Introduction

Social dilemmas are situations in which collective interests are at odds with private interests [1]. In other words, they describe situations in which the fully selfish and rational behavior leads to an outcome smaller than the one the individuals would obtain if they acted collectively. Social dilemmas create then a tension between private interests and public interests, between selfishness and cooperation. Classically, several different social dilemmas have been distinguished, including the Prisoner’s dilemma, Chicken, Assurance, Public Goods, the Tragedy of the Commons [2], and, more recently, the Traveler’s dilemma [3], [4]. Each of these games has been studied by researchers from different disciplines, as economists, biologists, psychologists, sociologists, and political scientists, because of the intrinsic philophical interest in understanding human nature and since many concrete and important situations, as pollution, depletion of natural resources, and intergroup conflict, can be modelled as social dilemmas.

The classical approaches explain tendency to cooperation dividing people in proself and prosocial types [5], [6], [7], [8], [9], or appealing to forms of external control [10], [11], [12], or to long-term strategies in iterated social dilemmas [13]. But, over the years many experiments have been accumulated to show cooperation even in one-shot social dilemmas without external control [14], [15], [16], [17], [18], [19], [20]. These and other earlier experiments [21], [22], [23], [24] have also shown that the rate of cooperation in the same game depends on the particular payoffs, suggesting that most likely humans are engaged in some sort of indirect reciprocity [25], [26] and the same person may behave more or less cooperatively depending on the payoffs. Consequently, the problem of making a predictive division in proself and prosocial types becomes extremely difficult, if not even impossible.

From these experiments, we can argue two conclusions: first, the observation of cooperation in one-shot social dilemmas without external controls suggests that the origin of cooperation relies in the human nature; second, the fact that the rate of cooperation depends on the payoffs suggests that it could be computed, at least approximatively, using only the payoffs. The word approximatively stands for the fact that numerous experimental studies have shown that cooperation is based on a number of factors, as family history, age, culture, gender, even university course [27], religious beliefs [19], and decision time [28]. Therefore, we cannot expect a theory able to say, given only the payoffs, the individual-level rate of cooperation in a social dilemma. We can expect instead a model predicting quite accurately population average behaviour using the mean value of parameters that could be theoretically updated at an individual-level.

In this article we make the first step in this direction: (1) we develop the first predictive model of cooperation; (2) we show that it explains a number of puzzling experimental findings that are not explained by the standard economic model, such as the fact that the rate of cooperation in the Prisoner’s dilemma increases when the cost-benefit ratio decreases, the rate of cooperation in the Traveler’s dilemma increases when the bonus/penalty decreases, the rate of cooperation in the Public Goods game increases when the pro-capite marginal return increases, the rate of cooperation in the Chicken game is larger than the rate of cooperation in the Prisoner’s dilemma with similar payoffs; (3) we show that it makes satisfactorily accurate quantitative predictions of population average behaviour in social dilemmas.

We mention that there are many other models that can be applied to explain deviation towards cooperation in social dilemmas, including the cognitive hierarchy model [29], the quantal level-k theory [30], the level k-theory [31], the quantal response equilibrium [32], the inequity aversion models [33], [34] and the noisy introspection model [35]. Nevertheless, all these models use free parameters and so they are not predictive, but descriptive.

The key idea behind the model is very simple: since experimental data suggest that humans have attitude to cooperation by nature, we formalize the intuition that people do not act a priori as single agents, but they forecast how the game would be played if they formed coalitions and then they act according to their most optimistic forecast.

We anticipate that forecasts will be defined by making a comparison between incentive and risk for an agent to deviate from the collective interest. This comparison leads to associate a probability to the event “agent defects”. As mentioned, we will show that this procedure works satisfactorily well in the prediction of population average behavior. The problem in passing to individual-level predictions is that the event “player defects”, given only the payoffs, is not measurable at an individual-level in any universal and objective sense and the dream is to use the factors mentioned above (family history, age, culture, incentives, iterations, etc.) to define parameters to update the measure of the event “player defects” at an individual-level. In fact, an attempt to extend the present model to iterated social dilemmas has been done in [36], leading to promising results: predictions tend to get close to experimental data as the number of iterations increases.

Even though our model is very general and can be applied to every symmetric game, we treat explicitly only four but very relevant and widely studied social dilemmas: the Prisoner’s dilemma, the Traveler’s dilemma, the Public Goods, and the Tragedy of the Commons. We begin with a short review of these games.

An Informal Description of the Model

Before introducing the model in general, we describe it informally in a particular case. Consider the Prisoner’s dilemma (recently experimented using MTurk in [20]) with monetary outcomes (expressed in dollars) . The idea is that players forecast how the game would be played if they formed coalitions. In a two-player game there are only two possible coalition structures: in the selfish coalition structure players are supposed to follow their private interests and in the cooperative coalition structure they are supposed to follow the collective interest. The analysis of these two coalition structures proceed as follows:

• In players follow their private interest and therefore, by definition, they play the Nash equilibrium . Since there is no incentive to deviate from a Nash equilibrium, each player gets 0.05 for sure and we say that the value of is 0.05 and write .
• To define the value of we argue as follows. If the players follow the collective interest, their largest possible payoff is 0.15 in correspondence to the profile of strategies . Since this profile of strategies is not stable (i.e., each player has a non-zero incentive to deviate from it), we introduce a probability to measure how likely such deviations are. To define this probability, we observe that:
  • the incentive to deviate from the collective interest is , since each player can get 0.20 - instead of 0.15, if she defects and the other cooperate;
  • the risk to deviate from the collective interest is , since each player can get only 0.05 instead of 0.15 if she follows her private interest but also the other one does the same.

We define the prior probability that a player abandons the coalition structure by making a sort of proportion between incentive and risk. Specifically, we define the probability that a player abandons to be . Now, note that the smallest payoff achievable by a player when she follows but the other player does not is the sucker payoff . Therefore, we define

The numbers and are interpreted as forecasts of the expected payoff for an agent playing according to and , respectively. Since and , the most optimistic forecast is in correspondence of the cooperative coalition structure . We use this best forecast to generate common beliefs or, in other words, to make a tacit binding between the players: to play only strategies which give a payoff of at least 0.10 to both players. More formally, we restrict the set of profiles of strategies and we allow only profiles , such that , for all . We define the cooperative equilibrium to be the unique Nash equilibrium of this restricted game.

From Fig. 1, it is clear that the cooperative equilibrium is in correspondence of the point in the red set that is closest to . This point can be computed directly by finding the smallest such thatthat is . Consequently, the cooperative equilibrium of this variant of the Prisoner’s dilemma is for both players. Notice that in [20] it has been reported that players cooperated with probability 58 per cent in one treatment and 65 per cent in another treatment and the over-cooperation in the second experiment was explained in terms of framing effect due to the different ways in which the same game was presented.

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Figure 1. The unit square represents the initial set of available profiles of strategies: player 1 can use all strategies , for all ; player 2 can use all strategies , for all .

The red set represents the set of allowed profiles of strategies in the restricted game.

https://doi.org/10.1371/journal.pone.0072427.g001

The Model

We now describe the general model. We recall that, motivated by the observation that attitude to cooperation seems to be intrinsic in the human nature, our main idea is to assume that players do not act a priori as single agents, but they forecast how the game would be played if they formed coalitions and then they play according to their most optimistic forecast. The only technical difficulty to formalize this idea is to define the forecasts. Following the example described in the previous section, they will be defined by assigning to each player and to each partition of the player set , interpreted as a possible coalition structure, a number which represents the expected payoff of player when she plays according to the coalition structure . This value will be indeed defined as an averagewhere represents the prior probability that players assigns to the event “players in abandon the coalition structure ” and is the infimum of payoffs of player when she plays according to the coalition structure and players in abandon the coalition.

This idea is very general and indeed, in a long-term working paper, we are developping the theory for every normal form game [37]. In case of the classical social dilemmas in consideration the theory is much easier, because of their symmetry.

• Symmetry. All players have the same set of strategies and for each player , for each permutation of the set of players and for each one has

(1)Coming to the description of the model, let be a symmetric game and denote the set of players, each of which has pure strategy set , mixed strategies and payoff function . We start by assuming, for simplicity, that and we will explain, at the end of this section, how the model generalizes to -player games.

A coalition structure is a partition of the set of players, that is a collection of pairwise disjoint subsets of whose union covers . Every set in the partition is called coalition. Given a coalition structure , we denote by the game associated to , whose players in the same coalition play as a single player whose payoff is the sum of the payoffs of the players belonging to that coalition. Call the set of Nash equilibria of the game . Now fix and let denote the other player. We denote by the maximal payoff that player can obtain leaving the coalition structure . Formally,(2)

will be called incentive of player to abandon the coalition structure .

Given a profile of strategies , a strategy is called -deviation from if .

We denote by the maximal loss that players can incur if she decides to leave the coalition structure to try to achieve her maximal possible gain, but also player deviates from the coalition structure either to follow her selfish interests or to anticipate player ’s deviation. Formally,(3)where runs over the set of Nash equilibria of and, for each such , runs over the set of strategies such that is maximized and runs over the strategies that are -deviations from either or . is called risk for player in abandoning the coalition structure .

We define the probability of deviating from the coalition structure by making a comparison between incentive and risk. There are certainly many ways to do such comparison. In this paper we use a quite intuitive and seemingly natural way to make it and, in future research, it would be important to investigate some others. Specifically, we define(4)and we interpret this number as prior probability that player assigns to the event “player abandons the coalition structure ”. Therefore is interpreted as prior probability that nobody abandons the coalition structure . Now, let be the infimum of payoffs for player if nobody abandons the coalition structure , that is the infimum of payoffs for player when each player plays according to a Nash equilibrium of , and let be the infimum of payoffs of player when she plays according to a Nash equilibrium of and plays a -deviation from a Nash equilibrium of . The value for player of the coalition structure is by definition

(5)Symmetry implies that , for all . Consequently, there is a coalition structure (independent of ) which maximizes . We use the number to define common beliefs or, in other words, to make a tacit binding among the players.

Predictions of the Model

Conclusions

Many experiments over the years have shown that humans may act cooperatively even in one-shot social dilemmas without forms of external controls and the rate of cooperation depends on the payoffs. This suggests that humans have attitude to cooperation by nature and therefore they do not act a priori as single players, as typically assumed in economics, but they forecast how the game would be played if they formed coalitions and then they play according to their most optimistic forecast.

We have formalized this idea assuming that each player makes an evaluation of the probability that another player abandons the collective interest to follow her private interest. This probability is defined by making a comparison between incentive and risk to deviate from the collective interest and gives rise to common beliefs that, mathematically, correspond to define a suitable restriction of the original game. On the one hand, this procedure seems qualitatively reasonable and we believe it provides a realistic picture of the mental processes that real subjects perform during the game. On the other hand, the formalization of this process, that is, the definitions of the risk, incentive, probabilities, and the induced game, is mathematically simple and seemingly natural but certainly deserves to be investigated better and possibly improved in future research.

However, the actual model makes us optimistic about this direction of research, being the first predictive model able to: (1) make satisfactorily accurate predictions of population average behavior in social dilemmas; (2) explain a number of experimental findings, such as the fact that the rate of cooperation in the Prisoner’s dilemma increases when the cost-benefit ratio decreases, the rate of cooperation in the Traveler’s dilemma increases when the bonus/penalty decreases, the rate of cooperation in the Public Goods game increases when the pro-capite marginal return increases, the rate of cooperation in the Chicken game is larger than the rate of cooperation in the Prisoner’s dilemma with similar payoffs.

The dream is to incorporate other components (as family history, age, culture, incentives, iterations, etc.) into the model in order to make individual-level predictions.