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\title{Steady steps versus sudden shifts: Cooperation in (a)symmetric linear and step-level social dilemmas}

\author{
Judith Kas\thanks{Economics of Consumers and Households Group; Wageningen
University and Department of Sociology, Utrecht; Currently at
Unit of Migration, Integration, Transnationalization, Social Science
Center Berlin (WZB). E-mail: judith.kas@wzb.eu.}
\and
David J. Hardisty\thanks{Sauder School of Business, University of British
Columbia, Vancouver.}
\and
Michel J. J. Handgraaf\thanks{Urban Economics Group, Wageningen University; 
Center for Research on Environmental Decisions (CRED) at Columbia
University, New York.}
}

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\begin{abstract}

Are groups of people better able to minimize a collective loss if there is
a collective target that must be reached or if every small contribution
helps? In this paper we investigate whether cooperation in social dilemmas
can be increased by structuring the problem as a step-level social dilemma
rather than a linear social dilemma and whether cooperation can be
increased by manipulating endowment asymmetry between individuals. In two
laboratory experiments using ‘Public Bad’ games, we found that that
individuals defect less and are better able to minimize collective and
personal costs in a step-level social dilemma than in a linear social
dilemma. We found that the level of cooperation is not affected by an
ambiguous threshold: even when participants cannot be sure about the
optimal cooperation level, cooperation remains high in the step-level
social dilemmas. We find mixed results for the effect of asymmetry on
cooperation. These results imply that presenting social dilemmas as
step-level games and reducing asymmetry can help solve environmental
dilemmas in the long term. 

\smallskip
\noindent
Keywords: cooperation, social dilemma, behavioral economics, environmental behavior
\end{abstract}

{\renewcommand{\thefootnote}{}

\footnotetext{

Support for this research was provided by the National Science Foundation,
grant SES-0951516 and the Social Sciences and Humanities Research Council
of Canada. The authors thank Andrew Lai, Kelly Yang, Yoonji Shim, and
Anthony Yam for their assistance in running the experiment.

Copyright: \copyright\ 2021.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

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\section{Introduction}


Many efforts have been made that aim to understand and increase cooperation
in order to solve social dilemmas. These dilemmas are often framed as
linear problems (e.g., Balliet, Li, MacFarlan \& Van Vugt, 2011), while
in reality many societal problems are not linear. In the current paper we
theorize the conditions under which cooperation may be higher in
non-linear dilemmas and we perform two incentive-compatible laboratory
experiments in which we test our theory. 

In linear social dilemmas each decision by every agent has a similar impact
on the outcomes of the dilemma: every extra euro donated, petition signed
and every kilometre driven less has an equal positive impact. However, in
reality many societal problems are not linear, but rather S-shaped, which
means that the expected outcome is probabilistically related to the
proportion of cooperation. For example, if a fish stock is threatened, not
every additional fish caught is considered equally harmful to the stock,
and each person cannot equally harm or protect the stock. Rather, the more
a population is depleted, the lower the capacity of the population to
reproduce itself and the larger the impact of catching an extra fish,
until the population is too small to reproduce itself (Myers, Rosenberg,
Mace, Barrowman \& Restrepo, 1994). Another example are ‘desire lines’,
paths created by pedestrians or vehicles through a patch of grass. If only
a few people walk on the grass there is no problem, but if more people
take the shortcut, the grass will get trampled. If a lot of people have
already walked on the grass, an additional person taking the shortcut does
not make a difference anymore, since the grass is dead anyway (Baron,
2008).

An extreme form of this probabilistic distribution is a step-level problem
in which contributions have no effect, except at a specific level of
cooperation (Abele, Stasser \& Chartier, 2010). As such, step-level
problems involve sudden rather than gradual shifts. Once a certain
threshold is reached, the outcomes change dramatically, but if the
threshold is not reached, no change occurs. An example of such a situation
is a situation where a group has to make a joint decision, e.g., the
decision to place solar panels on a shared roof. To get the majority vote,
some group members may have to spend time and effort to collecting
information and convincing others that it is a good idea. When they
succeed, the whole group benefits, but when they fail they do not benefit,
while they also bear the costs in terms of time and money.

The question of why some projects attract donations and supporters or
induce behavioural change, while others do not is topic of many research
papers and is a question of large societal relevance. In the current
research we study how the structure of a problem affects the ability for
groups to solve social dilemmas. Extensive research on both linear and
step-level social dilemmas has been done, for example in relation to
social value orientation (Attari, Krantz \& Weber, 2014; Balliet, Parks
\& Joireman, 2009), social identity (Simpson, 2006), the role of
uncertainty (Biel \& Gärling, 1995), membership fees (Bchir \& Willinger,
2013), social norms and efficacy (Attari et al., 2014) and the possibility
to punish (Cooper \& Stockman, 2002). However, we are not aware of any
direct empirical comparisons between these two problem types. In the
current paper we compare cooperation levels in a linear social dilemma
with a step-level social dilemma. We theorize why and under what
conditions the shape of the outcomes (relative to the total contributions)
matters. This helps us to form predictions about how different factors
will work out differently in the two dilemmas. We also test how this may
be moderated by asymmetry (where different people are endowed with
different amounts), which is found to be a complicating factor in
establishing cooperation (Zelmer, 2003).

In the first experiment we show that individuals cooperate more in
step-level social dilemmas than in linear social dilemmas and that
asymmetry between individuals in a group affects cooperation differently in
the two different dilemmas. We use ‘public bad’ games, in which groups only
pay a group cost when defection exceeds a certain threshold. In the second
experiment we explore the extent to which the higher cooperation level in
the step-level social dilemma is robust to ambiguity in the extent to which
each group member should cooperate to avoid exceeding the threshold.

\section{Theory}

\subsection{Game theoretic representation of linear and step-level social
dilemmas}

Social dilemma problems have been extensively studied using decision making
games: in experiments that resemble simplified social dilemma situations
individuals are asked to make a choice between their self-interest and the
group’s interest. The main paradigm that is used to study social dilemmas
are public goods games (in which players can cooperate by contributing to
a public good) and resource dilemma games (in which players cooperate by
not taking from a common resource pool; Van Dijk \& Wilke, 2000). However,
many societal problems do not resemble public goods or resource dilemmas,
but rather a ‘public bad’: the more CO\textsubscript{2} is emitted, the
worse it is for the climate, which is bad for everyone; the more people
litter, the worse for everyone, etc. The defecting behavior is here to
litter or to emit CO\textsubscript{2} and the cooperative behavior is to
\textit{not} perform those harmful behaviors. Because public bad games
have received relatively little attention in social dilemma research, the
current study explores the research questions in a public bad scenario.
Individuals receive tokens that they can either keep or put in a common
pool. Tokens that are kept incur costs for the individual, while tokens
given to the common pool may incur costs for the group but there are no
direct costs for the individual. Cooperation is here defined as the number of tokens kept, while defection is defined as the number of tokens deposited. Another reason to study this type of
social dilemmas is that findings from earlier research on public goods
games may not always be applicable to these situations where cooperating
means refraining from certain harmful behavior. While in public goods
games individuals with larger endowments are both wealthier and have more
power, this does not apply to the case of public bad games: individuals
who have a larger endowment have a larger influence on the group outcome,
but may also face larger costs without benefitting more from the group
outcome than individuals with a smaller endowment.

The linear social dilemma is represented by a Public Bad Game with a linear
relation between tokens given and group costs. In particular, we look at
behavior in a Public Bad Game that is framed as an environmental scenario.
For every pollution token deposited to the common pool, the group costs
increase with a fixed amount. We focus on situations in which the group costs
are equally split over all the actors and where the group costs per actor
are lower than the individual costs of keeping tokens. In the step-level
social dilemma, the relation between tokens deposited and group costs is
non-linear. Rather than paying per token in the common pool, the group
pays a fixed amount only when more tokens than a certain threshold are
given to the common pool, otherwise there are no group costs (but players
always have to pay the individual costs the for tokens they
keep). Individuals in groups will thus benefit from staying under the
threshold, analogous to a situation where fishers benefit from sustaining
the reproductive capacity of common fish stock. The
amount that needs to be paid is independent of the number of tokens extra
deposited above the threshold. (Therefore, once past the threshold, it is
rational for all players to put all pollution tokens into the common
pool.) 

% How is this analogous to CO2?

We use game-theoretical tools to describe how behavior differs between
linear and step-level Public Bad Games and to describe how asymmetry
between actors affects behavior differently in the two social dilemma
types. We assume that individuals have different motivations that drive
their behavior in these social dilemmas. While some individuals will be
motivated only to serve their own interest, other individuals may me more
altruistic. That means that they are more willing to give up their own
payoff to increase the payoff of others or reduce their costs.

\subsection{Cooperation in linear versus step-level social dilemmas}

We expect that individuals cooperate more in step-level social dilemmas
than in linear social dilemmas for two reasons. 

\subsection{Equilibria in linear social dilemmas}

First, in a linear social dilemma, there is only the Nash equilibrium.
Regardless of the choice of the other players, defecting always yields
superior outcomes for the self, because cooperation involves costs to the
cooperators that are not compensated with the benefits of a higher
cooperation level. Since the costs of keeping tokens are always higher
than the costs of depositing tokens to the common pool, self-interested
actors are expected to deposit all their (negative) tokens into the common
pool. This Nash equilibrium in which everyone defects is clearly not
Pareto efficient: if everyone would have cooperated, everyone would have
had a better outcome (Abele et al., 2010). 100\% cooperation is the Pareto
efficient solution, but this situation is not a Nash equilibrium, because
defecting is always payoff maximizing in linear social dilemmas. As a
consequence, the Pareto efficient solution in which everybody cooperates
is unstable (Abele et al., 2010): it is always tempting for individuals to
move towards defecting and thus move away from the optimal solution. If
people believe that others cannot resist the temptation to defect because
it gives them a higher payoff, they themselves may be less motivated to
cooperate. 

While purely self-interested individuals are not expected to cooperate in a
linear social dilemma, more altruistic individuals may be more motivated
to cooperate, since they do not only care about their own payoff, but also
about the payoff of others.

In a repeated social dilemmas, individuals may also cooperate if it induces
other players to respond in kind and cooperate as well. Earlier
experiments on linear social dilemmas indeed find that while some
individuals cooperate in linear social dilemmas, cooperation quickly
breaks down when there are no institutions such as punishments or
reputation systems to motivate self-interested individuals to cooperate
(Fehr \& Gächter, 2000). 

\subsection{Equilibria in step-level social dilemmas}

In step-level social dilemmas on the other hand, cooperating does not
necessarily lead to a reduction in personal payoff. If an actor expects
his or her cooperation to be critical for the group to stay under the
threshold, it may under certain circumstances be beneficial for the
self-interested individual to cooperate to avoid the group penalty for
exceeding the threshold. If the group penalty is sufficiently large and
the costs of keeping tokens sufficiently low, individuals may receive a
higher payoff if they keep some tokens than if they dump everything. When
the threshold is exactly reached there is a Nash equilibrium that is also
Pareto efficient (Abele et al., 2010). No one can move from cooperating to
defecting without harming the others \textit{and} themselves, because that
would mean that the threshold is no longer avoided and both the personal
and societal benefits would be smaller than if that person would have
cooperated. Similar to the linear social dilemma, more altruistic
individuals will be more likely than more selfish individuals to cooperate
if their contributions help the group.

The difference in contributions between the linear and the step-level
social dilemmas may depend on the level of the threshold (Rapoport \&
Suleiman, 1993). While the best strategy for self-interested individuals
in the linear social dilemma is to defect, cooperation increases their
personal payoff under certain conditions in the step-level dilemma. More
altruistic players are expected to cooperate more in the step-level social
dilemma when exceeding the threshold can be avoided only if they cooperate
more their ‘natural’ level of
cooperation, which is the number of tokens they would keep in the linear
social dilemma. If the threshold can be avoided by keeping fewer tokens
than they would keep in the linear social dilemma, altruistic individuals
may cooperate less in the step-level social dilemma than in the linear
social dilemma. 

A meta-analysis that includes 349 earlier studies on repeated linear public
goods games found an average cooperation level of 37.7\% (Zelmer, 2003).
In the current experiment we choose a threshold that can be avoided only if
subjects cooperate more than the average cooperation level in earlier
studies. We ask whether cooperation can indeed be higher in a step-level
game than in a linear game. Groups in step-level social dilemmas are
expected to minimize (costly) cooperation, while still staying under the
threshold.  They are not expected to cooperate (much) more than required
for staying under the threshold, so when staying under the threshold
requires less cooperation than the average cooperation rate in a linear
social dilemma, it is not possible to ask whether cooperation can be higher
in threshold social dilemmas than in linear social dilemmas. However,
staying under the threshold should also not require too much
cooperation. When staying under the threshold requires little cooperation,
it is less costly for individuals to cooperate to make sure the group as a
whole stays under the threshold. The more cooperation is required for
staying under the threshold, the lower the relative benefit of cooperating,
and the less likely that individuals will cooperate. A threshold that is
exceeded when groups deposit more than 50\% of their endowment is a
realistic target for groups and is also higher than the average cooperation
rate in linear games.

We hypothesize that with this threshold, cooperation will be higher in the
step-level social dilemma than in the linear social dilemma. 

\subsection{Anchor}

Another reason why we expect a difference in cooperation levels between
linear social dilemmas and the step-level social dilemmas applies only to
social dilemmas in which cooperation and defection are gradual (someone
can, e.g., cooperate for 75\% and defect for 25\%) rather than a dichotomous
cooperate or defect choice. Different individuals have different
perceptions of what an acceptable amount to cooperate is (e.g., Buchan,
Croson \& Johnson, 2004; Van Dijk \& Wilke, 1993). In linear social
dilemmas those individual differences will translate into different
cooperation rates by different persons, but step-level social dilemmas
have an anchor of what an acceptable amount to cooperate is and variety in
perceived acceptable amounts may therefore be relatively limited (Van
Dijk, de Kwaadsteniet \& De Cremer, 2009). The threshold indicates what
the right amount to cooperate is. If the threshold is exceeded when more
than 45 tokens are deposited and there are
three persons in a group, the fair amount to deposit is 15 for every
player, so in step-level social dilemmas it is easier for individuals to
decide how much they should cooperate. In contrast, linear social dilemmas
lack this reference point. If staying under the threshold in a step-level
dilemma requires more cooperation than the average cooperation rate in linear social dilemmas, and if
individuals try to avoid that threshold, cooperation will be higher in
step-level social dilemmas (Croson \& Marks, 2000; Suleiman \& Rapoport,
1992).

\paragraph{Hypothesis 1:} Individuals cooperate more in step-level social
dilemmas than in linear social dilemmas.

\subsection{Asymmetry within groups}

Diversity in endowments, influence or consequences of cooperation among
individuals in a group may affect behavior in both linear and step-level
social dilemmas. When these is heterogeneity in endowments, some
individuals have more tokens to keep or dump than others. While in the
symmetric case there may be ambiguity about how much similar individuals
should cooperate, asymmetry in endowments introduces a second source of
ambiguity: what should the contribution of each group member be, relative
to their endowment? Earlier studies on asymmetry in endowments show mixed
results: in a meta-analysis, Zelmer (2003) found that asymmetry in
endowments leads to lower cooperation in linear public goods games, while
in another meta-analysis of threshold public goods games, Croson and Marks
(2000) found no effect of asymmetry between actors. 

We discuss how the impact of (a)symmetry may be different for linear vs.\
step-level social dilemmas. We formulate two contrasting hypotheses. On
the one hand, we might expect the effect of asymmetry to be more negative
in linear social dilemmas than in step-level social dilemmas. While
defection is always better for the individual in the linear dilemma, this
is not always the case in step-level social dilemmas. If a person’s
cooperation is critical for staying under the threshold in the step-level
situation, (s)he should cooperate to maximize both his or her own benefit
and the collective benefit. In this case, cooperating overlaps with
self-interest and therefore individuals who possess higher endowments in
step-level social dilemmas may be likely to cooperate, even if group
members with fewer endowments cooperate relatively less. In linear social
dilemmas this logic does not apply, since it is not in the self-interest
of individuals with more endowments to compensate the lower cooperation of
individuals with fewer endowments, even if individuals with smaller
endowments cooperate less. Therefore, individuals who possess higher
endowments are expected to cooperate more in the step-level social dilemma
than in the linear social dilemma. This leads to the following hypothesis:

\paragraph{Hypothesis 2a:} There will be an interaction between (a)symmetry and
social dilemma type, such that the asymmetry has a more negative effect in
linear dilemmas than in step-level dilemmas.\smallskip

An alternative prediction comes from the logic of appropriateness (Weber,
Kopelman \& Messick, 2004). When deciding how much to cooperate or defect,
participants ask themselves ``what does a person like me do in a situation
like this?'' From this perspective, cues about the ``appropriate'' amount
to cooperate are critical. In the symmetric, step-level social dilemma, the
threshold (equally divided among participants) provides a natural reference
point for the appropriate amount to cooperate. In the other three
conditions, the asymmetry in endowments or lack of a threshold mean that
the ``appropriate'' amount to cooperate is less obvious, and people may
(selfishly) cooperate less as a result.

\paragraph{Hypothesis 2b:} There will be an interaction between (a)symmetry and
social dilemma type, such that the asymmetry has a more negative effect in
step-level social dilemmas than in linear social dilemmas.

\section{Study 1}

\subsection{Methods}
We conducted a laboratory experiment at the University of British Columbia
(Vancouver, Canada) in which 120 participants took part in one of the 16
experimental sessions (with 6 or 9 participants in each session, according
to the number that signed up).

\subsubsection{Design}

The participants played 20 to 40 rounds of a contextualized Public Bad Game
in groups of three participants.\footnote{The number of blocks played
  depended on the amount of time available. The participants did not know
  how many rounds or blocks they were going to play, although they could
  guess that each block would last for 10 rounds.} Three-person groups are
practical while still allowing for group dynamics that characterize
real-world problems.\footnote{ Groups of two to six individuals are
  commonly used in social dilemma games (Balliet, 2010; Sally, 1995;
  Zelmer, 2003).} The groups were shuffled after every block of 10
rounds. The experiment had a 2 (game type: linear vs.\ step-level) x 2
(symmetry: symmetric vs.\ asymmetric actors) between-participants
design. In the symmetric condition, all participants started each round
with 30 pollution tokens, while in the asymmetric condition one participant
per group received 20 tokens, one participant received 30 tokens and one
participant received 40 tokens at the beginning of each round.\footnote{ A
  ratio of two between the lowest and the highest endowment is commonly
  used (Rapoport \& Suleiman, 1993; Van Dijk \& Grodzka, 1992; Van Dijk \&
  Wilke, 1995).}  Participants were asked to imagine that they were the
owner of a company that was located at the shore of a lake, along with two
other businesses, and that these businesses together were responsible for
the maintenance of the lake. They were asked how many tokens they wanted to
keep (‘process in an environmentally friendly way’) and the number they
wanted to deposit to a common pool (‘dump in the lake’). For every token
kept, the participant paid costs of \$1 million. In the linear game, every
token deposited cost the group \$2 million.\footnote{ A multiplier (or
  Marginal Per Capita Return) of two is commonly used in social dilemma
  experiments (Croson \& Marks, 2000)} In the step-level game, the group
paid the costs of \$180 million only when they together deposited 46 tokens
(50\% of the total amount of tokens or more.\footnote{A threshold of 50\%
  of total possible cooperation/defection in step-level dilemmas is
  commonly used in social dilemma experiments (Croson \& Marks, 2000).}  If
they deposited 45 tokens or less, they did not have to pay the group
costs. The group costs were equally split over the group members. At the
end of the experiment, the participants received \$15 minus the \$0.10 for
every \$50 million cost in the game. Thus, the game was incentive
compatible. The average payment was \$11.45.

\subsubsection{Procedure}

Participants were recruited via posters in university buildings, messages
on Facebook and emails to subscribers of an email list. We used the
software Z-tree for playing the social dilemma game (Fischbacher, 2007).
Each session lasted for 50 minutes. 83 participants (69.2\%) were female,
age ranged between 18 and 60 (\textit{M}\textit{\textsubscript{age}} =
24.3, \textit{SD}\textit{\textsubscript{age}} = 7.8).

The participants were seated in a private cubicle with a computer, from
where they could not see the other participants. At the start of the
session, the experimenter read instructions out loud that informed the
participants about the duration and procedure of the study (Supplementary
materials). The participants read on-screen instructions explaining the
game and they answered three comprehension questions. If they failed to
give the right answer twice, the experimenter verbally explained the
steps to calculate the answers. All participants were able to correctly
answer the questions after getting the verbal explanation. Next, the
experimenter showed an example of the interface of Z-tree on the big
screen in front of the room, and all the participants were directed to the
starting screen of Z-tree on their individual computer terminals.

After playing the game, the participants answered five demographic
questions (gender, age, level and field of education and nationality) and
a number of questions on numeracy (Schwartz, Woloshin, Black \& Welch,
1997), environmental attitudes (Dunlap, Van Liere, Mertig \& Jones,
2000), social value orientation (Murphy, Ackermann \& Handgraaf, 2011),
consideration of future consequences (Strathman, Gleicher, Boninger \&
Edwards, 1994) and temporal discounting (Kirby, Petry \& Bickel, 1999). 

Two pilot sessions were conducted to make sure that the parameters we chose
led to variance in cooperation rates (i.e., avoiding ceiling and floor
effects).

In some groups reading the instructions and answering the comprehension
questions took longer than in other groups, and as the session time was
limited to 50 minutes, not every group played the same number of blocks.
On average fewer rounds (\textit{M} = 38.57, \textit{SD} = 3.53) were
played in the asymmetric than the symmetric conditions (\textit{M} =
30.26, \textit{SD = }8.26; \textit{F}(1, 118) = 53.08, \textit{p}
{\textless} .001). The reason for this is that participants in that
conditions took more time for reading the instructions and answering the
comprehension questions, probably because the asymmetric games were more
complex. One group played only half of the third block due to a lack of
time. 

\subsubsection{Analytical strategy}

To test the hypotheses, we performed a multilevel linear regression with
the average fraction of tokens kept by a participant in a block as the
dependent variable. (Thus, higher numbers indicate more cooperation.) We
included the type of game (linear or step-level), the symmetry of the
participant endowments (symmetric or asymmetric), and the interaction as
independent variables. 

Decisions are nested in participants, while participants and groups are
nested in session. Participants and groups are crossed: the same
participant was a member of multiple groups, because after every block the
groups were shuffled. We built the statistical model step-by-step and
compared the AIC and deviance of nested models to assess the model fit of
the nested models. First, we estimated an intercept only model
(\textit{AIC} = 38.92, \textit{Deviance} = 26.7, \textit{df} = 419).
Adding random intercepts of participants (\textit{AIC} 0.4 = $-$95.1,
\textit{${\Delta}$Deviance} = 127.8, \textit{p} {\textless} 0.001), groups
(\textit{AIC} = $-$156.8, \textit{${\Delta}$Deviance} = 63.7, \textit{p}
{\textless} 0.001), and sessions (\textit{AIC} = $-$168.8,
\textit{${\Delta}$Deviance} = 14.0 , \textit{p} {\textless} 0.001)
one-by-one, significantly improved the model fit. In the next step we
added the independent variables game type (linear or step-level), symmetry
of the participants, and the interaction between the two (\textit{AIC} =
$-$174.4, \textit{${\Delta}$Deviance} = 11.6, \textit{p} = 0.009). 

In a second model we also included the block number (1--4) as an independent
variable, as well as the interaction with the treatment variables
(\textit{AIC} = $-$200.9, \textit{${\Delta}$Deviance} = 34.5, \textit{p}
{\textless} 0.001). Allowing the slope of the block number to vary across
participants, groups and sessions did not improve the model fit (AIC =
$-$189.3.1, \textit{${\Delta}$Deviance} = 0.4, \textit{p} = 0.999). The
final model with random intercepts for participants, groups and sessions
and with game type, asymmetry, block number, and all interactions between
these three variables is reported in Table 1, models 1 and 2. 

In the step-level games, the fraction of tokens kept is not the only
measure of successful cooperation. We also explored if the extent to
which the groups managed to stay under the threshold differs across the
conditions. To do so, we ran a multilevel regression with the fraction of
rounds in which a group managed to stay under the threshold as the
dependent variable. Compared to the intercept only model (AIC = 83.7,
\textit{Deviance} = 12.7), the model with random intercepts for sessions
had a better fit (AIC = 72.5, \textit{${\Delta}$Deviance} = 53.8,
\textit{p {\textless} }0.001). In the next step we added asymmetry and the
block number (AIC = 70.7, \textit{${\Delta}$Deviance} = 7.8 , \textit{p} =
0.050). Allowing the slope of block to vary across sessions did not
improve the model fit (AIC = 74.6, \textit{${\Delta}$Deviance} = 0.1 ,
\textit{p} = 0.951). The final model with a random intercept for session
and asymmetry and block number as the independent variables is reported in
Table 1, model 3. 

\subsection{Results}

\subsubsection{Descriptive statistics}

Figure 1 shows the average number of tokens kept by participants in the
different conditions. Cooperation seems to start highest in the step-level
conditions, but strongly decreased in the step-level asymmetric condition.
In the other conditions cooperation was relatively stable over time. 

%Table 1
\begin{table}[H]
\caption{Results cross-classified multilevel regression on
fraction of tokens kept: mean and (s.e.).}
\hspace{-3ex}
\begin{tabular}{@{}L{2.3in}@{}C{1.3in}C{1.3in}C{1.3in}@{}}\toprule
Independent variables & Model 1 DV = fraction kept
 & Model 2 DV = fraction kept  & Model 3 DV = fraction under threshold\\\midrule
Constant  & 0.121  (0.043) & 0.222 (0.059) & 0.313 (0.160)\\\midrule
\textit{Main effects} &  &  &\\
Step-level (ref = linear) & 0.230$^{***}$ (0.059) & 0.070 (0.081) & $-$\\
Asymmetric (ref = symmetric) & 0.067 (0.063) & $-$0.006 (0.089) & 0.285 (0.235)\\
Block (1--4)
 & $-$ & $-$0.043$^*$ (0.018) & 0.080 (0.044)\\\midrule
\textit{Two-way Interactions} &  &  &\\
Step-level * Asymmetric & $-$0.114 (0.087) & 0.272$^*$ (0.121) & $-$\\
Steplevel * Block & $-$ & 0.066$^{**}$ (0.024) & $-$\\
Asymmetric * Block & $-$ & 0.030 (0.031) & $-$0.196$^{**}$ (0.073)\\\midrule
\textit{Three-way interaction} &  &  &\\
Steplevel * Asymmetric * Block & $-$ & $-$0.171$^{***}$ (0.041) &\\\midrule
\textit{Random effects } &  &  &\\
Variance intercept participant & 0.014 (0.117) & 0.023 (0.151) & $-$\\
Variance intercept group & 0.022 (0.149) & 0.009 (0.093) & $-$\\
Variance intercept session & 0.002 (0.047) & 0.002 (0.043) & 0.043 (0.207)\\\midrule
N & 420 & 420 & 76\\\bottomrule
\end{tabular}
\end{table}

\subsubsection{Regression results}

Table 1 contains the results of the regressions. Supporting hypothesis 1,
the participants kept more tokens in the step-level condition than in the
linear condition. There is no significant difference between the symmetric
and the asymmetric conditions, and there is no difference in the effect of
asymmetry in the linear game than in the step-level game, so we reject
both Hypotheses 2a and b. 

Over the blocks, cooperation slightly decreased in both linear conditions
(\textit{b} = $-$0.034, \textit{p} = 0.011), was more or less stable in the step-level
symmetric condition (\textit{b} = 0.023, \textit{p} = 0.19) and strongly decreased in the
step-level asymmetric condition (\textit{b} = $-$0.118, \textit{p} {\textless} .001).

%Figure 1
\begin{figure}[H]\centering
\includegraphics[clip=true,width=4in]{fig1.eps}
\caption{Average fraction of tokens kept across conditions and
blocks.}
\end{figure} 

Model 3 shows that in the step-level conditions, groups were equally likely
to stay under the threshold in the first block across the conditions, but
over the blocks this probability decreased in the asymmetric condition,
while it was stable in the symmetric condition. 

\subsection{Exploratory analyses}

In the theory section we discussed two mechanism that could explain the
difference between the linear and the step-level conditions. According to
our first mechanism, individuals in the step-level condition would have an
incentive to cooperate, because the costs of non-cooperation could exceed
the costs of cooperation. The willingness to cooperate more than a ‘fair
share’ (half of their endowment, or 15 tokens) can thus expected to be
higher in the step-level game if that was necessary for the group to stay
under the threshold.

To test if this indeed the case, we ran a regression with a binary
variable that indicates whether a subject kept more than half of their
endowment as the dependent variable. As an independent variable we
included a dummy variable that indicated if in the previous round a
contribution of more than half of a subject’s endowment was critical for
staying under the threshold. This thus depended on the number of tokens
kept by the group members of the subject. We added random intercepts for
subjects and groups and we controlled for block, round number (1-10) and
asymmetry. We first ran the model separately for the linear and the
step-level condition. After that we run the model on all subjects and we
add the interaction between game type and the dummy that indicated if a
higher contribution was critical. As expected, we find that individuals in
the step-level condition were more likely to keep more than half of their
tokens if necessary (\textit{b} = 0.093, \textit{t}(2044) = 6.082,
\textit{p} {\textless} 0.001), while we do not find evidence for that in
the linear games (\textit{b} = 0.024, \textit{t}(1675) = 1.384, \textit{p}
= 0.167). This difference in the effect between the linear and the
step-level game is significant (\textit{b} = 0.069, \textit{t}(3725) =
2.962, \textit{p} = 0.003).\footnote{ When we look at the tendency to keep
more than 15 tokens (instead of more than half of the endowment) the
findings slightly change. The effect of the previous round turns
significant in the linear games (\textit{b} = 0.052, \textit{t}(1675) =
2.738, \textit{p }= 0.003). However, the difference in the effect between
the linear and the step-level game remains large and significant
(\textit{b }= 0.113, \textit{t}(3725) = 4.509, \textit{p} {\textless}
0.001).} We thus find evidence that individuals are more willing to
cooperate in the step-level condition because it is in their own interest.

Based the self-interest argument, we also hypothesized that asymmetry
between individuals would have a smaller effect in step-level games (H2a).
In the linear games there would be no reason for individuals to cooperate
more if they have a larger endowment, because it is always in their
interest to defect. However, in the step-level game, it may be in the
interest of individuals with a large endowment to cooperate more in
absolute or relative terms, to insure that the group does not exceed the threshold.
We know that in the step-level condition subjects were more likely to keep
more than half of their endowment if that would have helped the group stay
under the threshold in the previous round. However, this mechanism can
explain only why asymmetry has a smaller effect on cooperation in the
step-level game if there is a difference in the frequency with which it is
necessary to keep more than half of one’s endowment. We tested this mechanism using
a regression in the step-level conditions with the need to keep more than
half of the endowment as the dependent variable and the asymmetry
condition as the independent variable. We again controlled for round and
block, and we included random intercepts for subjects and groups. The
results show that the situation that subject needs to keep more than half
of their endowment to avoid exceeding the threshold does not occur more
often in the asymmetric condition (\textit{b} = 0.018,
\textit{t}(2045) = 0.382, \textit{p }= 0.702). This result implies that
although subjects were willing to keep more tokens if necessary, the
frequency with which this was necessary was the same in the symmetric and
asymmetric step-level game. This means that the effect of asymmetry
on cooperation is not smaller in the step-level condition because it is in
the interest of the high-endowed to cooperate at a higher level.

The second mechanism that we discussed and that may explain why there is
more cooperation in the step-level condition than in the linear condition,
is that the threshold provides a clear reference point. The ambiguity
caused by a lack of a reference point is expected to decrease over the
rounds, as subjects over time get informed about the strategies of their
fellow group members. To test this mechanism, we studied variation in
contributions across the conditions and over the rounds, since ambiguity
is expected to lead to more variation in cooperation. Per group and per
round we calculated the standard deviation of the absolute number of
tokens and of the fraction of tokens kept by the three group members.
Figure 2 shows the average standard deviation within the groups over the
rounds

% figure 2
\begin{figure}[H]
\hspace{-2ex}
\includegraphics[clip=true,width=3in]{fig2a.eps}\qquad%
\includegraphics[clip=true,width=3in]{fig2b.eps}
\caption{Standard deviation of number of tokens kept (left panel)
and fraction of tokens kept (right panel) per round and condition.}
\end{figure} 

A regression with the standard deviation within each group per round as the
dependent variable and the treatment variables as independent variables and
with controls for the round and block number confirms that asymmetry in
endowments increases the standard deviation within groups (\textit{b} =
1.765, \textit{t}(1380) = 5.35, \textit{p} {\textless} 0.001) and that the
standard deviation decreases over the rounds (\textit{b} = $-$0.445,
\textit{t}(1380) = $-$7.98, \textit{p} {\textless} 0.001). However, we do
not find that there is more variance in the step-level condition than in
the linear condition (\textit{b} = 0.381, \textit{t}(1380) = 1.18,
\textit{p} = 0.240). The difference between the two conditions in
cooperation can thus not be explained by a difference in ambiguity with
regard to the appropriate level of cooperation. These findings are robust
to using the standard deviation in the fraction of tokens kept, instead of
in the absolute number of tokens kept.

We also argued that the additional ambiguity introduced by asymmetry is
expected to have a limited influence in the linear games, whereas it may
distort the positive effect of the clear threshold in linear games. We do
not find that asymmetry has a larger effect on the standard deviation per
group in the step-level condition than in the linear condition when we
include an interaction between game type and asymmetry (\textit{b} =
-1.027, \textit{t}(1379) = $-$1.57, \textit{p} = 0.117). When looking at the
standard deviation in the fraction of tokens kept we even find that the
negative effect of the asymmetry of endowments is larger in the linear
condition than in the step-level condition (\textit{b }= $-$0.069,
\textit{t}(1379)\textit{ = }$-$3.28, \textit{p} = 0.001). These findings
imply that we do not find support for the mechanism that the introduction
of asymmetry has a stronger effect in the step-level game.

\subsection{Discussion}

Supporting H1, there was more cooperation in the step-level social dilemma
than in the linear social dilemma. We argued that there could be two
mechanisms underlying this difference. First, in the step-level game it
may be in the self-interest of individuals to cooperate, while that is
never the case in the linear game. Second, the threshold provides a
reference point for the appropriate level of cooperation. We find evidence
for the first mechanism, but not for the second. 

We did not find an effect of asymmetry in endowments across group members in
the linear social dilemma. In the step-level social dilemma we found that
cooperation over time decreased strongly in the asymmetric game, but not in the
symmetric game. The difference in the effect of asymmetry on cooperation
between the linear and the step-level game was not significant, so we
reject Hypotheses 2a and 2b. To exclude the possibility that this the lack
of an interaction between game type and asymmetry was the result of two
distinct mechanisms with opposite effects, we made an effort to
empirically disentangle these mechanisms. We did not find evidence that
that asymmetry had a smaller effect in the step-level game because it was
in the self-interest of individuals to cooperate. Although we found that
asymmetry in general led to ambiguity, we did not find that this effect
was stronger in the step-level condition. The lack of the existence of the
interaction can thus not be explained by these two opposing mechanisms.

Another mechanism that could affect the interaction between game type and
asymmetry is that in the asymmetric step-level dilemma, it may not always
be in the self-interest of individuals with a large endowment to cooperate,
even if their cooperation is necessary for staying under the threshold. In some
cases the costs of full cooperation (i.e., keeping the full endowment) may
exceed the group costs divided over the individuals.  In the setup that we
used this was never the case: the individual costs of cooperation could not
exceed the individual share of the group cost.\footnote{ The maximum costs
  of cooperation were 40 tokens. The maximum costs of defection were 45
  tokens. The costs of cooperation could thus never exceed the costs of
  defection. }

This study had a number of limitations. First, the threshold level of 50\%
of possible contributions is a very focal level of contributions in social
dilemma games. This is a very salient threshold and it is easy for groups
to coordinate their contributions to reach this threshold. In a second
experiment we therefore asked whether the difference between step-level and
linear games is robust to different threshold levels.

Second, participants would always lose money by playing more rounds. Since
the duration of the experiment was fixed, in the first experiment
participants had an incentive to play slowly. In the second experiment we
addressed this limitation by calculating their payment with the average
number of tokens collected in all rounds, rather than the total number of
tokens.

In a second experiment we addressed these limitations and tested whether
the high cooperation levels observed in the step-level conditions of the
first experiment replicate. We also explored the extent to which the high
cooperation level found in the step-level social dilemmas is robust to the
specific level of the threshold chosen in the experiment. This explorations
provides insight in the mechanisms underlying the high cooperation level.

\section{Study 2}

In the first experiment we found that cooperation started off high but
quickly broke down in the step-level asymmetric social dilemmas. That result
suggests that perhaps groups can maintain a high cooperation level only if
the threshold is easy to coordinate on. In the second study we focus on
step-level social dilemmas, and vary whether the individuals in a group
are symmetric or not, to test whether the finding that asymmetry over time
decreases cooperation in the step-level social dilemma is robust.

Second, we test whether the large difference in cooperation rate between
the linear game and the step-level game in Study 1 is caused by the very
salient and easy to coordinate threshold that was used. In the second study
we use a more complicated threshold that does not allow the participants to
use a simple heuristic to decide how much they should cooperate to reach
the best outcome. The best outcome for the group occurs when the number of
tokens kept is exactly the amount that is required to stay under the
threshold. Every additional tokens that is kept incurs only costs for the
individual, while it does not help the group. For the individual it is best
to keep as few tokens as possible, while still staying under the
threshold. With a threshold that cannot be met when all group members keep
an equal amount of tokens, some individuals will have to keep more tokens
than others. For the individual it is attractive to be the one who keeps
least tokens. However, if all group members think in the same way, they
will all try to be the group member that keeps least tokens. The
consequence may be that groups more often exceed the threshold. Therefore,
we hypothesize that:

\paragraph{Hypothesis 3:} In a step-level social dilemma, there is less
cooperation when groups cannot exactly meet the threshold if all group
members cooperate an equal amount.\smallskip

If the high cooperation level in the symmetric step-level games can only be
maintained if all participants know how much they should cooperate to
reach the threshold and if this amount is the same for all group members,
cooperation should break down when this is not the case. Since both
symmetry within groups and an easy to coordinate threshold are necessary
conditions for this requirement, we expect that cooperation breaks down if
one of these two is lacking. We hypothesize that:

\paragraph{Hypothesis 4:} Both symmetry and an easy-to-coordinate threshold
are necessary to maintain a high cooperation level. Cooperation breaks
down if one of these elements is missing.

\subsection{Methods}

\subsubsection{Design and procedures}

The second study was conducted at the same university as Study 1 and the
procedure is similar to Study 1. 129 people participated in the experiment
in 19 sessions of 6 or 9 participants each. All groups played a step-level
game. We used a 2 (symmetric versus asymmetric) by 2 (easy-to-coordinate
versus difficult-to-coordinate threshold) between-participants design. The
easy-to-coordinate threshold was 45, just as in the first experiment. If
the participants together kept 45 tokens or more, they did not have to pay
the group threshold cost. In the difficult-to-coordinate condition
participants had to pay the group cost if they together kept 43 tokens or
less. 

The payments were similar to the first study, with the exception that
participants got paid according to the average number of tokens earned in
the game per round, instead of the total number of tokens, to avoid the
incentive to play slowly. On average participants received \$12.53. 

\subsubsection{Analytical strategy}

We analyzed the data in the same way as in Study 1. We first estimated an
intercept only model with the fraction of tokens kept as the dependent
variable (\textit{AIC} = $-$229.0, \textit{Deviance} = 18.9, \textit{df} =
509). Adding random intercepts of participants (\textit{AIC} = $-$321.5,
\textit{${\Delta}$Deviance} = 346.4, \textit{p} {\textless} 0.001) and
groups (\textit{AIC} = $-$379.2, \textit{${\Delta}$Deviance} = 59.7,
\textit{p} {\textless} 0.001) significantly improved the model fit. Adding
random intercepts for sessions did not improve the model fit (\textit{AIC
}= $-$377.2, \textit{${\Delta}$Deviance} = $-$0.0, \textit{p =}1.000). In the
next step we added the treatment variables (\textit{AIC} = $-$373.8,
\textit{${\Delta}$Deviance} = $-$0.6, \textit{p} = 0.896) and block and all
interactions (\textit{AIC} = $-$381.6, \textit{${\Delta}$Deviance} = 15.8,
\textit{p} {\textless} 0.003). Allowing the slope of block to vary across
participants significantly improved the model fit (\textit{AIC} = $-$387.5,
\textit{${\Delta}$Deviance} = $-$9.9, \textit{p} = 0.007). Allowing the
slope of block to vary across groups did not improve the model fit
(\textit{AIC} = $-$386.0, \textit{${\Delta}$Deviance} = $-$2.5, \textit{p} =
0.29). The final models with random intercepts for participants and groups
and with the treatment variables and block as the independent variables
and with random slopes for block for participants are reported in Table 2,
model 1 and 2.

We ran a multilevel regression with the fraction of round in which a group
managed to stay under the threshold as the dependent variable. Compared to
the intercept only model (\textit{AIC} =131.8, \textit{Deviance} = 21.1,
\textit{df} = 169), the model with random intercepts for sessions did not
have a better fit (\textit{AIC} = 133.1, \textit{${\Delta}$Deviance} =
$-$106.0, \textit{p} {\textgreater} 0.999). In the final model we added the
variables asymmetry and block number to the model without random
intercepts (\textit{AIC} = 136.1, \textit{${\Delta}$Deviance} = 0.2,
\textit{p} {\textgreater} 0.978). This model is reported in Table 2, model
3.

%Table 2
\begin{table}[H]
\caption{Results cross-classified multilevel regression on
fraction of tokens kept: mean and (s.e.).}
\hspace{-3ex}
\begin{tabular}{@{}L{2.3in}@{}C{1.3in}C{1.3in}C{1.3in}@{}}\toprule
Independent variables & Model 1 
DV = fraction kept  & Model 2 DV = fraction kept  & Model 3 DV = fraction under threshold\\\midrule
Constant  & 0.363 (0.026) & 0.415 (0.050) & 0.418 (0.128)\\\midrule
\textit{Main effects} &  &  &\\
Threshold = 47 (ref = 45) & 0.020 (0.037) & $-$0.011 (0.071) & 0.018 (0.181)\\
Asymmetric (ref = symmetric) & 0.010 (0.037) & 0.082 (0.071) & 0.255 (0.181)\\
Block & $-$ & $-$0.021 (0.016) & 0.016 (0.047)\\\midrule
\textit{Two-way Interactions} &  &  &\\
Threshold 47 * Asymmetric & $-$0.001 (0.054) & $-$0.172 (0.102) & $-$0.201 (0.261)\\
Threshold 47 * Block & $-$ & 0.012 (0.023) & 0.006 (0.066)\\
Asymmetric * Block & $-$ & $-$0.029 (0.023) & $-$0.117 (0.066)\\\midrule
\textit{Three-way interaction} &  &  &\\
\mbox{Threshold 47 * Asymmetric * Block} & $-$ & 0.070$^*$ (0.032) & 0.146 (0.096)\\\midrule
\textit{Random effects } &  &  &\\
Variance intercept participant & 0.013 (0.112) & 0.029 (0.170) & $-$\\
Variance intercept group & 0.010 (0.098) & 0.008 (0.091) & $-$\\
Variance slope Block participant & $-$ & 0.001 (0.037) & $-$\\\midrule
N & 510 & 510 & 170\\\bottomrule
\end{tabular}
\end{table}

\subsection{Results}

\subsubsection{Descriptive statistics}

Figure 3 shows the average number of tokens kept by the participants in the
different conditions. The average level of cooperation was similar in the
four conditions. In the easy-to-coordinate threshold conditions
cooperation seems to decrease slightly over time, while it was more or
less stable in the difficult-to-coordinate conditions. 

%Figure 3
\begin{figure}[H]\centering
\includegraphics[clip=true,width=4in]{fig3.eps}
\caption{Average fraction of tokens kept across conditions and
blocks.}
\end{figure} 

\subsubsection{Regression results}

Table 2 contains the results of the regressions. Model 1 shows that there
is no difference in the cooperation levels between the four conditions, so
we reject hypotheses 3 and 4. Model 2 shows that the fraction of tokens
kept does not change over the blocks in both symmetric conditions and the
asymmetric dilemma with an easy-to-coordinate threshold. In the asymmetric
condition with a difficult-to-coordinate threshold, cooperation increased
over the blocks. We do not find evidence for the hypothesis that asymmetry
in endowments decreases cooperation in step-level social dilemmas, nor for
the hypothesis that cooperation is lower when there is ambiguity with
respect to how much each participant should cooperate to exactly reach the
threshold. The average level of cooperation was similar to the average
level of cooperation in the symmetric step-level social dilemma of Study
1. There was no difference between the conditions in the number of times
groups managed to stay under the threshold (model 3).

Using a t-test at the level of the group we compared the cooperation levels
in the step-level games of Study 1 with those of Study 2. Overall,
cooperation is higher in the step-level games than in the linear games
(mean difference = 0.220, t(308) = 9.562, p {\textless} 0.001).
Cooperation in the step-level symmetric game in Study 1 was similar to the
average cooperation level in Study 2 (mean difference = 0.032, t(212) =
1.229, p = 0.221). The average cooperation level in the asymmetric
step-level game in Study 1 was lower than the average cooperation level in
 2 (mean difference = 0.093, t(200) = $-$3.024, p = 0.003). 

\subsection{Discussion}

In the second study we found no difference between symmetric and asymmetric
step-level social dilemmas. We also found that cooperation does not drop
if it is more difficult for group members to decide how much they should
cooperate to avoid going over the threshold. 

The average cooperation levels in the step-level dilemma in the second
study were similar to the cooperation levels in the symmetric step-level
games in Study 1. 

\section{General Discussion}

While many societal problems are framed as linear social dilemmas in
research and practice, the relation between cooperation and the outcome is
in reality often has a different shape. The aim of this paper is to explore
whether groups are better at minimizing collective losses when every small
contribution has an equal effect, or when there is there is a collective
target that must be reached. In a first laboratory experiment we compared
cooperation in linear social dilemmas with step-level social dilemmas. We
also investigated whether asymmetry between group members, which is often
found to be a complicating factor in solving social dilemmas, has a
different effect on cooperation in linear social dilemmas than in
step-level social dilemmas. We found that cooperation is higher in the
step-level social dilemma, but only when all group members are equal. In a
second study we further explored the conditions under which cooperation may
be higher in step-level social dilemmas. We varied the level of the
threshold to investigate whether cooperation can be high in step-level
social dilemmas only when there is no ambiguity in the amount each
participant should cooperate to exactly reach the threshold. In the second
experiment we found that neither asymmetry in endowments nor an ambiguous
threshold decrease cooperation.

Overall, these results suggest that cooperation can be higher in step-level
social dilemmas than in linear social dilemmas, and that this difference
may be robust to ambiguity with respect to the optimal cooperation level. The
difference between the two types of games seems to be caused by a
difference in the incentive structures for individuals. While in linear
social dilemmas the dominant strategy is to defect, that is not
necessarily the case in step-level social dilemmas. In the latter,
individuals can under certain conditions increase their personal payoff by
cooperating, if their cooperation is necessary to avoid the collective
loss, and if costs of the collective loss for the individual are larger
than the costs of cooperation. The level of cooperation in the second
experiment is comparable to cooperation in the symmetric step-level social
dilemma in the first experiment, and much higher than cooperation in the
linear social dilemmas of the first experiment. 

One question that remains is why cooperation broke down in the asymmetric
step-level social dilemma in the first study, but not in the second study.
One explanation for the contrasting finding is that the results of the
last blocks of the first study in that condition may not be reliable,
since only six groups reached the final blocks in that condition. 

This research suggests that people are more likely to cooperate in a social
dilemma when there is a single target that can be reached only when multiple
group members cooperate than when they know that every small contribution
has a small benefit to the group, regardless of the overall level of
cooperation. Future research should explore the robustness of this difference
to other differences in the structure of the social dilemma. One
important factor may be the level of the threshold.

When staying under the threshold does not require more cooperation than the
average level of cooperation in linear social dilemmas, the step-level
structure of the dilemma is not expected to increase cooperation. When
groups perceive the level of cooperation that is required to stay under the
threshold as too high, they may not be motivated to try to stay under it,
because the relative benefits of staying under it are smaller, while the
risk that the group still exceeds the threshold is larger. To better
understand drivers of cooperation and differences across individuals,
future studies could also include more measures on the motivations of
individuals to cooperate. Examples of constructs that might affect
cooperation are the perceived fairness of different cooperation levels,
perceived criticality and the perceived influence on other players.

Overall, this research suggests that groups are better able to maximize
group benefits when they face a large, sudden cost rather than small,
gradual costs of defection.

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