Collective Action Problems
An introduction to game theory and collective action problems
TLDR: All social interactions can be modelled and described using an apparatus of game theory. Some social interactions are structured such that each individual is motivated to act so as to undermine the interests of the collective, of which they are a member. These situations are called “collective action problems.” Several useful game-theoretic tools are introduced in order to help those wishing to understand the dynamics of these problems when they arise.
Intuitively, it may be thought that collectives comprised of rational and self-interested individuals would always be motivated to act so as maximise their shared interests. Of course, communities regularly fail to act in their common interests, however, the thought could still remain that these people are simply acting irrationally when doing so. In fact, prior to Olson’s (1965) revolutionary text, The Logic of Collective Action, this was the dominant understanding among philosophers, economists, and political theorists. What proved so radical about Olson’s work was the insight that ‘even if all of the individuals in a large group are rational and self-interested, and would gain if, as a group, they acted to achieve their common interest or objective, they will still not voluntarily act to achieve that common or group interest’ (Olson, 1965, p. 2). The primary reason for this is that there exists a class of social situations, which will henceforth be termed “collective action problems,” wherein individual and collective rationality can be thought to diverge (Bicchieri, 2017, p. 113). Ostrom writes how collective action problems ‘occur whenever individuals in interdependent situations face choices in which the maximisation of short-term self-interest yields outcomes leaving all participants worse off than feasible alternatives’ (1998, p. 1). What proves to be both so pernicious and interesting about these kinds of social situations is the separation of individual and collective rationality; each individual can be thought to be acting rationally so as to maximise their own interests, while simultaneously undermining the interests of the collective, of which they are a member (Hardin, 1968, p. 1244).
In an effort to achieve a clear sense of the structural challenges that collective action problems pose for both individuals and collectives it is instructive to first articulate and analyse the underlying anatomy and incentive framework of these dilemmas. It is often the case that many conceptual tools from game theory are employed for this purpose (Hardin, 1982; Kollock, 1998; Rasmusen, 2001). This is because a game-theoretic approach allows for the social dilemmas to be reduced to their basic structure and examined in terms of interdependent actions and outcomes. While a game-theoretic paradigm will be adopted in this piece, it should be noted that even within the domain of game theory there are several different conceptual lenses that may be applied to the study of collective action problems, each of which being more apt to reveal certain facets of the problems that alternative approaches tend to obscure. For instance, collective action has been examined at the social level of group organisation (Gilbert, 1992, 1990; Olson, 1965). Such analyses tend to emphasise the importance of group size and the amount of internal cohesion present inside of a community. For my purposes, however, I will focus on an individualistic, or agent-centred, conception of collective action (Bicchieri, 2017, 2006; Dawes, 1980; Ostrom, 2000, 1998). This approach is primarily concerned with the personal incentives of each actor inside of a collective action problem as they relate to others around them in the same situation.
An Introduction to Game Theory and the Prisoner’s Dilemma
All game-theoretic situations, which are often simply termed “games,” can be described using four essential elements (Rasmusen, 2001, p. 12). These being the players involved, the actions accessible to them, the outcomes of those actions (which may vary in terms of desirability and the probability of occurring), and the level of information available to each player. Each player ‘will devise plans known as “strategies” that pick actions depending on the information that has arrived at each moment’ (Rasmusen, 2001, p. 12). Strategies, therefore, are simply rules or heuristics that allow each agent to choose one action over the alternatives so as to attain their desired outcomes. The number of individuals in any given game-theoretic situation can range from one, to a potentially infinite number of actors. For the moment the discussion will be limited to those situations involving only two players, although upcoming sections will address cases involving a large number of players. As a starting point, it will be advantageous to introduce the most famous and widely known game-theoretic situation, the prisoner’s dilemma.
Albert Tucker, a mathematician who worked for the RAND Corporation in 1950, is credited with creating the tale of the prisoner’s dilemma at the behest of two of his colleagues, Merrill Flood and Melvin Dresher, who were themselves conducting a social experiment (Luce and Raiffa, 1957, pp. 94–5). Tucker’s story, and the social dynamics that it describes, has since been the focus of thousands of academic papers, and while there are many variations of the prisoner’s dilemma, the essential tale is as follows:
Two individuals are strongly suspected of committing a serious crime together. After being apprehended, police separate the two suspects and provide each with a decision. Each can choose to either confess to committing the crime, thereby implicating their partner in the act as well, or remain silent. If both suspects remain silent then they will each be charged with a lesser crime for which the police already have an overwhelming amount of evidence, resulting in a jail sentence of one year for each suspect. If one suspect confesses while the other remains silent, the police will allow the confessing individual to escape jail time. While at the same time the non-confessing partner will be sentenced to ten years in jail with the aid of their former partner’s confession. However, if both confess then they will be implicating one another and will each face five years in jail. After being separated the police explain, in full detail, the decision matrix that they are both presented with and each prisoner knows that the other faces exactly the same situation. It is also important to note that both individuals are assumed to be rational and that they are only concerned about minimising their own time in jail.
This dilemma can be represented in a two by two (two people and two actions) decision matrix below, where all possible outcomes are represented with the accompanying jail terms for each prisoner. The payoffs for each prisoner are represented in pairs (Prisoner 1, Prisoner 2); so for example, if Prisoner 1 decides to Confess whilst Prisoner 2 chooses to Remain Silent the outcome is represented in the bottom left quadrant with Prisoner 1 receiving no jail time while Prisoner 2 gets sentenced to ten years in jail.
For symmetric games, wherein each individual faces the same choice, it is often the case that only the row-chooser’s payoffs are represented. This representation helps to make each person’s preferences over the various outcomes as clear as possible.
The first detail to be cognizant of in the prisoner’s dilemma is that both actors are assumed to be acting so as to maximise the satisfaction of their preferences, which in this context is what is meant by individual rationality. Meaning that an agent is said to be individually rational if they act so as to maximise their own expected payoff (Bicchieri, 1993, p. 58; Liebrand, 1983, p. 136; Rasmusen, 2001, p. 12). In the prisoner’s dilemma individuals would prefer to serve as little time in jail as possible, and will consequently act so as to minimise their sentence length. Given such an incentive, each prisoner is motivated to confess, independently of how they anticipate their partner acting. Notice how Prisoner 1’s jail time is reduced if they confess independently of what Prisoner 2 decides to do. If Prisoner 2 remains silent Prisoner 1 can maximise their preferences by confessing (0 years in jail as opposed to 1); alternatively, if Prisoner 2 confesses Prisoner 1 can also maximise their preferences by confessing (5 years in jail as opposed to 10). Recall that this situation is symmetric, so confessing is also the best action for Prisoner 2. When this occurs, game theorists will call the strategy of always confessing a strictly dominant strategy (Rasmusen, 2001, p. 20). Clearly, if one action strictly dominates another in this manner it would be irrational for either prisoner to choose the non-dominant action. Meaning that if both prisoners acted individually rationally they would both confess and would each face five years in jail.
This state of affairs, wherein both prisoners confess, is not the optimal outcome for the collective however. Appreciating this requires the introduction of another economic concept of Pareto optimality. A situation is defined as being Pareto optimal, which is also known as Pareto efficient, ‘if it is the case that compared with it, no one’s utility can be raised without reducing someone else’s utility’ (Sen, 1993, p. 521). It is clear that the outcome where both prisoners confess is not Pareto optimal because there exists another outcome that would increase both prisoners’ utilities. This outcome occurs if both prisoners decide to remain silent, resulting in only one year in jail as opposed to five. Thus, the outcome achieved by both prisoners acting individually rationally leads to a Pareto inefficient outcome in the prisoner’s dilemma.¹ Such an outcome can be described as collectively irrational, because the collective has failed to maximise their preferences by reaching the Pareto optimal outcome.
What I hope is now obvious is that individual and collective rationality do not always coincide with one another. As mentioned above, the definition of a collective action problem, of which the prisoner’s dilemma is a paradigmatic example, is that individual and collective rationality diverge (Bicchieri, 2017, p. 113). The problem, or paradox as it is sometimes referred to, is that when both prisoners are acting individually rationally the collective reaches an irrational outcome, whereby both individuals spend five years in jail as opposed to only one if both had remained silent. This unfortunate, albeit interesting, consequence can be further elucidated with the concept of a Nash equilibrium, of which mutual confession in the prisoner’s dilemma is an example. A Nash equilibrium ‘specifies the set of strategies and beliefs such that each player’s strategy is optimal given [their] beliefs about other players’ strategies’ (Bicchieri, 1993, p. 58). Put differently, a strategy combination ‘is a “Nash equilibrium” if no player has [an] incentive to deviate from [their] strategy given that the other players do not deviate’ (Rasmusen, 2001, p. 26). Notice how a Nash equilibrium describes the conjunction of all agent’s strategies, a single strategy in isolation can not be a Nash equilibrium.
Famously, it has been proven that all games have at least one Nash equilibrium (Nash, 1950a, 1950b). There are, broadly speaking, two kinds of Nash equilibria: a pure and a mixed strategy Nash equilibrium. A pure strategy is one that ‘maps each of a player’s possible information sets to one action’ (Rasmusen, 2001, p. 66). This is in contrast to a mixed strategy that ‘maps each of a player’s possible information sets to a probability distribution over actions’ (Rasmusen, 2001, p. 66). For instance, a mixed strategy could dictate that an agent should adopt one action 30% of the time and another action 70% of the time. In summary, a pure strategy Nash equilibrium exists when the Nash equilibrium consists of only pure strategies, whereas a mixed strategy Nash equilibrium exists when the Nash equilibrium consists of mixed strategies. An example of a mixed strategy equilibrium will be explored in greater detail in the following section.
Finally, throughout the remainder of this piece the generic terms of “cooperation” and “defection” will be used to describe the two actions available to each individual in a game. There exist two conditions that distinguish the action of cooperation from that of defection.² Firstly, it must be the case that each individual’s utility is increased by others’ cooperative action. Which is to say that it must always be the case that one is made better off by increasing the number of others who cooperate. Secondly, the individual payoff received under universal cooperation must exceed the payoff received under universal defection. Put differently, the universal cooperative action must lead the collective to the Pareto optimal, collectively rational, outcome while universal defection achieves the opposite. Thus, in the prisoner’s dilemma, remaining silent will henceforth be referred to as cooperating, while confessing will be referred to as defecting. If either of these conditions do not hold, the two actions available may not be considered cooperation and defection.
Three Important Games for Collective Action Problems
Now that several relevant game-theoretical concepts have been introduced with the aid of the prisoner’s dilemma, it is appropriate to delimit the full space of collective action problems that will be addressed in this piece, all of which may be represented with two by two decision matrices. Of the 726 distinct two by two games that describe all possible ordinal preference rankings among payoffs for each individual (Guyer and Hamburger, 1968), there exist only three that may be defined as collective action problems. This is because there are four conditions for a two by two game to be representative of a collective action problem (Liebrand, 1983, pp. 124–5). The first of these is that the game must be symmetrical, meaning that both individuals in the game face the same decision between strategies (Liebrand, 1983, p. 125). Secondly, there must exist a strict preference ordering across all four outcomes, if there exists indifference between outcomes the collective action problem will not exist. The reason for this is that if one agent is indifferent between two outcomes this implies that they will, when trying to rationally maximise their expected payoff, be able to switch between pure strategies without cost. This feature, when paired with the symmetry condition, would remove any potential tension between individual and collective rationality. The removal of ties between outcomes results in only 78 distinct two by two games (Guyer and Rapoport, 1966, 1972, p. 412). Thirdly, the strategy that yields an individual their highest payoff must also result in a non-optimal outcome for their partner (Liebrand, 1983, p. 124). Finally, if both individuals act so as to maximise their own payoffs the collective receives a deficient, Pareto inefficient, outcome (Liebrand, 1983, p. 124). Given these four conditions there exist only three two by two game-theoretic situations that each describe how individual and collective rationality may diverge (Kollock, 1998, p. 187; Liebrand, 1983, p. 127). These three situations are known as the prisoner’s dilemma, the chicken game, and the stag hunt.
The prisoner’s dilemma, along with a specific instantiation of the problem, has already been described in detail above. For the sake of completeness, the decision matrix of the general prisoner’s dilemma is presented below.
Where: C > A > D > B and 2A > C + B
It is worth reiterating that in the prisoner’s dilemma defection dominates cooperation, and that the only Nash equilibrium that exists is the outcome wherein both individuals defect, resulting in a collectively irrational outcome.
The second two by two game that will be examined is the chicken game³, which has an associated hypothetical scenario, as follows:
Two teenage individuals begin driving their cars directly toward one another along a straight and level stretch of road. Both individuals have two options available to them; they may either swerve off of the road, thereby avoiding a head-on collision with their peer, or they may steel themselves and continue driving forward. If both parties fail to flinch and continue to drive toward one another, the resulting collision will result in the catastrophic loss of life for both individuals. If, however, one individual swerves while the other continues to drive straight, the swerver is steeped in shame and embarrassment while the steadfast individual basks in the glory of their courage. Alternatively, if both individuals swerve then both are revealed as cowards, although the embarrassment and shame is lessened, as they are not alone in their cowardice. Both individuals are individually rational and understand that they face an identical decision to that of their partner.
In this game, the cooperative action is to swerve, while driving straight ahead may be described as defection. The decision matrix for each driver is as follows:
Where: C > A > B > D
In this game there exist two pure strategy, and one mixed strategy, Nash equilibria. The two pure strategy Nash equilibria are the two outcomes where one individual swerves while the other drives straight ahead. The mixed strategy Nash equilibrium instructs each teenager to cooperate by swerving P% of the time and to defect by driving straight ahead (100 — P)% of the time (Sasidevan and Sinha, 2015, p. 3)⁴. Where P is given by:
Notice that, unlike in the prisoner’s dilemma, there is no dominant strategy, although there does exist a risk-dominant strategy, which is the strategy of swerving. Driving straight ahead is a risky strategy because the only way that it can reward an individual that adopts it is if their partner swerves. Because mutual defection is so costly, the range of payoffs for cooperating is smaller than the range of payoffs for defecting, making the cooperative action risk dominant (Myerson, 1991, p. 119). The collectively rational, Pareto optimal, outcome occurs when both parties cooperate by swerving. This outcome, however, is not a Nash equilibrium and both individuals will be motivated to drive straight if they believe that their partner will swerve. Herein lies the collective action problem; both individuals are motivated to defect by driving straight if they believe that the other will swerve. This of course raises the risk that both drivers will defect and drive straight into one another, leading to the worst outcome for all. Because of this, both individuals would prefer to adopt the opposite behaviour of their interaction partner, even if that means that they must defect and receive a lower payoff than their partner who cooperated.
This is exactly the opposite of what is observed in the stag hunt (Rousseau, 1984; Skyrms, 2004), which is also known as the assurance game. In this final two by two collective action problem both agents are motivated to coordinate their behaviour together. In order to appreciate this, consider the following vignette describing the social dilemma.
Suppose that there exist two hungry hunters. Each can decide to either hunt hare or hunt stag. If a hunter decides to hunt hare, they will be guaranteed to find and catch food independently of what the other hunter decides to do, although the amount of food they catch will depend on the other hunter’s strategy. If both hunters hunt hare they will be competing with each other for the same food source, meaning that each hunter catches fewer hare than they would have if they were the only hunter hunting hare. Alternatively, if a hunter decides to hunt stag they will only be able to catch and kill the animal if the other hunter accompanies them and also hunts stag. Meaning that if one of the hunters decides to hunt stag while the other chooses to hunt hare, the stag hunter will catch nothing and will not eat, while the hare hunter will catch multiple hare. Both hunters would prefer to hunt stag because the amount of food that a stag provides, even after an equal division, will exceed the amount that each would receive if they were the only one to hunt hare. Both hunters are individually rational and understand that the situation that they face is exactly the same as their partner’s.
The decision matrix for each hunter is as follows:
Where: A > C > D > B
In this game there exist two pure strategy Nash equilibria, the outcome where both hunters hunt stag and the outcome where both hunt hare. This is to say that neither hunter is unilaterally motivated to alter their strategy if they find themselves in either equilibrium. Prima facie, it may seem as though such a situation does not pose a social dilemma for both hunters at all. It is clear that both hunters will maximise their own, and the collective’s, utility if they can coordinate together and hunt stag. Furthermore, unlike in a prisoner’s dilemma, defection does not dominate cooperation, and because each hunter hunting stag is a Nash equilibrium, neither hunter is motivated to hunt hare if they are certain that the other will hunt stag. The problem, however, is one of trust and it is for this reason that the game is also known as the assurance game. Notice how hunting stag (cooperating) can be viewed as a risky strategy to adopt, in exactly the same way that driving straight ahead (defecting) was risky in the chicken game. While it is that case that one’s payoff can be maximised if they hunt stag, this only occurs if the other hunter also hunts stag. If only one hunter hunts stag, the lone stag hunter will go hungry while the hare hunter will sleep comfortably with a stomach full of hare. The strategy of hunting hare does not involve the same level of risk; either a lone hare hunter will catch as many hare as they can or both hare hunters will compete, but even so both will end their hunts with food. In this way, there is a smaller range in the possible payoffs for the hunters who hunt hare when compared with those that decide to hunt stag (Myerson, 1991, p. 119). It is for this reason that the strategy of hunting hare can be described as a risk-dominant strategy. Skyrms is quick to point out that simply because defection is a risk-dominant strategy does not imply that ‘rational players could not coordinate on the stag hunt equilibrium that gives them both a better payoff, but it is to say that they need a measure of trust to do so’ (2004, p. 3).
As mentioned above, the prisoner’s dilemma, the chicken game, and the stag hunt delimit the full scope collective action problems that exist given the paradigm of the two by two game (Kollock, 1998, p. 187; Liebrand, 1983, p. 127). These three games describe three different ways in which individual and collective rationality can diverge. Prior to describing the kinds of collective resources that are nested within such games, it will prove valuable to briefly contrast these problems with another species of two by two games: coordination games.
Coordination Games
A two by two coordination game is a situation where both individuals’ incentives are aligned and where there exist at least two Nash equilibria (Lewis, 1969, p. 24). Below are three examples of coordination problems:
Notice how the two Nash equilibria in a coordination game need not have equivalent payoffs. Furthermore, the two Nash equilibria need not result from both individuals undertaking the same action, many coordination games are characterised by situations where both parties must coordinate so as to undertake different actions. It may be thought that this definition of coordination games allows for the stag hunt to be classified as a coordination game, due to the fact that under the stag hunt there exist two Nash equilibria and both individuals will maximise their utility by cooperating together. However, this ignores the fundamental problem of the assurance game, which is that the risk-dominant strategy is to defect. In a coordination game there are no perverse incentives that can be thought to tempt rational individuals away from cooperating with their interaction partners. Coordination games do not represent collective action problems wherein individual and collective rationality diverge, they are simply games where two parties are incentivised to organise themselves so as to maximise both their individual and collective utility. A classic example of a coordination game in everyday existence is driving on the left (here in Australia) or the right side of the road.
Collective Resources
The payoffs that result for agents in any of the above games may be thought of as goods if the payoffs are positive, or harms if they are negative. This discussion will primarily focus on goods to the exclusion of harms, although the following classification along two axes may be applied to both kinds of resources equally. These two dimensions describe the excludability and subtractivity of the collective good in question (Ostrom et al., 1994, p. 6). The excludability of a good captures how difficult it is for beneficiaries, both current and future, to be prevented access to the potential rewards. For instance, some goods such as breathable air are very difficult to exclude people from, while other goods such as a haircut or a bottle of water are easily restricted. In contrast to this, the subtractivity of a good describes how much each person’s use of, or access to, the good may detract from the quality of the good itself. Take, as an example, a breaking news broadcast concerning a political event, the benefit of this broadcast to those that watch it is in no way diminished with each subsequent viewer. The same cannot be said of a library, this is because with each added person who enters and uses the library the goods that it provides are slightly reduced for everyone. Each additional person who enters the library makes it slightly busier, noisier, and reduces the number of books to browse through. Given these two axes there exist four kinds of goods, these being public goods, common-pool resources, toll goods, and private goods (Hess and Ostrom, 2007, p. 9; Ostrom et al., 1994, p. 7). Below is a table summarising each kind of good in terms of high or low excludability and subtractivity.
It should be noted that the two axes should be conceived of as continuous variables, meaning that some private goods are more subtractive or exclusive than others. Ostrom et al. emphasise this when they write how these categories ‘are similar to four large “continents” in the world of goods. [Where] each of these four types of goods differs from the other three, notwithstanding the substantial variety present within each broad type’ (1994, p. 7).
Before advancing there are two important details to note regarding goods that are generated in collective action problems. Firstly, it is only relatively non-excludable goods that are the focus of collective action problems. The reason for this is that those goods that are easily excludable, those being toll and private goods, may be easily regulated so that free-riding and tragedy of the commons scenarios may be avoided. Simply put, when a good is easily excludable those who don’t contribute, or who have already taken their fair share, may be refused further access at little cost to everyone else. However, when excludability is difficult, as is the case with public goods and common pool resources, this course of action is not feasible. Thus, ‘all collective action problems share the problem that excluding non-contributors to a collective benefit is non-trivial’ (Ostrom, 2003, p. 241). Secondly, all collective goods that are generated have an associated production function that maps contributions to the amount and quality of the good that is made widely available. The shape of the production function may take many forms, depending on the kind of good that is being generated (Hampton, 1987, pp. 247–50). For most goods the production function is monotonically increasing, meaning that as more people contribute to the good, the amount made available, or its quality, is increased. This is not always be the case, consider buskers performing on a busy street in the heart of a major city. A small number of buskers separated by some distance enhance the local environment considerably for all, although the inclusion of too many buskers on the street will, after a certain point, decrease the environmental quality. What’s more the collective good may be such that some threshold of contribution is required before the good is made available at all. For example, a bridge spanning a gorge that is only half built is of little practical use to anyone. With these notions in mind, Hampton contends ‘that understanding particular [collective action] problems depends upon understanding the relationship between the production structure of the collective good in question and the individual group member’s expected costs and benefits associated with its production’ (1987, p. 248). It is for this reason that this analysis has thus far focused on the underlying structure of collective action problems that may be represented using game-theoretic tools.
Two Dimensions of Collective Action Problems
In order to describe and model large-scale, or real-world, collective action problems the current schematisation of two by two games must be extended. The present analysis of collective action problems has, up until this point, been limited to an investigation involving two foundational assumptions, these being that there are two people involved and that they interact only once. However, in reality both of these considerations rarely hold true. To begin with, it is often the case that interactions between people are expected to occur more than once. For instance, two individuals bartering in a local market can reasonably assume that they will meet and interact in a similar situation again in the future. With this in mind, representing this exchange with a single two by two game played only once is a simplification, one that may obscure important elements of the interaction. The likelihood of a repeated interaction, along with the implications that this will have on each person’s decisions in the present interaction, may be crucial to understanding the social situation at hand. This notion of repeated interactions over time captures the temporal dimension that is present in many collective action problems. The second heretofore-undeveloped aspect of collective action problems is the fact that such situations often involve more than two people. Consider the actions necessary to avoid catastrophic climate change, or the number of people who vote in democratic elections. Clearly, attempting to model such large-scale interactions using matrices representing only two people is a major simplification. Expanding the analysis along both dimensions allows for a simple, yet explanatorily powerful, schematisation of four broad kinds of collective action problems defined in terms of repeated interactions and group size. This schematisation is presented in the table below.
Before examining how these extensions may alter the games introduced above it is necessary to make four additional remarks regarding this framing. Firstly, both group size and the number of rounds should again be thought of as continuous variables. Meaning that some interactions will continue indefinitely, while others may only last a relatively short amount of time. Equally, some collective action problems involve only a small community, while others may involve the entire population present on earth. This simple division into four kinds of collective action problems is artificial and simply represents a useful way of analysing the unique traits that different problems pose. Secondly, it may be wondered if expanding the strategy space to allow for alternative actions to either cooperation or defection constitutes a third dimension of analysis. The thought being that many social interactions involve far more than a dichotomous pair of options available to each agent. I grant that in the reality people interacting are faced with more than simply two options; however, incorporating such complexity will not prove useful in most cases. Instrumentally, the action of defecting in any given collective action problem may more adequately be thought of as a large set of actions that share the common trait that they are any action that is not the action of cooperating. Thirdly, the assumption of symmetry between agents could also be relaxed. Meaning that the assumption that each person’s contribution to, and extraction of, a collective good is equal to everyone else’s is often not the case. Fourthly, it has been assumed that each agent acts simultaneously, however, there are many situations and games where agents act sequentially, such as the ultimatum game (Thaler, 1988). Once again though, the inclusion of each of these realistic assumptions into the frameworks only acts to complicate the conceptual picture without a commensurate increase in understanding for the present purposes. In short, there exist many other dimensions that could be explored when classifying collective action problems, although for the present analysis only the temporal and group size dimensions will be examined in further detail.
One way in which two and n-person games differ is that the inclusion of additional agents often reduces the control that each agent has over the situation as a whole. Dawes (1980) describes three mechanisms by which this may occur. Firstly, in a two-person collective action problem the costs associated with defection are levelled exclusively on one’s interaction partner. However, ‘in most [large] social dilemmas in contrast, harm for defecting is diffused over a considerable number of players’ (Dawes, 1980, p. 182). Relatedly, under n-person collective action problems each individual’s actions may be anonymous, such a luxury is often not possible in a two-person game because agents are able to infer their partner’s actions given that they know how they personally acted and the outcome that resulted. Meaning that ‘in an n-person dilemma […] anonymity becomes possible and an individual can free-ride without others noticing her or his actions’ (Kollock, 1998, p. 191). This implies, finally, that each agent in a two-person context can single-handedly reward or punish their partner’s actions. Each agent is granted the ability to modify their partner’s actions through their choices alone, however, this is far less feasible in an n-person context (Dawes, 1980, p. 183). The common element of these three related features is that the inclusion of additional agents in the situation often, although not always, diffuses and reduces each person’s ability to alter the actions of others that they are interacting with.
Olson (1965) describes three further reasons why increasing group size may make large scale cooperation more difficult. Firstly, if the collective good in question does not scale in benefit with the number of people who have access to it, it follows that each individual’s incentive to defect increases with group size. The reason for this is that ‘the larger the group, the smaller the fraction of the total group benefit any person acting in the group interest receives, and the less adequate the reward for any group-oriented action’ (Olson, 1965, p. 48). Secondly, the costs associated with organising larger groups increase as the group size increases. Put simply, all other things being equal, it is easier to organise ten people together than it is to organise one hundred. Finally, Olson argues that as group size increases so too does the likelihood that the group is facing a more challenging social situation such as a prisoner’s dilemma. However, as Hardin (1982, p. 42) notes, this final claim rests on empirical evidence that will not be explored further. Thus, this final assertion is simply a generalisation that is worth keeping in mind, this being despite the fact that counter examples exist and are relatively easy to imagine (Hardin, 1982, p. 45; Ostrom, 1998, p. 15).
Now turning to the temporal dimension that has also been extended. Many collective action problems are characterised by repeated interactions over time with the same set of individuals. There are, broadly speaking, two forms of repeated interactions that can be specified in terms of the number of rounds undertaken, or the expected number to be undertaken. Where the definition of a round of a collective action problem is simply a single interaction of the relevant game situation. There exist situations involving a known finite number of rounds and there exist other circumstances where there could be a potentially infinite number of rounds. For the present purposes those situations involving a widely known finite number of rounds will not be explored in great detail. This is primarily due to the conceptual difficulty posed by the logic of backward induction,⁵ in addition to the fact that most social interactions, outside of artificial laboratory experiments, are not characterised by repeated interactions that are also known to be finitely repeated for a given number of rounds.
Therefore, repeated interactions will henceforth be treated as though there exists a positive probability lower than one that the situation will continue for another round. Such a formulation nicely avoids the pitfalls inherent in the logic of backward induction while also allowing for an expected number of interactions that is finite. The first important detail to note about these kinds of situations is that they allow for contingent strategies that are not possible in a single-shot collective action problem. A contingent strategy is any strategy that an individual adopts that prescribes an action that is dependent on the actions of their interaction partner from previous rounds. For instance, a very famous contingent strategy is known as “tit for tat.” The strategy of tit for tat simply instructs agents to cooperate at the first round of an iterated game and then to mimic, for all subsequent rounds, their partner’s action from the previous round (Axelrod, 1981, p. 311). Interestingly, in a series of computer tournaments conducted by Axelrod (1984), tit for tat was shown to be a very successful strategy in relation to other contingent strategies, winning almost all tournaments that it was placed in. Axelrod identifies four general principles that can account for tit for tat’s success in an iterated prisoner’s dilemma tournament. These being an avoidance of unnecessary conflict, a high propensity for both retaliation in the face of unprovoked aggression and forgiveness after conciliatory behaviour, and finally, that it was a predicable strategy that others can easily understand (Axelrod, 1984, p. 20). However, despite tit for tat’s success it can be demonstrated that there does not exist a single best strategy for any iterated game considered (Boyd and Lorberbaum, 1987; Garcia and van Veelen, 2018; Lorberbaum, 1994). It is simply the case that ‘what is best depends in part on what the other player is likely to be doing [… and this is turn] may well depend on what the [other] player expects you to do’ (Axelrod, 1984, p. 27).
Another interesting consequence of a repeated collective action problem is that the underlying game may be transformed simply as a result of the game repeating. In order to see how this may occur there are three important aspects that must be present. Firstly, it must be the case that the likelihood of a repeated interaction is high. Which is to say that the expected number of interactions must be high. Secondly, each agent must be able to adopt a kind of retaliatory, or threatening, contingent strategy that they could level against their interaction partner if defection is detected. In this way each individual may, for instance, threaten to defect for all future rounds if they are themselves the recipient of a defection. Thirdly, each agent must have a sufficiently low discount rate concerning future payoffs. Meaning that each agent must sufficiently value the payoffs for future rounds. In all cases what counts as “sufficiently” high or low is dependent on the relative payoffs of the basic interaction game. When all three of these elements are present it will be the case that each individual’s threatening strategy is able to alter the incentive structure for the current round of interaction. When this occurs ‘the future can therefore case a shadow back upon the present and thereby affect the current strategic situation’ (Axelrod, 1984, p. 12).⁶
This detail concerning repeated interactions is related to what is widely known as the “folk theorem.” Using the same logic and reasoning as above, the folk theorem demonstrates how any seemingly irrational or counter-productive strategy may be a Nash equilibrium in a repeated game. The key, once again, is that if the likelihood of the game repeating is sufficiently high while the discount rate of future payoffs is sufficiently low, the incentive structure of the game may be altered. If the game can be expected to continue for a sufficiently large amount of time, ‘a way can always be found to make one player willing to punish some other player for the sake of a better future, even if the punishment currently hurts the punisher as well as the punished’ (Rasmusen, 2001, p. 113). The folk theorem is often regarded as a disastrous finding for social scientists because it suggests that any strategy, no matter how irrational, may be proven effective in the sense that it can convince others to capitulate and consequently increase one’s expected payoff. While this is certainly the case, a simple, and realistic restriction may be included in the problem space that effectively eliminates the pitfalls presented by the folk theorem. If the discount rate for all agents is such that future events are not valued sufficiently highly it follows that most counterintuitive findings are easily avoided. Meaning that the mathematical oddity posed by the folk theorem is certainly worth keeping in mind when dealing with infinitely repeated games, or games that have a very high chance of repeating, although such a finding need not concern us in most cases.
The Structure of N-Person Collective Action Problems (Public Goods Games)
The inclusion of more than two players into the game-theoretic space allows for n-person variants of the three paradigmatic collective action problems introduced above. In such cases each person’s payoffs are a result of both their own contribution, and the overall level of group contribution, to a collective good.
Given that these situations are characterised by n people acting independently, a full decision matrix representing each possible outcome may not be practically included for all values of n. This is not to imply that an equation, or set of equations, may not be used to achieve the same outcome, whereby each individual’s payoff is both a function of their own, and the group’s, level of cooperation. However, another useful means of representing these situations is with what will henceforth be termed a “localised decision matrix,” which provides a “snap-shot” of the collective action problem for each individual in relation to the collective. Relatively few of these localised decision matrices are sufficient to describe the incentive structure of entire n-person collective action problems. Below is the localised decision matrix for an individual in relation to a collective of n people, m of which will defect (not including the individual facing the decision). Here m is any integer value between 0 and n-2. There will exist some ranking of the four payoffs, A through D, for the individual depending on both the number of defectors, and the social situation that they find themselves facing. The rankings of these payoffs, and the magnitude of these values, are sufficient to describe the decision that each individual will face in an n-person collective action problem given that the number of defectors has been defined. It is important to emphasise that the ordinal rankings of these four payoffs, in addition to the magnitude of these rankings, are dependent on the number of defectors in the social situation. Meaning that the rankings of A through D may change, depending on the social dilemma at hand, as m changes.
With this general form of the localised decision matrix in mind it is possible to define and model all three n-person collective action problems. Each set of localised decision matrices for the n-person collective action problems will also be accompanied by an associated graphical representation of the kind presented by Liebrand (1983, p. 130), Komorita (1976, p. 358), and Ostrom (1998, p. 3). Such graphs provide an instructive means of understanding both the overall structure of each n-person social situation, while also enabling a clear indication of what each localised decision matrix is describing. Or, perhaps more precisely, where on the graph each localised decision matrix is located.
Beginning with the n-person prisoner’s dilemma, which proves to be the simplest of the three n-person collective action problems. The reason for this is that a single localised decision matrix is able to describe each individual’s decision, independently of the level of overall group cooperation. This is because, for all levels of group cooperation, the action of defection dominates the action of cooperation. Meaning that independently to the number of defectors, the ordinal rankings of the four payoffs will always be: C > A > D > B. When described in these terms it is clear that only a single Nash equilibrium exists for the n-person prisoner’s dilemma, which is the situation of universal defection. This, however, is not the case for both the n-person chicken game and stag hunt situations.
In the case of the chicken game there exist three distinct situations that an individual may face given the level of overall group cooperation. The first case is characterised by there being a small proportion of the population who are cooperating, such that an individual deciding to defect can be described as being irrational, due to the fact that doing so results in a lowered overall utility when compared with cooperating. For the sake of clarity, recall that the number of cooperators is simply the total number of agents in the situation, n, minus the number of defectors, m. Furthermore, it will prove useful to define the point of intersection between the defection and cooperation curves as a tipping point, T, defined in terms of the number of cooperators (Bicchieri, 2017, p. 181). Therefore, when the number of cooperators is below this tipping point (in the sense that the level of group cooperation is not sufficiently large so as to reach the tipping point) an individual will face a decision where it is individually rational to cooperate. Which is to say that below the tipping point, cooperation dominates defection, and the ordinal rankings are: A > C > B > D. However, when there exist exactly the minimum number of cooperators necessary to reach the tipping point, the localised decision matrix that an individual will face is altered, and the rankings are: C > A > B > D. Thus, at the tipping point, where the payoff curves of defection and cooperation intersect, the localised decision matrix is comparable to that of a two-person chicken game. Finally, when the number of cooperators is above the tipping point the localised decision matrix is analogous to a two-person prisoner’s dilemma, wherein defection dominates cooperation, such that it is rational for an individual to defect. Meaning that the ordinal rankings are as follows: C > A > D > B.
In the n-person chicken game there exist two different kinds of Nash equilibrium that can be expected to occur at the tipping point between the curves of cooperation and defection. The first is mixed strategy Nash equilibrium where each individual will cooperate proportionally with the tipping point in relation to the overall number of participants, and defect otherwise. In this case, each individual will be motivated to cooperate with probability T/n and defect with probability 1- T/n. Alternatively, a subset of the population will always defect, while the remaining participants will always cooperate, such that the proportional size of each group means that the expected level of cooperation occurs at the tipping point. Both of these Nash equilibria are also stable Nash equilibria. An instructive way of thinking about the concept of stability is to imagine a slight shift in the system away from the equilibrium. If, after the shift, the system can be expected to return back to the initial equilibrium position, due to the incentives present for all agents, the equilibrium can be considered stable (McElreath and Boyd, 2008, p. 20). However, if the slight shift results in the system moving toward a different equilibrium position, it follows that the initial equilibrium was not stable. To see that both of these kinds of Nash equilibria are also stable Nash equilibria, notice that if an agent decides to alter their strategy and cooperate when they would have otherwise defected they will, in doing this, increase the overall level of group cooperation. When this occurs, the tipping point has been exceeded and it becomes individually rational for others to defect, thereby moving the collective back toward the stable Nash equilibrium. Alternatively, if an agent alters their strategy to defect when they would have otherwise cooperated, they also will be moving the collective away from the tipping point, such that cooperation becomes individually rational for others. Which will once again mean that the collective can be expected to return to the stable Nash equilibrium occurring at the tipping point.⁷ All of this demonstrates that the two forms of Nash equilibria at the tipping point of the n-person chicken game are stable. Meaning that it can be expected that a collective facing an n-person chicken game will cooperate such that the tipping point between the curves of cooperation and defection is reached.
However, the inverse is found on the n-person stag hunt. As was the case in the chicken game, the n-person stag hunt is characterised by three distinct situations that an individual may face given the level of overall group cooperation. In the first case, the number of cooperators is insufficient to exceed the tipping point, such that defection is individually rational. Which is to say that below the tipping point, defection dominates cooperation, and the ordinal rankings for the localised decision matrix are: C > A > D > B. However, when there exist exactly the minimum number of cooperators necessary to reach the tipping point, the localised decision matrix that an individual will face is altered, and the rankings are: A > C > D > B. This is because at the tipping point there will exist enough other cooperators such that it is individually rational to cooperate if one knows that enough others are also cooperating. Finally, when the number of cooperators exceeds the tipping point the decision matrix is once again altered and the rankings are as follows: A > C > B > D. Above the tipping point the structure of the social situation is such that it is individually rational to cooperate if one knows that enough others are also cooperating.
What all of this demonstrates is that in the n-person stag hunt it is individually rational to cooperate if one knows, or believes, that enough others will cooperate. Otherwise, it is rational to defect because at that level of group cooperation one perceives their situation to be akin to a prisoner’s dilemma. When conceived in these terms it becomes clear that there are two pure Nash equilibria in the n-person stag hunt, characterised by universal defection and universal cooperation. Both of these pure Nash equilibria are also stable Nash equilibria. Additionally, there also exist the same two forms of Nash equilibria that occurred at the tipping point in the chicken game. While both of these are Nash equilibria, because no agent is able to unilaterally alter their strategy in order to increase their expected payoff, they are nonetheless unstable Nash equilibria, and consequently they will not be considered in any greater length. This is because ‘if a Nash equilibrium is unstable [it is reasonable to] ‘expect actual players, for example, subjects in an experiment, not to play that equilibrium, or even be close to it’ (Benaïm et al., 2009, p. 1). For the sake of completeness, the Nash equilibrium in the n-person prisoner’s dilemma is also stable.
It is worth highlighting that while all of the graphs presented above have linear cooperation and defection curves, with a tipping point occurring at a cooperation level of 50%, this need not be the case. The curves for both cooperation and defection may take any form, linear or otherwise. All of the specifics of both curves are directly dependent on the qualities of the collective good being generated (Hampton, 1987, pp. 247–50). Furthermore, the point of intersection between both curves, when it exists, is also directly dependent on the qualities of the collective good in question. Meaning that in some n-person stag hunts the tipping point, T, will be exceeded with only 5% cooperation levels, while in other cases it may require a 95% level of cooperation to be exceeded. As already mentioned above, the key is that all collective goods have an associated production function that maps individual contributions to the amount and quality of the good that is made widely available. Finally, in all cases above it has been assumed that each individual can accurately ascertain the overall level of group contribution that exists, or will exist, when all agents act simultaneously, this is almost never true in real life. A persistent difficultly present in all of these situations is the fact that individuals do not know the overall level of group contribution that will exist when they act. A real challenge posed by the n-person stag hunt, for instance, is the fact that individuals do not know how many others will cooperate. Meaning that individuals in this collective action problem do not know which localised decision matrix they are facing, making the decision regarding how to act very difficult.
Resolving Collective Action Problems: Two Conditions of Success
Thus far, three collective action problems have been presented in terms of their underlying incentive structure. These social situations were initially introduced in a two-person and single-shot context, although the domain of inquiry has since been extended to include n-person and iterated variants of these problems as well. In order to finalise this introduction to collective action problems it will prove useful to define the productive resolution of these dilemmas.
The resolution of a collective action problem may be defined in many ways, although only two common methods will be examined here. The first, which may be termed “socially beneficial success,” is the outcome whereby all agents choose the action of cooperation over defection. This notion of success is conceptually similar to Rabin’s idea of “mutual-max,” wherein ‘outcomes are mutual-max when each person maximises the other’s material payoffs’ (1993, p. 1281). This is appropriately considered a resolution of a collective action problem because, given how cooperation has been defined above, the collective will have attained a Pareto optimal, collectively rational, outcome. Moreover, because all of the collective action problems considered are symmetric, socially beneficial success will also be a fair outcome, with each agent receiving an equivalent. The precise mechanisms for achieving this success will depend entirely on the specifics of the collective action being confronted. As an example, concerns regarding reputation (Kreps and Wilson, 1982), or employing punishment (Boyd and Richerson, 1992), have been demonstrated to be successful strategies that can alter the incentives of pre-existing collective action problems, thereby re-aligning individual and collective rationality.
The second condition of success, which is known as an “Evolutionary Stable Strategy” (ESS), captures the intuitive notion that for any strategy to be considered successful it must confer an adaptive fitness advantage to those who adopt it when compared with alternative strategies.⁸ In this way, a strategy that is an ESS against a competitor strategy can be considered successful because it will be capable of resisting invasion, and can therefore be expected to survive over time, provided that the competitor strategies invade alone (McElreath and Boyd, 2008, p. 43). It is important to recognise that when determining whether or not a strategy is an ESS against a competitor strategy, it is assumed that both strategies are socially mediated such that the evolutionary fitness of any strategy adopted is proportional to the payoffs that it confers to each individual. In order to determine if a strategy, A, is an ESS against a competitor strategy, B, it is necessary to consider the two pure equilibrium positions where only one strategy is present. At the two pure equilibrium positions all agents have adopted only one of the two strategies available. When considering these equilibria, if it is possible for the unadopted strategy to enter into the environment and succeed, in the sense that it receives a higher expected payoff with the strategy already present than the incumbent strategy does with its own kind, then it is clear that the pure equilibrium dominated by one strategy was not stable against invasion. There exist two conditions under which a pure equilibrium is stable, as shown below (Smith and Price, 1973, p. 17):
Strategy A is an ESS against strategy B if: V(A|A) > V(B|A)
Otherwise if: V(A|A) =V(B|A) and V(A|B) > V(B|B)
Where: V(B|A) = The expected payoff for a lone individual adopting strategy B, over all interaction rounds, when compared with a population full of others adopting strategy A.
In this way, if the expected payoff for strategy A interacting with another of its own kind is greater than the expected payoff for strategy B interacting with strategy A, it can be said that A is an ESS against B. Alternatively, if the expected payoff for strategy A interacting with another of its own kind is equal to the payoff for B interacting with A, and the expected payoff for A interacting with B is greater than the expected payoff for B interacting with another B, then it is also the case that A is an ESS against B. Notice how a strategy may only be an ESS in relation to a competitor strategy, this detail is worth keeping in mind because it means that while A may be an ESS against B, it may not be against another strategy C. Additionally, notice how the concept of an ESS can be thought of as a refinement of the concept of a Nash equilibrium. This is because there exists a ‘requirement that [an] ESS not only be a best response, but (a) that it have the highest payoff of any strategy used in equilibrium (which rules out equilibria with asymmetric payoffs), and (b) that it be a strictly best response to itself’ (Rasmusen, 2001, p. 126).
Given everything just outlined, I hope that the theoretical motivations for the inclusion of both conditions of success are clear. The thought is that a resolution of a collective action problem may be defined as either socially beneficial (i.e. Pareto optimal) and/or evolutionarily stable, meaning that it can prevent invasion by intruder strategies that invade alone.
Concluding Remarks
The full range of game-theoretic tools that may be brought to bear on collective action problems have not been examined within this piece. Many relevant ideas have been omitted for the sake of brevity, and interested readers are strongly encouraged to dive deeper if they are so inclined. I hope, however, that this piece has provided a solid foundation for those curious about game theory and its application to collective action problems.
As has been echoed throughout this piece, in order to understand the dynamics underlying collective action problems it is first necessary to understand the incentive structures governing these social interactions. Fundamentally, incentives drive all human behaviour, and game theory provides a theoretical and mathematical apparatus through which to view human incentives. Thus, the tools provided by game theory are indispensable for those wishing to conceptualise, model, or resolve any collective action problem that they might encounter.
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[1]: One ancillary detail to note regarding Pareto efficiency is that the Pareto optimal outcome may not be a fair outcome. Due to the fact that the prisoner’s dilemma is symmetrical the Pareto efficient outcome produces an equitable payoff for both prisoners. However, Sen writes how in general ‘Pareto efficiency is completely unconcerned with distribution of utilities (or of incomes or anything else), and is quite uninterested in equity’ (1993, p. 521).
[2]:Both of these conditions have been adapted from a classification of n-person prisoner’s dilemma problems introduced by Komorita (1976). However, Komorita also includes a third condition that only applies to prisoner’s dilemma situations, and is therefore not generalizable for the full range of games to be introduced in the following section.
[3]:The chicken game is also known as the Hawk-Dove game (McElreath and Boyd, 2008, p. 38) or the snowdrift game.
[4]:The mixed strategy Nash equilibrium is determined by finding the frequency of adopting each action that will equalise the expected payoff for each agent. Because each agent acts identically this frequency is wholly dependent on the relative value of each outcome.
[5]: Consider a finitely repeated prisoner’s dilemma involving five rounds of interaction. Given that both parties know that they will cease interacting after the cessation of the fifth round, neither individual is motivated to cooperate at the final round. Thus, on the final round both parties will defect, however, if both know this, and they do because they are both rational, they will reason that they are also not motivated to cooperate in the penultimate round either. This logic extends backward from the final round to the first round so that neither individual will be motivated to cooperate in any round (Axelrod, 1984, p. 10). Therefore, there is a sense in which a finitely repeated game where both parties know in advance the number of interactions that they will be involved in is equivalent to a game with only a single interaction round. This is the logic of backward induction, although, it relies on the fact that both individuals know how many finite rounds they will both play, without this qualification the logic of backward induction will not hold (Bicchieri, 1993, p. 96).
[6]: For example, it can be shown that if all of these conditions are met ‘then the shadow of the future transforms the two-person prisoner’s dilemma into the two-person stag hunt’ (Skyrms, 2004, p. 5).
[7]: If the curves for cooperation and defection are defined as FC(l) (the fitness for cooperating), and FD(l) (the fitness for defecting), in terms of l (the level of group cooperation). A Nash equilibrium is stable if Equation 1 holds at the point of intersection; alternatively if Equation 2 holds at the point of intersection, it is not a stable Nash equilibrium.
[8]:While the concept of an ESS originates from the field of biological game theory (Smith and Price, 1973), rather than from the rational choice and economics literature that has been the focus for much of this piece. This should not be thought of as a conceptual concern, primarily because the two fields are very closely related and there already exists a large amount of interplay between the two (Huttegger and Zollman, 2010).
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