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Journal of Economic Behavior and Organization
journal homepage: www.elsevier.com/locate/jebo
Research paper
From shocks to solidarity and superstition: Exploring the
foundations of faith✩
Aidin Hajikhameneh a,∗, Laurence R. Iannaccone b
aDepartment of Economics, San Jose State University, Dudley Moorhead Hall, 1 Washington Square, San Jose, CA 95112, USA
bInstitute for the Study of Religion, Economics, and Society, Chapman University, 338 N. Glassell, Orange, CA 92866, USA
A R T I C L E I N F O
JEL classification:
C91
C92
Z12
Z13
Keywords:
Uncertainty
Risk
Collective action
Prosocial behavior
Public goods game
Lab experiment
A B S T R A C T
Additive shocks can substantially increase cooperation in otherwise standard public goods game
experiments. We study shocks that randomly adjust players’ earnings by a fixed positive or
negative amount reported at the end of each round. These adjustments change neither the
return to players’ contributions nor the information about other group members. We compare
results across four treatments that employ the same group-level adjustment algorithm but frame
it differently, with pre-play descriptions that range from omitting all useful information to
accurately revealing its 50/50 random nature. In each treatment, overall contributions run about
50% higher than those obtained in the standard no-adjustment game. Contributions run higher
still, nearly 100% over baseline, in a treatment that individualizes the adjustments, truthfully
describing them as 50/50 random and separately calculated for each player. Our results contrast
with those of previous studies, which add risk to public goods games in ways that directly
interact with players’ contributions and typically reduce cooperation. Players’ contributions and
post-play feedback strongly suggest that our results trace back to a pair of deep-rooted impulses
that boost solidarity in response to external risk and rationalize the response with superstitious
thinking.
1. Introduction
Our species [has] a genetically evolved response to war and other shocks that likely operates in at least three ways. First, shocks cause
us to invest more heavily in the social ties and communities that we rely on. Second, [we] become more cooperative along normative lines.
[Third, shocks] increase people’s religious commitments and ritual participation. (Henrich,The WEIRDest People in the World, 2020, p
327.)
Since its 1984 debut, the linear public goods game of Isaac, Walker, and Thompson has served as the benchmark experimental
framework for analyzing free-rider problems and testing proposed solutions (Isaac et al.,1984). The resulting body of research
includes hundreds of replications, variations, and extensions.1One might imagine that the literature now covers all the major
determinants of real-world cooperation that fall within the game’s purview. But a significant source of solidarity appears to have
✩We thank Christopher Bader, Hernan Bejarano, Gabriele Camera, Gary Charness, Ananish Chaudhuri, Joshua Epstein, Daniel Houser, Joseph Henrich, Hillard
Kaplan, Michael McBride, Douglas Norton, Nathan Nunn, Ryan Oprea, Jared Rubin, Eric Schniter, and Nathaniel Wilcox for valuable comments and suggestions.
Figures were created with R and experiments were programmed using z-Tree (Fischbacher,2007).
∗Corresponding author.
E-mail addresses: aidin.hajikhameneh@sjsu.edu (A. Hajikhameneh), iannacco@chapman.edu (L.R. Iannaccone).
1The literature also includes hundreds of field experiments, simulations, and theoretical contributions. Working within the standard framework, scholars have
carefully measured the impact of changes in group size, number of rounds, endowment size, multiplier size, heterogeneity, and the demographic and personal
characteristics of players. Other experiments have modified the standard game and thereby measured the impact of communication among group members,
https://doi.org/10.1016/j.jebo.2024.106775
Received 7 May 2021; Received in revised form 25 June 2024; Accepted 8 October 2024
Journal of Economic Behavior and Organization 229 (2025) 106775
Available online 30 November 2024
0167-2681/© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
A. Hajikhameneh and L.R. Iannaccone
received little or no attention. The source is seemingly irrelevant risk: additive, observable, random, and unrelated to players’ actions.
It averages to zero, neither masks nor magnifies contributions, and provides no information about the players’ character or conduct.
As we will demonstrate in a series of otherwise standard public goods experiments, cheap noise raises contributions and net
earnings more effectively than previously studied forms of risk and uncertainty. Moreover, at least one form approaches the levels
of cooperation obtained in previously studied systems of sorting, exclusion, and punishment that require detailed and ongoing
information about individual players’ behavior. In Section 4, we argue that these results are best seen as products of two closely
linked responses to elevated external risk, both so deeply rooted in culture or human nature that they operate largely at subconscious
levels. The first response is increased solidarity; the second is superstition that rationalizes increased solidarity.
Our experiment was motivated by studies of superstition by psychologists, anthropologists, and evolutionary biologists. In nearly
all of this work, uncertainty, coincidence, and group interactions stand out as key determinants of socially significant superstitions.2
Across the board, higher stakes and higher risks (both probabilistic and Knightian) lead to more superstitions, more strongly
held superstitions, and more elaborate superstitions.3Socialization, communication, and social learning strengthen and sustain
superstitions, as do activities that require greater cooperation.4The literature also provides evidence, though largely indirect, that
seemingly irrational beliefs and rituals can benefit both individuals and groups – a major theme in ‘‘functionalist’’ interpretations
of religion and magic promoted by early 20th-century anthropologists and sociologists and the subject of much recent work in
evolutionary biology, anthropology, psychology, and economics (Malinowski,1948;Hayek ,[1989] 2011;Boyer,2001;Wilson,
2010;Henrich,2009;Atran and Henrich,2010;Norenzayan,2013;Norenzayan et al.,2016;Durrant and Poppelwell,2017;Purzycki
et al.,2018;Hayek,1989;Leeson,2012).
B. F. Skinner’s ([1948] 1992) famous study of ‘‘Superstition in the Pigeon’’ showed how coincidence could generate quasi-
superstitious behavior in a simple animal experiment. Unlike Skinner’s other experiments, which shaped animal behavior by
systematically rewarding specific actions, this experiment periodically rewarded pigeons with rewards regardless of what they did.
The pigeons nevertheless developed distinctive, durable, and idiosyncratic patterns of behavior based merely on what they happened
to be doing when the food appeared.5Subsequent experiments reported similar results with rats, monkeys, and other animals.6Later
experiments with young children and college students showed that chance correlations between actions and outcomes induced
idiosyncratic and ineffectual superstitions in humans based on erroneous inferences of cause and effect (Ono,1987;Wagner and
Morris,1987;Vyse,2013).
In light of this literature, we modified the standard public goods game by adding group-level earnings shocks that vary across
groups and over time. In each period, after players choose their contributions, the computer calculates an earnings adjustment for
each group – always equal to plus or minus half the original endowment. Each player then receives an on-screen report listing his
contribution to the group account, the group’s combined contribution, his personal and group account earnings, the computer’s
adjustment, and his total earnings. In practice, the odds of positive versus negative adjustments were always 50/50, but the
descriptive framing varied across treatments. One treatment’s pregame instructions explicitly stated that the computer’s adjustments
were 50/50 random; another treatment stated only that the adjustments were based on a complex sequence of calculations, and
two others truthfully noted that higher contributions might raise the probability of positive adjustments. Appendix Cdescribes the
adjustment algorithms in detail.
Standard economic theory and simple logic imply that additively separable shocks should have little or no effect, especially
when known to be random. The adjustment process is, after all, arithmetically equivalent to a post-play series of coin flips. Yet, in
each of the five adjustment treatments, average contributions and contribution paths substantially exceeded those of their standard
no-adjustment counterpart. Statistical analysis of players’ contributions and responses to post-game questions provide many more
insights. But a simple graph of contribution trends tells the basic story: see Fig. 1in Section 3below.
Though we are by no means the first economists to study the impact of risk in public goods experiments, we appear to be the first
to study the impact of arithmetically irrelevant risk. The experimental literature on risky public goods stretches back more than 30
years, with early contributions by Van Dijk and Grodzka (1992), Fisher et al. (1995), Dickinson (1998), Marks and Croson (1999),
and others. The experiments incorporate many forms of risk or uncertainty, including variability in the contribution multipliers,
provision thresholds, endowments, and information about the contributions of other group members – all of which can alter the
contributions of a rational money-maximizing player. Some studies vary the magnitude of risk, and some vary how or when it
resolves (e.g., by providing information about the risky quantities before or after players make their contributions). To facilitate
provision thresholds, unexpected restarts, punishment, reward, player sorting, and much more. For summaries of results from these and other variations and
extensions, see the excellent review papers by Ledyard (1995), Zelmer (2003), Villeval (2020) and Chaudhuri (2011,2016).
2The findings concern populations that range from preschool children to preliterate tribes. Methods include statistical analysis of surveys and quantitative
records, direct observation, ethnographies, interviews, and studies of historical records and literature. Activities include play, sports, romance, gambling,
examinations, warfare, and dangerous work.
3Professional baseball is a particularly striking example, where superstitions are ubiquitous (very high-risk) pitching, less prevalent in (moderate-risk) hitting,
and largely absent in (low-risk) fielding (Bleak and Frederick,1998;Gmelch,1971).
4Superstitions are far more prevalent in team sports than individual sports Vyse (2013). Palmer (1989, p. 65) likewise notes the ‘‘clear correlation between
the degree of taboo and magic associated with each type of fishing and the number of men who must cooperate’’ in studies of superstition among contemporary
commercial fishers and in Malinowski’s classic study of in the ocean versus lagoon fishing among preliterate Trobriand Island tribes (Malinowski,1948,1918).
5Though Skinner only called this outcome asort of superstition, he emphasized its similarity to human superstitions in which ‘‘accidental connections between
a ritual and favorable consequences suffice to set up and maintain the behavior in spite of many unreinforced instances.’’
6As noted by Davis and Hubbard (1972), many of the reported ‘‘superstitions’’ arose in standard conditioning experiments. In contrast, their own study of
rats was a true analog of Skinner’s experiment with both fixed and variable reinforcement schedules.
Journal of Economic Behavior and Organization 229 (2025) 106775
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A. Hajikhameneh and L.R. Iannaccone
comparisons, nearly all the experiments employ forms of risk or uncertainty that, for any given set of players’ actions, yield the same
expected payoffs as the corresponding baseline model, and they usually achieve this neutrality by replacing the baseline model’s
parameters with offsetting pairs of equally probable values.
Past studies have typically found that risk or uncertainty has a weakly negative impact on overall contributions. Similar results
occur in experiments that incorporate risk or uncertainty in other social dilemma games, including prisoner’s dilemma, stag-hunt,
coordination, trust, and common resource pool games (Dickinson,1998;Killingback et al.,1999;Gangadharan and Nemes,2009;
Levati et al.,2009;Levati and Morone,2013;Fischbacher et al.,2014;Bjork et al.,2016;Stoddard,2013;Stoddard et al.,2015;
Aksoy and Krasteva,2020;Bejarano et al.,2021;Vesely et al.,2017;Krawczyk and Le Lec,2016;Butera et al.,2020).
Increased contributions are, in fact, so uncommon that when Butera and List (2017) ran an experiment in which multiplier
uncertainty sometimes raised contributions, they launched a special replication project with help from colleagues at three other
universities. When the project failed to replicate the original findings, Butera et al. (2020, p 25) concluded that ‘‘Knightian
uncertainty likely has a limited impact on cooperation, corroborating the existing [traditional VCM] approach of focusing on strategic
uncertainty to study public goods.’’ Citing the results of several different studies, Théroude and Zylbersztejn (2020, p 405) likewise
note ‘‘the robustness of the standard patterns of cooperation not only across different domains of (noncompetitive) risk but also
across different domains of (noncompetitive) randomness: risk, ambiguity, and uncertainty.’’ They conclude that these results provide
‘‘strong support for the ‘gold standard’ approach to studying human cooperation by means of the VCM paradigm, in which the MPCR
from the public good is deterministic, homogeneous, and publicly known.’’
Our results suggest that these conclusions do not apply as broadly as previously imagined. Indeed, it may turn out that adding
seemingly irrelevant forms of risk or uncertainty change outcomes in a great many experimental settings.
2. Experiment design and treatments
The public goods game provides a natural starting point for experiments on superstition – partly because of its simple structure
and trade-offs, but also because of its repeated play in stable groups and the ease with which uncertainty could be introduced. Our
treatments were all variants of the standard (repeated linear) public goods game. Sessions involved 16 players seated at computer-
equipped cubicles, randomly assigned to anonymous unchanging 4-person groups, and playing for 20 periods. At the start of every
period, each player received a 10-token endowment and, by entering a number between 0.00 and 10.00, chose how much to
contribute to the group’s joint account. Any remaining tokens went into the player’s personal account, and group investments were
then pooled, multiplied by 1.6, and divided equally among the group’s members.
The Baseline Treatment: Our baseline treatment was a standard public goods game with payoffs determined entirely by the
contributions of group members. Hence, for any given period, 𝑡, player 𝑖’s earnings could be written 𝜋𝑖= (𝐸−𝑔𝑖) +𝜇 𝐺𝑖where
𝐸denotes the player’s endowment, 𝑔𝑖his own contribution, 𝐺𝑖the group’s total contribution, and 𝜇the contribution multiplier
(known at the ‘‘marginal per capita return’’ or MPCR). 𝐸= 10 tokens (each worth 5 cents), and 𝜇= 1.6∕4.
The Five Adjustment Treatments: In all other treatments, each period included a second stage. After players entered their
contributions, the computer made ‘‘earnings adjustments,’’ adding or subtracting an amount equal to half the endowment. Hence,
the payoffs in all non-baseline treatments had the form 𝐸−𝑔𝑖+𝜇 𝐺𝑖+𝐴𝑖, with 𝐴𝑖= ±5 tokens. Binary additive adjustments alter
the standard game’s structure in a way that introduces the minimal amount of uncertainty consistent with psychologists’ studies of
superstition and the minimal amount of group interaction consistent with anthropologists’ studies of supernaturalism. Appendix A
provides the instructions for all treatments, and Appendix Cspecifies all forms of the adjustment algorithm.
In each of the five adjustment treatments, random number calculations generated adjustments with 50/50 odds of being ±5.
The direct contribution of adjustments to earnings was equivalent to a post-game increment of half the 20 endowments times the
number of heads minus tails in a series of 20 coin-flips. Hence, the adjustments were arithmetically irrelevant insofar as they (a) did
not depend on players’ actions, (b) neither magnified nor masked players’ contributions, and (c) had a zero expected mean of zero
and small standard deviation averaging less than 4% of players’ total earnings).
Four of the five adjustment treatments employed group-level adjustments. In practice, these treatments differed only with respect
to the way their pre-play instructions described the adjustments.
1. Random: The Random treatment’s pre-play description explicitly stated that each adjustment would be random with a 50/50
chance of being ±5, that all members of one’s group would receive the same adjustments, and that the other groups would
receive adjustments based on separate 50/50 calculations.
2. Unknown: The Unknown treatment employed the same ±5 adjustment algorithm but did not note its random nature. Hence,
the otherwise identical instructions replaced well-defined risk with ambiguity (aka ‘‘Knightian uncertainty’’).7
In the third and fourth treatments, the adjustment algorithm could take different forms. The instructions stated that ‘‘You may
wish to think of the computer as an invisible observer that monitors everyone’s behavior in stage 1 and then chooses all the group
adjustments in stage 2." The instructions also noted that a program would run at the start of the experiment to select but not reveal
7Section 4describes results from a second version of Unknown treatment analyzed in Hajikhameneh (2025). It differed from the first version in just one way:
The pre-play instructions truthfully stated that after the last period, players would be asked to comment on any patterns they had observed in the computer’s
adjustments or any relationships between investments; that some of the comments would be with future participants; and that if any of their own feedback was
chosen for sharing, they would receive an additional payment equal to the future participants’ average earnings.
Journal of Economic Behavior and Organization 229 (2025) 106775
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Table 1
Contribution statistics - All treatments.
Group contributions (% of endowment)
Treatment Mean SE Median N-obs
Baseline 27.5% 1.35% 23.1% 320
Random 41.3% 1.11% 40.0% 320
Unknown 40.6% 1.12% 38.1% 320
Absolute 42.1% 1.23% 40.6% 320
Relative 44.9% 1.11% 43.9% 320
Individual 54.0% 1.21% 57.5% 320
the computer’s ‘‘personality". In both treatments, the first personality was ‘‘randomly oriented’’ and would choose adjustments based
on a 50/50 random number calculation, whereas the second chose adjustments that were partly based on players’ contributions.8
Whereas the first and second adjustment treatments were designed to assess the impact of risk versus uncertainly, the third and
fourth were designed to assess the impact of narrative frames that described the adjustment process as something akin to a system
of rewards and punishments.
3. Absolute: The treatment’s instructions truthfully noted that the second personality was ‘‘percentage oriented’’ and would
steadily raise the probability of a +5 adjustment from 20% to 80% as the group contributed a larger percentage of its total
endowment. For the sake of brevity, we will refer to this as the Absolute treatment.
4. Relative: The treatment’s instructions truthfully noted that the second personality was ‘‘comparison oriented’’ and would
steadily raise the probability of a +5 adjustment from 20% to 80% as the group contributed more relative to the average
contribution of the other three groups in their session.
To determine whether the shared nature of adjustments promoted cooperation among group members, we included a final treatment
that invoked the adjustment algorithm for each player rather than each group.
5. Individualized: The treatment’s instructions truthfully stated that the computer would make separate ±5 adjustments to the
earnings of each player based on separate 50/50 random calculations. In all other respects, the Individualized treatment was
identical to the Random treatment.
Each session concluded with a risk elicitation task and a questionnaire. The questionnaire asked subjects about their gender,
college year and major, decision-making styles, supernatural beliefs, religious involvement, and perceptions about the adjustment
process (Appendix A.2). The risk elicitation task was based on the paired lottery method of Holt and Laury (2002).9Each treatment
included four sessions with 16 subjects per session. Subjects received a $7 show-up fee, their 20-period cumulative earnings, and
payment for the risk elicitation task. Total payments ranged from $15.00 to $29.00 and averaged $21.50. All subjects were students
at a mid-sized private university, and none participated in more than one session.
3. Results
Our study was exploratory, motivated by ethnographies, surveys, and experiments conducted by social scientists whose theories
and methods differ from those of mainstream economics. Although our experimental design and treatments grew out of specific
questions and conjectures, the results do not qualify as tests of precise predictions. For work of this nature, it seems best to begin
by simply listing our main results.10 We turn to additional results in Section 4and weigh the evidence for and against alternative
explanations. Appendices Dand Eshow that the claimed results stand up to detailed statistical analysis.
Fig. 1shows the path of group contributions as a share of group endowments for each treatment – the standard (zero-adjustment)
Baseline in black, the four group-level adjustment treatments in solid colors, and the Individualized adjustment treatment in dotted
red. Table 1lists the overall contribution means, standard errors, and medians for each treatment calculated at the group level.
Note the strength and consistency of the main results.
1. The contribution paths for all four treatments with group-level adjustments are similar, with differences in average
contributions across the four treatments that are both substantively small and statistically insignificant.
8The instructions did not reveal that the program selected the random personality with probability 0.99. Not surprisingly, all sessions ended up using the
random personality.
9Subjects observe two sequences of 10 paired lotteries: Lottery A, independent of the sequence, has a probability distribution of (1∕2,1∕2) in winning the
prize pair of ($1, $3); lottery Boffers the prize pair of ($0.1, $5); and the probability of winning $5 is increasing in the sequence. The number of times a
subject chooses lottery Bprovides a measure of risk aversion. For details, see Appendix A.3.
10 For the record, we anticipated that contributions would be lowest in the Baseline, equally low in the Random treatments, and significantly higher in the
Unknown treatment. The Individualized treatment was designed to test the hypothesis that group-level adjustments strengthened group identity – see Section 4.1.4.
We thought contributions would likely run higher in the Absolute and Relative treatments than in the Unknown treatment.
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Fig. 1. Average contributions - All treatments.
2. Both individually and collectively, the average contributions and contribution paths for the four group-level adjustment
treatments are substantially (and significantly) higher than their Baseline counterparts.11 Overall, their average contributions
exceed the Baseline by a factor of 1.54.
3. The path of average contributions in the Individualized treatment lies well above the paths of all four group-level adjustment
treatments. Overall contributions in the Individualized treatment exceed the group-level treatment average by a factor of
1.28, and the ratio is 1.31 for Individualized versus Random – the two treatments where players are explicitly told that the
adjustments are based on 50/50 calculations.12
4. First-period contributions in the Individualized treatment exceed those of other treatments even though players make these
contributions before experiencing any adjustments.13 The Individualized treatment players contributed more than 65% of
their endowments in the first period, compared to 52% for Baseline players and 55% for players in the four treatments with
group-level adjustments.
5. In all treatments, average contributions decline over time. But the relative decline (as measured by the change in log odds
ratios, or decline relative to initial or average contributions) is substantially larger for the baseline treatment. Likewise, the
final period decline is substantially larger for the baseline treatment.
6. In all adjustment treatments, additive shocks to players’ earnings affect contributions in ways that differ dramatically from
those induced by previously studied forms of public goods risk or uncertainty.
11 Pairwise Mann–Whitney–Wilcoxon (MWW) tests of group contribution means yielded a p-value of 0.02 for Baseline versus Unknown and p-values of 0.00 for
Baseline versus Random, Absolute, and Relative. Note also that only one of the 16 Random treatment groups contributed less overall than the median Baseline
group. In contrast, only one Baseline group gave more than the median Random group (and that Baseline group was an obvious outlier with contributions
averaging 38.2 of 40 possible tokens, whereas the second highest Baseline group averaged just 13.9, and the remaining Baseline groups averaged 9.8.
12 The MWW p-values for pairwise comparisons of the group contribution means are 0.00 (Individualized vs. Baseline), 0.02 (Individualized vs. Unknown),
0.05 (Individualized vs. Absolute), 0.11 (Individualized vs. Relative), and 0.03 (Individualized vs. Random). The MWW p-value is 0.01 for Individualized vs. all
four group-level adjustment treatments combined.
13 For MWW pairwise comparisons of first-period contributions, the p-values are 0.00 (for Individualized vs. Baseline), 0.01 (Individualized vs. Unknown), 0.35
(Individualized vs. Absolute), 0.01 (Individualized vs. Relative), 0.06 (Individualized vs. Random), and 0.01 for Individualized vs. Random, Unknown, Absolute,
and Relative combined. For first-period contributions, we ran MWW tests based on the contributions of individuals rather than groups because nothing yet links
players to their other group members.
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Summary: People respond very differently to additive shocks than to previously studied forms of risk or uncertainty. Numerous
experiments spanning more than 25 years have investigated other forms of public goods game risk or uncertainty, including shocks
to the multipliers, endowments, and contribution thresholds. These have almost always reduced contributions. The reductions are
usually small, often insignificant, and frequently sensitive to small changes in the treatments. Appendix Fprovides a visual summary
of the effects that risk or uncertainty have on contributions relative to their (red highlighted) baseline counterparts in seven repeated
linear public goods experiments. As noted in the introduction, similar forms of risk and uncertainty also reduce cooperation in other
social dilemma experiments, including those based on prisoner’s dilemma, stag-hunt, common pool resource, coordination, and trust
games.14
4. Explaining the results
The contrast between our results and those of the existing literature calls for explanation, and all the more so given that our
work was motivated by psychological experiments and anthropological field studies. Below, we argue that our results are best seen
as products of two closely linked responses to elevated external risk, both so deeply rooted in culture or human nature that they
operate largely at subconscious levels. The first response is increased solidarity; the second is superstition that rationalizes increased
solidarity. We arrived at this conjecture only after a long process that included analyzing results with the simple methods of Section 3
and the statistical methods reported in Appendices Eand D, analyzing results from additional experiments reported in Hajikhameneh
and Iannaccone (2023) and Hajikhameneh (2025),15 reviewing numerous economic experiments involving risk and uncertainty and
numerous studies across the social sciences on superstition and supernaturalism, and receiving detailed feedback on earlier versions
of this paper from colleagues, referees, and editors.
The conjecture must remain tentative until the experimental results are proven robust, replicable, and generalizable. Even then,
there would be little reason to adopt it if more conventional explanations suffice. So consider our results in light of established
experimental findings, particularly those concerning public goods and risk or uncertainty.
4.1. Conventional explanations
4.1.1. Confusion, distraction, or cognitive load
One might ask whether our adjustment treatments raise contributions by inducing distraction, confusion, or skepticism. Public
goods experiments can be sensitive to small changes in design, conditions, or expectations. A short pause mid-way through the
sequence of rounds can reset contributions to their initial levels (Chaudhuri,2018). As Cookson (2000) demonstrates, ‘‘apparently
superficial changes in presentation may have strong and replicable effects on experimental findings, even when care is taken to
make the language and presentation of instructions as neutral as possible.’’
Several facts suggest that problems of this nature had minimal impact. Compared to other public goods experiments involving risk
or uncertainty, our 50/50 additive shocks are easier to describe, observe, and understand. In typed, post-play feedback, our subjects
expressed neither confusion nor skepticism but rather described the experiment as interesting, straightforward, and enjoyable. Just
one of the 128 random-treatment subjects questioned whether adjustments were truly random. The baseline and adjustment sessions
were all completed within 45 minutes. Subjects never asked for clarification when given the chance to do so after the experimenter
read the instructions. The greatest difference in average contributions among the five adjustment treatments did not arise in response
to the uncertain nature of the Unknown, Relative, or Absolute treatments. Rather it arose between the Random and Individualized
treatments, both of which described the adjustments as 50/50 random.
4.1.2. Experimenter demand
Might our treatments have raised contributions because the instructions or the mere presence of adjustments led subjects to
believe that we, the experimenters, wanted them to raise their contributions? Problems of this sort, known as ‘‘experimenter
demand,’’ can certainly bias experimental outcomes (Zizzo,2010).
Here again, there are reasons to suspect that experimenter demand effects were small and probably much smaller than those
associated with other variations on the public goods game. Again, consider the two treatments that explicitly described the
adjustments as 50/50 random. These risk-centered instructions were short and simple, with no hint of a desired response, and
we ourselves did not expect a response. We thought it more likely that contribution paths and levels would mirror those of the
zero-adjustment Baseline treatment. Insofar as the path might deviate from Baseline, previous experiments suggested that a fall in
contributions was more likely than a rise. Moreover, the additive nature of the adjustments neither suggested nor admitted any way
to capitalize on the adjustments, even if subjects had quasi-magical faith in their ability to predict them.
In contrast, consider the many public goods experiments that reveal what each member contributed while also providing ways
to reward, punish, or expel them. Or consider experiments that introduce risk together with a costly means of reducing it. There is
14 We are not claiming that increased cooperation has never occurred among the scores of studies that add risk or uncertainty to otherwise standard social
dilemma experiments. See, for example, Stoddard (2017), Aksoy and Krasteva (2020), Théroude and Zylbersztejn (2020). But the exceptions are rare and, to
the best of our knowledge, always appear alongside seemingly similar treatments that yield decreased contributions.
15 From April 2017 through May 2022, we ran more than 60 sessions with 1056 university students for three different shocks and superstition projects. See
Appendix Bfor details.
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little doubt that subjects in these experiments know what they are expected to do, and the only question is whether they will do it
more in some treatments than others.
Our treatments also provide a built-in check on the magnitude of experimenter demand similar to the ‘‘weak’’ and ‘‘strong’’
treatments approach suggested by DeQuidt et al. (2019). Suppose subjects view the presence of adjustments as cues to appropriate
behavior. In that case, the Unknown treatment certainly makes it easier for subjects to imagine that the experimenters’ have
programmed the adjustment algorithm to reward the appropriate behavior. The Relative and Absolute treatments go further still by
explicitly stating that the computer’s ‘‘personality’’ may be programmed to reward higher contributions. But average contributions
and contribution paths turned out to be statistically indistinguishable across all four group-level adjustment treatments.
4.1.3. Risk or ambiguity aversion
As noted in Section 2, the experiment included a risk preference task based on the paired lottery method of Holt and Laury
(2002). In practice, however, the impact of risk aversion was substantively small and statistically insignificant in the GLS and
Tobit regressions reported in Appendices Dand E. And insofar as aversion to risk or ambiguity influenced players, it would have
weakened rather than strengthened our central result. Even in the many past experiments where risk or uncertainty did reduce
overall contributions, authors have rarely interpreted the outcome as rooted in risk or ambiguity aversion, and reviews of the
literature on public goods experiments make little or no mention of risk or ambiguity aversion Ledyard (1995), Zelmer (2003),
Chaudhuri (2011,2016).
4.1.4. Identity effects
Shared identity can greatly increase group solidarity, and many experiments attest to the ease with which group identities
form around shared attributes and experiences (Chen and Li,2009). Shared threats, especially those attributed to a common
enemy, can radically increase cooperation and self-sacrifice (McAdam,1986;Gambetta,2005). Could our adjustment mechanism
be raising contributions by causing each group’s members to view the computer as their common enemy? The adjustments did
amount to arbitrary reward and punishment, and every member received the same adjustments while other groups received separate
adjustments.
We used the Individualized treatment address this questions. If shared adjustments strengthen group identity, then the
Individualized adjustments should reduce this effect. Recall that this treatment truthfully stated that the computer would make
separate ±5 adjustments to the earnings of each player based on separate 50/50 random calculations. In contrast, the instructions
for each of the four group-level adjustment treatments truthfully stated that the computer would make the same adjustment to each
member of the group.
But far from lowering contributions, the Individualized treatment exceeded their group-adjustment counterparts by 28% and
the baseline by more than 96% — differences that are substantively large and statistically significant. And even if individualized
adjustments did somehow manage to enhance group identity more effectively than group-level adjustments, one is hard-pressed to
explain why either adjustment boosted contributions far more effectively than treatments designed explicitly to foster strong group
identity, such as Eckel and Grossman (2005).16
4.1.5. Risk sharing
Just as saving can smooth consumption when income varies over time, so sharing can smooth consumption when income varies
among individuals. Development economists have documented consumption smoothing responses to both individual and collective
output shocks in agricultural villages (Townsend,1994;Fafchamps and Lund,2003;De Weerdt and Dercon,2006). A lab experiment
by Kaplan et al. (2012) found that food sharing routinely emerged in high-variance areas but not their low-variance counterparts
when subjects individually foraged across a varied virtual landscape. The experiment’s results are especially noteworthy because
foraging, grouping, and sharing decisions were endogenous, and although participants could communicate by text, they could not
make binding agreements.
The risk-sharing story is, however, a poor match for our experiment because additive adjustments provide no direct means
of sharing output.17 A true analog would enable players to pool their adjustments or overall earnings rather than contributions.
Moreover, the benefits from sharing are greatest when an uninsured negative shock seriously harms an individual or subset of
individuals within the group. Nothing like that happens here, and cumulative risk in all versions of the game is minimal because
the sum of twenty independent ±5 shocks is almost certain to be small relative to overall earnings.
If risk sharing drives our results, it would seem to do so through predispositions rather than rational calculation. This is, in fact,
the realm of our proposed explanation.
16 Eckel and Grossman (2005) compare contributions across six treatments that build off a linear VCM baseline quite similar to ours (with 15 periods, an
MPCR multiplier of 0.4, and 5 players per group). Their identity treatments included a (complete anonymity) baseline VCM, two other ‘‘weak’’ identity treatments
in which players could see but not interact with the other members of their groups, and three ‘‘strong’’ identity treatments with a pregame activity that required
face-to-face interaction within one’s group. Overall contributions in the two (non-baseline) weak treatments did not exceed those of the baseline. Contributions in
the three strong treatments exceeded those of the weak treatments and baseline by 30%. In contrast, our relatively weak Random treatment raised contributions
by 50% and our Individualized treatment raised contributions by 96%.
17 For any given sequence of member contributions ranging over any number of periods, the adjustments do not affect what a member earns from his/her
own endowments and contributions, nor does the shift from group to individual shocks change the expected mean or variance of the earnings members derive
directly from their own adjustments.
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4.2. The proposed explanation
Consider the conjecture that increased contributions stem from two closely linked responses to elevated external risk – solidarity
and superstition – both operating primarily through impulses and predispositions.
Begin by noting two qualifications. First, and for reasons addressed below, the conjecture concerns external/environmental risk
rather than the internal/strategic risk resulting from the actions of others. Second, it concerns solidarity, hence commitment within
groups rather than generalized trust, altruism, or cooperation.
4.2.1. Risk-induced solidarity
Support for the solidarity conjecture comes from many sources, starting with the fact that conventional explanations fail to
anticipate or explain the results. This failure is particularly surprising given the simple nature of the treatments, the magnitude of
the observed effects, the consistency of the effects across treatments, and the size of the experimental literature that works off the
same baseline model.
An impulse or predisposition can explain why, in most treatments, players’ average contributions exceed their baseline
counterparts in the first period, before they receive any adjustments or any information about their fellow members’ contributions.18
Further support comes from the consistency of contributions across all four group-level adjustment treatments. Average
contributions and contribution paths were nearly identical whether the group-level adjustment algorithm was described as totally
random, unknown, or possibly programmed to reward higher contributions. This consistency, which we certainly did not anticipate,
suggests that players responded more strongly to the mere existence of the risky environment than they did to information about
whether their own actions might change the risk.19 Moreover, as we explain below, regression analysis of players’ contribution
patterns and responses to post-play questions, as well as their communication among group members in the related experiments of
Hajikhameneh and Iannaccone (2023) show that players did respond to the different descriptions of the adjustment process, albeit
in other, more subtle ways.
Mathematical models and multi-agent simulations suggest the means by which the conjectured predisposition might have
evolved, enhanced group and individual fitness, and withstood less cooperative types or mutations.20
Online Appendix describes a multi-agent framework within which the proposed predisposition can survive, spread, resist
invasion, and increase both individual and group welfare. The model combines spatial, evolutionary, and population dynamics and
thus falls within the class of multi-agent models that Epstein (1998, p. 38) calls demographic games. Epstein analyzes a demographic
Prisoner’s Dilemma in which pure defectors and pure cooperators interact over a 2-dimensional world and shows that clusters
of cooperative agents can emerge, endure, and predominate even though agents interact randomly, every interaction yields PD
payoffs, players have no memory of past interactions, and defectors cannot be distinguished from cooperators.21 Our multi-agent
framework extends Epstein’s model to include a wider range of games, including linear and non-linear public goods games with
N players, all 2-player binary choice games, mixed strategies, varying rates of spatial mobility, varying distances of interaction,
continuous variation in cooperation rates, evolving rates of cooperation, and shocks that can vary in magnitude, frequency, region.
For a seemingly realistic range of parameter values, clusters of agents with higher rates of cooperation survive and sometimes
dominate when interactions are localized. More importantly, adding positive and negative shocks to the payoff functions can cause
the share of cooperators to increase further still and support the survival of agents with higher rates of cooperation.22
Spatial dynamics are key to the outcomes. Results collapse to the full-defection, standard Nash equilibria if the probability
of interaction among agents is independent of their proximity. But where the probability of interaction correlates with proximity,
clusters of cooperative agents emerge and flourish as cooperative agents interact with cooperative neighbors (and thereby live longer
and produce more offspring). Conversely, clusters of defecting agents tend to decline, and this self-defeating tendency limits their
tendency to displace cooperators in mixed regions. The same logic applies when we replace a population of pure cooperators and
pure defectors with a population of high and low cooperators (where high versus low can depend on probabilistic rates of cooperation
or quantifiable levels of cooperation). Moreover, with cooperation a matter of degree, overall rates of cooperation depend on the
18 For MWW pairwise comparisons of first-period contributions, the p-values are 0.00 (for Baseline vs. Individualized), 0.55 (Baseline vs. Unknown), 0.06
(Baseline vs. Absolute), 0.97 (Baseline vs. Relative), 0.00 (Baseline vs. Random), and 0.01 for Baseline vs. Individualized, Random, Unknown, Absolute, and
Relative combined. For first-period contributions, we ran MWW tests based on the contributions of individuals rather than groups because nothing yet links
players to their other group members.
19 The magnitude and consistency of effects carry over to closely related experiments reported in Hajikhameneh and Iannaccone (2023) and Hajikhameneh
(2025).
20 The central challenge facing any claim regarding cooperative predispositions deeply rooted in culture or human nature is demonstrating that such a
predisposition could be the product of evolutionary forces. It is not enough to show that the trait is functional in the sense of raising welfare among those
who have it; the trait must also be resistant to invasion and mutation. The same challenge confronts cooperative predispositions in PD, PGG, and other social
dilemma games.
21 Gibaud (2016) provides a mathematical analysis of Epstein’s demographic prisoner’s dilemma, and identifies the parametric conditions under which zones
of cooperation emerge, and Dorofeenko and Shorish (2002) derive similar results for a one-dimensional variant of the model with agents scattered along a line.
Other mathematical papers show that cooperation remains viable under fairly weak conditions for spatial versions of the prisoner’s dilemma, public goods game,
stag-hunt, and other social dilemma games (Szolnoki et al.,2011;Schweitzer et al.,2002;Nowak and May,1993;Brandt et al.,2003;Killingback et al.,1999;
Yang and Zhang,2021;Huang et al.,2015;Helbing et al.,2010;Gibaud,2016).
22 Note that neither Epstein’s demographic PD nor our demographic VCM require repeated interactions, nor do they admit strategies like tit-for-tat that require
agents to recall individual agents, or recognize agent types, or condition current actions on past experience.
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Table 2
Rates of repeated contributions following a positive or negative adjustment.
Repeated contributions𝑡
Including 0 and 10 Excluding 0 and 10
Treatment Negative adjustment𝑡−1 Positive adjustment𝑡−1 Negative adjustment𝑡−1 Positive adjustment𝑡−1
Random 32% 40% 20% 27%
Absolute 28% 46% 14% 33%
Relative 28% 41% 19% 30%
Unknown 26% 38% 21% 31%
Individual 31% 43% 18% 26%
Total 29% 42% 18% 30%
game’s underlying parameters and tend to increase in times or places of greater (additive) risk. The conjectured predisposition can
thus be modeled as the product of an evolutionary process.
One might argue that spatial models better suit the era of hunter-gatherers than the present-day world of large-scale civilizations.
But this long premodern period is precisely when most distinctly human predispositions evolved.23 Moreover, Axelrod et al. (2002),
Santos et al. (2006), and others demonstrate that cooperation can also emerge and endure in settings where social networks take
the place of spatial clusters.
4.2.2. Superstition
Our results suggest that external/environmental risk also fosters superstitions. Some of these superstitions can be attributed to
well-known cognitive biases, including the tendency to ‘‘see" patterns (such as trends, turning points, alternation, or clusters) in
random phenomena (Kahneman et al.,1982;Vyse,2013, chap 4). Hence, when the experiment’s post-play survey asked subjects if
they had noticed adjustment patterns, many noted streaks of positive or negative adjustments.
Other superstitions can be attributed to the conditioned response that Skinner ([1948] 1992) described in his classic animal
experiment. When a pigeon is intermittently rewarded with food, it tends to repeat the idiosyncratic behaviors that it happened to
be doing when the food appeared. Skinner speculated that the same tendency leads humans to adopt idiosyncratic and ineffectual
rituals in gambling, athletics, and other risky activities.24 The same tendency appears in our experiments: a positive adjustment in
one period significantly increases the probability that a player will make the same contribution in the following period. Table 2
shows that across all treatments repetitions occur far more frequently following positive adjustments, and all the differences are
statistically significant.25 26 Whether consciously or unconsciously, players act as if they believe that higher contributions yield higher
adjustments, even when told the adjustments are random. Appendix Eprovides additional analysis, including GLS regressions on
the determinants of repetition.
Although conditioning can explain repetition, it sheds little light on the overall increase in contributions relative to the Baseline.
After all, the adjustment treatments punish as often as they reward. Conditioning likewise fails to account for the substantially higher
contributions in the Individualized treatment, where adjustments have the same ±5 magnitude and 50∕50 probability as in all other
treatments. Nor does conditioning help us understand why any group-level treatment raises contributions in the first period, before
any adjustments occur, much less why an even larger effect appeared in the first period of the Individualized treatment. First-period
effects cannot be modeled as responses to game-related outcomes. Standard cognitive biases are similarly limited.
A third class of superstitions, based on rationalization rather than conditioning or misperception, can fill this crucial gap. People
rationalize when they provide sincere but incorrect reasons for impulses or actions based on motives they fail to understand or prefer
not to acknowledge. Numerous psychological experiments and case studies reveal that rationalization is a fundamental feature of
human behavior and, more surprisingly, routinely works to move people’s conscious beliefs and actual behavior in the direction
of their unconscious rationalizations (Loftus,1992;Loftus and Pickrell,1995;Hall et al.,2010,2013;Johansson et al.,2005).27
Although rationalizations routinely reverse the standard order of rational action (by deriving preferences from behavior, rather
23 The value of risk-induced solidarity was immense in early human and hunter-gatherer times when primitive technologies and small-scale societies meant
that routine environmental shocks might easily wipe out an entire tribe. Again, the intuition is straightforward: Assume that group members can often discern
when they enter a period or place of larger than normal potential shocks (due to seasonal variation, weather patterns, periods of disease, changes in animal or
insect populations, changes in the character or number of nearby groups, etc.). Assume that a certain level of cooperation is sufficient to ensure group survival
under normal circumstances but less likely to do so in a high-shock period. These forces do not suffice to boost the cooperation level in standard evolutionary
models or standard game theory because defectors consistently outperform cooperators. But they work in spatial settings where negative shocks can kill off
clusters of defectors, and they work even more effectively if positive shocks raise relative rates of reproduction among clusters of cooperators.
24 Although Skinner and other psychologists characterize many human superstitions as the result of inadvertent operant conditioning, they can also be modeled
as byproducts of adaptive learning strategies and natural selection (Beck and Forstmeier,2007;Foster and Kokko,2009).
25 The table’s left side columns report on repetitions among all contributions; the right side columns report on repetitions of contributions that are neither zero
(the Nash equilibrium) nor ten (the social optimum) so as not to confuse repetitions caused by +5 adjustments with repetitions caused by persistent free-riding
or unconditional cooperation.
26 Using Table E.2 of Appendix E, Wald tests yield 𝑝 <0.000 for all treatments except Individual for which 𝑝 <0.01.
27 Within economics, cognitive dissonance is the form of rationalization most frequently invoked by theorists and experimentalists. See, for example, the heavily
cited papers by Akerlof and Dickens (1982), Rabin (1994), and Konow (2000).
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than vice versa), recent scholarship has begun to explore the many ways in which the human propensity to rationalize may be
adaptive and fitness-enhancing.28
A seemingly superstitious tendency to mistakenly perceive random adjustments as rewards for higher contributions appears
even more clearly in post-play feedback from subjects who played the Unknown treatment with an added pregame instruction. In
a two-generation study of adjustment effects, Hajikhameneh (2025) truthfully told the first-generation subjects that after the game
ended, they would be asked if they observed any patterns in the computer’s adjustments or any relationships between people’s
investments and the computer’s adjustments.29 Three results stand out: First, only 12 of the 64 players described the adjustments
as random or unrelated to player contributions. Second, a much larger number, 21 of the 64, claimed that higher contributions led
to more positive adjustments or that lower contributions led to more negative adjustments. Third, and most striking of all, no one
claimed that lower contributions led to more positive adjustments or that higher contributions led to more negative adjustments.
The remaining players claimed to be unsure of the relationship or described complex patterns that defy classification.30
Finally, note that superstitions and risk-induced solidarity may help to explain why players contributed more in the Individualized
treatment than its group-level Random counterpart. Superstitions that link personal actions to an outcome will probably operate
more strongly when the outcome does not depend on the actions of others.31 Alternatively, insofar as subjects act rationally subject
to the erroneous belief that contributions influence adjustments, their incentive to free ride will be greater when any such influence
relates to the actions of the entire group rather than them alone. The same argument applies to a risk-sharing impulse discussed in
Section 4.1 above.
5. Conclusions
We have shown that additive shocks boost cooperation in a series of otherwise standard public goods game experiments. Relative
to the no-shock baseline, overall contributions increased by about 50% when a group’s members always received the same shock
– and this was true whether the 50/50 random nature of the shocks was known, unknown, or possibly linked to cooperation. The
increase grew to 96% when players were truthfully told that each would receive adjustments based on separate random number
calculations. Although the treatments described the shocks differently, players’ earnings always equaled those of a standard public
goods game supplemented by a separate payment that equaled half their overall endowment times the share of heads minus tails
in twenty simulated coin flips. Hence, players who correctly perceived the nature of the adjustment process had no clear reason to
behave differently than in the standard game, and zero contributions remained the money-maximizers’ Nash equilibrium.
The results are nothing like those obtained in public goods experiments that incorporate risk or uncertainty in ways that directly
interact with players’ actions. In those experiments, which change contribution multipliers, endowments, contribution thresholds,
or information about the actions of other players, the impact of risk or uncertainty is almost always negative, inconsistent, or
statistically insignificant.
If we had to trace our results to a single source, we would posit a predisposition to pull together in the face of large exogenous
shocks. We suspect this impulse evolved in premodern settings where deviations from normal patterns of weather, disease, wildlife,
and the behavior of other groups could substantially change rates of survival. But our results suggest that a second source interacts
with the first – a predisposition to rationalize one’s own non-rational impulses and behavior, which in this case promote false causal
claims that justify increased solidarity. This superstitious response can explain the additional jump in contributions when players
know that they each receive separate shocks because any causal inference now links the shocks more strongly to one’s own behavior.
Superstition may also explain why contributions fail to increase in public goods experiments where risks directly change the
return to contributions or the information about other players. Studies by psychologists, anthropologists, sociologists, and historians
show that superstitions abound where outcomes are largely determined by forces that cannot be controlled or understood. The lack of
real and relevant knowledge impels people’s explanation-seeking minds to invent stories that offer solutions to unsolvable problems.
28 Cushman (2020) convincingly argues that rationalization evolved because it ‘‘extract[s] information from the [evolved, adaptive] non-rational processes that
influence our behavior’’ and thereby supplies ‘‘useful fictions’’ that promote fitness-enhancing beliefs and desires. Yong et al. (2021, p. 790) describe six major
adaptive benefits of rationalization and cite religions and ideologies as examples of the many group-level rationalizations [that can] function as a powerful basis
for collective solidarity, direction, and cooperation’’. See also Kyle Stanford et al. (2020), Levy (2020), Railton (2020). And note that additively separable utility
shocks play a key role in the game theoretic model and analysis of Levy and Razin (2012), which links shocks to cooperation via religions that promote false
but potentially beneficial beliefs that cooperation increases the probability of positive versus negative shocks.
29 To encourage detailed and clearly worded feedback, players were truthfully told that some comments would be shared with future participants, and if any
of their comments were included, they would receive an additional payment equal to the average of what those future participants earned. The opportunity
to provide potentially lucrative responses to post-game questions appears not to have influenced the subjects’ game-playing behavior (as measured by average
contributions, the path of average contributions, and regressions relating contributions to adjustments and player attributes).
30 In this variant of the experiment and in all others, including variants that incorporated text-based conversation among group members, no player ever
referred to the experimenters or programmers. Instead, their texts consistently treated the computer as an entity in its own right. While this does not rule out
‘‘experimenter demand’’ effects, it strongly suggests that our subjects found it natural to talk about the computer as an independent agent, though they knew it
was merely code running on a machine and programmed by humans.
31 A psychological phenomenon known as ‘‘the illusion of control’’ may further contribute to this effect, since it can influence behavior that precedes any
actual outcomes and should operate more strongly if subjects see their personal actions as a major input into the process. However, the illusion cannot by
itself explain why subjects employ higher contributions to control the outcome (as opposed to lower contributions, lucky number contributions, etc.). Moreover,
despite the sizeable psychological literature on the illusion (Langer,1975;Thompson et al.,1998;Presson and Benassi,1996;Stefan and David,2013), economic
experiments provide little evidence of the effect (Filippin and Crosetto,2016;Klusowski et al.,2021).
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Real and observable links between players’ actions and uncertain outcomes may therefore limit the impulse to act as if an external
agent is monitoring, rewarding, or punishing one’s behavior.
The experiments of Skinner and others show that the simple superstitions induced by coincidental rewards are largely
idiosyncratic. But evolutionary forces all but guarantee that superstitions which benefit both individuals and groups are more likely
to survive and spread – shaping traditions of belief and behavior that, in turn, shape cultures and possibly even genes. Our results
may provide insight into the historical prevalence of superstitions and supernaturalism, including the ‘‘big god’’ religions that many
economists, anthropologists, biologists, and psychologists now view as supporting the transition to large-scale societies. 32
Before asserting any of these conjectures with confidence, the underlying experimental results must be proved robust, replicable,
and generalizable. Some evidence for replicability already exists in the consistent sign and magnitude of the cooperation jump
induced by all four group-level treatments plus several others analyzed in Hajikhameneh and Iannaccone (2023) and Hajikhameneh
(2025). Some additional evidence may be found in the consistency of results across sessions run in 2017, 2018, and 2022
(Appendix B), as well as the fact that regression analyses reveal no large or consistently significant contribution effects linked
to individual attributes, including gender, risk preferences, decision styles, political orientation, college major, college year,
supernatural beliefs, or religious involvement. More will be learned from replications run at different labs and much more still
from replications in the field and across cultures.
If the results do replicate, we would then need to determine how much they depend on specific features of the game: the size
of shocks relative to endowments, the binary nature and sequential independence of the shocks (which makes superstitions easy to
test and reject), and the extreme limits on players’ capacity to create a culture of belief (though we have made some first steps in
treatments that add text-based communication within groups or across generations of players). We would especially want to know
whether additive shocks enhance cooperation in related games, such as the prisoner’s dilemma, stag hunt, common-pool resource,
and coordination. After all, the critical question is whether risks and rationalizations combine with other pervasive features of human
thought and action to sustain cooperation in a wide range of real-world settings.
Economists may need to pay more attention to the importance of seemingly irrelevant forms of risk and uncertainty. The world
is filled with both, and existing theory may radically understate their importance. If adding the equivalent of coin-flip shocks to the
public goods game can consistently raise cooperation, then simple shocks may similarly alter outcomes in other experiments and
everyday life.
We are, in any case, left with the remarkable fact that within the much-studied realm of the public goods game, additive
uncertainty seems to work its magic at no cost to subjects or experimenters, with no communication among members, and without
information about the characteristics or contributions of individual members. The critical question is whether the magic also works
in real-world settings, particularly those where previously studied systems of costly signaling, punishment, exclusion, or reward do
not.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Appendix A. Experiment instructions
This section provides the experiment instructions as it was presented to the subjects. Various treatments differ in the combi-
nation of adjustment mechanisms and their descriptions. In what follows, we, therefore, offer the general instructions plus the
treatment-specific description of the adjustment mechanism.
A.1. Instructions
This experiment is a study of group and individual behavior. Everyone in this experiment is receiving exactly the same
instructions. You will earn $7 simply for participating in this experiment. But if you follow the instructions and make careful
decisions, you can earn a significant amount of additional money. You will be paid at the end of the experiment, privately and
in cash.
During the experiment your earnings will be measured in tokens. Each token is worth 5 cents, so at the end of the experiment
your tokens will be converted to dollars at the rate of 20 tokens per dollar.
It is important that you remain silent and not look at other people’s work during the experiment. If you talk, laugh, or exclaim out
loud you will be asked to leave and will not be paid. If you have questions or need assistance, raise your hand and an experimenter
will come to you.
32 See Durkheim ([1912] 1965), Henrich and Muthukrishna (2021), Norenzayan (2013), Norenzayan et al. (2016), Stark (2005), Wade (2009), and
Wilson (2010).
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A.1.1. Your group
Each participant will be randomly assigned an ID number and randomly assigned to a four-person group. The groups will remain
the same throughout the entire experiment. There are 16 participants in today’s experiment, so there will be 5 groups in all with 5
members in each group. Because you will be known only by your random ID number, there will be no way for anyone (including
experimenters) to know which group you are in or what decisions you make.
A.1.2. Your decisions
The experiment includes 20 decision-making periods, and each period consists of 2 stages. At the start of each period you will
receive 10 new tokens.
Stage 1 Instructions: [All Treatments]
In the first stage of each period, you must decide how many of your 10 new tokens you want to invest in your group’s joint
account versus how many you want to invest in your own personal account. The tokens you invest in your personal account will
be directly added to your earnings. (For example, if you invest 5 tokens in your personal account then you will earn 5 tokens from
your personal account.) Earnings from your group’s joint account will depend on the total number of tokens that you and the other
members of your group invest in the joint account. Each member will earn 0.5 times the total number of tokens invested in the joint
account. (For example, if each member of your group invested 6 tokens in the joint account, then the total joint investment would
be 25 tokens, and each member would earn 0.5 ×25 tokens, which equals 9.6 tokens.) To make your investment decisions, simply
enter a number between 0.00 and 10.00 indicating the amount you want to invest in your group’s joint account. The computer will
automatically invest the rest of your 10 tokens in your personal account. (For example, if you enter the number 6.0, then 6 of your
tokens will go into your joint account and the computer will put your remaining 5 tokens into your personal account.)
Stage 2 Instructions: [Random Treatment]
In the second stage of each period, the computer will make adjustments to the earnings of all the participants in the experiment.
The adjustments will be either +5 or −5 tokens. The same adjustment will be made to the earnings of every member in your group,
but the members of other groups may receive different adjustments. The computer will choose the adjustment for your group based
on a random number calculation, so that there will be a 50% chance that 5 tokens are added to the earnings of everyone in your
group and a 50% chance that 5 tokens are subtracted from every member’s earnings. The computer will calculate a different random
number for each of the other three groups in the experiment.
Stage 2 Instructions: [Unknown Treatment]
After the members of your group have made their investment decisions, the computer will make an adjustment to your earnings.
The adjustment will be either +5 or −5 tokens. The same adjustment will be made to the earnings of every member in your group,
but the members of other groups may receive different adjustments. The computer will choose the adjustment for your group based
on a complex sequence of calculations. The computer will choose a separate adjustment for each of the other three groups based
on a separate sequence of calculations.
Stage 2 Instructions: [Relative Treatment]
In the second stage of each period, the computer will make adjustments to the earnings of all the participants in the experiment.
The adjustments will be either +5 or −5 tokens. The same adjustment will be made to the earnings of every member in your group.
But the members of other groups may receive different adjustments.
You may wish to think of the computer as an invisible observer that monitors everyone’s behavior in stage 1 and then chooses all
the group adjustments in stage 2. The observer’s personality will be selected by a program that runs when we begin the experiment.
There are two possible personalities, but you will not be told which one the program ends up selecting.
One personality is randomly oriented. This type of observer always chooses your group’s adjustment based on a 50/50 random
number calculation. So, there will always be a 50% chance that 5 tokens are added to the earnings of everyone in your group and
a 50% chance that 5 tokens are subtracted from every member’s earnings. The observer will calculate a different random number
for each of the other three groups in the experiment.
The other personality is comparison oriented. This type of observer always chooses your group’s adjustment based on the number
of tokens that your group invests in its joint account compared to the investments made by the other groups. If your group’s
investment equals the average investment made by the other groups, there will be a 50% chance that 5 tokens are added to the
earnings of everyone in your group and an 50% chance that 5 tokens are subtracted from every member’s earnings. If your group
invests almost nothing compared to the other groups’ average, your chance of a+5 adjustment will be nearly 20%. If you group
invests much more than the other groups’ average, your chance of a+5 will be nearly 80%. And as the relative size of your group’s
investment grows from much less to much more than that of the other groups, your chance of getting a+5 adjustment will steadily
grow from 20% to 80%. But no matter how little or how much your group invests, the chance of getting +5 is never less than 20%
nor greater than 80%.
The observer’s personality will remain unchanged throughout the experiment, so your adjustments in the second stage of each
period will all be randomly oriented or else they will all be percentage oriented.
Stage 2 Instructions: [Absolute Treatment]
In the second stage of each period, the computer will make adjustments to the earnings of all the participants in the experiment.
The adjustments will be either +5 or −5 tokens. The same adjustment will be made to the earnings of every member in your group.
But the members of other groups may receive different adjustments.
You may wish to think of the computer as an invisible observer that monitors everyone’s behavior in stage 1 and then chooses all
the group adjustments in stage 2. The observer’s personality will be selected by a program that runs when we begin the experiment.
There are two possible personalities, but you will not be told which one the program ends up selecting.
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Table A.1
Table 1.
You Mbr 2 Mbr 3 Mbr 5 Explanations
Investment in joint acct: 6 3 1 10 𝑇 𝑂 𝑇 𝐴𝐿 = 6 + 3 + 1 + 10 = 20
Investment in personal acct: 5 7 9 0 10− Investment in joint account
Earnings from joint acct: 8 8 8 8 Each member earns 0.5 ×𝑇 𝑂 𝑇 𝐴𝐿
Earnings from personal acct: 5 7 9 0 Same as personal acct. investment
Adjustment: +5+5+5+5 50/50 chance of being +5 or −5
Overall earnings for period: 17 20 22 13 =Earnings from both accounts
plus Computer’s adjustment
One personality is randomly oriented. This type of observer always chooses your group’s adjustment based on a 50/50 random
number calculation. So, there will always be a 50% chance that 5 tokens are added to the earnings of everyone in your group and
a 50% chance that 5 tokens are subtracted from every member’s earnings. The observer will calculate a different random number
for each of the other three groups in the experiment.
The other personality is percentage oriented. This type of observer always chooses your group’s adjustment based on the
percentage of tokens that your group invests in the joint account. If your group invests 20% of its tokens or less, there will be
a 20% chance that 5 tokens are added to the earnings of everyone in your group and an 80% chance that 5 tokens are subtracted
from every member’s earnings. If your group invests 80% or more in the joint account, there will be an 80% chance of getting a+5
adjustment. And as the percentage of invested tokens grows from 20% to 80%, your chance of getting a+5 adjustment will steadily
grow from 20% to 80%. But no matter how little or how much your group invests, the chance of getting +5 is never less than 20%
nor greater than 80%. (Notice that because each group has 50 tokens in all, 20% of the total is 8 tokens and 80% is 32.)
The observer’s personality will remain unchanged throughout the experiment, so your adjustments in the second stage of each
period will all be randomly oriented or else they will all be percentage oriented.
Stage 2 Instructions: [Individualized Treatment]
In the second stage of each period, the computer will make separate adjustments to the earnings of all the participants in the
experiment. The adjustments will be either +5 or −5 tokens. The computer will choose each adjustment based on random number
calculations, so that there will be a 50% chance that 5 tokens are added to your earnings and a 50% chance that 5 tokens are
subtracted from your earnings. The computer will perform separate random number calculations for each participant, so you will
each have a separate 50/50 chance of a+5 or −5 adjustment.
A.1.3. Your earnings
At the end of each period the computer will show you how much you invested in each account, the total amount that was invested
in your group’s joint account, and how much you earned from each account. You will not see how each of the other members in
your group divided their tokens between their own personal accounts and the group’s joint account.
[Baseline Treatment:] Your overall earnings in each period will be your earnings from the joint account plus your earnings
from your personal account.
[All Other Treatments:] Your overall earnings in each period will be your earnings from the joint account, plus your earnings
from your personal account, plus the computer’s adjustment.
A.1.4. Examples for the random treatment
Table A.1 shows what happens if you invest 6 tokens in your group’s joint account, while the second member of your group
invests 3 tokens, the third member invests 1, and the fourth invests 10, and the computer randomly makes a+5 adjustment for
your entire group. The total number of tokens invested in the joint account is 6+3+1+10 =20, and 0.5 ×20 =8. So each member
earns 8 tokens from the joint account. You yourself earn an additional (10-6) =5 tokens from your personal account, the second
member earns an additional 7 tokens, the third member earns an additional 9 tokens, and the fourth member earns an additional
0 tokens. The computer’s randomly adjustment adds another 5 token to each member’s earnings. So you earn 17 tokens overall, 8
from the group’s account, 5 from your personal account, and 5 from the adjustment. As you can see in the bottom row of the table,
the other members earn 20, 22, and 13.
Table A.2 shows what happens if you invest 3.50 tokens in your group’s account, while the second member of invests 6.50 tokens
in the group account, the third invests 0, and the fourth invests 9, and the computer randomly makes a−5 adjustment for your
entire group. The total amount invested in the group’s account is 3.5+6.5+0+9=19, and 0.5 ×19 =7.6. So each member earns
7.6 tokens from the group account. You yourself earn an additional 6.5 tokens from your personal account. So you earn 9.1 tokens
in all, 7.6 from the group’s account, 6.5 from your personal account, and −5 from the random adjustment. The other members earn
6.1, 12.6, and 3.6.
Table A.3 shows what happens if you invest all 10 of your tokens in your group’s account, while the other three members do
the same, and the computer randomly makes a+5 adjustment. Because the total group investment is 50 and 0.5 ×50 =16, you
earn 21 overall, and the other members earn the same.
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Table A.2
Table 2.
You Mbr 2 Mbr 3 Mbr 5 Explanations
Investment in joint acct: 3.5 6.5 0 9 𝑇 𝑂 𝑇 𝐴𝐿 = 3.5 + 6.5 + 0 + 9 = 19
Investment in personal acct: 6.5 3.5 10 1 10− Investment in joint account
Earnings from joint acct: 7.6 7.6 7.6 7.6 Each member earns 0.5 ×𝑇 𝑂 𝑇 𝐴𝐿
Earnings from personal acct: 6.5 3.5 10 1 Same as personal acct. investment
Adjustment: −5−5−5−5 50/50 chance of being +5 or −5
Overall earnings for period: 9.1 6.1 12.6 3.6 =Earnings from both accounts
plus Computer’s adjustment
Table A.3
Table A.3.
You Mbr 2 Mbr 3 Mbr 5 Explanations
Investment in joint acct: 10 10 10 10 𝑇 𝑂 𝑇 𝐴𝐿 = 10 + 10 + 10 + 10 = 50
Investment in personal acct: 0 0 0 0 10− Investment in joint account
Earnings from joint acct: 16 16 16 16 Each member earns 0.5 ×𝑇 𝑂 𝑇 𝐴𝐿
Earnings from personal acct: 0 0 0 0 Same as personal acct. investment
Adjustment: +5+5+5+5 50/50 chance of being +5 or −5
Overall earnings for period: 21 21 21 21 =Earnings from both accounts
plus Computer’s adjustment
A.2. Questionnaire
The following questions were asked of all subjects. Question response categories follow the question in italics or in itemized lists.
•What is your major?
–Math, Engineering, or the Physical Sciences
–Business or Economics
–English, Foreign Languages, or Classics
–Humanities
–Other
•What is your Gender? [Male, Female]
•How well do the following statements fit your own decision-making style?
–When making important decisions I focus on facts and logic [Always, Never]
–When making important decisions I trust my feelings and intuition [Always, Never]
–When making important decisions I consult with religious or spiritual leaders [Always, Never]
•How strongly do you agree or disagree with the following statements?
–Some places really are haunted by spirits [Strongly agree, Agree, neutral, Disagree, Strongly Disagree]
–Some people can use the power of their minds to heal other people [Strongly agree, Agree, neutral, Disagree, Strongly
Disagree]
–Some people can use the power of their minds to ‘‘see’’ into the future. [Strongly agree, Agree, neutral, Disagree, Strongly
Disagree]
–Would you describe yourself as a ‘‘spiritual’’ person? [Very spiritual, Somewhat spiritual, Not at all spiritual]
–Would you describe yourself as a ‘‘religious’’ person? [Very religious, Somewhat religious, Not at all religious]
•Which of the following best describes your current religion? [Christian, Jewish, Muslim, Buddhist, Hindu, Other, No religion]
•In the past year, about how often have you attended religious services?
–Almost every week.
–About two or three times per month.
–About once each month.
–Several times each year.
–Once or twice each year
–Never or almost never.
•When you were growing up, around age 11 or 12, how often did you usually attend religious services? [Same response categories
as previous question]
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•The instructions for the experiment were clear and easy to follow [Strongly agree, Agree, Disagree, Strongly disagree]
•Thank you for completing this experiment. We value your feedback, so please use the following text box for comments or
suggestions
The following questions only were asked only in the Unknown treatment.
•Did you notice any patterns in the computer’s adjustments [Yes, No]
•If you think you noticed any patterns in the computer’s adjustments, please describe them briefly. (Type ‘‘no pattern’’ if you
did NOT notice any patterns.)
The below questions only were asked only in the Absolute and Relative treatments.
•Did you notice any patterns in the computer’s adjustments?
•If you think you noticed any patterns in the computer’s adjustments, please describe them briefly. (Type ‘‘no pattern’’ if you
did NOT notice any patterns.)
•How strongly to you agree or disagree with the following statements about the computer’s adjustments?
–Positive adjustments were more likely when we invested more in our group account [Strongly agree, Agree, neutral,
Disagree, Strongly Disagree]
–Positive adjustments were less likely when we invested more in our group account [Strongly agree, Agree, neutral, Disagree,
Strongly Disagree]
–Positive adjustments were more likely when we invested particular amounts (such as 12, 25, or 32 tokens) [Strongly
agree, Agree, neutral, Disagree, Strongly Disagree]
–Positive adjustments were unrelated to our investments in the group account [Strongly agree, Agree, neutral, Disagree,
Strongly Disagree]
–Positive adjustments tended to run in streaks [Strongly agree, Agree, neutral, Disagree, Strongly Disagree]
–Positive and negative adjustments tended to alternate back and forth [Strongly agree, Agree, neutral, Disagree, Strongly
Disagree]
–Positive and negative adjustments were basically just random [Strongly agree, Agree, neutral, Disagree, Strongly Disagree]
–My investment decisions influenced the computer’s adjustments [Strongly agree, Agree, neutral, Disagree, Strongly Disagree]
A.3. Risk preference elicitation instructions
In the questions that follow, you are going to be asked to make ten decisions. Each decision will be between Option A and Option
B. One of the ten choices you make will be randomly selected to determine your earnings for this part of the experiment.
Options Your choice
A B
$1 or $3 each with probability 1∕2 $0.1 with probability 9∕10 or $5 with probability 1∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 8∕10 or $5 with probability 2∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 7∕10 or $5 with probability 3∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 6∕10 or $5 with probability 5∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 5∕10 or $5 with probability 5∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 5∕10 or $5 with probability 6∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 3∕10 or $5 with probability 7∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 2∕10 or $5 with probability 8∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 1∕10 or $5 with probability 9∕10 A or B
$1 or $3 each with probability 1∕2 $0.1 with probability 0∕10 or $5 with probability 10∕10 A or B
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Appendix B. Treatment dates
Date Treatment Session
2017-04-10 Baseline 1
2017-04-11 Baseline 2
2017-04-11 Random 1
2017-04-12 Baseline 3
2017-04-12 Random 2
2017-04-12 Random 3
2017-04-18 Baseline 4
2017-04-18 Watchful 1
2017-04-18 Watchful 2
2017-04-19 Random 4
2017-04-19 Watchful 3
2017-04-19 Watchful 4
2017-04-20 Watchful (com) 1
2017-04-20 Watchful (com) 2
2017-04-24 Baseline (com) 1
2017-04-24 Random (com) 1
2017-04-24 Random (com) 2
2017-04-26 Baseline (com) 2
2017-04-26 Watchful (com) 1
2017-04-28 Baseline (com) 3
2017-04-28 Random (com) 3
2017-05-01 Watchful 5-value (com) 1
2017-05-03 Watchful 5-value (com) 2
2017-05-04 Watchful 5-value (com) 3
2017-05-11 Baseline (com) 4
2017-12-04 Unknown 1
2017-12-04 Unknown 2
2017-12-06 Unknown 3
2017-12-06 Unknown 4
2018-03-12 Individualized Random 1
2018-03-12 Individualized Random 2
2018-03-13 Individualized Random 3
2018-03-13 Individualized Random 4
2018-05-03 Unknown (com) 1
2018-05-04 Unknown (com) 2
2018-05-09 Unknown (com) 3
2018-11-12 Absolute 1
2018-11-12 Absolute 2
2018-11-12 Relative 1
2018-11-13 Absolute 3
2018-11-13 Relative 2
2018-11-14 Absolute 4
2018-11-14 Relative 3
2018-11-14 Relative 4
2022-02-16 Unknown (multi-gen 1) 1
2022-02-16 Unknown (multi-gen 1) 2
2022-02-17 Unknown (multi-gen 1) 3
2022-02-17 Unknown (multi-gen 1) 4
2022-03-29 Unknown (multi-gen, 2a) 1
2022-03-30 Unknown (multi-gen, 2t) 1
2022-04-01 Unknown (multi-gen, 2a) 2
2022-04-01 Unknown (multi-gen, 2t) 2
2022-04-04 Unknown (multi-gen, 2a) 3
2022-04-04 Unknown (multi-gen, 2t) 3
2022-04-20 Watchful rep (com) 1
2022-04-20 Watchful rep (com) 2
2022-04-21 Watchful rep 1
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Date Treatment Session
2022-04-21 Watchful rep (com) 3
2022-05-04 Unknown (multi-gen 2t, sac) 1
2022-05-05 Unknown (multi-gen 2t, sac) 2
2022-05-06 Unknown (multi-gen 2a, sac) 1
2022-05-10 Unknown (multi-gen, 2a sac) 2
2022-05-10 Watchful rep 2
2022-05-11 Watchful rep 3
2022-05-12 Unknown (multi-gen, 2t sac) 3
2022-05-16 Unknown (multi-gen, 2a sac) 3
Key to Treatment Types:
Baseline =Standard VCM (i.e., 20-round repeated linear public goods game)
Random =VCM +adjustments, with algorithm described as 50/50 random
Unknown =VCM +adjustments, with algorithm described as complex calculations
Watchful =VCM +adjustments, with algorithm described as complex calculations that take account of all players’ contributions
Watchful 5-value =replication of Watchful with adjustments that equal −5, −2, 0, +2, or +5.
Watchful rep =replication of the Watchful with slightly different adjustment algorithm.
Absolute =Random or else an algorithm that favors higher contributions
Relative =Random or else an algorithm that favors groups with higher -ranking contributions
(com) =group members can converse via text before periods 1, 5, 10, and 15.
(gen-1) =players know they will be asked to provide comments for future players after the game.
(gen-2x xxx) =players receive input from the gen-1 players
Appendix C. Adjustment algorithm
C.1. Adjustment algorithm in the random and unknown treatments
In the second stage of each round 𝑡, the computer adjusts player earnings by adding either a high or low value, 𝐿= 5or 𝐻= −5.
Adjustments may vary across groups but are equal within groups. Groups with the highest contributions are ranked 1, and all other
groups are ranked 2:
𝑅𝑎𝑛𝑘𝑗 𝑡≡{1𝑖𝑓 𝑓 𝐺𝑗 𝑡=𝑚𝑎𝑥{𝐺.𝑡}
2𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (1)
Groups also receive scores between 0 and 1 based on their contribution 𝐺𝑗 𝑡and a random number 𝑢𝑗 𝑡uniformly distributed uniformly
over the unit interval::
𝑆 𝑐 𝑜𝑟𝑒𝑗 𝑡≡(𝐺𝑗 𝑡+𝑢𝑗 𝑡) −⌊𝐺𝑗 𝑡+𝑢𝑗 𝑡⌋(2)
The groups’ adjustments are then based on their ranks and scores:
𝐴𝑑 𝑗 𝑢𝑠𝑡𝑚𝑒𝑛𝑡𝑗 𝑡≡{𝐻 𝑖𝑓 (𝑅𝑎𝑛𝑘𝑗 𝑡= 1𝑎𝑛𝑑 𝑆 𝑐 𝑜𝑟𝑒𝑗 𝑡≥𝛼)𝑜𝑟 (𝑅𝑎𝑛𝑘𝑗 𝑡= 2𝑎𝑛𝑑 𝑆 𝑐 𝑜𝑟𝑒𝑗 𝑡≤𝛽)
𝐿 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (3)
The net effect is that groups with the highest contributions receive the high-value adjustment with probability (1 −𝛼)
and the other groups receive the high-value adjustment with probability 𝛽. For (1 −𝛼)> 𝛽, the computer operates more and
more like a ‘‘god" that rewards generosity or commitment to the group as (1 −𝛼)increases relative to 𝛽. For 𝛽 >(1 −𝛼), the computer
operates more and more like a ‘‘god" that rewards selfishness or personal commitment as 1 −𝛼increases relative to 𝛽. The ‘‘god’s"
conduct becomes more and more nearly capricious (i.e., random with respect to group commitment) as (1 −𝛼)approaches 𝛽.To
keep the focus on Malinowski’s theory of supernaturalism, Skinnerian superstition, and the impact of socialization, we set
(1 −𝛼) =𝛽= 0.5.
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C.2. Adjustment algorithm in the absolute and relative treatments
In the second stage of each period, the computer adjusts each subject’s earnings by adding either a high or low value, 𝐻= 5
or 𝐿= −5. Within each group 𝑗, all players receive the same adjustment, but adjustments vary across groups. We designed the
adjustment algorithm so that simple parameter changes would yield different types of "gods".
Details: Each group of receives an independently calculated adjustment. Let 𝐴𝑗 𝑡denote that adjustment that Group 𝑗receives
in period 𝑡. The probability that this adjustment takes the high value is then defined as follows:
𝑃(𝐴𝑗 𝑡=𝐻)≡𝑏0𝜋0+𝑏1[𝜋1+ (1 − 2𝜋1)𝐹1(𝐺𝑗 𝑡
𝐺∗)]+𝑏2[𝜋2+ (1 − 2𝜋2)𝐹2(𝐺𝑗 𝑡
𝐺𝑗+
𝐺−𝑗 𝑡)](4)
The 𝐹𝑖functions can be any strictly monotonic function with 𝐹𝑖(0) = 0and 𝐹𝑖(1) = 1, but for simplicity we let each 𝐹𝑖be the
identity function. The values 𝜋𝑖≤0.5and (1 −𝜋𝑖)≥0.5determine the minimal and maximal probabilities of receiving a high
adjustment. In our experiments, we set 𝜋0= 0.5and 𝜋1=𝜋2=.20. Hence, the probability of a high adjustment is never less than
20% nor greater than 80%.
𝐺𝑗 𝑡denotes the total investment by group 𝑗in period 𝑡,
𝐺−𝑗 𝑡denotes the average investment for all other groups during period
𝑡, and 𝐺∗denotes the maximum possible group investment (=𝑛𝐸 for groups of size 𝑛and endowment 𝐸). The weights 𝑏𝑖are
non-negative and sum to one. Hence the probability of group 𝑗getting a high adjustment is the weighted average of a fixed weight,
group 𝑗’s investment relative to the maximum possible investment, and group 𝑗’s investment relative to the average investment of
the other groups.
Group 𝑗thus receives the high adjustment in period 𝑡if a random number calculated by the ztree function Random( ) is less than
the probability threshold determined by the P function above. For just two adjustment levels, the probability of the low adjustment
is 1 −𝑃. Separate adjustments for all other groups are calculated in the same manner.
To obtain a purely random adjustment mechanism, we set 𝑏0= 1and 𝑏1=𝑏2= 0. The probability of a high adjustment is then
always 0.5 (or whatever value we choose for 𝜋0):
𝑃(𝐴𝑗 𝑡=𝐻)≡0.5
To obtain an adjustment mechanism based solely on the group’s absolute contributions, we set 𝑏1= 1and 𝑏0=𝑏2= 0. For
𝜋1= 0.2, the probability of a high adjustment then ranges from 0.2to 0.8as the group’s contribution ranges from zero to 100% of
the maximum possible contribution (and equals 0.5 whenever the group contributes half of the maximum).
𝑃(𝐴𝑗 𝑡=𝐻)≡[0.2 + 0.6( 𝐺𝑗 𝑡
𝐺∗)]
To obtain an adjustment mechanism based solely on the group’s relative contributions, we set 𝑏1= 1and 𝑏0=𝑏2= 0. For
𝜋1= 0.2, the probability of a high adjustment then ranges from 0.2(when the group contributes zero and at least one other group
contributes more than zero) to 0.8(when the group’s contribution is non-zero and all other groups contribute zero). [Note: We
define 𝐹2(0∕0) ≡0.5, so the probability of a high adjustment is 0.5whenever group 𝑗’s contribution equals the average contributed
by the other groups, even when all groups contribute zero.]
𝑃(𝐴𝑗 𝑡=𝐻)≡[0.2 + 0.6(𝐺𝑗 𝑡
𝐺𝑗+
𝐺−𝑗 𝑡)]
Relatively straightforward changes in Eq. (4) yield a wide array of adjustment mechanisms. For example, assigning non-zero
values to both 𝑏1and 𝑏2yields adjustment mechanisms that depend on both relative and absolute contributions. The adjustment
mechanisms can likewise be extended further to depend on both current and past contributions.
Appendix D. Regression analyses of contributions
The following GLS and Tobit regressions omit the Baseline treatment in order to focus on the effect of adjustments. GLS standard
errors are clustered at the group level, and Tobit standard errors are obtained by bootstrapping. The Random treatment is the
omitted category (see Tables D.1 and D.2).
Appendix E. Determinants of repeat contributions
Below, we analyze player responses to positive versus negative adjustments. Recall that the theory of operant conditioning
predicts that subjects will tend to repeat behaviors followed by positive reinforcement and will tend to avoid behaviors followed
by negative reinforcement. Given that all adjustments were +5 or −5, we can only estimate the effects of positive versus negative
reinforcement. We do so by measuring rates of repetition and rates of inertia.
Repetition Effects: Table E.1 lists mean, standard error, and number of repeat contributions across the treatments. Our repetition
measure excludes contributions of zero (the Nash equilibrium) and ten (the social optimum) so as not to confuse repetitions caused
by +5 adjustments with repetitions caused by persistent free-riding or unconditional cooperation. Note that repeat investments in
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Table D.1
GLS analysis of contributions.
(1) (2) (3) (4)
Contribution Contribution Contribution Contribution
Absolute 0.085 0.096 0.092 0.134
(0.519) (0.528) (0.529) (0.522)
Relative 0.359 0.412 0.433 0.606
(0.497) (0.512) (0.516) (0.512)
Unknown −0.069 −0.028 −0.025 −0.056
(0.527) (0.539) (0.535) (0.523)
Individual 1.274∗∗ 1.304∗∗ 1.319∗∗ 1.456∗∗∗
(0.517) (0.531) (0.533) (0.555)
Positive adjustment𝑡−1 −0.045 −0.056 0.065
(0.084) (0.073) (0.141)
Risk −0.526 −0.528
(0.647) (0.646)
Female −0.230 −0.231
(0.261) (0.261)
Period −0.113∗∗∗ −0.113∗∗∗
(0.013) (0.013)
Absolute ×Positive adjustment𝑡−1 −0.082
(0.262)
Relative ×Positive adjustment𝑡−1 −0.335∗
(0.191)
Unknown ×Positive adjustment𝑡−1 0.070
(0.195)
Individual ×Positive adjustment𝑡−1 −0.274
(0.196)
Intercept 4.126∗∗∗ 4.054∗∗∗ 5.665∗∗∗ 5.603∗∗∗
(0.352) (0.362) (0.486) (0.492)
Observations 6400 6080 6080 6080
Standard errors clustered by group in parentheses.
∗p<0.1, ∗∗ p<0.05, ∗∗∗ p<0.01.
The Random adjustment treatment is the omitted category.
both the Random and Unknown adjustment treatments are nearly twice as common as repeats in the no-adjustment Baseline. In
contrast, the difference in repetition rates for Random versus Unknown is small and statistically insignificant.33
To better understand the determinants of repeat investments, we regress a measure of repeated contributions repetitions on
treatment type, gender, and risk preference. The NoChange indicator variable equals one if the player made exactly the same
contribution in the current and preceding period. We, again, exclude contributions of 0 and 10 tokens.
The regression clusters standard errors by group and includes player-specific random effects. In all four regressions, we see
subjects are much more likely to repeat their previous contribution after a positive adjustment. Hence, for the entire sample of
contributions, subjects repeat their contributions about 30% of the time following a+5 adjustment but only 20% of the time
following a−5 adjustment (see Table E.3).
Appendix F. Contribution paths from some past studies
For each study, we have highlighted the deterministic baseline game’s contribution path in red. We chose this studies based on
the fact that they employed baseline VCM’s very similar to the present experiment and also included a figure showing the time path
of average contributions for each treatment. Though they should not be viewed as a representative sample of all experiments that
add risk or uncertainty to an otherwise standard repeated linear VCM, they do illustrate the tendencies that we and others have
noted.
33 The two-sided rank sum Wilcoxon test p-values are 0.00 for both Baseline versus Random and Baseline versus Unknown, but the corresponding 𝑝-value is
0.37 for Random versus Unknown.
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Table D.2
Tobit analysis of contributions.
(1) (2) (3) (4)
Contribution Contribution Contribution Contribution
Absolute 0.059 0.066 0.059 0.141
(0.563) (0.631) (0.647) (0.606)
Relative 0.403 0.480 0.490 0.718
(0.582) (0.538) (0.590) (0.603)
Unknown −0.030 0.026 0.017 0.025
(0.506) (0.466) (0.519) (0.565)
Individual 1.665∗∗∗ 1.700∗∗∗ 1.703∗∗∗ 1.870∗∗∗
(0.547) (0.640) (0.635) (0.657)
Positive adjustment𝑡−1 −0.072 −0.085 0.099
(0.088) (0.092) (0.190)
Risk −0.553 −0.556
(0.822) (0.932)
Female −0.130 −0.132
(0.367) (0.410)
Period −0.147∗∗∗ −0.146∗∗∗
(0.014) (0.014)
Absolute ×Positive adjustment𝑡−1 −0.160
(0.325)
Relative ×Positive adjustment𝑡−1 −0.443∗
(0.242)
Unknown ×Positive adjustment𝑡−1 −0.005
(0.248)
Individual ×Positive adjustment𝑡−1 −0.332
(0.266)
Intercept 3.980∗∗∗ 3.885∗∗∗ 5.820∗∗∗ 5.724∗∗∗
(0.395) (0.382) (0.570) (0.677)
Observations 6400 6080 6080 6080
Bootstrapped standard errors in parentheses.
∗p<0.1, ∗∗ p<0.05, ∗∗∗ p<0.01.
The Random treatment is the omitted category.
Table E.1
Repeat contribution rates.
Treatments Repeat contributions
(excluding 0 and 10)
Mean SE N
Baseline 0.13 0.11 807
Random 0.23 0.13 1019
Absolute 0.23 0.13 1001
Relative 0.25 0.13 1049
Unknown 0.26 0.13 1110
Individual 0.22 0.13 980
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A. Hajikhameneh and L.R. Iannaccone
Table E.2
Contribution effects of positive vs. negative adjustments. Excluding 0 and 10.
(1) (2) (3) (4)
No change No change No change No change
Positive adjustment𝑡−1 0.112∗∗∗ 0.112∗∗∗ 0.078∗∗∗ 0.078∗∗∗
(0.014) (0.014) (0.022) (0.022)
Absolute −0.002 −0.051 −0.051
(0.034) (0.035) (0.033)
Relative 0.009 −0.011 −0.011
(0.036) (0.039) (0.039)
Unknown 0.027 0.018 0.016
(0.033) (0.037) (0.036)
Individual −0.012 −0.019 −0.020
(0.031) (0.035) (0.035)
Absolute ×Positive adjustment𝑡−1 0.103∗∗∗ 0.103∗∗∗
(0.040) (0.040)
Relative ×Positive adjustment𝑡−1 0.038 0.038
(0.035) (0.035)
Unknown ×Positive adjustment𝑡−1 0.018 0.018
(0.033) (0.033)
Individual ×Positive adjustment𝑡−1 0.014 0.014
(0.043) (0.043)
Risk −0.093∗∗
(0.043)
Female −0.005
(0.023)
Intercept 0.180∗∗∗ 0.175∗∗∗ 0.192∗∗∗ 0.235∗∗∗
(0.011) (0.026) (0.028) (0.031)
Observations 5159 5159 5159 5159
Dependent variable indicates no change in contribution from previous period.
Random is treatment omitted category.
Standard errors in parentheses. Subjects clustered by group.
Regressions exclude cases with contribution =0 or 10.
Table E.3
Contribution effects of positive vs. negative adjustments.
(1) (2) (3) (4)
No change No change No change No change
Positive adjustment𝑡−1 0.119∗∗∗ 0.119∗∗∗ 0.079∗∗∗ 0.080∗∗∗
(0.013) (0.013) (0.013) (0.013)
Absolute 0.013 −0.034 −0.036
(0.042) (0.043) (0.042)
Relative −0.011 −0.040 −0.026
(0.043) (0.048) (0.047)
Unknown −0.035 −0.052 −0.044
(0.038) (0.039) (0.039)
Individual 0.012 0.006 0.016
(0.045) (0.043) (0.042)
Absolute ×Positive adjustment𝑡−1 0.094∗∗ 0.093∗∗
(0.036) (0.037)
Relative ×Positive adjustment𝑡−1 0.058∗0.058∗
(0.034) (0.034)
Unknown ×Positive adjustment𝑡−1 0.035 0.035
(0.026) (0.026)
Individual ×Positive adjustment𝑡−1 0.013 0.013
(0.039) (0.039)
Risk −0.022
(0.066)
Female −0.116∗∗∗
(0.032)
Intercept 0.294∗∗∗ 0.298∗∗∗ 0.318∗∗∗ 0.397∗∗∗
(0.014) (0.032) (0.031) (0.047)
Observations 6080 6080 6080 6080
Dependent variable indicates no change in contribution from previous period.
Random is treatment omitted category.
Standard errors in parentheses. Subjects clustered by group.
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A. Hajikhameneh and L.R. Iannaccone
Appendix G. Supplementary data
Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jebo.2024.106775.
Data availability
Data will be made available on request.
References
Akerlof, G.A., Dickens, W.T., 1982. The economic consequences of cognitive dissonance. Am. Econ. Rev. 72 (3), 307–319.
Aksoy, B., Krasteva, S., 2020. When does less information translate into more giving to public goods? Exp. Econ. 23 (4), 1148–1177.
Atran, S., Henrich, J., 2010. The evolution of religion: How cognitive by-products, adaptive learning heuristics, ritual displays, and group competition generate
deep commitments to prosocial religions. Biol. Theory 5 (1), 18–30.
Axelrod, R., Ford, G.R., Riolo, R.L., Cohen, M.D., 2002. Beyond geography: Cooperation with persistent links in the absence of clustered neighborhoods. Pers.
Soc. Psychol. Rev. 6 (4), 341–346.
Beck, J., Forstmeier, W., 2007. Superstition and belief as inevitable by-products of an adaptive learning strategy. Hum. Nat. 18 (1), 35–46.
Bejarano, H., Gillet, J., Rodriguez-Lara, I., 2021. Trust and trustworthiness after negative random shocks. J. Econ. Psychol. 86, 102422.
Bjork, L., Kocher, M., Martinsson, P., Nam Khanh, P., 2016. Cooperation Under Risk and Ambiguity. Tech. rep., Working Papers in Economics, No 683, Department
of Economics, School of Business, Economics and Law at University of Gothenburg.
Bleak, J.L., Frederick, C.M., 1998. Supersititious behavior in sport: Levels of effectiveness and determinants of use in three collegiate sports. J. Sport Behav. 21
(1), 1.
Boyer, P., 2001. Religion Explained: The Evolutionary Origins of Religious Thought. Basic Books.
Brandt, H., Hauert, C., Sigmund, K., 2003. Punishment and reputation in spatial public goods games. Proc. R. Soc. B 270 (1519), 1099–1104.
Journal of Economic Behavior and Organization 229 (2025) 106775
23
A. Hajikhameneh and L.R. Iannaccone
Butera, L., Grossman, P.J., Houser, D., List, J.A., Villeval, M.-C., 2020. A New Mechanism to Alleviate the Crises of Confidence in Science-With An Application
to the Public Goods Game. Tech. rep., National Bureau of Economic Research.
Butera, L., List, J.A., 2017. An Economic Approach to Alleviate the Crises of Confidence in Science: With an Application to the Public Goods Game. Tech. rep.,
National Bureau of Economic Research.
Chaudhuri, A., 2011. Sustaining cooperation in laboratory public goods experiments: a selective survey of the literature. Exp. Econ. 14 (1), 47–83.
Chaudhuri, A., 2016. Recent advances in experimental studies of social dilemma games. Games 7 (1), 7.
Chaudhuri, A., 2018. Belief heterogeneity and the restart effect in a public goods game. Games 9 (4), 96.
Chen, Y., Li, S.X., 2009. Group identity and social preferences. Amer. Econ. Rev. 99 (1), 431–457.
Cookson, R., 2000. Framing effects in public goods experiments. Exp. Econ. 3 (1), 55–79.
Cushman, F., 2020. Rationalization is rational. Behav. Brain Sci. 43, e28.
Davis, H., Hubbard, J., 1972. An analysis of superstitious behavior in the rat. Behaviour 1–12.
De Weerdt, J., Dercon, S., 2006. Risk-sharing networks and insurance against illness. J. Dev. Econ. 81 (2), 337–356.
DeQuidt, J., Vesterlund, L., Wilson, A.J., 2019. Experimenter demand effects. In: Handbook of Research Methods and Applications in Experimental Economics.
Edward Elgar Publishing.
Dickinson, D.L., 1998. The voluntary contributions mechanism with uncertain group payoffs. J. Econ. Behav. Organ. 35 (4), 517–533.
Dorofeenko, V., Shorish, J., 2002. Dynamical Modeling of the Demographic Prisoner’s Dilemma. Working Paper, Institut für höhere Studien.
Durkheim, E., 1912 1965. The Elementary Forms of the Religious Life [1912]. na.
Durrant, R., Poppelwell, Z., 2017. Religion, Crime and Punishment: An Evolutionary Perspective. Springer.
Eckel, C.C., Grossman, P.J., 2005. Managing diversity by creating team identity. J. Econ. Behav. Organ. 58 (3), 371–392.
Epstein, J.M., 1998. Zones of cooperation in demographic prisoner’s dilemma. Complexity 4 (2), 36–48.
Fafchamps, M., Lund, S., 2003. Risk-sharing networks in rural Philippines. J. Dev. Econ. 71 (2), 261–287.
Filippin, A., Crosetto, P., 2016. Click ‘n’roll: no evidence of illusion of control. Economist 164 (3), 281–295.
Fischbacher, U., 2007. z-Tree: Zurich toolbox for ready-made economic experiments. Exp. Econ. 10 (2), 171–178.
Fischbacher, U., Schudy, S., Teyssier, S., 2014. Heterogeneous reactions to heterogeneity in returns from public goods. Soc. Choice Welf. 43 (1), 195–217.
Fisher, J., Isaac, R.M., Schatzberg, J.W., Walker, J.M., 1995. Heterogenous demand for public goods: Behavior in the voluntary contributions mechanism. Public
Choice 85 (3), 249–266.
Foster, K.R., Kokko, H., 2009. The evolution of superstitious and superstition-like behaviour. Proc. R. Soc. Lond. Ser. B: Biol. Sci. 276 (1654), 31–37.
Gambetta, D., 2005. Can we make sense of suicide missions? In: Making Sense of Suicide Missions. Oxford University Press, Oxford, pp. 259–300.
Gangadharan, L., Nemes, V., 2009. Experimental analysis of risk and uncertainty in provisioning private and public goods. Econ. Inq. 47 (1), 146–164.
Gibaud, S., 2016. Spatialized Evolutionary prisoner’s dilemma: Homogenization and propagation of chaos. arXiv preprint arXiv:1609.05036.
Gmelch, G., 1971. Baseball magic. Transaction 8 (8), 39–41.
Hajikhameneh, A., 2025. Uncertainty, superstition, and endogenous group formation: an intergenerational experiment.
Hajikhameneh, A., Iannaccone, L.R., 2023. God games: An experimental study of uncertainty, superstition, and cooperation. Games Econom. Behav. 139, 88–116.
Hall, L., Johansson, P., Tärning, B., Sikström, S., Deutgen, T., 2010. Magic at the marketplace: Choice blindness for the taste of jam and the smell of tea.
Cognition 117 (1), 54–61.
Hall, L., Strandberg, T., Pärnamets, P., Lind, A., Tärning, B., Johansson, P., 2013. How the polls can be both spot on and dead wrong: Using choice blindness
to shift political attitudes and voter intentions. PLoS One 8 (4), e60554.
Hayek, F.A., 1989. In: Bartley III, W.W. (Ed.), The Fatal Conceit.
Hayek, F.A., 1989 2011. The Fatal Conceit: The Errors of Socialism, vol. 1, University of Chicago Press.
Helbing, D., Szolnoki, A., Perc, M., Szabó, G., 2010. Punish, but not too hard: how costly punishment spreads in the spatial public goods game. New J. Phys.
12 (8), 083005.
Henrich, J., 2009. The evolution of costly displays, cooperation and religion: Credibility enhancing displays and their implications for cultural evolution. Evol.
Hum. Behav. 30 (4), 244–260.
Henrich, J., 2020. The WEIRDest People in the World: How the West Became Psychologically Peculiar and Particularly Prosperous. Penguin UK.
Henrich, J., Muthukrishna, M., 2021. The origins and psychology of human cooperation. Annu. Rev. Psychol. 72 (1), 207–240.
Holt, C., Laury, S., 2002. Risk aversion and incentive effects. Amer. Econ. Rev. 92 (5), 1644–1655.
Huang, K., Wang, T., Cheng, Y., Zheng, X., 2015. Effect of heterogeneous investments on the evolution of cooperation in spatial public goods game. PLoS One
10 (3), e0120317.
Isaac, R.M., Walker, J.M., Thomas, S.H., 1984. Divergent evidence on free riding: An experimental examination of possible explanations. Public Choice 43 (2),
113–149.
Johansson, P., Hall, L., Sikstrom, S., Olsson, A., 2005. Failure to detect mismatches between intention and outcome in a simple decision task. Science 310 (5745),
116–119.
Kahneman, D., Slovic, S.P., Slovic, P., Tversky, A., 1982. Judgment Under Uncertainty: Heuristics and Biases. Cambridge University Press.
Kaplan, H.S., Schniter, E., Smith, V.L., Wilson, B.J., 2012. Risk and the evolution of human exchange. Proc. R. Soc. Lond. Ser. B: Biol. Sci. rspb20112614.
Killingback, T., Doebeli, M., Knowlton, N., 1999. Variable investment, the continuous prisoner’s dilemma, and the origin of cooperation. Proc. R. Soc. B 266
(1430), 1723–1728.
Klusowski, J., Small, D.A., Simmons, J.P., 2021. Does choice cause an illusion of control? Psychol. Sci. 32 (2), 159–172.
Konow, J., 2000. Fair shares: Accountability and cognitive dissonance in allocation decisions. Amer. Econ. Rev. 90 (4), 1072–1091.
Krawczyk, M., Le Lec, F., 2016. Dictating the risk: experimental evidence on giving in risky environments: comment. Amer. Econ. Rev. 106 (3), 836–839.
Kyle Stanford, P., Thomas, A.J., Sarnecka, B.W., 2020. Rationalization may improve predictability rather than accuracy. Behav. Brain Sci. 43, e49.
Langer, E.J., 1975. The illusion of control. J. Pers. Soc. Psychol. 32 (2), 311.
Ledyard, J.O., 1995. Public goods: A survey of experimental research. In: The Handbook of Experimental Economics. Princeton University Press, pp. 111–194.
Leeson, P.T., 2012. Ordeals. J. Law Econ. 55 (3), 691–714.
Levati, M.V., Morone, A., 2013. Voluntary contributions with risky and uncertain marginal returns: the importance of the parameter values. J. Public Econ.
Theory 15 (5), 736–744.
Levati, M.V., Morone, A., Fiore, A., 2009. Voluntary contributions with imperfect information: An experimental study. Public Choice 138 (1), 199–216.
Levy, N., 2020. Rationalization enables cooperation and cultural evolution. Behav. Brain Sci. 43.
Levy, G., Razin, R., 2012. Religious beliefs, religious participation, and cooperation. Am. Econ. J. Microecon. 4 (3), 121–151.
Loftus, E.F., 1992. When a lie becomes memory’s truth: Memory distortion after exposure to misinformation. Curr. Dir. Psychol. Sci. 1 (4), 121–123.
Loftus, E.F., Pickrell, J.E., 1995. The formation of false memories. Psychiatr. Ann. 25 (12), 720–725.
Malinowski, B., 1918. 53. Fishing in the trobriand islands. Man 18, 87–92.
Malinowski, B., 1948. Magic, Science and Religion, vol. 23, Beacon Press Boston.
Marks, M.B., Croson, R.T., 1999. The effect of incomplete information in a threshold public goods experiment. Public Choice 99 (1), 103–118.
McAdam, D., 1986. Recruitment to high-risk activism: The case of freedom summer. Am. J. Sociol. 92 (1), 64–90.
Norenzayan, A., 2013. Big Gods: How Religion Transformed Cooperation and Conflict. Princeton University Press.
Journal of Economic Behavior and Organization 229 (2025) 106775
24
A. Hajikhameneh and L.R. Iannaccone
Norenzayan, A., Shariff, A.F., Gervais, W.M., Willard, A.K., McNamara, R.A., Slingerland, E., Henrich, J., 2016. The cultural evolution of prosocial religions.
Behav. Brain Sci. 39.
Nowak, M.A., May, R.M., 1993. The spatial dilemmas of evolution. Int. J. Bifurcation Chaos 3 (01), 35–78.
Ono, K., 1987. Superstitious behavior in humans. J. Exp. Anal. Behav. 47 (3), 261–271.
Palmer, C.T., 1989. The ritual taboos of fishermen: An alternative explanation. MAST. Marit. Anthropol. Stud. 2 (1), 59–68.
Presson, P.K., Benassi, V.A., 1996. Illusion of control: A meta-analytic review. J. Soc. Behav. Pers. 11 (3), 493.
Purzycki, B.G., Henrich, J., Apicella, C., Atkinson, Q.D., Baimel, A., Cohen, E., McNamara, R.A., Willard, A.K., Xygalatas, D., Norenzayan, A., 2018. The evolution
of religion and morality: a synthesis of ethnographic and experimental evidence from eight societies. Relig. Brain Behav. 8 (2), 101–132.
Rabin, M., 1994. Cognitive dissonance and social change. J. Econ. Behav. Organ. 23 (2), 177–194.
Railton, P., 2020. Rationalization of emotion is also rational. Behav. Brain Sci. 43.
Santos, F.C., Pacheco, J.M., Lenaerts, T., 2006. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl. Acad. Sci. 103
(9), 3490–3494.
Schweitzer, F., Behera, L., Mühlenbein, H., 2002. Evolution of cooperation in a spatial prisoner’s dilemma. Adv. Complex Syst. 5 (02n03), 269–299.
Skinner, B.F., 1948 1992. "Superstition" in the pigeon. J. Exp. Psychol. [Gen.] 121 (3), 273.
Stark, R., 2005. The victory of reason. In: How Christianity Led to Freedom, Capitalism, and Western Success. New York.
Stefan, S., David, D., 2013. Recent developments in the experimental investigation of the illusion of control. A meta-analytic review. J. Appl. Soc. Psychol. 43
(2), 377–386.
Stoddard, B., 2013. Uncertainty in Payoff-Equivalent Appropriation and Provision Games. University of South Dakota Working Paper.
Stoddard, B., 2017. Risk in payoff-equivalent appropriation and provision games. J. Behav. Exp. Econ. 69, 78–82.
Stoddard, B.V., et al., 2015. Probabilistic production of a public good. Econ. Bull. 35 (1), 37–52.
Szolnoki, A., Szabó, G., Czakó, L., 2011. Competition of individual and institutional punishments in spatial public goods games. Phys. Rev. E 84 (4), 046106.
Théroude, V., Zylbersztejn, A., 2020. Cooperation in a risky world. J. Public Econ. Theory 22 (2), 388–407.
Thompson, S.C., Armstrong, W., Thomas, C., 1998. Illusions of control, underestimations, and accuracy: a control heuristic explanation. Psychol. Bull. 123 (2),
143.
Townsend, R.M., 1994. Risk and insurance in village India. Econometrica 539–591.
Van Dijk, E., Grodzka, M., 1992. The influence of endowments asymmetry and information level on the contribution to a public step good. J. Econ. Psychol. 13
(2), 329–342.
Vesely, S., Wengström, E., et al., 2017. Risk and Cooperation: Experimental Evidence from Stochastic Public Good Games. Working Paper 2017:3, Department
of Economics, School of Economics and Management, Lund University.
Villeval, M.C., 2020. Public goods, norms and cooperation. In: Handbook of Experimental Game Theory. Edward Elgar Publishing.
Vyse, S.A., 2013. Believing in Magic: The Psychology of Superstition-Updated Edition. Oxford University Press.
Wade, N., 2009. The Faith Instinct: How Religion Evolved and Why It Endures. Penguin.
Wagner, G.A., Morris, E.K., 1987. ‘‘Superstitious’’ behavior in children. Psychol. Rec. 37 (4), 471–488.
Wilson, D., 2010. Darwin’s Cathedral. University of Chicago Press.
Yang, L., Zhang, L., 2021. Environmental feedback in spatial public goods game. Chaos Solitons Fractals 142, 110485.
Yong, J.C., Li, N.P., Kanazawa, S., 2021. Not so much rational but rationalizing: Humans evolved as coherence-seeking, fiction-making animals. Am. Psychol. 76
(5), 781.
Zelmer, J., 2003. Linear public goods experiments: A meta-analysis. Exp. Econ. 6, 299–310.
Zizzo, D.J., 2010. Experimenter demand effects in economic experiments. Exp. Econ. 13 (1), 75–98.
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25
