain origins and enjoy certain relations, all of which is common knowledge (that is, public or transparent) among agents. What usually varies between definitions concerns the reasons or causes of intentions and the way in which they are intertwined and mixed with other mental attitudes.
Indeed, authors have suggested idiosyncratic definitions of cooperation, due to the absence of strong constraints (other than adequacy with intuitive cases) as to what would count as a correct one. 3 According to Gilbert’s original account (1989), agents first form public conditional intentions, which become unconditional once they are common knowledge. 4 As we just saw, Bratman (1992, 1993, 1999) adopts a dynamic framework and suggests that mutually consistent plans of action are necessary. Miller (2001) highlights the crucial role of collective goals (themselves left undefined), that is, of reasons why agents act. Tuomela (2007) emphasizes the constructive nature of the goal, that is, the fact that it is been collectively built and accepted by agents beforehand. This guarantees that agents act according to the “we-mode,” without which cooperation would degenerate into a kind of joint action with mutual helping. 5 This very brief overview shows that the main possible ingredients of cooperation include intentions, beliefs, goals, and reasons to act. 6
One salient limitation of the usual accounts is that authors identify sufficient conditions for cooperation because their analyses stem from examples of egregious cooperation. 7 Different intuitions thus lead to noticeably different definitions. By contrast, Regan’s analysis, although rarely mentioned in the literature, focuses on necessary conditions for cooperation. Regan asks which cases could lead to the failure of cooperation and, by doing so, enumerates some of its essential ingredients, although without providing a full definition.
Overall, no extant definition of cooperation focuses on its necessary aspects. Indeed, any of the previous analyses could apparently be weakened without making cooperation disappear: as we will see, there could be cooperation without any agreement, detailed plan of action, or collectively built goal. One aim of this paper is to define the minimal cooperation arising in such cases, that is, to pinpoint the conceptual core of cooperation.
The second, related aim of the paper is to show that the rationality of some cooperative actions does matter to their definition. By contrast, usual definitions typically assume that for a cooperative action to be rational is a desirable but optional property at best, by no means a necessary one. 8 Adding rationality constraints does not lead to stronger or fuller forms of cooperation. More generally, the origin of the components of cooperation, whether rational or not, is often deemed irrelevant. I argue that this is debatable at least in the case of minimal cooperation.
Overall, we are faced with two issues: (1) Classical definitions of cooperation are too strong and exclude some legitimate cases; (2) they fail to take into account rational justifications of cooperation. The plan of the paper is as follows. Section 2 introduces cases of minimal cooperation of which no satisfying definition exists, and it suggests that they can be analyzed from preexisting classical collective concepts, namely, collective goals and a weakened form of common knowledge. Section 3 highlights the usually underrated relevance of game theory when defining collective intentions and claims that an adequate game theoretic equilibrium should be used as a guide toward a definition of minimal cooperation. Section 4 then builds the analytic definition itself, emphasizing its rational features. Section 5 discusses several key characteristic of this definition, which allows it to escape several criticisms and which allows me to cash out the conceptual distinction between the theorist’s and the agents’ concept of cooperation. I conclude by answering several concrete questions as to what can and cannot be called cooperation in general.
2. Minimal Cooperation, Mass Actions
This section provides three examples of what I call minimal cooperation: demonstrations, flash mobs, and anonymous experimental games. It aims to show that cases of minimal cooperation are not uncommon and that human beings are usually quite successful in such settings; so a definition of minimal cooperation is needed at least for the same reasons that classical definitions were.
Imagine a group of a hundred individuals who all wish that a demonstration be a success, for which at least 50 participants are needed. Participation happens to be costly; agents cannot communicate beforehand and have no idea who the other participants will be. There is no prior reason to think that enough agents will participate. Now suppose that all agents decide to participate: the demonstration is a success. Is it a case of cooperation? I suggest it is indeed. All agents have the same collective goal, although not everyone’s participation would have been needed to realize it. However, according to classical definitions, this is not cooperation, because the intentions to participate did not stem from the knowledge that others intended to act but from an estimation of their tendency to participate. 9
Is such a case unrealistic? After all, there is no communication between participants, although it seems to be part of most mass actions. However, this is misguided. As a second example, consider a flash mob: individuals gather in a public place and accomplish simultaneously an identical action before breaking up instantly. In flash mobs, most participants do not know each other. 10 One may also decide to participate after reading about a planned flash mob on an Internet website, before knowing any of the possible participants. Most often, a flash mob has no other motive than its result—a salient, noticeable collective effect, which happens to be unexpected, fun, or quirky. Flash mobs are a textbook case of minimal cooperation: 11 individuals cannot base their decision to participate on each other’s individual intentions or identities. They can rely on little more than on the message source (how public is it? what are the chances that a random reader decides to participate?) 12 and on the past success of such events. 13
A third kind of example is provided by experimental game theory. For decades (Rapoport and Chammah 1965), and recently with a renewed vitality (Camerer 2003), experiments on prisoner’s dilemmas, public good games, and the like have allowed theorists to gather data about human cooperative behavior, here defined by the presence of a mutual benefit. In some of these experiments, agents face a one-shot (i.e., nonrepeated) social dilemma in which agents are anonymous and cannot communicate. They must decide to cooperate in the absence of any direct information about others—all they know is the number of potential participants. I claim that when agents choose an action leading to a mutual benefit in such situations, there can be minimal cooperation in the same sense as in the demonstration and flash mob examples. Indeed, these are paradigm examples of cooperation for game theorists.
Are not these situations just too bare to be called cooperative? Rational choice theory distinguishes between decision and interaction problems (respectively dealt with by decision theory and game theory). In the former, agents are isolated: they must take a decision by considering external events on which they have no influence. In the latter, agents interact and each agent’s situation depends on how others see them. Information can be exchanged, and all rational strategies must be stable, that is, mutually depend on each other and form a coherent whole. However, in demonstrations, flash mobs, and one-shot anonymous social dilemma, there hardly is any mutual dependence between individual decisions. So agents seem to face quasi-decision problems, in which they function like automata and apply decision routines which treat others as parts of the external environment. This is at odds with what it intuitively means to cooperate: to act together with someone, to face her and adjust to her actions.
However, intuitions must sometimes be resisted. Indeed, minimal cooperation is a limiting case in which interactions should be as weak as possible. Classical definitions do not challenge intuition, precisely because they focus on sufficient characteristics. On the contrary, when looking for minimal cooperation, one naturally reaches the border that separates cooperative actions from merely collective ones. Predictably, intuition thus becomes less and less able to discriminate between cases. In other words, the lower limit of cooperation is at the limit of our intuition. In what follows, I provide reasons to think that the structure of minimal cooperation is still close enough to that of classical definitions to be considered as bona fide cooperation.
3. Goals and Beliefs
As shown in the introduction, classical definitions meet few constraints other than that of adequacy to intuitive cases. There are additional constraints on minimal cooperation though. Classical definitions share a common structure and are based on a certain list of ingredients; were the definition of minimal cooperation extremely different, there would be no reason to think that it is cooperation rather than a lesser kind of collective action. In other words, a definition of minimal cooperation should meet two continuity constraints, concerning reference and structure; that is, it should respectively refer to concepts akin to those used in classical definitions, and it should share their overall structure. This would indicate that cooperation, whether in minimal or classical cases, is a unified kind. 14
To define minimal cooperation requires gluing various pieces together, a full justification of which is beyond the scope of a single paper. For the sake of unification, it will be sufficient to describe two key ingredients for minimal cooperation that are also present in classical definitions, namely, collective goals and common knowledge—although in a weakened form. 15 Thanks to these, the definition of minimal cooperation will turn out to be structurally similar to classical definitions.
3.1. The Necessity of Collective Goals
This section explains why minimal cooperation cannot do without the concept of a collective goal, by considering examples in which the presence or absence of a collective goal determines whether there is cooperation or not.
Chwe (2001) showed that most products advertised during the U.S. Super Bowl (the most widely watched event of the year) can be called social products, that is, products whose attraction increases with the number of people consuming them, either for social (e.g., beer) or compatibility (e.g., computers) reasons. At first glance, this merely constitutes a problem of coordination. Following Lewis’s (1969) famous analysis, what advertisements accomplish is render one option more salient than others, which in turn increases the chances that people will favor it. However, this is best described as a problem of cooperation, because agents always have the option to consume nothing. Agents would prefer that most of them consume the same good; if not, they had rather consume nothing than be only a few to consume the same good. Still, such mass consumption of an identical good could only be a kind of behavioral cooperation (reaching a mutual benefit), that is, of collective action but not of cooperative action in the intentional sense emphasized so far.
However, the nature of the collective action depends on the reasons for which agents have such preferences. One can prefer to consume what others consume because she would be better off and/or because the whole group would be better off or prefer that the whole group of consumers to make similar choices because it benefits the group. 16 When most agents act for such a reason, there is room for a collective goal, which is considered by many theorists 17 as a crucial element for cooperation. In other words, all things being equal, the presence or absence of collective reasons to act can mark the border between cooperative and collective action. The reasons are collective in two respects: their content concerns the group, and they are collectively held by most agents (in a way to be clarified in the next section).
Let us consider a second example. 18 In a city, girls prefer to dress similarly by wearing miniskirts. This is a mere collective action, based on imitation and on the shared belief that enough other girls dress this way. In particular, there is no collective goal here: girls try to coordinate without considering a good coordination as beneficial to their group. However, if girls dressed this way in the group’s interest, for instance to make the presence of the girl group manifest in the city, the action would be cooperative. Without this collective goal, there can be no more than a collective action based on a shared reason.
Is the presence of a goal necessary to cooperation? It is straightforward to check that each example of minimal cooperation introduced in the previous section contains collective goals. In demonstrations or flash mobs, participants typically do not decide to join when many others join as well just because it is preferable to doing nothing. They care for the consequences of the collective actions, whether immediate (the effect of a flash mob) or not (a possible change of policy triggered by a demonstration). Moreover, they believe that enough other participants have a similar goal. Both the content and epistemic structure of the goal are collective.
What about anonymous one-shot interactions? In such cases goals are salient: they emerge from the situation. In a social dilemma, the presence of a mutual benefit that Pareto-dominates another outcome 19 constitutes a possible collective goal. Although the literature contains no precise analysis of when collective preferences can arise from combinations of individual preferences or payoffs, in many situations this is obvious enough. 20 Certain configurations of intertwined individual preferences are such that possible collective goals can be perceived, and understood as being perceivable by others, by agents. Collective goals can naturally emerge from situations in which no collective preference preexists.
Overall, collective goals are necessary to minimal cooperation, even if not collectively built. Indeed, as all other crucial elements highlighted by theorists (e.g., commitments, plans of action) are absent in such situations, collective goals are the only way to ensure a cooperative dimension.
3.2. Weakening Common Knowledge
What makes minimal cooperation weaker than classical cooperation is not the nature of collective goals but their collective epistemic characteristics, more precisely what agents know or believe about each other’s goals.
All classical definitions of cooperation have a similar epistemic structure: they provide a list of ingredients (beliefs, goals, intentions, etc.) and add to them a condition that they be common knowledge among agents. For X to be common knowledge among agents means that it is public and is usually described in the following way: everyone knows X, everyone knows that everyone knows X, everyone knows that everyone knows that everyone knows, and so on. It has been argued (Lewis 1969; Aumann 1976; Paternotte 2011) that a more realistic definition should be based on the concept of a public event: X is common knowledge among agents if there is an event or fact E such that E is public (whenever E is the case, everyone knows that it is) and that knowing E entails knowing X. However, there seems to be no common knowledge in minimal cooperation, 21 because the previous definitions run into two obstacles.
First, as we have seen, minimal cooperation does not presuppose that the set of agents is common knowledge, which is necessary for any further common knowledge. 22 Second and more important, if agents have never met before deciding to act, they typically do not get information from a unique public event but from multiple private sources (think of advertisements or pamphlets). As a consequence, the concept of common knowledge should be at worst abandoned, at best weakened.
Fortunately, the existing literature already provides solutions consistent with the latter option: several alternate versions of common knowledge allow it to escape the aforementioned criticisms. First, common knowledge of facts can be defined without presupposing a background common knowledge of agents’ identities. Gold and Sugden (2007) recently introduced T-conditional common knowledge (where T is a subset of a group S), in which rather than asking that agents know what specific others know, they know what other members of T know (without having to know who in S is actually a member of T). One agent may not know the identities of the group members while still knowing that being a member has consequences on someone’s knowledge. In particular, if the members of a group are those who perceived a particular goal, assuming that they have some knowledge related to the goal is not problematic.
As a consequence, the content of the common knowledge is modified. In classical cooperation, there is common knowledge among participants of their individual intentions to act: this is possible because by definition, being a participant implies having decided to act. In minimal cooperation, as there is only common knowledge among possible participants, it can only be of tendencies to participate (more on this in section 4.2).
As to the second problem, common knowledge also turns out to be formally definable without a public event. Cubitt and Sugden (2003) noticed that there is no major conceptual consequence when replacing a public event E, which entails knowledge of itself, with several typical events Ei that each entails the knowledge of the others. If an agent watches an advertisement often enough, she will know that others watching the same kind of shows as her are likely to have seen it too. Watching a commercial indicates that others have seen or will see another occurrence of it. Common knowledge does not necessarily arise from publicity; typicity can do the job as well, 23 in which case we speak of indirect common knowledge.
Overall, the kind of common knowledge needed for minimal cooperation is doubly weakened: it is conditional and indirect—whereas classical common knowledge is unconditional and direct. It arises from disjoint information sources and disconnected agents who still share some relevant properties. That a common knowledge clause can be salvaged in cases of minimal cooperation is an additional hint that the latter is still legitimate cooperation, rather than mere behavioral collective action.
To summarize so far, minimal cooperation is characterized by collective goals that are not collectively built and are weakly common knowledge. This makes it conceptually close to classical cooperation, which also contains goals and common knowledge, although in a stronger guise.
4. Minimal Cooperation and Rationality
For all their similarities, there are significant differences between classical and minimal cooperation: for in the latter, rationality plays a key role, and the content of the individuals’ mental states is not others’ intentions but their tendencies to act.
4.1. Collective Intentions and Game-Theoretic Equilibria
I first highlight the similarity between collective intentions and game-theoretic equilibria, which will later help defining the former in cases of minimal cooperation. In game theory, an interaction is modeled as a game, defined by a set of agents, for each agent a set of available actions and a preference relation between the possible combinations of actions. 24 What actions can be called rational depends on whether they can be part of an equilibrium: it consists of a set of strategies (one for each agent) that are mutually optimal, that is, that maximize the agent’s expected utility given her beliefs about the state of the world and the strategies of others. In other words, an equilibrium is a set of mutually justifying strategies. Many different types of equilibria exist, depending on the parameters taken into account in the expected utility maximization calculation.
Although equilibria pervade the whole of game theory, few people have noticed the conceptual similarity they share with analytical definitions of cooperation. This resemblance stems from works that were in full swing during the 1985–1995 decade, 25 showing that it is possible to find epistemic conditions that guarantee that agents act in accordance with a given equilibrium. 26
Such epistemic characterizations of equilibria are strikingly similar to collective intentions. Choices of actions correspond to individual intentions, calculations to reasons to act . . . and part of these conditions must be common knowledge in both cases. In particular, a collective intention seems fundamentally stable, as agents’ individual intentions at least partially justify each other. However, there are two differences: while epistemic characterizations guarantee that individuals rationally choose certain strategies, collective intentions are compatible with certain choices, which can be rational or not.
An epistemic characterization that guarantees an outcome usually entails that there is a collective intention, but the converse does not hold. This is consistent with the fact that collective action theorists usually consider rationality to be a welcome characteristic of a cooperative action but not a necessary one. Definitions of cooperation usually contain conditions of the form: an agent intends to do her part partly because of a given reason—such definitions specify which mental attitudes are causes of or reasons for others but by a process left unspecified. By contrast, this process is obvious in game theoretic equilibria: a mental attitude is reason for a choice if it partly explains how this choice maximizes expected utility. However, a process independent from any maximizing rationality may see the job through as well. 27
Why bother about game-theoretic equilibria in the first place, if they do not constrain collective intentions? The reason is that they show how strong they need to be. If one finds a game-theoretic equilibrium that leads to a given collective outcome in a certain context, then the collective action should not constrain the agents’ mental states any more strongly. What is enough for rational cooperation is more than enough for cooperation tout court. This is particularly useful in our case. Suppose we find an equilibrium that makes it rational to cooperate in situations such as those described in section 2 (to recall: demonstrations, flash mobs, anonymous one-shot interactions) and characterized in section 3. Then it cannot be denied that nothing more than collective goals, weakened common knowledge, and beliefs about tendencies to act is needed for a collective intention—as it is enough for a rational collective intention that guarantees the outcome.
Accordingly, the next section precisely identifies such an equilibrium, which then allows us to proceed to a definition of minimal cooperation.
4.2. Bayesian Nash Equilibria
How can cooperation be rationally justified? This has long been a worry for game theorists. The problem is to explain why agents can choose to cooperate when another action is individually preferable, as experiments show that a significant proportion of individuals do indeed cooperate (Camerer 2003). Most existing accounts focus on punitive mechanisms or reputation effects in repeated games (Kreps and Wilson 1982, Osborne and Rubinstein 1994). However, cooperation in one-shot anonymous games without communication is more problematic—the only classical Nash equilibria in prisoner’s dilemma is that of noncooperation.
However, there are other kinds of Nash equilibria. One solution is to suppose that individuals can adopt various preferences, including social ones: they may sometimes prefer what is best from a group’s point of view. 28 When several types or preferences are possible, the set of agents’ real preferences cannot possibly be common knowledge, but the tendencies, or propensities, to adopt these preferences can be. This is what is formalized in Bayesian Nash equilibria, in which agents have several possible types (several sets of preferences, or utility functions) and prior probabilities to be of any given type. Strategies are then assigned to types rather than agents; agents choose the strategy that maximizes their expected utility given the probabilities that others are of different types and the strategies of those types. In a prisoner’s dilemma with two types (individualists and social agents), there exist Bayesian Nash equilibria that allows for mutual cooperation, 29 provided that the probability for an agent to have a social type is high enough. In other words, introducing uncertainties as to agents’ types can make cooperative strategies rational.
Bayesian equilibria do fit cases of minimal cooperation: agents do not need to know the choices or intentions of others but only the probability (or propensity) that they form intentions depending on their type (the goal they adopt). Individual intentions must be stable with regard to other possible individual intentions rather than to other individual intentions themselves. These possible intentions in turn depend on propensities of goal adoption and on the salience of various groups. 30 A definition of minimal cooperation needs to eliminate the reference to individual intentions: this is just what Bayesian Nash equilibria do. 31
That cooperative strategies can be part of Bayesian equilibria ensures just what we needed: that collective intentions formed in our cases of minimal cooperation can be viable with collective goals, weakened common knowledge, and reference to propensities to act. 32
4.3. The Role of Rationality
I now turn to the role that rationality plays in minimal cooperation. Definitions and explanations of cooperation are usually developed independently. In particular, as seen earlier, rationality is but an optional feature for cooperation. For instance, it can be demanded that the agents’ choices of action be consistent with their preferences, that their beliefs be formed “rationally,” or that their mutual contribution to collective goals be common knowledge. That rationality is only an incidental property makes sense: imposing some rationality constraints on classical cooperation is likely to lay it open to accusations of unrealism, since it could apply only to cognitively sophisticated-enough agents.
By contrast, rationality is crucial to minimal cooperation, for the following reasons. When collective goals are not collectively built, and in the absence of any prior interaction, salient reasons to participate become essential to estimating the degree with which a goal could be adopted. In minimal cooperation, an agent first notices a possible goal, then estimates how salient it is—that is, how likely it is that others perceive it too. Rational calculations thus are the indications as to whether a goal is attainable. In other words, the only salient reasons to act are rational ones.
In classical cooperation, agents can exchange information, make promises and agreements, elaborate plans of action, in a word influence each other, and foster goal adoption in various ways. In minimal cooperation, collective goals are barely salient; agents are isolated and racked by general uncertainty. If participating is irrational, and without knowing anything about other possible participants, why would one suppose that they will actually participate? By contrast, when rational reasons to act are still available, and in the absence of others, they become predominant.
Still, this shows only that rational considerations are important to minimal cooperation rather than essential—they are only one of the many guises under which reasons to cooperate can arise, which does not justify their presence in a definition of cooperation. However, the main argument is as follows: if acting cooperatively was not rational in a highly uncertain situation, then agents would have strong reasons not to act cooperatively and, in particular, not to believe that others will do so either. The rationality of a cooperative action does not guarantee cooperation, but its irrationality precludes it. Demanding that minimal cooperation be rational must be seen as a guarantee that reasons to not cooperate are absent. When agents cannot directly extract from others good reasons to cooperate, reasons to not cooperate are insurmountable.
However, explicitly referring to rationality in an analytical definition of cooperation raises some issues. Including the detailed maximization calculations of agents is out of the question. We saw that definitions do not usually state the precise links between intentions to act and the reasons leading to their formation. This makes sense because the more detailed the definition, the higher the risk that it unduly excludes some legitimate cases of cooperation. Referring to propensities of goal adoption is an acceptable middle ground: it mentions the relevant parameter without imposing a specific kind of reasoning.
Moreover, excluding references to precise calculations also has positive consequences. First, cognitive limitations are allowed; it would be absurd to assume that agents involved in mass actions carry out complex probabilistic calculations. Rather, they are likely to use heuristics, such as equating probabilities to adopt a goal with the proportion of agents likely to adopt it. We often tune our cooperative behavior on the estimated number of participants, which allows us to predict more easily the success or failure of a collective action. A satisfying definition of minimal cooperation should be loose enough to allow for such shortcuts.
The definition also tolerates other kinds of mistakes from the agents. For instance, different agents may perceive different groups. This means that even agents who adopt the same group goal may rationally choose to participate but for slightly different reasons (that is, different calculations) as they make different hypothesis as to other agents’ goal adoptions. I think it is correct to say there is still cooperation in such cases. When generalized uncertainty rules, to demand that all agents perceive the situation in identical ways would unduly decimate legitimate cases of cooperation. Agents cooperate as long as they act for correct reasons—reasons that are strong enough not to be overruled by neglected ones.
5. Minimal Cooperation
5.1. Definition
This section finally defines minimal cooperation by interlocking definitions of more elementary ingredients, weakening collective goals, and integrating a rational aspect. In what follows, relevant groups must be identified with caution. I distinguish between the set G of agents, who benefit from the realization of the collective goal; the set P of agents, who perceive the goal; and the set C of agents, who adopt the goal. 33
(pCG) B is a possible collective goal for a group G in situation S if and only if
B can be realized by members of G and at least two members of G are needed to realize it.
B is simultaneously realized for all members of G.
The realization of G is preferable to it not being realized for all members of G acting in the interest of the group.
Either B has been made salient before the situation S occurs, or it emerges from the structure of individual preferences of members of G (it at least provides a mutual benefit).
Clauses (a), (b) and (c) are P-conditional common knowledge among members of G, where P is the set of members of G having perceived G such as defined by (a) and (b).
Clauses (a), (b) and (c) describe necessary properties of a collective goal, either a preexisting or an emergent one: (a) is elementary; (b) specifies that it has a collective nature; and (c) indicates the kind of collective preference it induces. Clause (d) does not specify the precise configuration of preferences from which a collective goal can emerge, which would necessitate a paper of its own. In this regard, the definition is still incomplete. Still, the preference structure mentioned in (d) should be such that any emerging goal has the properties (a)-(c). Clause (e) may be “empty” in the sense that it does not demand that anyone actually perceive the goal. However, if someone perceives it, she knows that others who perceive it know about it.
As such, (pCG) merely describes the fact that a situation contains a collective goal. It entirely belongs to the presuppositions of cooperation. Even if (e) contains epistemic elements, it does not require any specific epistemic state to be the case since they are conditional epistemic states. The next definition introduces the epistemic dimension.
(dCG) B is a distributed collective goal for a subset C of G if and only if
B is a possible collective goal for G.
Every member of $C$ adopts B as a goal in S and intends to contribute to its realization.
Every member of $C$ maximizes her expected utility by accomplishing her part of B depending on the propensities for any member of G to perceive the goals present in S and to adopt them.
(A) and (C) are P-conditional common belief in G, where P is the set of members of G having perceived B.
(B) is the case partly because of (C) and (D).
A distributed collective goal is a possible collective goal that has been perceived and adopted by a set of agents. To adopt it, an agent has to believe that it is realizable given the propensities that others also adopt it—there must be a rational reason to adopt it. Scrutinizing Bayesian Nash equilibria reveals that to adopt a group goal does not automatically entail that it is rational to choose the cooperative action: it crucially depends on how likely others are adopt it as well. If the goal was not compatible with any equilibrium, it would be deemed unrealizable and consequently abandoned. Clauses (C), (D), and (E) merely translate into a nonformal language the characteristics of Bayesian Nash equilibria based on probabilistic group goal adoption.
(wCI) Agents of a group C have a weak collective intention to realize B if and only if
Every member of C has C for her goal and intends to do her part of it.
Every member of C maximizes her expected utility by accomplishing her part of B depending on the propensities for any member of G to perceive and adopt B as a goal in S.
Every member of C believes that other members of C have a propensity to act according to B.
(ii) and (iii) are G-conditional common belief (for a group G, containing C).
(i) partly because of (ii), (iii), and (iv).
To have a weak collective intention amounts to intend to act in accordance with a group equilibrium for a common reason. A distributed collective goal can be understood as the superposition of a possible collective goal and a weak collective intention. The conditions are thus similar to that of (dCG), except that no constraints are imposed upon the structure of B and on the reason to act. Agents may have a weak collective intention to act for the benefit of anyone or on the basis of any principle or ideology. The new clause (C) ensures that the reason to act be public (previously, this was automatically entailed by the definition of a possible collective goal); and (D) that there is conditional common belief for any group G: it is sufficient that B be salient for members of C. Again, nothing guarantees that members of C can actually realize B, as long as they adopt it and reckon that it can be realized with a sufficient probability. No more can be demanded, as agents who act for the same reason may ignore who they are. In other words, a group can have a distributed collective goal or a weak collective intention and still fail to cooperate.
The final definition essentially follows from the previous ones:
(mC) Members of a group C (successfully) minimally cooperate to realize B if and only if
Members of C can realize B.
Members of C want to realize B together.
B is a collective distributed goal for C.
Members of C realize B together in accordance with and partly because of the weak collective intention they have in virtue of (3). 34
Clause (1) ensures that members of $C$ can realize B. Clause (2) asks for a clarification. As agents do not know each other, how can they want to realize something together? This only means that every member of $C$ wants B to be realized by members who are acting in the group’s interest, that is, for an adequate reason. This condition expresses the fact that if an agent learns after the fact that B has been realized although a number of agents were only pretending to cooperate and had ulterior motives independent of B, she would not think there has been cooperation. However, this does not mean that the subjective perception that agents have of cooperation is necessary for cooperation, for only the desire that the goal is realized in such a way is necessary. I come back to this point later, when I discuss the concept of naive cooperation.
Last, note that no additional condition about epistemic states is necessary, as the definition of a distributed collective goal already takes care of the agents’ epistemic states.
5.2. Consequences
Minimal cooperation is defined through interlocking, conceptually distinct layers: a collective goal, coupled with a weak collective intention, constitutes minimal cooperation.
The prevalent concept here is that of distributed collective goal. In classical definitions, a goal cannot be modified after having been adopted. Here, however, participants are in a state of global uncertainty such that the goal can be abandoned if no group equilibrium justifies it. Collective goals thus pervade the whole of cooperation: their presence triggers preference change; their salience calibrates the agents’ beliefs; and their confirmation or abandonment determines which choices of action are made. Collective goals come close to being the only public information that is available to the agents. In this sense, minimal cooperation is essentially teleological.
Minimal cooperation is not as weak as it seems though. Full cooperation corresponds to Tuomela’s we-mode: the collective building of a group goal, with agents fully endorsing the group’s interests. At the other end of the spectrum, I-mode cooperation describes agents following their subjective preferences (which can be group oriented or even altruistic), in which there are only mutually compatible individual goals. Minimal cooperation, which contains distributed collective goals, is beyond the I-mode.
According to the previous analysis, an agent who adopts a group goal does not just adopt any utility function but one that follows from the collective goal. Group utility maximization in group equilibria is nothing else than the counterpart in the economic vernacular of the intention to realize a group goal; the agent tends to reason according to an equilibrium because it strives to realize the goal. Moreover, the goal is the primary cause of the cooperation, as its success depends on it being adopted. Minimal cooperation has elements of the we-mode, such as adopting group preferences, and the agents adopt these preferences for group reasons, that is, the presence of a collective goal. However, in the absence of any contact with others, one cannot be said to be acting as a full group member. 35
6. Discussion
6.1. Circularity
Classical analyses of collective concepts frequently face two kinds of circularity issues. First, some definitions of collective intentions markedly rely on conditional intentions, in the following manner: a group’s collective intention to realize X is equivalent to the set of the members’ individual intentions to do their part of X provided that others do the same. The obvious circularity problem is that there is no fundamental reason why members should ever form unconditional intentions to act, since it can be formed only if others have unconditional intentions, whereas everyone has only mutually dependent conditional intentions. 36 Imagine two competitors managing to agree to help each other, through the intervention of a third party. Each competitor accepts to sign on the condition that the other also signs, not wanting to risk committing for nothing in return. This situation will be a deadlock as long as conditional intentions to sign do not become unconditional intentions. The problem is thus to justify the transition from a set of interdependent conditional intentions to a set of unconditional, “pure” intentions. Note that this is not a circularity of definition but one of explanation that renders the definition obsolete. Just as an individual intention should directly lead to an action, a collective intention should directly lead to a collective action. But the circularity of explanation prevents this.
Usual remedies consist in describing pragmatic ways in which agents escape it: by finally deciding to act unconditionally, by promising to do it, or thanks to more or less cognitive mechanisms, such as when an indication that others intend to act is sufficient to trigger the transition to an unconditional intention (Tuomela 2007). However, the definition of minimal cooperation contains no conditional intention: agents decide to act from the prior belief in the propensity that others may. The circularity of explanation is thus avoided; this is an advantage of getting rid of individual intentions within agents’ reasons to act.
However, there is a second, more stringent circularity problem, which appears when some conditions within the definition of a concept refer to this very concept. 37 For Gilbert (1989), individuals will form a plural subject doing X if and only if they jointly committed to do X together. For Tuomela, a necessary condition for agents to intentionally accomplish X together is that they act in accordance with and because of their joint intention to do X together (Tuomela 2007, 125, 144). The definition of minimal cooperation has a similar profile, since it refers to the participants’ will to act together. Circularity, and even triviality, would arise if the meaning of “to act together” somehow implied cooperation.
Can definitions survive this kind of circularity? According to Gilbert, they can if analytic, sophisticated definitions of cooperation contain only references to simple, naive conceptions of cooperation, such as those that a normal human being as opposed to the theorist would have. Tuomela follows a similar line when he remarks that cases of elementary joint action—for instance when realized by children aged less than a year and even animals—should not demand that agents possess a sophisticated concept of cooperation. I agree with these answers. However, there is now a need for a definition of naive cooperation, without which the analysis would not be complete. I partly anticipated this objection by explaining after the definition what “to act together” meant; the next section expounds more precisely what is meant by naive cooperation.
6.2. Kinds of Cooperation
How can the daily, naive concept of cooperation be defined? The previous definitions adopted the point of view of an omniscient theorist, who has access to every action, mental state, and causal links and can consequently unambiguously decide whether a given interaction is cooperative. However, the agents ignore most of this. Naive cooperation is the phenomena that appear to be cooperation from their incomplete point of view.
Imagine several agents acting simultaneously. Before the action is realized, no agent can know whether there will be cooperation. Even if an agent adopts a goal and knows everyone else’s propensity to adopt it as well, she cannot access any intention to act, cannot know whether the goal has been perceived and adopted by anyone else. Once the action is realized, everyone’s actions and intentions are known; 38 all know whether the collective goal has been attained. However, the reasons for which others acted remain opaque.
In games of incomplete information, it is well known that the same action can be chosen by agents of different types or with different preferences. For instance, an agent may cooperate if she has adopted a group goal but also if she wants to deceive others into thinking she has, to better reap benefits from them in the future. How do agents form their beliefs about others? In a dynamic framework, agents update their beliefs: from the observed actions, the list of optimal actions for every type of agent and the prior probabilities that agents are of any possible type, all posterior probabilities can be deduced. Depending on their values, different types of perceived cooperation are possible.
Ideally, agents should conclude there was cooperation only when posterior probabilities that any agent adopted the relevant group goal are 1. This is a case of conspicuous cooperation. It does not quite nail down naive cooperation though. Consider a viewer learning that a demonstration was a success, in terms of the number of participants: she will have a tendency to see this as a manifest sign of successful cooperation between participants and of the cohesion of the organizing group. Still, it is quite possible that a significant number of agents participated for individual reasons, which are unrelated to the collective goal: to see how the demonstration went, to spend a day with friends, or even to eat for free. 39 Moreover, cooperation could be apparent even though only a fraction of the rank-and-file members actually participated.
There are two sorts of cases. First, when the agents’ actions cannot be observed individually, there will appear to be cooperation simply when the collective goal is attained. With hindsight, the same football team could be described as united or divided on the mere basis of a match’s score. These are cases of behavioral cooperation: the realization of a given collective goal.
Second, when individual actions have been observed, another approximation seems possible. Rather than to demand certainty as to the agents’ adoption of a group goal, it should be sufficient to ask that the posterior probabilities of goal adoption be higher than those of any other possible type. There would be cooperation whenever the best explanation of the agents’ actions is that they adopted the team goal. A participant may conclude that a demonstration is not really cooperation if she sees others mostly caring about eating, dancing, and having fun . . . and that it is cooperation if participants really seem to care about the demonstration. This is a case of plausible cooperation. It will sometimes coincide with conspicuous cooperation, for instance when the only possible explanation of a collective action is that agents adopted the group goal, as in a one-shot anonymous prisoner’s dilemma. According to the analytical definition of minimal cooperation, only conspicuous cooperation can be cooperation. However, plausible cooperation is a good approximation of it from the subjective point of view of normal agents or when the observed date is too scarce.
Now going back to the circularity issue, the condition that agents must want to act together in the minimal cooperation definition means nothing else than the following: agents must at least want that the collective goal be realized and possibly that agents act to realize it. Consequently, the definition of minimal cooperation does not contain intentions to cooperate. When they act, agents intend to do their part because they believe that enough other potential members will do theirs for the goal to be realized. After having acted, agents will perceive the collective action as cooperation if it is plausible enough that participants acted for the correct group reasons. The definition thus refers not to intentions to cooperate but to intentions to play a part towards a collective goal and to desires that it be realized by members who do adopt this goal.
This analysis also applies to more specific cases. For instance, an agent can unintentionally favor cooperation if her individual action contributes to the collective goal without her perceiving it (and the associated group). One can also cooperate while believing there is no cooperation going on, for instance when mistakenly believing that the collective goal could be attained thanks only to the actions of some non participants. She would then do her part so that the goal is realized while mistakenly believing this is not cooperation, because she took some participants to act for the wrong reasons; however, there is cooperation from the theorist’s point of view. As a consequence, common belief that there is cooperation is not needed for cooperation to happen.
Last, let us appreciate again how crucial collective goals are to cooperation. Whether they emerge or preexist, they are a prior condition for cooperation. However, they also provide indications that a collective action is really cooperative. Since they only indicate plausible cooperation, there is room for mistake. However, the possibility of such errors may actually be an asset for the success of cooperation: if agents believe that cooperation happens more often than it actually does, it can favor subsequent beliefs that a group is cohesive and efficient, makes group more manifest and goal adoption more likely, and so on. In other words, the imperfect observability of cooperation may boost its success.
7. Conclusion
I have defined minimal cooperation. It applies in situations that contain a manifest collective goal (salient in and possibly emerging from the situation itself), in which a group of agents realize it by choosing their actions in accordance with an equilibrium. Agents do not need to know each other beforehand, to communicate or even to perceive each other when they make their choice. The concept of collective goal lies at the heart of this definition; without it, there would only be a “parallel” collective action.
As we have seen, such a definition helps solve some of the traditional philosophical problems related to cooperation: definition circularity and reference to an unanalyzed, naive form of cooperation. Parallel to the sophisticated conception of minimal cooperation, there is a more natural or intuitive one, mostly based on the realization of a collective goal. It acknowledges that apparent success of cooperation is often a posterior criterion leading to the belief that there actually was cooperation. This can be called behavioral cooperation or even plausible cooperation in dynamic situations. Such a naive conception explains people’s proclivity to postulate the existence of groups.
The concept of minimal cooperation now allows us to answer precisely several questions concerning the nature of cooperation: 40
- Is there cooperation if an agent acts cooperatively but misidentifies her cooperative action? No. To act in the group’s interest while failing to correctly identify the collective goal entails that cooperation fails—the agent did not fulfill her part. Even if “acts cooperatively but misidentifies her cooperative action” is understood as “accomplishes her part for the wrong reasons,” cooperation still fails because all participants must act for the right (group-based) reasons.
- Must agents be aware that they are cooperating? No. They must be aware that they act in the group’s interest or to realize a collective goal. Without information about others, when they take their decision, they cannot know whether a sufficient number of agents have decided to fulfil their part as well, even less whether it is for the right reasons.
- Must agents have a common plan of action? No. First, they obviously do not need a plan of action agreed on beforehand. In a sense, they can still act in accordance with a plan: they know that their actions must be part of a combination of actions that realizes the collective goal and that they must be part of a group equilibrium. However, there is no guarantee that others are following the same “plan.”
- Must the intentions to act be conditional on the belief in other agents’ intentions? No. Agents do not need to know other agents’ intentions to act. They base their decision on the only tendencies, or propensities, that others form such intentions. As a consequence, intentions to act are not conditional, in the sense that they would only become proper intentions after noticing the other agents’ similar intentions.
-Must everyone’s reasons to act be identical? Yes. It is necessary that agents fulfil their part because they adopt the group goal. However, possible reasons to act do not have to be common knowledge, whether direct or indirect. Some agents might have perceived only part of the possible reasons to act of others: if they act in the group’s interest, they know that others may act for the same reason. At this point, it matters little whether the detailed list of other possible groups varies depending on agents.
Footnotes
Acknowledgements
I thank Samir Okasha and Raimo Tuomela for their useful comments.
Declaration of Conflicting Interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Arts and Humanities Research Council of the UK, Grant No. AH/F017502/1, which I gratefully acknowledge.
1
2
This point will become important in section 2.
3
We will see later in section 3 that two further constraints can be found in the specific case of minimal cooperation.
4
5
In what follows, I refer to these accounts as “classical accounts” of cooperation or “classical definitions.”
6
Other concepts seem conceivable, such as agreements or promises; however, because they only reinforce, or lead to certain individual beliefs and intentions, they should not be considered fundamental. To put it differently, they describe ways in which ingredients necessary for cooperation can appear.
7
Consider Bratman’s (1993) recurrent examples of singing a duet or painting a house together and 
walking together, which she calls “a paradigmatic social phenomenon.”
9
Moreover, in this case there is no common knowledge of the set of participating members, since members do not even know their identities before the demonstration. More on this later.
10
Although they may know some of them, since a flash mob is usually triggered by collective e-mails, the forwarding of which is encouraged.
11
Or at least they used to be. Nowadays, flash mobs are much better organized and always imply communication between many participants beforehand.
12
Indeed, the more popular the idea of flash mobs became, the more likely they were to attract participants.
13
The claim that flash mobs are cases of cooperation may be challenged when there exists a triggering message containing instructions, which reveals a preexisting level of organization that contradicts the emerging aspect. However, an initial message merely describes what the cooperative action actually is. Agents who receive it only have to choose whether to follow the instructions (participate) or not. Moreover, any communication arising between participants after the cooperative action has started has no influence on this prior decision.
14
In addition, this shows that there is no more reason to resist the minimal definition than the classical ones. This helps put classical and minimal cooperation on an equal footing.
15
In this section, I speak indifferently of minimal cooperation or of the corresponding minimal collective intention and thus temporarily neglect the distinction introduced in section 1.
16
Of course, this is exceptional in cases of mass consumption but less so for medium-sized groups. Think of a hall of residence in which students have to choose their computer. They could prefer to have computers with similar characteristics because it would be beneficial to the whole community (e.g., whether when working together or playing online games in a network).
17
See for instance Tuomela (2000, 2007) or .
18
19
That is, which is preferable for all agents and strictly preferable for at least one. Theorists seem to tacitly consider that a possible collective goal is an outcome that Pareto-dominates another outcome that is an equilibrium of the game. Assessing this description would require a different paper. However, it is enough for my purpose to note that there is widespread consensus on this point.
20
For instance, everyone agrees on what outcomes can be called mutually beneficial, or considered as a possible group goal, in games such as the prisoner’s dilemma or the stag hunt.
21
This is why mass actions are garden-variety cases of minimal cooperation: the more numerous the participants, the less likely it is that their set is known beforehand and, thus, that there can be any common knowledge between them.
22
23
Chwe’s example of a unique commercial broadcast during a massively watched event is rather an exception than the rule.
24
Preferences are often represented numerically by utility functions.
25
26
This should not be conflated with the converse claim: no one would argue that agents who act in accordance with, say, a Nash equilibrium necessarily have to be in a determined epistemic state. Rather, some precise sets of epistemic states necessarily lead to equilibrium-following choices. For instance, showed that whenever the structure of the game and the agents’ rationality are shared knowledge (known by everyone) and their choice of action is common knowledge (publicly known), they will necessarily act according to a Nash equilibrium. (A set of strategies form a Nash equilibrium when for any agent, changing her strategy would decrease her expected utility, the other agents’ action being fixed.)
27
Imagine an imitation mechanism that may make agents act similarly to what they believe others will do, whatever their personal interest is.
28
Indeed, experiments show that agents do display various preferences and can thus be of different types: they can strive to maximize not only their own utility (individualists) but also that of other agents (altruists), the average of all utilities (social agents), or their differences (competitors) or even minimize these differences (fair agents).
29
More precisely, equilibria in which it is rational to cooperate for social agents and rational to defect for individualistic ones.
30
Bayesian Nash equilibria can be interpreted as a formalization of group identification, that is, the faculty for an agent to perceive herself as a member of a group rather than as an isolated individual. Cooperative behavior is known to be favored when group identity is made salient, which can be accomplished in various ways (Tajfel et al. 1971; Tajfel 1973; Dawes 1980; Orbell and Dawes 1981; Dawes et al. 1988; Brewer and Gardner 1996). In particular, group identification can be seen as at least implying the adoption of a group goal or of group preferences (see De Cremer and Van Vugt 1999; ). If psychological realism cannot be a necessary condition for a conceptual definition, it still is a welcome feature, as cooperation is supposed to be reachable for human beings with normal cognitive abilities.
31
, 74ff) provides a definition in which participants only need to believe that others will participate with some likelihood. However, he provides no analysis as to the origin or nature of this likelihood (it could be merely a behavioral condition or express the absence of mistakes from participants) and also does not consider this as full-fledged cooperation.
32
A refinement is needed though. In Bayesian Nash equilibria, the set of possible groups is exogenous: it is common knowledge among agents from the beginning. In the group identification literature, experimenters manipulate the perception of groups by agents to trigger a group identity; the more salient a group, the more likely agents are to identify with it. In other words, the set of perceived possible groups can be altered. Consequently, the definition to come must keep clear the distinction between possible types/groups (from the theorist’s vantage point) and perceived types/groups (relative to the agents). Propensities to perceive a group must be added to propensities to identify with a group or to adopt group goals. The difference will become clear in section 5.
33
Or equivalently identify with G; see note 31.
34
Recall that the definition of a distributed collective goal already contains a weak collective intention.
35
36
This problem is mentioned, among others, by Tuomela (2007, 97-98). A good illustration of such a definition is given by talk of sets of quasi-dispositions (i.e., of conditional dispositions) to act.
37
38
I here exclude the possibility of mistakes.
39
40
Note that all these questions bear upon necessary conditions of cooperation—elements that should be part of cooperation. It is only natural to answer them by referring to minimal cooperation.