Conditional cooperation and confusion in public-goods experiments

Experimental Methods.

For an overview of our experimental design and treatment orders, please see SI Appendix. We included 24 players in each of our three sessions. All decisions were anonymous, and our players did not know who else was in their group. The mean self-reported age of our players was 39.7 ± 1.9 y (SEM), ranging from 18 to 74 y, and 38/34 reported that they were female/male.

We first gave our participants the instructions (SI Appendix) and standard control questions (SI Appendix), both copied verbatim as much as possible from ref. 2. We then told the players that before they would play with people, they would first be playing in a special case with computer players. The full text is available in SI Appendix, but the key text is “Before you begin, you are going to play this game in a special case. In this special case, you will be in a group of just you and the COMPUTER”; “The computer will pick the decisions of the other 3 players. The computer will pick their decisions randomly and separately (so each computer player will make its own random decision)”; and “You are the only real person in the group, and only you will receive any money.” In contrast to a previous study (34), our initial game instructions did not mention playing with computer players, to prevent the possibility of our players being distracted during the instructions on the game’s payoffs or during the control questions by thoughts of computer players. All players had to click an on-screen button with the words, “I understand I am only playing the computer” before they could proceed.

After explaining that they would be playing with computers and that nobody else could benefit from their contributions, we then explained that they would be playing the strategy method, using the same instructions copied verbatim as much as possible from ref. 2 (SI Appendix). Again, we only told our players about the strategy method after the initial instructions and control questions, and after we told them they would be playing with computers (instructions copied and as much as possible from ref. 2). This way their strategies were not merely recording the decisions of players that had already made up their mind of how to strategically respond to human players. Instead, they represented behavior in response to a choice that had no social consequences and therefore should not reflect any social preferences or concerns other than how to maximize personal income (“You are the only person in the group, and only you will receive any money”; SI Appendix). To prevent any learning, we did not provide our players with any information on their earnings from this game.

After the strategy method, which we used to categorize our participants as has been done previously (1⇓⇓⇓⇓–6), we then had our participants play a typical public-goods game, again with computers. In the typical public-goods game, players have to decide simultaneously on a single contribution with no knowledge of each other’s actions (SI Appendix). This way we could compare behavior in the strategy method with behavior in the typical public-goods game setting where conditional responses to the behavior of other players are not possible. We then had our participants continue to play the typical public-goods game, but now with people, for six rounds (SI Appendix). We provided no feedback during these six rounds; therefore, they were essentially a one-shot game that prevented signaling, reciprocity, and learning and therefore minimized any order effects (51⇓–53), “You WILL NOT receive any information about the decisions of the other players, nor about your earnings in these rounds, nor will anyone else at any time except for the experimenter after the experiment.” (SI Appendix).

The first two treatments, with computers, are diagnostic tests of understanding and follow on naturally from the control questions used by others (1, 2). More specifically, our measure was of who did and did not maximize their earnings when there were no social concerns and provided a baseline measure of income maximization behavior. We could then compare our measure of income maximization with behavior in the subsequent public-goods game with people, where there were social concerns, rather than simply attributing all failures of income maximization to social preferences as has been done before (1, 2, 35, 59).

We did not counterbalance the order of our treatments because we wanted to first classify our participants on their ability to maximize their personal income before allowing our participants to play with humans. Overall our treatment order provided a logical progression (49, 50), and there was no reason to suspect treatment order effects as in all cases we provided no feedback and therefore any treatment-order effects should be minimal. We are also not arguing for any differences in behavior between treatments that could be correlated with or confounded by experience or time. Furthermore, we can compare our results in our opening treatments of games with computers to the highly stylized results of prior published research on public-goods games. As we show below (SI Results, Table 1, and Table S1), our distribution of social types is statistically equivalent to the prior research. In addition, as we also show below (SI Results) our players’ behavior in the unconditional game with computers is well in line with previous research. Our players contributed 39% of their endowment, which is very similar to the results reported in other studies with humans (2).

Statistical Methods.

Below is a glossary of the terms we use in the results.

• • FET: Fisher’s exact test for analyzing two-way contingency tables of categorical data.

• • LM: Linear model. Used when modeling continuous data (e.g., mean contributions across six rounds of play) with one data point per individual.

• • GLM: Generalized linear model with binomial-logit link. Used when modeling binary or proportional data (e.g., correct/incorrect answer; or contributions from 0 to 20 MU) that came from a single observation per individual (e.g., contributions to the computers in the one-shot game). Parameter estimation used the Hybrid method, with the Pearson χ² scale parameter, and a model-based estimator of the covariance matrix. Effect statistics used the Likelihood ratio χ² and profile-likelihood confidence intervals.

• • GLMM: Generalized linear mixed model with binomial-logit link. Used when modeling binary or proportional data that came from repeated measures of the same individuals. We specified a first-order auto-regressive covariance structure for the residuals to control for repeated measures of the same individuals across rounds. Degrees of freedom were varied across tests according to the Satterthwaite approximation. Significance tests used model based covariances.

The Distribution of Behavioral Types and Altruism Toward Computers.

Refs. 1 and 2 were the first to classify players on the basis of their pattern of contributions in the strategy method. They separated their players into four categories, defining three social types, termed conditional cooperators, humped/triangle cooperators, and free riders. Anyone they could not classify into these categories was described as other (1) or unclassifiable/confused (2). We classified our behavioral types according to this classification scheme but with the addition of two more types. Below we outline the criteria for all of the above categories and also how we categorized our players.

Conditional cooperators (n = 36; 50%) are defined by the method of ref. 1 that classifies conditional cooperators as those that have a contribution schedule that has a significant and positive Spearman’s rank correlation with the mean contribution of the three groupmates. In total, 36 of our 72 participants met these criteria at the P < 0.01 level (players 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 18, 23, 27, 29, 31, 32, 33, 35, 36, 39, 45, 46, 48, 49, 51, 56, 57, 58, 64, 67, 69, 70, and 72; SI Appendix). Two other participants (players 21 and 40; SI Appendix) also had significant correlations at the conventional P < 0.05 level (P = 0.013 and 0.020 respectively) but we did not classify them as conditional cooperators; instead, they were humped and unclassified (see below), respectively.

Humped cooperators (n = 4; 6%) (also termed triangle cooperators, e.g., in ref. 2) are defined as those who both “increase their contributions with the contribution of others up to a point” and then “decrease their own contributions the more others contribute” (2, p. 542). We found that four of our players displayed such a pattern (players 21, 30, 42, and 62; SI Appendix), first increasing and then decreasing, on average, after an idiosyncratic threshold point, with a peak contribution when their computerized groupmates contributed 16, 10, 7, and 7 MU, respectively.

Negative cooperators (n = 3, 4%) are shown by a further three of our players (players 26, 50, and 65; SI Appendix) who exhibited a pattern that was not evident in ref. 1 or described in ref. 2, but is similar to both the above two patterns. We termed these participants as negative cooperators, because they showed significantly conditional cooperation (significant Spearman’s rank correlation), but with a negative correlation. When comparing the statistical distributions of our study with previously published studies, we needed to have the same number of categories, so we decided to categorize these negative cooperators as humped cooperators. This recategorization was because all of them can be described by the same rule as humped cooperators, increasing contributions and then decreasing contributions the more others contribute after a point (2, p. 542), if one considers their idiosyncratic turning point to be 0 MU.

Unconditional cooperators (n = 5; 7%) were any player that exhibits a constant positive contribution regardless of what their groupmates contribute. Ref. 1 classified their one such player as other. It is not clear if ref. 2 classified such players as confused/unclassifiable or did not find any such participants. We found five (7%) such players (players 19, 24, 55, 60, and 63; SI Appendix).

Noncooperators (n = 15; 21%) are quite simply those that contribute 0 MU regardless of what their groupmates contribute and thus are in one sense similar to the above unconditional cooperators. These players in other studies are typically termed free riders, as they metaphorically enjoy the benefits of public goods such as a publicly funded transport service, without contributing to the costs. This is the payoff-maximizing strategy in the typical public-goods game and thus should be exhibited by all participants, regardless of social preference, when playing with computers, if they understand the game. We found 15 such players (players 1, 11, 20, 22, 25, 28, 37, 43, 44, 47, 52, 59, 61, 66, and 71; SI Appendix). We also had nine players (12.5%) that we chose not to categorize (unclassified; players 12, 17, 34, 38, 40, 41, 53, 54, and 68; SI Appendix).

As explained in the main text, the distribution of our different behavioral types when playing with computers is strikingly similar to that previously observed when individuals are playing with other humans. In comparison with refs. 1 and 2, we find no difference between all three distributions (χ² test: χ²₍₆₎ = 6.4, P = 0.384); no difference between our results and the first study (1) (χ²₍₃₎ = 4.2, P = 0.240), or the latter study (2) (χ²₍₃₎ = 3.8, P = 0.288), or compared with a combination of both the first and the latter study (χ²₍₃₎ = 5.2, P = 0.156).

We primarily compared our distribution to those reported by refs. 1 and 2 as these were the pioneering studies that developed the method we replicated. However, there are also other published studies measuring the frequency of conditional cooperators, primarily for the purpose of cross-cultural comparison. We therefore also compared our distribution of behavioral types to a collection of prior published distributions generated using the strategy method of refs. 1 and 2.

Including this study, we found, in total, eight studies detailing a total of 14 distributions from various locations using a total of 708 players. Two of the studies were with computerized groupmates and 12 with human groupmates (Table S1). In the 12 studies with humans, the frequency of conditional cooperators ranged from 42% to 63% for 11 of them, with one study reporting 81%, and ours was 50%. The frequency of noncooperators (free riders) ranged from 2% to 36%, and ours was 21%. The overall mean frequencies for each type for all human and computerized studies are reported in Table S1.

To compare the reported distributions for the 12 human studies with the two computerized studies, we used a χ² test on the combined absolute distributions of all human studies compared with both computer studies, and we found no significant difference (χ² test: χ²₍₃₎ = 2.6, P = 0.458; Table S1). For completeness, we also tested the distribution of each type separately, using FET, and found no significant differences for any type (conditional cooperators: P = 0.470; humped cooperators: P = 0.600; unclassified: P = 0.602; free riders: P = 0.168) (Table S1).

Play with Computers Predicts Play with Humans.

We found that our players again cooperated with computers, contributing on average 39% of their endowment (7.7 MU, median = 8 MU, mode = 0 MU), a result in line with findings in other studies with humans (2). They then cooperated likewise with humans (mean = 41% of endowment, 8.1 MU, median = 8 MU, mode = 0 MU). As reported in the main text, we also found that the behavioral type in the strategy method with computers significantly predicted the contributions in an unconditional game with both computers and humans. The results in the main text are based on the classification scheme of ref. 2, which has the four categories presented in Fig. 2 (noncooperator, conditional cooperator, humped cooperator, and unclassified). The results are qualitatively the same when we use our classification scheme presented in Fig. 1, which has six categories (the same as ref. 2 plus negative cooperators and unconditional cooperators). The six-way classification scheme significantly predicted the one-shot contribution with computers (computers: GLM: F_5,66 = 4.9, P = 0.001) and all six rounds with humans (the complete absence of any feedback on the behavior or earnings of the players converted the six games with humans into essentially one-shot encounters), regardless if we ran a LM of each player’s mean contribution over six rounds (F_5,66 = 4.3, P = 0.002) or a GLMM of all six contributions, controlling for individual (F_5,95 = 5.5, P < 0.001). The six-way classification scheme also significantly predicted contributions for four of the six human rounds analyzed separately (R1: F_5,66 = 2.6, P = 0.032; R2: F_5,66 = 3.3, P = 0.010; R3: F_5,66 = 4.3, P = 0.002; R4: F_5,66 = 1.8, P = 0.134; R5: F_5,66 = 1.6, P = 0.172; R6: F_5,66 = 5.7, P < 0.001; Fig. S1). There was also no significant interaction between behavioral type and groupmate type (computer or humans), suggesting that all types treated humans and computers alike (GLMM testing for an interaction effect on contributions between behavioral type and nature of groupmates: F_5,473 = 0.470, P = 0.799).

Furthermore, at the individual level, contributions made in one-shot games with humans were correlated with and showed no significant difference to contributions in the one-shot game with computers. This significant similarity was true for the first round with humans (mean difference = 0.57 MU, correlation = 0.73, P < 0.001; paired t test t₍₇₁₎ = 0.96, P = 0.339), for each and every round, and for the mean contribution across all six rounds (mean difference = 0.38 MU, correlation = 0.78, P < 0.001; paired t test t₍₇₁₎ = 0.73, P = 0.471; Table S2).

As described in the main text, we also compared whether individuals differed in how they adjusted their behavior conditionally, depending on whether they were playing computers or humans. At the same time that they made their contribution decision, we asked all individuals what they expected the mean contribution of their groupmates would be. We did this to test whether (i) players’ contributions in the unconditional rounds are a function of their expectations about their groupmates’ mean contributions; (ii) their functions differed when playing with computers or humans; and (iii) their functions in the unconditional rounds corresponded to and were consistent with their behavior in the strategy method.

First, as in previous studies, and reported in the main text, we found that contributions were positively correlated with the amount that they expected their human group mates to contribute, confirming that one does not necessarily need to incentivize the reporting of expectations (GLMM on six rounds of data, contribution ∼ expectation: F_1,405 = 152.9, P < 0.001, β = 0.210 ± 0.017; Fig. S2). This correlation was true overall and separately for each of the six rounds with humans (GLM round 1: F_1,70 = 59.1, P < 0.001, β = 0.32 ± 0.054, R²_adj = 0.49; GLM round 2: F_1,70 = 51.4, P < 0.001, β = 0.26 ± 0.044, R²_adj = 0.50; GLM round 3: F_1,70 = 37.1, P < 0.001, β = 0.23 ± 0.047, R²_ad = 0.41; GLM round 4: F_1,70 = 41.6, P < 0.001, β = 0.21 ± 0.038, R²_adj = 0.46; GLM round 5: F_1,70 = 50.8, P < 0.001, β = 0.22 ± 0.039, R²_adj = 0.50; GLM round 6: F_1,70 = 49.3, P < 0.001, β = 0.23 ± 0.040, R²_adj = 0.46; note R²_adj are from LMs rather than GLMs that control for the binomial error structure).

Second, as we reported in the main text, we also found qualitatively the same relationship between contributions and expectations when playing with computers (GLM: F_1,70 = 17.0, P < 0.001, β = 0.17 ± 0.046, R²_adj from a linear model = 0.22; Fig. S2), and analyzing all of the data together shows that quantitatively this correlation did not significantly depend on whether groupmates were computers or humans (GLMM: interaction between nature of groupmates and expectations, F_1,486 = 2.5, P = 0.116, difference in β = 0.05 ± 0.034; Fig. S2). The above results were robust to various forms of analysis. The overall result remained qualitatively the same if we changed from model-based covariances to using robust covariances instead (F_1,96 = 1.1, P = 0.296). The interaction also remained nonsignificant if we compared play with computers with each round of play with human groupmates separately (GLMM, computer vs. human round 1: F_1,80 = 1.0, P = 0.330, difference in β = 0.039 ± 0.040; computer vs. human round 2: F_1,83 = 0.4, P = 0.520, difference in β = 0.025 ± 0.039; computer vs. human round 3: F_1,92 = 0.0, P = 1.000, difference in β = 0.000 ± 0.044; computer vs. human round 4: F_1,94 = 0.1, P = 0.720, difference in β = 0.016 ± 0.045; computer vs. human round 5: F_1,86 < 0.1, P = 0.885, difference in β = 0.006 ± 0.042; computer vs. human round 6: F_1,76 < 0.1, P = 0.918, difference in β = 0.004 ±0.040).

We also repeated the above analyses, comparing contributions as a function of expectations between the round with computers and the six rounds with humans, but separately for noncooperators (n = 15) and conditional cooperators (n = 36). Neither conditional cooperators nor noncooperators differ in how they do or do not condition their contributions on their expectations depending on the nature of their groupmates (GLMM: interaction between expectations and groupmates on contributions, conditional cooperators: F_1,243 = 0.001, P = 0.973; noncooperators: F_1,101 = 3.418, P = 0.067).

For completeness, we also compared the extent to which individuals contributed more or less than they expected their groupmates to contribute and compared whether this differed depending on playing with humans or computers. For instance, did those that contributed a little less than they expected of their human groupmates (selfish biased conditional cooperation; refs. 1 and 2) do the same when playing computers? This allowed us to ask if behavior differed when playing with computers or humans while controlling for expectations; for example, maybe players cooperated with computers but became more generous with humans? For the sake of simplicity, we merely compare the first, or final, round of play with humans with the round of play with computers, using a paired-samples t test to control for individual.

On average, individuals contributed 1.1 (first round) or 1.5 MU (final round) less than they expected of their human groupmates and 2.3 MU less than they expected of the computer. These decisions were not significantly different in magnitude (paired t tests, first round: t₍₇₁₎ = 1.9, P = 0.062; final round: t₍₇₁₎ = 0.9, P = 0.375). Repeating the above process for the behavioral types reveals the same results for the conditional cooperators (paired t tests, first round: t₍₃₅₎ = −0.1, P = 0.936; final round: t₍₃₅₎ = 0.4, P = 0.657); humped cooperators (paired t tests, first round: t₍₆₎ = 0.3, P = 0.746; final round: t₍₆₎ = −0.6, P = 0.601); and the unclassified players (paired t tests, first round: t₍₁₃₎ = −1.2, P = 0.258; final round: t₍₁₃₎ = 0.2, P = 0.813). Noncooperators, in contrast, do show a significant difference, contributing 7.8 MU less than their expectations of the computer, but only 3.0 (first round) or 3.5 MU (final round) less than their expectations of humans (paired t tests, first round: t₍₁₄₎ = −2.6, P = 0.020; final round: t₍₁₄₎ = −1.9, P = 0.08). However, this is not because they contributed more to humans (they tended to contribute 0 MU regardless) but because they had higher expectations of their computerized groupmates (mean = 9.1 MU) than their human groupmates (mean = 5.3 MU; GLMM: F_1,99 = 9.5, P = 0.003), leading to a larger discrepancy.

Third, we then investigated whether their contributions in the unconditional rounds corresponded to and were consistent with their behavior in the strategy method. Methodologically, if a participant contributed more in the one-shot game than they did in the strategy method for the corresponding expectation, then we scored them as having a positive discrepancy and vice versa (Fig. S3). For the sake of simplicity, we only analyzed behavior from the first round of play with humans.

When comparing individual contributions, as a function of expectations, with behavior in the strategy method, nearly 50% of the 72 participants were perfectly consistent, contributing the identical amount in the one-shot encounter as they did in the strategy method for the corresponding expectation. This was true regardless if their groupmates were computers (34/72, 47%) or humans (33/72, 46%, FET: P = 1.000) and also suggests affective forecasting was not a serious problem (Fig. S3).

We found that our participants, when playing with either computers or humans, were equally likely to be consistent with their prior decision in the strategy method. We also found that those that deviated, also deviated equally, in both absolute terms (mean absolute discrepancy; vs. computers: ±2.46 MU and vs. humans: ±3.29 MU, paired t test: t₍₇₁₎ = 0.83333, P = 0.074) and in net terms (mean net discrepancy; vs. computers: +0.9028 MU, and vs. humans: +0.9028, paired t test: t₍₇₁₎ = 0.0, P = 1.0). These identical net discrepancies (+0.9028) were not significantly different from 0 MU, either when playing with computers (LM: intercept, F_1,71 = 3.085, P = 0.083) or, more importantly, when playing with humans (LM: intercept, F_1,71 = 2.014, P = 0.160; Fig. S3).

In summary, although around 50% of players are not perfectly consistent when transferring from the strategy method to the one-shot game, their shift in behavior does not depend on their group-mates being computers or humans, and they show no prosocial shift, being equally likely to become either more or less favorable to their human groupmates (mean net discrepancy of 0.90 MU not significantly different to 0 MU; Fig. S3).

Conditional Cooperators Misunderstand the Game.

After we had our participants first play the strategy method with computers, then a one-shot game with computers, and then six one-shot games with humans, all without feedback to prevent any learning about the game’s payoffs or the propensities of other players, we then asked them a question to test whether they understood the game. Specifically, the question tested whether they knew that the payoff-maximizing strategy was independent of what other players contributed, or did they incorrectly believe that the game’s payoffs were interdependent. The question read “In the game, if a player wants to maximize his or her earnings in any one particular round, does the amount they should contribute depend on what the other people in their group contribute?” We allowed them to answer either yes/sometimes/no/unsure. Note as this was a multiple-choice format, some players may have answered correctly merely by chance. The full breakdown of results is available in Table S3, with responses separated by behavioral type as measured in the preceding strategy method with computerized groupmates.

As the classification scheme for different types can be made more or less complex, we have analyzed the responses using several different classification schemes, using logistic regression (GLM with binary-logit link) to test whether players responded correctly (chose no) or not (chose yes/sometimes/unsure). In all cases, the behavioral types correlated with the answers to our question on understanding (Table S4). Our most complex classification scheme (expansive) is represented in Fig. 1. It contains six behavioral types; the four from ref. 1, plus our two extra, which are negative cooperators and unconditional cooperators. This scheme significantly predicts if a player will answer our question about the nonstrategic nature of the game (GLM: χ²₍₅₎ = 15.3, P = 0.009). The next most complex uses the same criteria as ref. 1 but also includes negative cooperators (F01 + negative), a phenomenon not occurring in ref. 1 and not documented in ref. 2. Again this scheme significantly predicts correct answers or not (GLM: χ²₍₄₎ = 14.8, P = 0.005). The next most complex classification scheme (F01) is simply that of ref. 1. Here we had to choose which category is most suitable for the negative cooperators; as mentioned above, we chose to classify them as humped cooperators because humped cooperators also show a negative correlation, after an idiosyncratic point, between contributions and the contributions of others. Again this scheme significantly predicts understanding (GLM: χ²₍₃₎ = 12.9, P = 0.005).

More simply, we have a three-way classification scheme, which combines conditional and humped cooperators into one category compared with noncooperators and unclassified. Again this scheme significantly predicts understanding of the game (three-way, GLM: χ²₍₂₎ = 12.9, P = 0.002). Finally, we have two simple, binary classification schemes. First, do players cooperate (contribute) or not? The division of cooperators and noncooperators significantly predicts understanding (GLM: χ²₍₁₎ = 11.9, P = 0.001). As does our final scheme, which simply classifies players as making constant contributions (noncooperators and unconditional cooperators) or inconstant contributions (conditional, humped cooperators, and unclassified). The constant players may or may not cooperate but show no form of conditionality in any sense and thus act as though they believe the game’s payoffs are not interdependent. Again, this classification scheme significantly predicts the correct response to the question testing their beliefs and understanding about the game’s payoffs (GLM: χ²₍₁₎ = 12.1, P = 0.001).

Finally, evidence that many players were uncertain in how to play the game comes from the lack of stability in contributions exhibited by our players in the preceding six rounds of play with human groupmates. Such instability suggests a degree of bet hedging by many players. In these rounds, which had no information feedback and thus no learning about the game or one’s groupmates could occur, only 21 (29%) of the 72 players remained constant, and 13 of these were noncooperators (contributed 0 MU). On average, the 72 players changed their contributions by ±3.2 MU from the previous round (mean absolute changes for rounds 2, 3, 4, 5, and 6, respectively: 2.4, 3.3, 3.8, 3.3, and 3.2 MU). On average, the mean range in contributions (maximum − minimum contribution) was 7.3 MU, with 10 (14%) players exhibiting the full range possible (0–20 MU).

Standard Control Questions Fail to Control for Understanding.

The control questions are available in SI Appendix. We have sufficient experience of running public-goods game experiments to confirm the difficulty in getting all players to answer questions correctly in a timely manner. An experimenter nearly always has to get involved, either in person or on-screen, and try to help without helping too much, as is clear from the following quotes; “After completion of the questionnaire, the questions were publicly solved. Any remaining questions were answered in private.” (3, p. 176); “Subjects who showed problems in understanding the task were given assistance.” (4, p. 89); “Once all participants had completed the exercises, the experimenter solved them in public. Any remaining questions that the subjects had were then answered in private.” (6, p. 151); “if any participant repeatedly failed to answer correctly, the experimenter provided an oral explanation.” (50, p. 200) (bold emphases added. An anonymous reviewer also claimed that in ref. 2: “Before the start of the experiment subjects had to read the instructions and to answer control questions and were instructed again when necessary”).

Clearly such processes can influence the participants and allow participants through that have not independently answered all questions. For this reason, we allowed our players to answer as they wished, without any further instruction. This allowed us to identify where each participant went wrong, to identify sources of misunderstanding, and to quantify, for the first time to our knowledge, how well people respond to such questions.

It is not always clear if a player answered the control questions correctly. In the strictest sense, 16 players (22%) answered all 10 questions, pertaining to the four scenarios, perfectly. The breakdown per scenario was as follows: scenario 1 (two questions), 47 of 72 (65%) players answered correctly; scenario 2 (two questions), 39 (54%) players answered correctly; scenario 3 (three questions), 23 (32%) players answered correctly; and scenario 4 (three questions), 37 (51%) players answered correctly. These correct response rates differ significantly (χ² test: χ²₍₃₎ = 16.6, P < 0.001). However, some participants answered in ways that could be interpreted as correct based on certain ambiguities in the questions. For example, in scenario 1, nobody contributes, so nobody makes a profit on his or her initial endowment of 20 MU. Some players incorrectly entered 0 MU, perhaps meaning a gain of 0 MU relative to the endowment, although the question asked them to calculate the “total income (in MU).” Six players appeared to correctly calculate the returns from the public project, but either forgot or thought that one does not add the saved, noncontributed, MU. To be generous, one could also categorize these players as passing the test, to give a total of 22 players (31%) that answered the questions correctly.

As a robustness check of the results in the main text, we analyzed our data from another, larger, study conducted in the same laboratory. This study had 216 players and was very similar in that the players had near identical instructions and control questions that were copied verbatim from ref. 35 instead of ref. 2. Ref. 35 uses eight control questions that are an identical subset covering the same four scenarios of ref. 2 but with only two subquestions instead of three for scenarios 3 and 4. These players then also played a one-shot unconditional game with the computer and the six-stage one-shot unconditional game with humans and answered our control question on whether the game is interdependent or not (but in the postexperiment questionnaire rather than in the experiment). The main difference was that they did not complete a strategy method game.

Of these 216 players, 78 (36%) correctly answered all eight questions, and of these, only 41 (53%) maximized their income (contributed 0 MU) when playing with computers. These 78 players contributed no more on average when playing with humans (5.4 MU) than they did when playing the computer (4.6 MU, paired t test: t₍₇₇₎ = 1.0, P = 0.311). Regarding our control question on whether the game is interdependent or not, we found that only 29 (37%) of the 78 correctly answered that the game is not interdependent. However, 21 of these 29 (72%) maximized their income when playing with the computer, significantly more than compared with the 20 of the 49 (41%) that answered our question incorrectly (FET: P = 0.0097). Thus, our results are robust when analyzing a larger sample than presented in the main text. This larger sample confirms that answering the standard control questions does not guarantee that players understand the game and that our control question significantly helps explain behavior even among those that answered all of the standard control questions correctly.

Comprehenders Are Not Cooperators.

We conducted paired t tests between individual contributions to the computer and contributions to humans for those that could be deemed to understand the game (comprehenders). We examined the behavior of three types of players that could all be argued to be comprehenders. First, all of the individuals that contributed 0 MU in both the strategy method and in the one-shot game with the computer (n = 13 of 72, 18%) and thus could be argued to best comprehend the game as they demonstrated they knew the income-maximizing strategy. Of course some of these could be false positives as some players may not fully understand the game but still contribute 0 MU for various reasons. Second, the individuals that answered all of the standard control questions correctly and thus have previously been assumed to understand the game (n = 16, 22%) (1⇓⇓⇓⇓–6). Third, the individuals that passed our beliefs test and arguably understood the essential dilemma of the game (n = 21, 29%). Again some of these may be false positives, for two reasons: (i) as the test was a multiple-choice question meaning some could guess the correct answer; and (ii) some players may correctly think that the decisions of one’s groupmates are irrelevant to one’s attempts to maximize earnings but incorrectly believe that to contribute is always profitable.

In all three cases, we find that comprehenders are no more cooperative with humans than they are with computers. First, the 13 players that maximized their income when playing computers did not contribute significantly more than 0 MU in any individual round with humans (Table S5). The mean contributions by round were as follows: 1.6; 0.6; 1.0; 2.1; 1.9; and 0.0 MU for the six rounds, respectively. None of these were significantly different from 0 MU (paired t tests: R1, t₍₁₂₎ = 1.3, P = 0.224; R2, t₍₁₂₎ = 1.0, P = 0.337; R3, t₍₁₂₎ = 1.3, P = 0.227; R4, t₍₁₂₎ = 1.5, P = 0.168; R5, t₍₁₂₎ = 1.4, P = 0.175; R6, t₍₁₂₎ = 0.0, P = 1.0), nor across all six rounds combined (paired t test: t₍₁₂₎ = 1.957, P = 0.074).

Second, the 16 players that answered all of the standard control questions correctly showed no prosocial bias toward humans, giving just as much to computers (4.8 MU) as they did to humans in each and every round. They did not contribute significantly more than 4.8 MU in any individual round with humans. The mean contributions by round were as follows: 5.3; 5.7; 4.6; 6.4; 6.2; and 4.3 MU for the six rounds, respectively. None of these were significantly different from 4.8 MU (paired t tests: R1, t₍₁₅₎ = 0.4, P = 0.708; R2, t₍₁₅₎ = 0.7, P = 0.482; R3, t₍₁₅₎ = 0.2, P = 0.815; R4, t₍₁₅₎ = 0.8, P = 0.450; R5, t₍₁₅₎ = 1.0, P = 0.338; R6, t₍₁₅₎ = 0.6, P = 0.562) or across all six rounds combined (paired t test: t₍₁₅₎ = 0.6, P = 0.529).

Third, the 21 players that answered our control question correctly also showed no prosocial bias toward humans. They did not contribute significantly more than the 5.1 MU they gave to the computers in the first round in five of the six subsequent individual rounds with humans. The mean contributions by round were as follows: 7.0; 6.6; 6.7; 7.1; 7.6; 6.4; and 6.9 MU for the six rounds, respectively. Only one of these was significantly different from 5.1 MU (paired t tests: R5, t₍₂₀₎ = 2.1, P = 0.048), and this would no longer be the case if we were to control for multiple testing (e.g., with a Bonferroni correction). None of the other rounds were significantly different (R1, t₍₂₀₎ = 1.4, P = 0.186; R2, t₍₂₀₎ = 1.4, P = 0.183; R3, t₍₂₀₎ = 1.5, P = 0.139; R4, t₍₂₀₎ = 1.4, P = 0.188; R6, t₍₂₀₎ = 1.3, P = 0.224) or across all six rounds combined (paired t test: t₍₂₀₎ = 1.7, P = 0.098).

Measuring Motivations.

We assumed that players playing with computers are trying to maximize their income as much as players are ever motivated to do so. We have then inferred that because behavior is unchanged when shifting from playing with computers to humans that players are equally motivated to try and maximize personal income when playing with humans. One may be tempted to ask, why not simply ask the players what their motivations were? Therefore, in a postgame questionnaire, conducted while the payments to our players were prepared, we asked our players about their motivations during their games with humans.

As this was a hypothetical questionnaire, players could make their responses appear prosocial without paying a cost to do so, contrary to typical economic decisions in experiments measuring social preferences. Therefore, responses to this question could be argued to provide a lower bound of self-interested motivations among our players. We asked, “What was your most important motivation in the games with real people? Please select the answer that best describes your motivations?” (numbers chose shown in brackets).

• Making myself the maximum money possible (36)

• Making other people the maximum money possible (3)

• Making everyone the maximum money possible (13)

• Making the group the maximum money possible (11)

• Making myself more money than other people (1)

• Other (8)

We consider options 1 and 5 to reveal a self-interested motivation (n = 37), whereas options 2–4 are prosocial (N = 27, or 35 if including others). Despite the opportunity to appear prosocial at no cost, and in contrast to the low frequency of noncooperators in our incentivized strategy method (n = 15, 21%), the most common response, chosen by 50% of players, was a self-interested motivation, “making myself the maximum money possible” (n = 36 of 72). Table S6 reveals how these responses depended on behavioral type.

Why Might Players Think That Contributing to the Public Good Is a Profitable Decision?

It has been claimed that the income-maximizing strategy is “simple to figure out” (13). However, this claim cannot be asserted; it has to be scientifically ascertained. We argue that there are several reasons why players may reach mistaken conclusions about the game. Although we do not wish to criticize the difficult job of making instructions for economic games, we think it possibly significant that the instructions we copied from ref. 2 used the word invest when describing contribution decisions. The word invest has clear financial connotations to do with risk and gain (as confirmed in a postgame survey by ref. 34), suggesting that returns are not fixed and may depend on other factors, such as what other people choose to invest. Also, the fact that the experimenter then offers players the opportunity to explicitly condition on the behavior of others would only further reinforce this belief (27, 28). Furthermore, the instructions specify that each MU contributed returns 0.4 MU, but this does not clearly specify that they incur a cost of −0.6 MU, and judging from the answers to the control questions, it seems that some participants thought that each MU contributed returned 1.4 MU (i.e., the 0.4 MU return was pure profit). Evidence of this potential confusion comes from a study showing that cooperation is significantly reduced in the public-goods game when players are explicitly informed that they “lose money on contributing” (63). Finally, one reason that people may contribute is that they may think contributing something is a necessary requirement to benefit from the public good (e.g., paying to become a member of a club/group in the world). These mixed reasons may explain why many of our players varied their contributions during the six rounds of play with humans. Such within-individual variation suggests some players used a bet-hedging strategy in response to a risky or uncertain decision environment.

However, why would the standard control questions (SI Appendix) not prevent these erroneous beliefs? An examination of the specific questions will show that the first two scenarios do little other than reinforce these false beliefs. Specifically, they show that if no one contributes, then no one makes any additional money (scenario 1), but then conversely if everyone contributes fully, then everyone makes a lot of money (scenario 2). The only hint that contributions are costly comes from scenario 3, whereby players can see that for the case where the rest of the group invests 30 MU, to contribute 0 MU is better than 10 or 15 MU. However, scenario 4 then shows them that they benefit the more others contribute, specifically if they invest 8 MU, they do better if the others contribute 22 MU instead of 12 or 7 MU. Tellingly, players struggle the most with scenario 3, with only 32% of player answering correctly, compared with 65%, 54%, and 51% for scenarios 1, 2, and 4, respectively. This significant difference means that if experimenter assistance is provided, it is most likely to be for scenario 3, which is the only scenario that demonstrates there is a social dilemma.

Why Might Players Show a Correlation Between Contributions and Either Expectations of Others or the Actions of Others?

Again, without wishing to criticize the instructions, they were perhaps not ideal, as they specifically state to the participants that they can “condition their contribution on that of the other group members” (2). In addition, the input screen asks them to enter their “conditional contribution to the project” (2). This suggestive language implies that this is something they may wish to consider doing and therefore may act as a demand characteristic for the game (27, 28).

There are also other reasons why contributions may correlate with either expectations or the decisions of others, without reflecting concerns for fairness or inequity aversion (59). Conditional cooperators are considered to have social preferences because their decisions correlate with their knowledge and beliefs about the contributions of their groupmates. However, this correlation does not automatically imply social preferences. If we relax the assumption that players perfectly understand the game, with full confidence, then there are three alternative reasons, none of which require any social preferences, as to why this correlation may exist: (i) if a player thinks that “contributing X MU is best,” then it is reasonable for them to conclude that many others will think likewise, making their expectations match their own behavior, rather than vice versa; (ii) if a player is uncertain but knows that others are contributing X MU, then it is reasonable for them to conclude that others know better what to do and to copy their behavior in the hope of a better payoff, making their behavior match their knowledge of other players’ behavior; and (iii) if a player thinks that the income-maximizing response to what others contribute changes depending on what they contribute, then they will in some sense condition their contributions on these other contributions. This will lead to their behavior matching both their knowledge and expectations of others, even when others are merely computers playing randomly, as evidenced by our results.

A revealing insight from the strategy method is the presence of humped and negative cooperators. These players reduce their contributions the more their groupmates contribute. This is not consistent with a sense of fairness or social preferences. It is consistent with the appropriate strategy for a threshold, or nonlinear, public-goods game (62). In such games, a self-interested player can benefit by compensating for the lack of contributions by others players. Clearly, in such games, the income-maximizing strategy does depend on knowing the behavior of others.

It is probably reasonable to consider that some of these behaviors may be unique artifacts of the strategy method. For instance, negative and humped cooperators may not occur in unconditional versions of the game, although they might. Due to their rarity, it would require a huge sample size to capture enough of them to detect their behavior in the unconditional game. It would also require repeated games that control for learning, through the omission of feedback between rounds, to sample enough decisions from each individual. However, conditional cooperators are much more common and show consistency between play in the unconditional games and the strategy method. This means that even if the strategy method inflates the frequency of apparent conditional cooperators, concerns about the misinterpretation of their motives are not just restricted to experiments using the strategy method.