We consider an environment where players are involved in a public goods game and must decide repeatedly whether to make an individual contribution or not. However, players lack strategically relevant information about the game and about the other players in the population. The resulting behavior of players is completely uncoupled from such information, and the individual strategy adjustment dynamics are driven only by reinforcement feedbacks from each player's own past. We show that the resulting "directional learning" is sufficient to explain cooperative deviations away from the Nash equilibrium. We introduce the concept of k−strong equilibria, which nest both the Nash equilibrium and the Aumann-strong equilibrium as two special cases, and we show that, together with the parameters of the learning model, the maximal k−strength of equilibrium determines the stationary distribution. The provisioning of public goods can be secured even under adverse conditions, as long as players are sufficiently responsive to the changes in their own payoffs and adjust their actions accordingly. Substantial levels of public cooperation can thus be explained without arguments involving selflessness or social preferences, solely on the basis of uncoordinated directional (mis)learning. Cooperation in sizable groups has been identified as one of the pillars of our remarkable evolutionary success. While between-group conflicts and the necessity for alloparental care are often cited as the likely sources of the other-regarding abilities of the genus Homo [1, 2], it is still debated what made us the "supercooperators" that we are today [3, 4] . Research in the realm of evolutionary game theory [5-10] has identified a number of different mechanisms by means of which cooperation might be promoted [11, 12] , ranging from different types of reciprocity and group selection to positive interactions [13] , risk of collective failure [14] , and static network structure [15, 16] . The public goods game [17] , in particular, is established as an archetypical context that succinctly captures the social dilemma that may result from a conflict between group interest and individual interests [18, 19] . In its simplest form, the game requires that players decide whether to contribute to a common pool or not. Regardless of the chosen strategy by the player himself, he receives an equal share of the public good which results from total contributions being multiplied by a fixed rate of return. For typical rates of return it is the case that, while the individual temptation is to free-ride on the contributions of the other players, it is in the interest of the collective for everyone to contribute. Without additional mechanisms such as punishment [20] , contribution decisions in such situations [18, 19] approach the free-riding Nash equilibrium [21] over time and thus lead to a "tragedy of the commons" [22] . Nevertheless, there is rich experimental evidence that the contributions are sensitive to the rate of return [23] and positive interactions [13] , and there is evidence in favor of the fact that social preferences and beliefs about other players' decisions are at the heart of individual decisions in public goods environments [24] . In this paper, however, we shall consider an environment * Electronic address: hnax@ethz.ch † Electronic address: matjaz.perc@uni-mb.si where players have no strategically relevant information about the game and/ or about other players, and hence explanations in terms of social preferences and beliefs are not germane. Instead, we shall propose a simple learning model, where players may mutually reinforce learning off the equilibrium path. As we will show, this phenomenon provides an alternative and simple explanation for why contributions rise with the rate of return, as well as why, even under adverse conditions, public cooperation may still prevail. Previous explanations of this experimental regularity [18] are based on individual-level costs of 'error' [25, 26] . Suppose each player knows neither who the other players are, nor what they earn, nor how many there are, nor what they do, nor what they did, nor what the rate of return of the underlying public goods game is. Players do not even know whether the underlying rate of return stays constant over time (even though in reality it does) because their own payoffs are changing due to the strategy adjustments of other players, about which they have no information. Without any such knowledge, players are unable to determine ex ante whether contributing or not contributing is the better strategy in any given period, i.e., players have no strategically relevant information about how to respond best. As a result, the behavior of players has to be completely uncoupled [27, 28] , and their strategy adjustment dynamics are likely to follow a form of reinforcement [29, 30] feedback or, as we shall call it, directional learning [31, 32] . We note that, in our model, due to the one-dimensionality of the strategy space, reinforcement and directional learning are both adequate terminologies for our learning model. Since reinforcement applies also to general strategy spaces and is therefore more general we will prefer the terminology of directional learning. Indeed, such directional learning behavior has been observed in recent public goods experiments [33, 34] . The important question is how well will the population learn to play the public goods game despite the lack of strategically relevant information. Note that well here has two meanings due to the conflict between private and collective interests: on the one hand, how close will the population get to playing the Nash equilibrium, and,
Another related work studying cooperation in public goods games uses a version of reinforcement learning called directional learning to (mis)learn and achieve more cooperative outcomes deviating from Nash Equilibria REF .
Directional learning and the provisioning of public goods
Sci. Rep. 5 (2015) 8010
Economics Physics Computer Science Biology Medicine