The shadow of the future promotes cooperation in a repeated prisoner’s dilemma for children

Cooperation among genetically unrelated individuals can be supported by direct reciprocity. Theoretical models and experiments with adults show that the possibility of future interactions with the same partner can promote cooperation via conditionally cooperative strategies such as tit-for-tat (TFT). Here, we introduce a novel implementation of the repeated Prisoner’s Dilemma (PD) designed for children to examine whether repeated interactions can successfully promote cooperation in 10 and 11 year olds. We find that children cooperate substantially more in repeated PDs than in one-shot PDs. We also find that girls cooperate more than boys, and that children with more conduct problems cooperate less. Finally, we find that children use conditional cooperation strategies but that these strategies vary by gender and conduct problem rating. Specifically, girls and children with few conduct problems appear to follow an altruistic version of win-stay, lose-shift (WSLS), attempting to re-establish cooperation after they had defected. Boys and children with more conduct problems appear to follow a Grim strategy, defecting for the duration after the partner defects. Thus we provide evidence that children utilize the power of direct reciprocity to promote cooperation in strategic interactions and that, by late elementary school, distinct strategies of conditional cooperation have emerged.


Pilot testing
The interface we used for the PD was new and untested; therefore, we conducted pilot tests with three 4 th grade classrooms. The initial repeated game design had a probability of continuation of 1/6, represented by a die roll on screen after each round.
The children were confused by this aspect of the game and teachers confirmed that the children at this age do not comprehend probabilities. Therefore, we modified the repeated game to be fixed length. We pilot tested this version of the repeated game and the one-shot games with different 4 th grade classrooms and tested the cartoon video used to introduce the game.

Procedure
Children were first asked if they would like to play the game (an approved assent procedure in addition to parent consent). Two children opted not to participate. Children then watched a short video introducing the interface and describing the basic features of either the one-shot or the repeated game, depending on condition: i) children would not know who their partner in the game was; ii) they would play a series of games with a different partner each time; iii) boys played with boys and girls played with girls (where applicable); iv) players would earn points during the game that they could use to "purchase" prizes on a different day. In the video, children in the one-shot condition were told that they would play multiple, single interaction games, changing partners for each game. Children in the repeated condition were told that they would play several 6round games, playing with the same partner for each game and then changing to a new Next, children did practice trials with the live interface to ensure that they understood the consequences of each combination of decisions. Once children had logged into the system, the experimenter instructed all of the children to click the push or pull button (order counterbalanced for the repeated and one-shot classes); then all to click the other button. Next, the experimenter instructed half the class to click push, and half to click pull; then the opposite; and last two again dividing the class differently, after research assistants checked each child's payoffs, to ensure that all had experienced all four possible payoffs.
Children were then shown the prizes that could be earned in order to incentivize them to take their decisions seriously. They were allowed to ask questions before the testing began and were told that talking was not allowed once the study started. The experimenters returned about one week after the testing session to allow children to select prizes based on the number of points earned -the "prices" of the prizes were scaled so that all children received at least one prize.
Before starting the testing session, barriers were set up on tables to reduce interaction between the children and prevent them from seeing each other's screens.
Children were reminded that there was no talking allowed once the study began. The  (Table 2). 1,790 decisions were analyzed.

Strengths and Difficulties Questionnaire
The SDQ was sent home to parents along with the consent forms. Fifty-eight parents completed the SDQ (91%). The children in our game were from a typical classroom sample, and we did not have clinical diagnoses for the children. However, the SDQ scoring site (http://www.sdqinfo.org/) notes that for community samples, 10% of the children typically fall in the abnormal range of the scale. This was the case for children in our sample on the Conduct Problems scale, and these children constitute the High Conduct Problems group. See Table S1 for the full questionnaire with the Conduct Problems items in bold.

Gender and Conduct Problems
To facilitate interpretation of the coefficients in the final model we calculated the odds ratios for the main predictors. Cooperation was lower for males so we used the reciprocal odds ratio (1/exp(coeff)). This showed that girls were 2.49 times more likely to cooperate than boys. Higher conduct problem scores also predicted lower cooperation, but Conduct is a continuous variable so we used 1-exp(coeff) to calculate the marginal decrease. This showed that a one point increase in conduct problems score predicted a 15% decrease in the likelihood of cooperation.

Cooperation over rounds
Cooperation declined over the testing sessions in both the one-shot and the repeated conditions, but overall cooperation remained higher in the repeated games (Figure S1.) Figure S1. Cooperation over 30 rounds of play in the one-shot and repeated games.