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Journal of Economic Interaction and Coordination (2023) 18:463–490
https://doi.org/10.1007/s11403-022-00373-6
REGULAR ARTICLE
Group contest in a coopetitive setup: experimental
evidence
Hubert János Kiss1,2 ·Alfonso Rosa-Garcia3·Vita Zhukova4
Received: 21 April 2021 / Accepted: 3 October 2022 / Published online: 15 October 2022
© The Author(s) 2022
Abstract
We study experimentally cooperation in group contests under a new sharing rule that
captures elements of coopetitive setups that are often characterized by the tension
between cooperation and competition. It introduces an allocation of the obtained prize
which is inversely proportional to individual efforts. We use it to study if the pervasive
over-expenditure observed in group contests remains even when individual effort is
extremely disincentivized and compare its effects with the egalitarian sharing rule.
Participants in our experiment make more effort with the egalitarian than with the
inverse proportional rule, but we document a sizeable over-expenditure even with the
inverse proportional rule. We find that contribution in a public goods game is positively
associated with effort in the group contest. Social value orientation, risk attitudes,
competitiveness, or other personality traits do not predict behavior consistently.
Hubert János Kiss gratefully acknowledges financial support from the National Research, Development
Innovation (NKFIH) under project K 119683, and from the Hungarian Academy of Sciences, Momentum
Grant No. LP2021-2. The authors are grateful for the financial support from the Spanish Ministry of
Science and Innovation under the projects ECO2017-82449-P (Hubert J. Kiss) and ECO2016-76178-P
and PID2019-107192GB-I00 (AEI/10.13039/501100011033) (Alfonso Rosa-Garcia).
BHubert János Kiss
kiss.hubert@krtk.hu
Alfonso Rosa-Garcia
alfonso.rosa@um.es
Vita Zhukova
vzhukova@ucam.edu
1KRTK KTI, Tóth Kálmán u. 4, Budapest 1097, Hungary
2Corvinus University of Budapest, F˝ovám tér 8, Budapest 1093, Hungary
3Department of Economics, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain
4Department of Business, Universidad Católica San Antonio de Murcia, Campus de Los Jerónimos,
s/n, Guadalupe, 30107 Murcia, Spain
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464 H. J. Kiss et al.
Keywords Competitiveness ·Egalitarian sharing rule ·Group contest ·Inverse
proportional sharing rule ·Public goods game ·Risk attitudes ·Social value
orientation
JEL Classification C72 ·C92 ·D70 ·D72 ·H41
1 Introduction
Group contests are ubiquitous, including rent-seeking and lobbying, innovation tourna-
ments and R&D races or sports competitions. One of the main finding of the literature
on group contests is that the actual effort of individuals who participate in such con-
tests tends to be significantly higher than the Nash equilibrium prediction. Since the
effort generally does not affect the size of the prize to be gained in the contest and,
hence, is wasteful, over-expenditure leads to large welfare losses. A large part of the
literature aims at explaining this phenomenon by investigating how different elements
of group contests and individual characteristics affect observed behavior in laboratory
experiments. We rely on this literature to study group contests in coopetitive setups,
where the group members pursuing a common goal have at the same time opposing
individual objectives. A considerable share of studies in the management literature on
coopetition is concerned about the tension between value creation (cooperation) and
capture (competition) (see, for instance, Bengtsson and Kock 2014; Bouncken et al.
2015). Such tensions may go against cooperation, and we attempt to come up with a
model and experimental design that exhibit this feature.
We introduce and analyze a new sharing rule (that we call inverse proportional) that
disincentivizes more intensely participants from contributing to the group performance
than known sharing rules (e.g., egalitarian or proportional). With the inverse sharing
rule, the prize obtained in the case of winning is allocated to each individual of the
winner group in an inversely proportional way to the effort exerted by the individual.
For instance, if member A made half of the effort of member B in the winning group,
member A receives twice as much from the prize than member B. Importantly, the
sharing rule that we introduce here induces that zero effort becomes the dominant
strategy for each individual (in contrast to the known sharing rules that predict a
positive effort). The main aim of this work is to study the behavior under this rule and
to investigate if over-expenditure persists even if zero effort is a dominant strategy
for the individual. As a secondary objective, in the experiment we explore individual
characteristics (risk, social and competitive preferences) of the participants to see
how these associate with observed choices in the group contest to shed light on the
significance of individual heterogeneity, another common finding in the literature.
The inverse proportional sharing rule may seem at first puzzling as it does not
reward cooperation.1However, it may emerge in coopetitive setups that suffer from
the tension between value creation and capture, supported by many examples in the
1Colasante et al. (2019) and Rapoport and Bornstein (1987) show that competition may foster cooperation
through solving the free-rider problem.
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Group contest in a coopetitive setup: experimental… 465
management literature.2A simple economic explanation of such tensions can be the
resource constraint, as firms use the same resources for cooperation and competition.
This feature is captured by the inverse proportionality sharing rule in a group contest
context: The more an agent spends on cooperation, the less resources she has to obtain
a higher share of the prize (if her group wins it). Consider two examples that illustrate
the previous ideas. One is the case of cycling races, in which often two or more cyclists
break away from the peloton. To maintain their distance, the cyclists have to cooperate,
that is, all of them have to be in front and to take the lead that requires a lot of effort as
the leader has to struggle more with air resistance than the followers who are sheltered
from wind.3On the other hand, these cyclists also compete because each of them
wants to win the competition and arrive first at the finish line. To win, they need to
economize on efforts so they are interested in less cooperation. The more cooperative
a cyclist is, the more likely it is that the escape is successful and they arrive first at
the finish line, but at the same time more cooperation decreases his chance of winning
the race. A similar dilemma is faced by cyclists in the pursuit squad, who would
form the other group competing in the contest. Consider also the example of a set of
firms that are using one of several available standards in the market, but at the same
time they compete for customers. The high-definition optical disk format competition
is a suitable illustration. As HDTV televisions became popular in the mid-2000s, a
need emerged for an inexpensive storage medium that should succeed the established
DVD format. Leading electronic companies (e.g., Sony, Panasonic, Samsung) founded
Blu-ray Disc Association to develop and license the homonymous technology and to
promote business opportunities for this standard. A similar association arose led by
Toshiba (and supported by Microsoft) to advance a competing format called HD DVD.
Members of both associations cooperated in the development of the respective format
(the prize being a higher market size for the format), but at the same time competed
to sell players compatible with the given format to customers. Firms in such consortia
dedicate part of their own resources to the development of the standard, while they
still have to compete for customers against their allies in the consortium. These firms
face, therefore, a trade-off, the more they contribute to the success of the format,
the less resources they have to increase the probability of capturing a larger share of
the market.4When group members cooperate to win the contest, while at the same
time they also compete, any contribution to the group objective is decreasing the
opportunities to become successful in the within-group competition.5
2Kim and Parkhe (2009) show how competing similarity (e.g., geographical market coverage or product
market coverage) hinders, while cooperating similarity (e.g., societal culture or management practices)
fosters coopetition. Hence, tension may arise when competing similarity is too high relative to cooperat-
ing similarity. In a similar vein, Park et al. (2014) show that the right balance between competition and
cooperation is beneficial for firm performance, while imbalances may lead to tensions.
3Aerodynamic drag can be reduced up to 50% for cyclists that are not in front (Blocken et al. 2018).
4These situations are also present in lobbies of industry associations: Members contribute resources for
lobbying efforts, but at the same time they compete with each other directly. Further examples are chambers
of commerce and federations of entrepreneurs or schemes of protected designation of origin (which promote
and protect names of quality agricultural products of a given region).
5Clearly, the trade-off between using resources for cooperation or competition may be more acute in some
environments, possibly due to factors like competitive intensity, network externalities or market uncertainty
(Ritala 2012).
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466 H. J. Kiss et al.
Our inverse proportional sharing rule captures these types of incentives that involve
the trade-off between cooperation and competition due to limited resources. The theo-
retical analysis shows that individuals maximize their payoff making zero effort to the
group performance, independently of the actions of the rest of individuals.6This is a
strong prediction that we test in the laboratory, analyzing both if this new sharing rule
has an effect on effort as well as if over-expenditure remains even in the extreme case
where zero effort is always optimal from an individual point of view. In two sessions,
both with 14 four-member groups, we investigate behavior in group contest with egal-
itarian and inverse proportional sharing rule. Moreover, after the group contest, our
participants played additional games to measure their social attitude, risk preferences,
cooperativeness and competitiveness, as well as an extensive questionnaire capturing
personality traits with the aim to better understand what drives their choices in the
group contest.
We find a sizeable over-expenditure with the inverse sharing rule, although zero
contribution is optimal for each individual. Participants’ efforts remain relatively high
even at the end of the experiment, after 20 rounds. However, we find a sizable and
significant difference in effort between the egalitarian and the inverse proportional rule,
effort being much lower under the latter rule, as expected. When relating behavior in
group contest with decisions in other situations, we find that contributions in a public
good game are positively associated with effort in the group contest, independently
of the sharing rule. Competitiveness, measured á la Niederle and Vesterlund (2007)
explains part of the behavior. Finally, we do not find any consistent effect of social
value orientation or risk aversion.
The rest of the paper is organized as follows. In Sect. 2, we present the theoretical
model upon which our main hypothesis is based. Then, in Sect. 3we briefly review
the literature and formulate our hypotheses. Section 4describes the experiment. In
Sect. 5, we present our findings and Sect. 6concludes.
2Model
There are two groups (A and B) with NAand NBmembers, respectively, that compete
to win a contest and receive a prize υ. In our case groups have 4 members (NA=NB
= 4) and the prize is 4000 tokens. Players in both groups choose simultaneously and
independently a level of effort xiA and xjB that is irreversible.
The performance of group A/B, denoted as XA/XB, is a function of all individual
efforts within the given group. We use perfect substitution of effort, so
fA(x1A,...xNAA)=
NA
i=1
xiA.(1)
This function describes situations in which the performance of the group hinges on
the joint effort of all individuals of the group. As of costs of effort, for simplicity we
6Moreover, zero effort is also the social optimum.
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Group contest in a coopetitive setup: experimental… 467
assume homogeneous linear costs, that is ciA(x)=cjB(x)=xfor all i=1, ..., NA
and j=1, ..., NB.
The probability of winning the contest depends on the relative performance of the
groups. A contest success function (CSF) determines the probability that group A wins
the prize:
pA(XA,XB)=(XA)r
(XA)r+(XB)r(2)
We use the lottery CSF, so r=1. In this case higher performance implies higher
winning probabilities.
Our main modification is in the way the prize is split. A general way to capture
the division of the prize is the following. In the case group A wins the prize, player
ireceives a share of the prize which is defined by the following sharing rule Nitzan
(1991)
siA(x1A,...xNAA)=αA
NA
+(1−αA)xiA
NA
i=1xiA
.(3)
In words, share αAof the prize is split equally among all members of the winning
group, and the rest (1 −αA) is divided according to relative effort. Egalitarian division
occurs if αA=1, while if αA=0, then the prize is split in proportion to relative effort.
In egalitarian treatment we use αA=1. For the other treatment, we propose a new
sharing rule that states that the share of the prize that a member receives is inversely
proportional to relative effort. More concretely, the less effort a member makes, the
more she receives proportionally from the prize.7We define the inverse proportional
share in the following way:
siA(x1A,...xNAA)=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1
xiA
NA
j=1
1
xjA
if xjA >0,∀j∈[1,...,NA],
1
card (xjA=0)if xiA =0,
0ifxiA >0 and card(xjA =0)>0,
(4)
where card(xjA =0)denotes the cardinality of players in group A who made zero
effort. In words, if each player in group A makes a positive effort then player ireceives
the inverse proportional share. If some member in group A makes zero effort, the
inverse proportional share would not be well-defined. But by the logic of the sharing
rule, in that case players who make zero effort share the prize equally among them.8
To illustrate the effect of sharing rules consider the following example. Suppose a
group with subjects Ann, Bob, Chloe and Dan who made the following efforts: 1,2,4
and 4, respectively. In the egalitarian treatment, upon winning the contest all of the
members would receive 1
4of the prize. However, the inverse proportional rule gives
four/two times as much share from the prize to Ann/Bob, than to Chloe and Dan who
7Choi et al. (2016) propose a different possibility that yields an unequal split of the prize in a group contest
that also exhibits internal conflict. In their theoretical model, power asymmetry determines who wins the
internal conflict. Hausken (2005) also has a theoretical model that features within- and between-group
conflict.
8Note that neither the general formulation of the sharing rule by Nitzan (1991) nor other approaches to
the sharing rule like Balart et al. (2017) are able to capture the inverse proportional rule.
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468 H. J. Kiss et al.
receive the same share. The formula above yields the following shares: 1
2,1
4,1
8and 1
8,
respectively, to Ann, Bob, Chloe and Dan.
Given perfect substitution in the performance function, homogeneous linear costs,
a lottery contest success function (r=1), equal group sizes (NA=NB=N) and the
egalitarian sharing rule, the expected payoff of a risk-neutral player i in group A can
be written as:
πiA(x1A,...xNA
,XB)=N
i=1xiA
(N
i=1xiA +N
j=1xjB)
1
Nυ−xiA (5)
The first term of the expected payoff represents the expected benefit of contributing
effort xiA. Higher effort of player iin group A increases the probability of group A
winning the contest, but yields the same share of the prize ( 1
N) for all members of
group A (if this group wins the contest). Cost is captured by the second term. Hence,
there is an inherent tension, because player ihas an incentive to cooperate with other
members of her group, but given the cost of cooperation, there is also an incentive to
free ride.
In this setup, with the egalitarian sharing rule there is a unique equilibrium group
effort, and if we assume the ‘fair share’ of each individual effort to be the same, then
we obtain9
x∗
1A=.. =x∗
iA =.. =x∗
NA =υ
4N2.9(6)
If we use the inverse proportional sharing rule, then the expected payoff of a risk-
neutral player iin group A if all efforts are positive can be written as:
πiA(x1A,...xNA
,XB)=N
i=1xiA
(N
i=1xiA +N
j=1xjB)
1
x1A
NA
i=1
1
x1A
υ−xiA (7)
Under the inverse proportional sharing rule, there is a unique equilibrium group
effort that coincides with the individual effort:
x∗
1A=.. =x∗
iA =.. =x∗
NA =0.(8)
In Appendix A.2 we show that given this rule, zero effort is the dominant strategy
for each individual. It means that the optimal decision is always to make a zero con-
tribution, independently of the decision of the rest of individuals. Note that under this
rule, the Nash equilibrium in dominant strategies coincides with the socially efficient
outcome: Since there is a fixed prize in the contest, it is socially efficient not to make
any effort. While this efficient outcome is common in group contests, as far as we
know the inverse proportional sharing rule is the first rule in this context in which
individual optimal action and social optima coincide.
The previous predictions are based on the following assumptions: (i) players have
identical valuations about the payoffs and winning the contest; (ii) players maximize
their individual expected payoff without regard to the team’s interest; and (iii) players
9Appendix A.1 contains the details of the proof following the line of reasoning by Katz et al. (1990).
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Group contest in a coopetitive setup: experimental… 469
are risk-neutral. Hence, deviations from the predictions may be due (at least partly) to
the fact that these assumptions do not hold.
3 Literature review and hypotheses
In this section, first we review the main findings of the group contest literature that are
related to our theoretical predictions, then we briefly speak about the role of members’
heterogeneity in effort in the framework of group contests. After that, we formulate
our hypotheses.
3.1 Over-expenditure of effort
A general finding in the group contest literature both in the field (e.g., Erev et al.
1993) and in the laboratory (e.g., Nalbantian and Schotter 1997; Van Dijk et al. 2001)
is that with proportional or egalitarian sharing rule contests between groups lead to
high individual effort and little free riding. More recent experimental studies (Abbink
et al. 2010; Ahn et al. 2011; Bhattacharya 2016; Brookins et al. 2015; Cason et al.
2012,2017; Chowdhury et al. 2016; Eisenkopf 2014;Keetal.2013,2015; Leibbrandt
and Sääksvuori 2012; Sheremeta 2011) also consistently find that average effort level
(though often showing a declining pattern) is significantly higher than the equilibrium
prediction.10
Several explanations for the over-expenditure of effort have been proposed. Pure
joy of winning explains part of the over-expenditure in individual contests (Cason
et al. 2018; Price and Sheremeta 2011,2015; Sheremeta 2010). Bounded rationality,
both in individual (Fallucchi et al. 2013) and group contests (Chowdhury et al. 2014;
Lim et al. 2014), as well as relative payoff maximization (Mago et al. 2016), also have
some explanatory power when we try to understand over-expenditure.
A different set of explanations involves social preferences. In social dilemma and
collective action games, participants often contribute more to the public account than
predicted by standard game theory, (see, for instance, Chaudhuri 2011). Theories based
on social preferences like altruism (e.g. Andreoni 1990), fairness (e.g. Rabin 1993)or
inequality aversion (e.g. Bolton and Ockenfels 2000; Fehr and Schmidt 1999)offer
potential explanation for such behavior and it is natural to think that such social prefer-
ences may be at work toward other members of the group.11 There is also evidence that
different individuals have different tendencies to cooperate, some individuals acting
as conditional cooperators, and others as free-riders. Conditional cooperation has been
found to be a key factor in sustaining high contributions in public good games (Ostrom
2000; Keser and Van Winden 2000; Thöni and Volk 2018), and there is evidence that
it plays also a role in group contests (Kiss et al. 2020). Relatedly, social identity the-
10 Sheremeta (2013) reports based on 30 studies that the median over-expenditure is 72%.
11 A related explanation (called parochial altruism) goes further and besides altruism toward in-group
members claims the existence of hostility toward members of the rival group. However, such hostility
toward out-group subjects is not very common (Abbink et al. 2012; Halevy et al. 2008; Yamagishi and
Mifune 2016). Chowdhury et al. (2021) find that in-group preferences boost concern about individual
payoff, while the presence of an out-group enhances the interest in the group payoff.
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470 H. J. Kiss et al.
ory (Tajfel and Turner 1979) that proposes that a strong group identity may blur the
differences between individual and group interests is another potential explanation.
Chowdhury et al. (2016) provides experimental evidence that social identity is in fact
important to understand the effort level that players choose in group contest.
Over-expenditure in group contest and the possible causes of it have been studied
in environments where equilibrium decisions imply a positive effort, but we are not
aware of any study that investigates the case when no effort is a dominant strategy for
the subjects.
3.1.1 Heterogeneity in behavior
While the theoretical predictions are mostly based on symmetry, in real life individuals
differ in a myriad of ways that may affect their behavior. In most experimental studies
on group contest some degree of heterogeneous behavior can be observed. For instance,
in Abbink et al. (2010), on average, the most contributing group member expends
three times more effort than the least contributing member.12 We do not only observe
heterogeneity on the individual level, but also on the group level. In Abbink et al.
(2010), for instance, the most competitive group made six times more effort than the
least competitive group.
Besides the principal aim to see if behavior is different under the egalitarian and
the inverse proportional rule, another objective of this study is to investigate which
individual characteristics and to which extent associate with differences in individual
behavior. Sheremeta (2018) pursues the same goal studying individual contests. He
elicits loss and risk aversion, cognitive abilities and impulsiveness. He shows that
cognitive abilities affect overbidding, but impulsiveness has an even larger predictive
power.
3.2 Hypotheses
Prior to running the experiment, we registered it at the Open Science Foundation
(https://osf.io/93aus/).13
Hypothesis 1 (egalitarian vs. inverse proportional prize sharing): Effort will be
significantly lower in the treatment with the inverse proportional prize sharing than in
the treatment with the egalitarian rule.
Note that even the egalitarian rule (used, for instance, in Nalbantian and Schotter
1997; Abbink et al. 2010,2012; Ahn et al. 2011; Sheremeta 2011; Cason et al. 2012,
2017) provides incentives for free riding, because a participant would receive the same
12 Theoretically, the existence of asymmetric equilibria may explain why we observe different levels of
effort (see, for instance, Baik 1993,1994).
13 In the pre-registration we formulated two sets of hypotheses. Hypothesis 1 refers to the treatment effect
and conjectures lower effort in the inverse proportional treatment than in the egalitarian treatment. We
complement them by testing also if efforts remain positive under the inverse proportional rule. In the
second set of hypotheses (Hypotheses 2–5), in an exploratory analysis we study how different individual
characteristics may affect individual effort in the group contest, and they are conjectured based on the
insights provided by the literature. Here we lump together the second set of hypotheses into one conjecture.
In the registration we used a slightly different wording, i.e., instead of effort we used the term contribution.
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Group contest in a coopetitive setup: experimental… 471
share of the prize even upon making zero effort as other members of the group who
exert a positive level of effort. Evidence of such possible free riding is provided by the
comparison of the egalitarian and the proportional sharing rule. Amaldoss et al. (2000),
Gunnthorsdottir and Rapoport (2006) and Kugler et al. (2010) find that the proportional
sharing rule leads to higher individual efforts than the egalitarian rule. Since the inverse
proportional sharing rule disincentivizes effort even more than the egalitarian rule, we
expect that under the former rule individual efforts will be significantly lower than
under the latter one.
Hypothesis 2 (positive effort in inverse proportional prize sharing): Effort will
be significantly different from zero in the treatment with the inverse proportional prize
sharing rule.
Given the persistent presence of over-expenditure, we expect to see it also in our
experiment, even though positive individual contributions are extremely disincen-
tivized. With this hypothesis, we want to test if the group contest structure promotes
effort per se, even when there are no incentives to make any effort from an individual
point of view.
We also want to explore how over-expenditure is associated with individual char-
acteristics.
Conjectures (level of effort and individual characteristics):
– The higher the individual social value orientation, the higher the effort in the group
contest.
– The more risk averse an individual is, the less effort she makes in the group contest,
ceteris paribus.
– The higher the contribution in the public goods game, the higher the effort in the
group contest.
– Competitiveness is associated to effort in the group contest.
Based on the literature about social value orientation, altruist and prosocial par-
ticipants should tend to contribute more to the performance of the group than
individualistic or competitive subjects. Balliet et al. (2009) and Bogaert et al. (2008)
show that social value orientation correlates with cooperation in social dilemmas,
altruist and prosocial individuals behaving in a more cooperative manner than indi-
viduals classified as individualists or competitors. Field studies also reveal that proself
individuals behave differently from prosocial individuals.14
Contributing to the group performance is risky because winning the prize depends
on how much effort the other members make and on the total effort of the other group.
These are factors beyond the control of the subjects and represent a source of uncer-
tainty. Not contributing to the group performance lowers the probability of winning
the prize but increases the amount of earnings because the money not contributed is a
certain earning for the participants. Since due to the contest nature effort is even more
risky than in a simple public goods game, we hypothesize that anything else being
equal more risk aversion associates with less effort. There are several studies that show
that risk attitudes correlate with contribution in public goods game (Sabater-Grande
14 For instance McClintock and Allison (1989) show that prosocial individuals exhibit more helping behav-
ior than proself ones. The same is found in case of pro-environmental initiatives (Joireman et al. 2001)or
choice between public transportation vs. commuting by car (Van Vugt et al. 1995,1996).
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472 H. J. Kiss et al.
and Georgantzis 2002; Lange et al. 2007; Charness and Villeval 2009; Gangadharan
and Nemes 2009), more risk averse individuals contributing less.
It is natural to think that those participants who are more cooperative in the public
goods game will be more cooperative in the group contests as well, independently
of the sharing rule. In the group contests that we study there is a tension between
individual and group interests. Contributing more to the group performance increases
the probability of winning, but lowers the individual earning, ceteris paribus.This
conflict is more pronounced in the case of the inverse proportional treatment than in
the egalitarian case. Contribution to the public account in the public goods game is
regarded as a proxy for cooperativeness (e.g., Chaudhuri 2011). Peysakhovich et al.
(2014) show convincingly that on the individual level decisions in different cooperation
games are strongly correlated.
Regarding competitiveness, note that competition is present on two levels. Groups
compete against each other, but on the individual level there is a competition between
the members of the same group. If the group competition motive dominates, then the
individuals who are more competitive are expected to make more effort. If the com-
petition on the individual level is stronger than the group competition, then we expect
the opposite to happen. Thus, competitiveness seems relevant to explain behavior in
our design, although it is not clear which direction should dominate.
4 Experiment
The experiment consisted of 2 sessions, each corresponding to an experimental treat-
ment characterized by a different sharing rule. Each session had 5 phases and a
debriefing questionnaire. In both treatments, phase 1 corresponded to the group con-
test, while later phases represented experimental games to gather information about
the participants’ characteristics.
We made clear to participants that
– during the phases they would earn tokens that at the end would be converted into
Euros;
– the exchange rate may change between phases, but in any case more tokens implied
more Euros15;
– their final payoff would be the sum of two payoffs (performance in phase 1 and
in one of the other 4 phases, randomly chosen by the computer) plus the show-up
fee;
– after the 5 phases there would be a questionnaire and after finishing the question-
naire they would be paid in private.
In the group contest phase, first groups of four subjects were formed randomly
and anonymously. There were 14 groups of 4 individuals in each session. This phase
consisted of 20 rounds during which groups were fixed. Each group played against
another group, called rival group, such that the pair of rival groups was also fixed
across all experimental rounds (as, for instance in Abbink et al. 2010).
15 We changed the exchange rates between tasks in a way that in expected terms the payoffs in the different
phases be equal.
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Group contest in a coopetitive setup: experimental… 473
At the beginning of each round, each subject received 1000 tokens that she could
use to buy competition tokens for her group, one competition token costing one token.
The tokens not used for buying competition tokens remained on the account of the
subject. Subjects knew that the same conditions applied to the other members of the
group.
After each round, the amount of the competition tokens a group had accumulated
determined the chance of winning the contest. More concretely, imagine groups A and
B. Suppose that members of group A/B have bought 1000/2000 competition tokens.
Then, group B had twice as high probability of winning the contest (2/3) than group
A (1/3).16
The group that won the contest received a prize of 4000 tokens. In the egalitarian
treatment, each member of the winning group received the same amount of the prize,
that is 1000 tokens. The payoff of any member of the group is the sum of the tokens
that the participant did not use to buy competition tokens plus an equal share of the
prize (if her group wins).17 In the inverse proportional treatment, the less competition
tokens a member of the winning group bought, the higher is her share from the prize.18
In both treatments at the end of each round, each participant obtained the following
information:
– the number of competition tokens that she bought;
– the total number of competition tokens that the group accumulated;
– the total number of competition tokens that the rival group accumulated;
– whether the group the participant belongs to is the winner group;
– individual’s payoff in the round, expressed in tokens.
In both treatments, earnings in this phase consisted of the sum of the payoffs of 5
randomly chosen rounds (as, for instance, in Chowdhury et al. 2014,2016). Subjects
knew that their earnings would be converted into Euros at the end of the experiment
at the following exchange rate: 1000 tokens = 1.2 euros. At the end of the phase, each
participant was provided the information on the five rounds randomly chosen to be
paid to the participant and on individual’s payoff per phase in tokens and in euros.
The rest of the phases was the same in both treatments. In phase 2 we measured social
preferences using the social value orientation Murphy et al. (2011). Phase 3 consisted
of a one-shot play of the public goods game. Phase 4 served to elicit risk attitudes using
16 A wheel of fortune determined which group won the contest in the following way. Following the previous
example, two- third of the wheel would belong to group B and the rest to group A, and after spinning the
wheel the winner is the group over whose territory the pointer of the wheel stopped. We made clear that the
probability of winning increased in the number of competition tokens.
17 For instance, if a participant uses 350 tokens from her initial endowment of 1000 tokens to buy compe-
tition tokens and her group wins the contest, then her final payoff is (1000 −350)+1000 =1650 tokens.
If the other group wins, then her payoff is just 1000 −350 =650.
18 In the instructions we explained the division of the prize in this treatment using the following exam-
ple. Assume a group that consists of subjects A,B,C and D who have bought 100, 200, 400 and 400
competition tokens, respectively. Suppose that this group wins the contest. Subjects A/B/C/D would
receive 2000/1000/500/500 tokens from the prize. In turn, it implies that in this round the payoff of A
is (1000 −100)+2000 =2900, the payoff of B is (1000 −200)+1000 =1800, while C and D would
receive (1000 −400)+500 =1100 tokens, both. As in the other treatment, if the other group wins, then
the group receives no prize and the final payoff equals the initial endowment minus the tokens used to buy
competition tokens, that is A/B/C/D receives 900/800/600/600 tokens.
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474 H. J. Kiss et al.
the bomb risk elicitation task (Crosetto and Filippin 2013). In phase 5, we measured
participants’ competitiveness using the Niederle–Vesterlund experimental procedure
(Niederle and Vesterlund 2007). Finally, at the end of the experiment, subjects had
to fill in a debriefing questionnaire. We gathered socio-demographic information and
biological features. We explain each phase in more detail in Appendix B.
4.1 Procedures
We ran the experiment in July, 2018 in the laboratory of LINEEX (Valencia, Spain).
Fifty-six individuals (corresponding to 14 four-member groups) participated in each
session.19 In the egalitarian and inversely proportional treatment 39.3% and 66.1% of
the subjects were females, respectively.
Sessions lasted about two hours and participants earned on average 18 Euros. There
were subjects studying Economics or Business, but 36% of the participants studied
social sciences, 22% engineering and architecture, 16% health sciences and 5% arts
and humanities.
5 Results
5.1 Egalitarian and inverse proportional sharing rule
Figure 1shows the histograms of individual efforts in the two experimental treatments.
In the inverse proportional treatment, effort is clearly more skewed to the left than in the
egalitarian treatment, resulting in overall less effort. Two-sample Wilcoxon rank-sum
test rejects (p<0.001) the equality of the efforts in the two treatments.20
Figure 2shows the average effort per period in the two treatments. Visual inspection
strongly suggests a treatment effect, as in the egalitarian treatment the average effort is
considerably higher in each period (except in period 1) than in the inverse proportional
treatment.
The average effort in the egalitarian treatment is 332.3, while in the inverse propor-
tional treatment it is 124.8, the former being more than 2.5 times larger than the latter.
The difference persists over time, efforts being consistently lower in the inverse pro-
portional treatment. In the first five rounds, the average effort is equal to 353.3/183.8 in
the egalitarian/inverse proportional treatment, the former being 92% more; for rounds
6 to 10 the same numbers are 371 and 129.5 (186% more); for rounds 11 to 15 the
average efforts are 319.1 and 100.2 (218% more); and for rounds 16 to 20 they equal to
303.8 and 92.3 (229% more). The two-sample Wilcoxon rank-sum test confirms that
there is a significant difference in individual efforts between treatments (p<0.001),
and the difference is also significant for all periods (except period 1) between the two
treatments (p<0.001 in all cases).
19 In the absence of previous findings on the effect of the inverse proportional sharing rule, we could not
determine the appropriate sample size using power calculations. We followed closely the design imple-
mented in Abbink et al. (2010), including the number of groups.
20 Matching group per experimental treatment is the unit of observation for the nonparametric test.
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Group contest in a coopetitive setup: experimental… 475
Fig. 1 Histogram of individual efforts in the treatments
Fig. 2 Average individual effort per period in the treatments (horizontal lines indicate theoretical predic-
tions)
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476 H. J. Kiss et al.
Fig. 3 Heterogeneity in efforts within and between groups (95% confidence interval)
In both treatments we observe a marked downward trend. We check for the presence
of a downward trend in both treatments through a within-subject analysis by comparing
decisions in the first and last five rounds. A statistically significant lower effort in the
last 5 rounds is found relative to the first 5 rounds in both treatments (Wilcoxon signed-
rank test, pvalue = 0.085/<0.001 in the egalitarian/inverse proportional treatment).
The theoretical symmetric prediction for the egalitarian treatment is 62.5. Given
average effort of 332.2, we have an 431.5% over-expenditure in the egalitarian
treatment, on average. Similarly, the average effort of 124.8 tokens in the inverse
proportional treatment is much higher than the predicted zero effort. Hence, although
effort in the inverse proportional treatment is significantly lower, than in the egalitar-
ian treatment, we observe over-expenditure even in the inverse proportional treatment.
Moreover, we find that this high level of effort remains even at the end of the experi-
ment. Observing a decrease in the effort across rounds in each experimental treatment,
we test differences between predicted and actual effort exerted by participants in the
last 5 rounds, which is 303.9 and 94.9 in egalitarian and inverse proportional treat-
ment, respectively. t-test shows statistically significant differences in average effort
with respect to the predicted effort of 62.5 in egalitarian and 0 in the inverse propor-
tional treatment (p<0.001 in both treatments), which implies the presence of the
over-expenditure in both treatments even at the end of the group contest phase.
The above descriptive statistics and tests strongly suggest that there is a difference in
the behavior of participants between treatments. Subjects in the egalitarian treatment
on average made considerably more effort than participants in the inverse proportional
treatment. Furthermore, in both treatments we document a sizable over-expenditure.
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Group contest in a coopetitive setup: experimental… 477
The previous findings are silent about the heterogeneity within and between groups.
It is natural to ask to which extent the treatment effect is general or due to some
groups with extreme efforts, and if the sharing rule influences variance of behav-
ior within groups. Figure 3indicates that the treatment effect is general as most of
the group efforts in the egalitarian treatment are clearly above the group efforts in
the inverse proportional treatment.21 In both treatments, there is considerable het-
erogeneity between groups. Within groups the heterogeneity seems to be higher in
the egalitarian treatment, but it can be due to a higher level of efforts overall. If we
adjust for the level of effort and calculate the coefficient of variation, then the average
coefficient of variation in the egalitarian treatment (0.843) is lower than in the inverse
proportional treatment (1.210).22 Overall, the inverse proportional sharing rule leads
to less effort in general and is associated with more varied behavior.
Besides the treatment, we are also interested in the association between individ-
ual effort and economic preferences. Table 1shows average individual effort in both
treatments measured in phase 1 according to the terciles of cooperation measured in
phase 3, and risk tolerance measured in phase 4. For instance, 216/87 in the top left
cell indicates that participants who were the least risk tolerant and least cooperative
contributed on average 216/87 tokens in the egalitarian/inverse proportional treatment.
In line with the previous descriptive statistics, in all instances efforts in the egalitarian
treatment are larger than efforts in the inverse proportional treatment. Moreover, hold-
ing constant risk tolerance, we see that more cooperative participants tend to make
more efforts in both treatments, in line with our corresponding hypothesis. However,
the association between risk tolerance and individual effort is not clear, in contrast to
our conjecture.23
5.2 Regression analysis
To investigate the effect of the sharing rule and the different individual characteristics,
we proceed with a regression analysis. The variable Treatment is a dummy variable
which equals 1 if the observation comes from the inverse proportional treatment.
Prosocial is a binary variable which is equal to 1 if the individual is classified as a
prosocial one in the social value orientation task. Risk attitude is represented by the
variable Risk preferences that measures the number of boxes the participant decided to
take out of the store in the bomb risk elicitation task. The higher this number, the more
risk-tolerant a subject is. From the public goods game, the variable Contribution indi-
cates the amount of tokens contributed to the common project and is a natural measure
of cooperativeness: the higher the amount, the more cooperative is the participant.
Competitiveness is a dummy variable with value 1 if the subject chose the tourna-
21 If we compute average group efforts, the two-sample Wilcoxon rank-sum test confirms that there is a
significant difference in group efforts between treatments (p<0.001).
22 Considering the coefficient of variation on the group level, the two-sample Wilcoxon rank-sum test
confirms that the coefficient of variation is lower in the egalitarian treatment (p<0.06).
23 Note that while in the case of risk tolerance the intervals of the terciles are very similar in both treatments,
we see sizable differences in cooperativeness. It may be due to the fact that these preferences were measured
after the group contest, so experiences in the group contest may have affectedcooperativeness. Nevertheless,
it is still true that cooperativeness and effort in the group contest are positively associated.
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478 H. J. Kiss et al.
Table 1 Individual effort in treatments according to cooperation and risk tolerance
Risk tolerance Cooperation
Tercile 1 Tercile 2 Tercile 3
Egalitarian/inverse Egalitarian/inverse Egalitarian/inverse
(≤200)/(≤100)(200 <≤450)/(100 <≤250)(450 <)/(250 <)
Tercile 1
Egalitarian/Inverse 216/87 297/141 463/182
(≤40)/(≤35)(256)/(124) (177)/(119) (325)/(138)
Tercile 2
Egalitarian/Inverse 276/134 300/129 319/146
(40 <≤50)/(35 <≤50)(312)/(194) (212)/(134) (250)/(164)
Tercile 3
Egalitarian/Inverse 287/67 451/92 462/194
(50 <)/(50 <) (220)/(151) (332)/(82) (332)/(244)
The numbers in parentheses below the treatments’ names indicate the values of the corresponding terciles. For example, for cooperation ≤200 indicates that in the lowest
tercile participants contributed less or equal to 200 units in the public goods game
Each cell provides the mean and the standard deviation (in parentheses) of individuals’ effort per experimental treatment
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Group contest in a coopetitive setup: experimental… 479
ment payment instead of the piece-rate payment in the competition task (Niederle and
Vesterlund 2007).
In Fig. 2we see a clear declining pattern of the effort, so we control for that
by introducing the variable Per i o d indicating the period that the decision was made
in. We also consider the possibility that effort made in the previous period may be
correlated with effort in the current period, so we include the variable Effort (t−1)
in the specifications (3)–(5). To account for past decisions, we include the variables
Group (-i’s) total effort (t−1)and Rival group’s total effort (t−1)that indicate the
total effort of the participant’s group (excluding own effort that is captured already by
Effort (t−1)) and the total effort of the rival group in the previous period, respectively.
These variables control partially for the probability of having won the contest in the
last period.
We also consider a wide range of controls grouped into socio-demographics, lin-
guistic and cognitive ability and personality characteristics.24 In all the regressions
standard errors are clustered on the group level.
In Table 2, in specifications 1–3 we see a clear treatment effect (significant at 1%),
indicating that in the inverse proportional treatment participants make less effort. In
specifications 4 and 5, when we introduce interaction terms between treatment and our
variables of interest, the coefficient of Treatment decreases moderately, and it ceases
to be significant, indicating that the interaction terms absorb the effect of treatment.
As expected, the variable Per i o d is negative and significant, reflecting the down-
ward trend in effort. The lack of significance in the interaction term reveals that this
effect is not substantially different between treatments. Own previous effort (Effort
(t−1)) is consistently significant and positive, indicating that participants who make
a great effort in a period tend to do so in the next one as well. When interacted with
treatment, the coefficient is negative and significant, showing that overall the posi-
tive correlation in own effort between subsequent periods is weaker in the inverse
proportional treatment. The variable capturing the effort of the other group members
(Group (-i’s) effort (t−1)) is consistently significant and negative, showing that a
major total effort by the other group members tends to make the participant exert less
effort. The interaction with treatment is consistently significant, positive and of com-
parable magnitude, indicating that Group (-i’s) effort (t−1)does not affect effort in
the inverse proportional treatment. The total effort of the rival group in the previous
period (Rival group tot. effor (t−1)) has a similar effect as the total effort of the other
group members: the higher the total effort of the rival group in t−1, the lower the
participant’s effort in period t. The corresponding interaction term indicates that in
the inverse proportional treatment this negative association is more than compensated
by an opposite effect.
Turning to the measured preferences, prosociality as captured by the social value
orientation is not associated with the participants’ efforts, even the sign of the coef-
24 Socio-demographics comprises variables like female, age, body mass index, academic degree, digit
ratio, breadwinner, breadwinner’s employment, number of siblings, and hours worked per week. Linguistic
and cognitive ability refers to participant’s command of more than two languages and cognitive abilities
were measured through cognitive reflection test. Related to personality, we consider personality traits as
agreeableness, conscientiousness, extraversion, neuroticism and openness from the BIG5 and subject’s
self-reported self-esteem and happiness. See Appendix G for a more detailed discussion on these variables.
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480 H. J. Kiss et al.
Table 2 Determinants of individual effort (Random effect panel linear model)
(1) (2) (3) (4) (5)
Treatment (=1 if Inverse) −200.365∗∗∗ −175.695∗∗∗ −161.729∗∗∗ −133.597 −140.575
(20.437) (23.247) (18.497) (88.205) (88.308)
Prosocial (SVO) −6.215 −12.094 2.358 2.757
(22.800) (16.984) (26.475) (25.243)
Risk pref.s (Bomb task) 0.756 0.379 1.562 1.024
(0.969) (0.610) (0.961) (1.121)
Contribution (PGG) 0.180∗∗ 0.109∗∗ 0.119∗∗ 0.126∗∗
(0.074) (0.049) (0.056) (0.063)
Competitiveness (Niederle and Vesterlund 2007)−10.647 −5.214 −41.937∗∗ −40.366∗∗
(23.954) (14.428) (20.234) (19.032)
Period −4.388∗∗∗ −3.099∗−3.145∗
(1.018) (1.862) (1.867)
Effort (t−1)0.376∗∗∗ 0.395∗∗∗ 0.373∗∗∗
(0.068) (0.072) (0.075)
Group(-i’s) effort (t−1)−0.051∗∗∗ −0.065∗∗∗ −0.062∗∗∗
(0.018) (0.021) (0.022)
Rival group tot. effor (t−1)−0.022∗−0.051∗∗∗ −0.044∗∗∗
(0.012) (0.013) (0.012)
Treatment*Prosocial −3.522 −13.401
(38.787) (33.823)
Treatment*Risk pref.s −2.062∗−1.383
(1.231) (1.457)
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Group contest in a coopetitive setup: experimental… 481
Table 2 continued
(1) (2) (3) (4) (5)
Treatment*Contribution −0.004 −0.021
(0.081) (0.080)
Treatment*Competitiveness 63.077∗∗∗ 58.618∗∗
(23.844) (23.599)
Treatment*Period −2.686 −2.601
(2.629) (2.633)
Treatment*Effort (t−1)−0.180∗∗ −0.161∗∗
(0.073) (0.069)
Treatment*Group(-i’s) effort (t−1)0.053∗∗ 0.047∗
(0.027) (0.027)
Treatment*Rival group effort 0.079∗∗∗ 0.072∗∗∗
(0.019) (0.018)
Socio-demographics (9 variables) YES YES YES YES YES
Linguistic and cogn. ability (3 variables) NO NO NO NO YES
Personality traits (7 variables) NO NO NO NO YES
Constant −17.267 6.476 176.713 167.431 409.226
(279.534) (306.172) (187.549) (188.513) (256.069)
Observations 2240 2240 2128 2128 2128
Standard errors clustered on the group level in parentheses. ∗p<0.10, ∗∗ p<0.05, ∗∗∗ p<0.01
Socio-demographics: Female, Age, Body Mass Index, Academic degree, 2-4DR(left hand),Number of siblings
Breadwinner, Breadwin.’s employm., Work (hrs/week). Linguistic and cogn. ability: Reflective, (Ir)reflecitve
Number of languages. Personality traits: Agreeableness, Conscientiousness, Extraversion, Neuroticism
Openness, Happiness (degree), Self-esteem
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482 H. J. Kiss et al.
ficient is changing. If we interact this variable with treatment, we do not see any
significant effect either. Risk tolerance exhibits a consistent positive association with
effort, more risk-tolerant subjects making more effort, but the coefficient fails to be
significant. The interaction between risk attitudes and treatment does not reveal any
consistently significant association. Cooperativeness proxied by contribution in the
public goods game has a consistent and significant positive coefficient in all specifica-
tions, indicating that more cooperative individuals tend to make more effort in group
contest, ceteris paribus. The effect of cooperativeness is not contingent on treatment,
as indicated by the interaction term. Turning to competitiveness, in specifications
(4) and (5), competitiveness and the interaction term involving competitiveness are
significant and of comparable magnitude, but with opposite sign, suggesting that in
the egalitarian treatment competitive participants tend to make less effort, but in the
inverse proportional treatment there is no difference in effort as a function of com-
petitiveness.25 Appendix D.4 shows that including a dummy variable of whether the
group won the contest in period t−1 does not change the findings, as neither the
dummy, nor the corresponding interaction term is significant in any specification.
In Appendix F we carry out the same analysis, but separately for each treatment.
This analysis reveals that cooperativeness has a markedly lower effect in the egalitarian
treatment than in the inverse proportional treatment. Prosociality, risk attitudes and
competitiveness do not show a consistently significant association with effort in these
regressions. Own effort in the previous period has the same effect as in Table 2,butthe
magnitude is considerably lower in the inverse proportional treatment. The group’s
and the rival group’s total effort in the previous period have similar coefficients as
before in the egalitarian treatment. However, the coefficients of these variables fail to
be significant in the inverse proportional treatment.
From the rest of the variables that we used as controls, among personality traits, we
find that conscientiousness tends to have a negative effect on the effort, while agree-
ableness associates in the same way with effort in the egalitarian treatment (see Table
11 and Table 12 in Appendix G, respectively). As for the set of socio-demographic
variables, we do not observe consistent patterns that hold consistently across the treat-
ments.
6 Conclusion
Over-expenditure in group contests is a puzzling phenomenon that attracted the
attention of many scholars. In this paper, we investigated if a new sharing rule
(called inverse proportional) that theoretically strongly disincentivizes effort elimi-
nates over-expenditure or not. The sharing rule is based on the idea that in coopetitive
environments where members of the same group cooperate and compete with each
other at the same time face a resource constraint: More resources spent on cooperation
lowers the capacity to compete. Therefore, the sharing rule assumes that less cooper-
25 Our measure of competitiveness may depend on the beliefs of the individual about her own performance
relative to the members of the group. To take it into account, in Appendix E we estimate the same model
controlling for the beliefs of the individual about her performance, and the effects of competitivenessremain
similar.
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Group contest in a coopetitive setup: experimental… 483
ation proportionally increases the capacity to capture a larger share of the prize. In a
laboratory experiment, we compare behavior under this new rule to behavior observed
when the prize is shared in an egalitarian way. In line with the theoretical prediction,
we document that less effort is made under the inverse proportional rule than under
the egalitarian one. However, still we observe a substantial over-expenditure that does
not vanish even at the end of the experiment.
To understand better the behavior of the participants we also measured economic
preferences. We see that more cooperative participants choose a higher level of effort
in both treatments. Competitive preferences have some effect in the inverse propor-
tional treatment: more competitive participants contributing larger amounts from their
endowment. Social and risk preferences do not seem to have any effect.
Supplementary Information The online version contains supplementary material available at https://doi.
org/10.1007/s11403- 022-00373- 6.
Acknowledgements We are grateful for comments and suggestions from seminar participants at the Uni-
versidad Autónoma de Madrid and the Universidad de Murcia (Spain). We are especially indebted to Roman
Sheremeta for some initial advice on the experimental design, Lisa Vesterlund for sharing her z-tree code
for the competitiveness task and Antoni Rubí Barceló for extensive comments.
Funding Open access funding provided by ELKH Centre for Economic and Regional Studies.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included
in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If
material is not included in the article’s Creative Commons licence and your intended use is not permitted
by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the
copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
A Theoretical predictions
In this section we derive the theoretical predictions of the model.
A.1 Egalitarian case
Given perfect substitution in the performance function, homogeneous linear costs, a
lottery contest success function (r=1), equal group sizes (NA=NB=N) and the
egalitarian sharing rule, the expected payoff of a risk-neutral player i in group A can
be written as:
πiA(x1A,...xNA
,XB)=N
i=1xiA
(N
i=1xiA +N
j=1xjB)
1
Nυ−xiA (9)
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484 H. J. Kiss et al.
As in Katz et al. (1990), we use the Nash equilibrium to find the solution. Assuming
a regular interior solution, the first-order condition for members of group A is
∂πiA
∂xiA
=N
j=1xjB
(N
i=1xiA +N
j=1xjB)2
1
Nυ−1=0(10)
Similarly, for group B
∂πjB
∂xjB
=N
j=1xiA
(N
i=1xiA +N
j=1xjB)2
1
Nυ−1=0 (11)
In a symmetric equilibrium, x1A=x2A.. =xiA =.. =xNA and x1B=x2B.. =
xjB =.. =xNB, the previous conditions become
∂πiA
∂xiA
=xjB
N2(x2
iA +2xiAxjB +x2
jB)υ−1=0(12)
and ∂πjB
∂xjB
=xiA
N2(x2
iA +2xiAxjB +x2
jB)υ−1=0 (13)
These first-order conditions imply that in equilibrium x∗
iA =x∗
jB. As a consequence,
we obtain that
x∗
1A=.. =x∗
iA =.. =x∗
NA =υ
4N2(14)
A.2 Inverse proportional case
In the proof, we will use the following additional notation:
XA−i=
jA,j=i
xj
By our definition the inverse proportional rule implies the following expected pay-
offs:
πixi,A,XA−i,XB=XA
XA+XB
x−1
i,A
jAx−1
j
v−xi,Aif i,A>0 (15)
πi(0,XA−i,XB)=XA−i
XA−i+XB
vif $XA−i>0 (16)
The second line specifies that if all other members of group A make a positive effort
while member i makes zero effort, then in case group A wins the contest, member i
will receive the whole prize.
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Group contest in a coopetitive setup: experimental… 485
Next, we prove that the individual payoff upon making a positive effort is less than
the payoff related to zero effort, conditional on all the other members of the group
making a positive effort:
πixi,AXA−i,XB−πi(0,XA−i,XB)=XA
XA+XB
x−1
i,A
jAx−1
j
v−xi,A−XA−i
XA−i+XB
v
=⎡
⎣
XA
(XA+XB)
x−1
i,A
jAx−1
j−XA−i
(XA−i+XB)
⎤
⎦v−xi,A
=⎡
⎣
XA·x−1
i,A·(XA−i+XB)−XA−i·(XA+XB)jAx−1
j
(XA+XB)jAx−1
j(XA−i+XB)
⎤
⎦v−xi,A
=⎡
⎣
XA·x−1
i,A·XA−i+XA·x−1
i,A·XB−XA−i·XA·jAx−1
j−XA−i·XB·jAx−1
j
(XA+XB)jAx−1
j(XA−i+XB)
⎤
⎦v−xi,A
=⎡
⎣
XB−XA−i·XA·jA,j=ix−1
j−XA−i·XB·jA,j=ix−1
j
(XA+XB)jAx−1
j(XA−i+XB)
⎤
⎦v−xi,A
=⎡
⎣
1−XA−i·jA,j=ix−1
j XB−XA−i·XA·jA,j=ix−1
j
(XA+XB)jAx−1
j(XA−i+XB)
⎤
⎦v−xi,A
=⎡
⎣
1−jA,j=ixj·jA,j=ix−1
j XB−XA−i·XA·jA,j=ix−1
j
(XA+XB)jAx−1
j(XA−i+XB)
⎤
⎦v−xi,A<0(17)
since jA,j=ixj·jA,j=ix−1
j>1 always.
Hence, we proved that
πixi,AXA−i,XB−πi(0,XA−i,XB)<0,
so making zero effort yields always a higher payoff than making a positive effort if
the other members of the group make a positive effort.
We turn now to the case when another member of the group makes zero effort. In
this case, making a positive effort would yield the loss of the effort, because if the
group wins the contest, then the member with zero effort obtains the whole prize, while
in case of losing the contest the effort is lost anyway. Therefore, if another member
of the group makes zero effort, then making zero effort yields a payoff that is higher
than the payoff related to a positive effort.
The same argument applies also if nobody in the group makes a positive effort.
Thus, we have shown that to make zero effort leads always to higher payoffs than
to make a positive effort in the case of the inverse proportional sharing rule.
B Experimental phases after contest game
In phase 2 we measured social preferences using the social value orientation. There are
various ways to measure social value orientation, we followed Murphy et al. (2011).
123
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486 H. J. Kiss et al.
Participants were randomly paired and each of them had to make 6 decisions. In
each decision, subjects saw 9 pairs of payoff allocations, each allocation containing
a payoff for the decision-maker and a payoff for her co-player. The decision-maker
had to choose in all 6 decisions her preferred joint payoff distribution. Choices can
be scored to come up with a single score. Social value orientation conceptualizes four
idealized orientations:
– altruists maximize the allocation for the other party;
– prosocial individuals tend to maximize their own payoffs, but care also about the
other player’s payoff;
– subjects with individualistic tendencies are not concerned about the other player,
they just maximize their own payoffs;
– competitive individuals attempt to maximize their own payoffs, but at the same
time also minimize the other player’s payoff.
The score achieved after choosing the 6 allocation allows to classify participants in
one of the above categories. Participants knew that if at the end of the experiment
this phase would be chosen for payment, then the computer would pick one of the six
decisions and would randomly choose one of the participants in each pair (called the
elector) and the allocation chosen by the elector in the given decision would be paid.
The exchange rate used for payment was 1 token = 0.02 Euros.
Phase 3 consisted of a one-shot play of the public goods game. We aimed to mea-
sure cooperativeness with this game. We presented the most widely used format of
the public goods game, with four players, each of them endowed with 1000 tokens.
Participants had to decide how much of the endowment to assign to a public account,
knowing that everybody in the group would receive 40% of the total amount assigned
to the public account (that is, marginal per capita return = 40%). The earning of a
subject in this phase consisted of the amount received from the public account plus
the amount not assigned to the public account. The exchange rate used for payment
was 1 token = 0.02 Euros.
Phase 4 served to elicit risk attitudes using the bomb risk elicitation task (Crosetto
and Filippin 2013).26 Participants were presented the following situation. There was
a store with 100 boxes, one of them containing a bomb. The bomb could be in any
of the boxes with the same probability. Subjects had to decide how many boxes they
wanted to take out of the store. For each box taken out that did not contain the bomb,
they received money (1 token), but if the bomb happened to be in one of the boxes that
had been taken out, then their payoff in this task was zero. We explained that if, for
example, they decide to take out 7 boxes and the bomb is in box 42, then they would
earn 7 tokens. However, if a participant decides to remove 56 boxes from the store and
the bomb is in box 51, then her earning is zero. We informed participants that if at the
end of the experiment this task is the payoff-relevant task, the exchange rate would be
1 token = 0.1 Euros.
In phase 5, we measured participants’ competitiveness using the Niederle–
Vesterlund experimental procedure (Niederle and Vesterlund 2007). The only modifi-
26 Crosetto and Filippin (2016) compare four risk elicitation methods, among them the bomb risk elicitation
task. They show the pros and cons of each method and argue convincingly that the bomb risk elicitation
task is appropriate to distinguish between subjects based on their risk attitudes.
123
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Group contest in a coopetitive setup: experimental… 487
cation compared to the original study is that we used the slider task (Gill and Prowse
2011,2018) instead of adding up numbers as experimental task. Following Niederle
and Vesterlund (2007), this phase consisted of 4 subphases. In subphase 1, subjects
performed the slider task (positioning as many sliders on the number 50 as they could
in one minute) knowing that they would be paid on a piece-rate basis, earning 1 token
for each correctly positioned slider. We also explained that if at the end of the experi-
ment this task would be chosen for payment, then the exchange rate would be 1 token
= 0.15 Euros. In subphase 2, before playing again the slider task, groups of 4 members
were formed randomly and we informed participants that they would be paid as in a
tournament. More concretely, we told them that only the member of the group with the
highest number of correctly placed sliders receives a payoff, but she receives 4 tokens
for each correctly positioned slider.27 We used the same exchange rate as before. We
also informed subjects that they would not know the result of the tournament until
the end of the phase. In subphase 3, we explained that they would perform again the
slider task and that they could choose the way to be compensated: piece-rate payment
as in subphase 1 or tournament payment as in subphase 2. Hence, we have a binary
classification: participants are either competitive (if they choose the tournament) or not
competitive (if they choose the piece-rate scheme). As before, we explained that we
would not tell them the result of the tournament until the end of the phase. We applied
the same exchange rate here as in the previous subphases. In subphase 4, participants
were not required to perform the task again, but could earn money by choosing an
incentive scheme (piece rate vs. tournament) to be applied to their performance in sub-
phase 1. We reminded participants about the number of correctly positioned sliders in
subphase 1. We used the same exchange rate as in the previous subphases. At the end
of this subphase, participants were asked to evaluate their own performance relative
to other members of the group in subphase 1 and in the subphase 2. We incentivized
participants for this task. This subphase allows us to see what participants believed
about their relative performance.
At the end of the experiment, subjects had to fill in a debriefing questionnaire. We
gathered socio-demographic information (age, gender, educational attainment, field of
study, knowledge of languages, number of siblings, education and employment of the
breadwinner in the family, factors related to family income) and biological features
(height, weight, dexterity: left- vs right-handed and the digit ratio).28 We also measured
cognitive abilities with a 5-item version of the cognitive reflection test (Frederick 2005;
Toplak et al. 2014). We used a 10-item version of the Big Five test (Rammstedt and
John 2007) to elicit personality traits and the Rosenberg test (Rosenberg 1965)to
measure self-esteem. We also asked participants if they were happy in general.
27 We also explained that in case of a tie, the computer randomly would select one of the members with
the highest number of sliders correctly positioned.
28 We gathered information on participants’ digit ratio since we believed that there might be some rela-
tionship between digit ratio and decisions in the group contest. There is a growing literature indicating that
digit ratio associates with several individual characteristics, such as competitiveness, social preferences and
risk aversion among other domains of economic interest (Brañas-Garza et al. 2013,2018; Garbarino et al.
2011; Millet and Dewitte 2006; Pearson and Schipper 2012). Brañas-Garza et al. (2019) report that digit
ratio has explanatory power in contests.
123
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488 H. J. Kiss et al.
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