1 Introduction

Many situations in social and economic life are characterized by rivalry and conflict between two or more competing parties. Warfare, socio-political conflicts, political elections, lobbying, R&D competitions, and promotion tournaments, are all examples of inter-group conflicts in which groups spend scarce and costly resources in order to compete with other groups. Within each competing group, group members may differ with respect to a variety of characteristics such as preferences, resources, wealth, productivity, or motivation, which, in turn, can affect their ability and willingness to compete. Acknowledging that such within-group heterogeneity is the rule rather than the exception, a straightforward implication is that competing groups are rarely identical, and contests are typically not symmetric.

Examples abound. For instance, countries competing for access to natural resources or geopolitical influence will typically (if not always) differ regarding the degree of diversity in society, such as the distribution of income, education, or other sociodemographic characteristics. In the domain of organizations, firms often rely on interfirm alliances to compete with other firms or alliances (for example in the context of developing new products). Such alliances can be cross-function when partners contribute diverse and complementary resources, or same-function when firms have similar competencies (Amaldoss and Staelin Reference Amaldoss and Staelin2010). The resulting competition can then be either symmetric (between two cross-function alliances or two same-function alliances) or asymmetric (between a cross-function and a same-function alliance). Similarly, within-firm contests between groups can occur in the context of performance-contingent payment schemes, such as paying bonuses to the best performing group(s) to increase productivity (Nalbantian and Schotter Reference Nalbantian and Schotter1997; Bandiera et al. Reference Bandiera, Barankay and Rasul2013). Within-group heterogeneity in such settings is only natural, as group members can have different skills or abilities. It follows that the competing groups themselves are also not necessarily similar, resulting in asymmetric competition.Footnote ¹

Despite these rather natural applications, the bulk of previous literature on contests has focused on situations in which symmetrical agents or groups compete against each other (see Dechenaux et al. Reference Dechenaux, Kovenock and Sheremeta2015; Sheremeta Reference Sheremeta2017, for overviews). There is, however, a respectable (and growing) number of studies that have investigated the effects of various types of asymmetries, including group size (Abbink et al. Reference Abbink, Brandts, Herrmann and Orzen2010; Ahn et al. Reference Ahn, Isaac and Salmon2011), wealth (Rapoport et al. Reference Rapoport, Bornstein and Erev1989; Hargreaves Heap et al. Reference Hargreaves Heap, Ramalingam, Ramalingam and Stoddard2015), sharing rules (Kurschilgen et al. Reference Kurschilgen, Morell and Weisel2017), and the availability of communication (Cason et al. Reference Cason, Sheremeta and Zhang2017) and punishment opportunities (Sääksvuori et al. Reference Sääksvuori, Mappes and Puurtinen2011).

We contribute to this literature by experimentally testing how heterogeneity in players’ ability to contribute to the group’s success affects competition. In our systematic analysis, we study the effect of heterogeneity both within and between groups. For ease of exposition, we will use the terms homogeneous and heterogeneous to describe within-group structures (i.e., whether group members are similar or not), and symmetric and asymmetric to capture the relationship between the groups (i.e., whether competing groups are similar or not). In our setting, a high-ability person is more efficient in converting her effort to a contribution to the group than a low-ability person. This means that the marginal productivity, or the contribution/effort ratio, of the high-ability person is higher: for each unit of invested effort, a high ability player contributes more to the group than a low ability player. Heterogeneity in this respect is only natural, as some group members may be stronger, smarter, or have better task-specific capabilities than others. As an illustration, think of a group of salespersons, with one member who is more talented, experienced, or is endowed with a more densely populated sales territory, than the others. The high ability individual has a higher marginal productivity in the sense that even if all group members exert the same effort (in terms of, e.g., hours worked or energy expenditure), she will contribute more to the group’s success than her less able peers.

Previous studies modeled ability, or related concepts, in different ways. Sheremeta (Reference Sheremeta2011b) manipulated players’ valuation of the prize, i.e., some group members derive a higher payoff than others when the group wins the contest.Footnote ² Ryvkin (Reference Ryvkin2011), in a theoretical model, and Brookins et al. (Reference Brookins, Lightle and Ryvkin2015), in an experiment, consider within-group heterogeneity based on players’ cost of effort; for some players, investing effort towards the group’s success is costlier than for others. From an individual, self-interested, perspective, high abilities (as operationalized by us), high prize valuations, and low effort costs are all expected to increases players’ inclination to contribute to their group. However, if players care not only about their own payoff, but also about the distribution of payoffs within the group (e.g., have a preference for equality) these different ways of introducing heterogeneity may differ in how they affect contributions. The reason is that when abilities are heterogeneous, equal efforts lead to equal payoffs; high-ability players’ inclination to contribute is at odds with equality. When valuations or effort costs are heterogeneous, equal efforts result in un-equal payoffs; group members without higher prize valuations, or with lower effort costs, must contribute more than others for payoffs to be equal. Whether these differences in the modeling of ability matter behaviorally is ultimately an empirical question, which the current study can help answer.

We use a laboratory experiment to investigate the role of heterogeneity in abilities within and between groups. The major advantage of using a laboratory experiment is that it allows to tightly distinguish between a player’s ability and her effort choice. In the field this is almost impossible to achieve, since typically only performance—which is a function of both effort and ability (and noise)—can be observed. The laboratory setting further allows us to exogenously manipulate the composition of players within groups, which circumvents complicating factors such as self-selection that emerge in most field settings where groups form endogenously.

As a workhorse for studying group contests, we follow previous literature by using an experimental version of Tullock’s contest game (Tullock Reference Tullock, Buchanan, Tollison and Tullock1980) in which two groups compete for a prize that is divided equally among all members of the winning party (Katz et al. Reference Katz, Nitzan and Rosenberg1990). We study this basic decision situation in three different treatments, in which we systematically vary the heterogeneity both within and between groups. In the first treatment, we study (commonly explored) symmetric contests between two homogeneous groups, in which all group members in both groups are equally able to compete. To study the pure effect of within-group heterogeneity, in the second treatment both competing groups are equally heterogeneous. Specifically, each group consists of one low-ability, one medium-ability, and one high-ability player. Importantly, we hold the average ability of group members constant compared to homogeneous groups, which consists of three medium-ability players. In the third and last treatment, we focus on the most interesting and natural situation in which the two competing groups differ from each other. To provide a clean comparison to the first two treatments, and to be able to investigate how conflict engagement depends on the opponent’s group type (while holding constant the own group type), we examine an asymmetric contest between a homogeneous group and a heterogeneous group.

This set of treatments further distinguishes our study from the above-mentioned studies by Sheremeta (Reference Sheremeta2011b) and Brookins et al. (Reference Brookins, Lightle and Ryvkin2015). In both studies competing groups are always heterogeneous to a certain degree; in the latter, contests are always asymmetric. In contrast, our design includes the natural benchmark, featured in the bulk of the literature, of contests involving completely homogeneous groups as well as symmetric contests between heterogeneous groups. Including these situations allows to better identify and separate the role of within- and between-group heterogeneity.Footnote ³

We find that while heterogeneity per se has no discernable impact on the degree of competition, asymmetry between groups leads to an intensification of conflict, with homogeneous and heterogeneous groups winning the contest equally often. One reason for this result is that players in heterogeneous groups contribute to the success of the group much more equally than predicted by a number of theories (see Sect. 2), which agree in stating that only high-ability players should contribute, while low- and medium-ability players should free-ride.Footnote ⁴

Intra- and inter-group dynamics—the way individuals condition their behavior on the past behavior of others—help to further explain these results. We find that for homogeneous groups, asymmetric contests lead to increased conditional cooperation among group members, while no such effect is observed for heterogeneous groups. Furthermore, in heterogeneous groups players of different abilities condition their effort on that of their ‘relevant’ peer (see Croson et al. Reference Croson, Fatas and Neugebauer2005, Reference Croson, Fatas, Neugebauer and Morales2015; Kölle Reference Kölle2015). Low-ability players react to efforts by the medium-ability player but largely ignore the ones by the high-ability player. Medium-ability players, in contrast, react to both low- and high-ability players. The latter, in turn, react to the behavior of their medium-ability group member, but only in symmetric contests; when the contest is asymmetric they become “unconditional cooperators”, as their contribution is not driven by the effort levels of their group members.

Finally, we show that asymmetric contests are not only more intense, but also more volatile, suggesting that facing a group that is different from your own can lead to an increase in strategic uncertainty. As a result of the increased intensity and volatility, asymmetric contests have two detrimental effects for the participants as they decrease individual earnings and increase payoff inequality within groups. Taken together, our results show that heterogeneity in abilities significantly affects contest behavior only in the most natural setting where heterogeneity exists both within and between groups.

The rest of the paper is organized as follows. Section 2 presents the general setup of our group contests as well as theoretical predictions. In Sect. 3 we describe our experimental design and procedures in more detail. Section 4 summarizes our results. In Sect. 5 we discuss our findings and provide some behavioral rationale for the observed effects. Section 6 concludes.

2 Model and predictions

Consider the following inter-group contests for public goods by Katz et al. (Reference Katz, Nitzan and Rosenberg1990). Let there be n = 2 groups of $m\ge 1$ risk-neutral players each, competing to win a prize $\mathrm{mV}$ . All group members are endowed with the same amount of resources (e.g., time) w, which they can either use to help the own group win the contest, or they can use it for themselves (e.g., for an alternative private activity). Let $e_{i,j}\ge 0$ represent the effort (resources) spent by player i in group j. Importantly, players may differ in their ability to contribute to their group activity; a high-ability player is better able than a low-ability player in converting her effort to an actual contribution to the group. That is, her contribution to the group output is given by $x_{i,j}=e_{i,j}·{\alpha }_{i,j}$ , where ${\alpha }_{i,j}$ is the ability of player i in group j determining her marginal productivity of effort. We assume that individual group members’ efforts are perfect substitutes, i.e., total effort of group j is given by $E_j={\sum }_{i=1}^m{e}_{i,j}$ and the total contribution of group j is given by $X_j={\sum }_{i=1}^m{\alpha }_{i,j}{e}_{i,j}$ . Following Tullock (Reference Tullock, Buchanan, Tollison and Tullock1980), the probability of group j winning the contest and securing the prize is given by the following contest success function:

$p_j\left(X_1,X_2\right)=\left(\begin{array}{cc}\frac{X_j}{X_1+X_2}& if{X}_1+X_2>0\\ \frac{1}{2}& otherwise.\end{array}\right)$

When a group wins, each member of the winning group receives an equal share of the prize, $V$ .Footnote ⁵ The expected payoff of player i in group j is then given by

${\pi }_{i,j}\left(x_{i,j},X_1,X_2\right)=w-e_{i,j}+\frac{X_j}{X_1+X_2}V$

In our experiment (see Sect. 3), we consider two different type of groups, which can be either homogeneous or heterogeneous in the composition of the ability of their members. It follows that three types of inter-group contests can arise: two symmetric contests where both groups share the same internal composition of their members’ ability (either homogeneous or heterogeneous), and one asymmetric contest with a homogeneous group competing with a heterogeneous one. The different contests may lead to different predicted outcomes. We describe below the resulting equilibria predicted under three distinct assumptions: players maximize either their own earnings; players maximize the joint-profit of their own group; or players maximize the difference in earnings between their own and the competing group.

Under the standard assumption of purely self-interested individuals, following Konrad (Reference Konrad2009), if all players in all groups are equally able to contribute to the group output, i.e., ${\alpha }_{i,j}$ is the same for all players (symmetric homogeneous contest), there is a unique equilibrium prediction for the total group effort that is the same as in a two-player contest and equal to $E_1=E_2=\frac{V}{4}$ . Theory, however, remains silent about the behavior of individual group members: any combination of efforts that add up to $\frac{V}{4}$ constitutes an equilibrium. When both groups are heterogeneous but identical (symmetric heterogeneous contest), the prediction about the group effort level does not change, but there is a clear-cut prediction for the individual efforts. As contributions to the group are perfect substitutes and costs of effort are linear, in equilibrium only the member with the highest ability in each group should exert effort, while the other group members should free ride (see Baik Reference Baik2008, for a similar result when group members differ with regard to the evaluation of the prize).

When groups differ with respect to the most able group member (asymmetric contests), the group contest reduces to an asymmetric contest between the most able group members in each group. In the following, we denote the ability of the most able member within each group by ${\overline{\alpha }}_j$ . The solution is then given by simultaneously solving the two reaction functions of the optimal efforts of the two players, yielding $E_1=\frac{{\left({\overline{\alpha }}_1{\overline{\alpha }}_2{E}_{2}V\right)}^{\frac{1}{2}}-{\overline{\alpha }}_2{E}_2}{{\overline{\alpha }}_1}$ and $E_2=\frac{{\left({\overline{\alpha }}_1{\overline{\alpha }}_2{E}_{1}V\right)}^{\frac{1}{2}}-{\overline{\alpha }}_1{E}_1}{{\overline{\alpha }}_2}$ . By denoting $\alpha ={\overline{\alpha }}_1/{\overline{\alpha }}_2$ as the relative ability between the most able player of both groups, the resulting unique symmetric Nash Equilibrium is given by $E_1=E_2=\frac{\alpha V}{{\left(1+\alpha \right)}^2}$ , which for $\alpha \ne 1$ is strictly smaller than $\frac{V}{4}$ . As a result, effort is predicted to be lower in asymmetric than in symmetric contests (see also Fonseca Reference Fonseca2009, for a similar result investigating heterogeneity between players in individual contests).

Experimental evidence from group contests typically shows a departure from standard predictions, with a general tendency of over-dissipation (see Sheremeta Reference Sheremeta2017, for an overview). Several explanations have been put forward to explain such over-investments by groups. Some studies have argued that agents derive some nonmonetary joy of winning (Sheremeta Reference Sheremeta2010), while others explain this finding by assuming that players make mistakes and are only boundedly rational (Lim et al. Reference Lim, Matros and Turocy2014). Further studies argue that agents are motivated not only by self-interest, but also care about the payoffs of others. In particular, agents may be motivated by joint profit maximization, i.e., they strive to maximize the sum of payoffs within the own group (Leibbrandt and Sääksvuori Reference Leibbrandt and Sääksvuori2012), or by relative payoff maximization/parochial altruism (i.e., the display of altruism towards in-group members along with hostility towards out-group members (Bernhard et al. Reference Bernhard, Fischbacher and Fehr2006; Choi and Bowles Reference Choi and Bowles2007; Abbink et al. Reference Abbink, Brandts, Herrmann and Orzen2010)), leading them to maximize the difference between the own and the other group’s payoff. Applying the last two concepts to our setting, the following qualitative predictions across treatments can be derived.

When players try to maximize joint payoffs, the objective function is given by $mw+\frac{X_j}{X_1+X_2}mV-E_j$ . In this case, in symmetric contests the predicted effort is equal to $E_1=E_2=\frac{\mathrm{mV}}{4}$ , while in asymmetric contests aggregate effort is predicted to be $E_1=E_2=\frac{\alpha mV}{{\left(1+\alpha \right)}^2}$ . Hence, while compared to the standard prediction of purely selfish players effort levels are predicted to be higher, it still holds that effort should be lower in asymmetric than in symmetric contests, and that in heterogeneous groups only high ability players should exert any effort.

If, instead, group members are motivated by parochial altruism, they strive to maximize the difference between their own and the other groups’ payoff, ( $\frac{X_1}{X_1+X_2}mV-E_1)-\left(\frac{X_2}{X_1+X_2}mV-E_2\right)$ . In this case, the total effort exerted by each group in symmetric contests will be $E_1=E_2=\frac{\mathrm{mV}}{2}$ , which is higher than predicted efforts in asymmetric contests, given by $E_1=E_2=\frac{2\alpha mV}{{\left(1+\alpha \right)}^2}$ . Both of these effort levels are higher than the ones predicted under the assumption of pure self-interest and joint payoff maximization. Yet, as before it holds that in heterogeneous groups only high ability players should be active, while in homogeneous groups any combination of efforts leading to the predicted aggregated effort is an equilibrium.

To summarize, while the three different assumptions about agents’ objective function lead to level differences in the predicted effort levels, they all share some common qualitative characteristics. First, irrespective of the type of contest (symmetric or asymmetric) homogeneous and heterogeneous groups are predicted to exert the same level of aggregate effort. Second, within heterogeneous groups only the member with the highest ability should exert any positive effort (the others should free ride), while in homogeneous groups there is a continuum of optimal effort combinations. Third, total effort is predicted to be higher in symmetric compared to asymmetric contests.

In our experiment (see below) we empirically test each of these predictions. With regard to the predicted effort levels within heterogeneous groups, we expect behavior to deviate from theory as it predicts an extreme distribution of labor, which, in turn, creates substantial inequality within groups. There is now ample evidence from a variety of contexts that many people care about the relative distribution of outcomes (see e.g., Fehr and Schmidt Reference Fehr and Schmidt1999; Bolton and Ockenfels Reference Bolton and Ockenfels2000; Sobel Reference Sobel2005; Fehr and Schmidt Reference Fehr and Schmidt2006). Applied to our context, if players dislike inequality within groups they have an incentive to match their group members’ efforts, as payoff equality can only be obtained if all group members exert the same level of effort, irrespective of their ability. For heterogeneous groups, this is in stark contrast to the theoretical predictions above which state that only the high ability player should contribute. Hence, if players are also motivated by fairness considerations within groups, we should expect a more equal distribution of labor in heterogeneous groups. In Sect. 5 we address this point more formally and discuss how inequity concerns can rationalize our empirical findings.Footnote ⁶

3 The experiment

Our experimental game is based on the model described above. Participants were randomly divided into three-person groups (m = 3). Each group was then matched with another, randomly selected, group (n = 2), to compete for a prize in 45 consecutive periods using a partner-matching protocol. We chose this fixed matching protocol as many field settings are characterized by repeated interactions among the same agents. Furthermore, we chose a rather long repetition of 45 rounds because previous research has documented pronounced learning effects in group contests (Fallucchi et al. Reference Fallucchi, Renner and Sefton2013), and we were not only interested in contest behavior in the short-run, as most previous literature, but also in how heterogeneity and asymmetry affect competition in the long-run, once participants had sufficient time to learn.

The prize was 300 points, to be shared equally among the members of the winning group, irrespective of their individual efforts (i.e., for each member V = 100). In each period, each group member received an endowment of 100 tokens (w = 100), which they could either use for their own private consumption or use to exert effort $e_i\in \left(0,1,\dots ,100\right)$ to increase the probability of the group winning the contest. There were three types of players, low-ability, medium-ability, and high-ability, which differed in the effectiveness of their effort. Homogeneous groups consisted of three medium-ability players, and; heterogeneous groups consisted of one low-ability player, one medium-ability player, and one high-ability player.Footnote ⁷ Each token spent by a low-, medium-, or high-ability player yielded a contribution of one, two, or three to the group, respectively (i.e., ${\alpha }_{low,j}=1$ , ${\alpha }_{medium,j}=2,{\alpha }_{high,j}=3$ ). Player types were assigned randomly and remained constant throughout the experiment. Importantly, both homogeneous and heterogeneous groups have the same total endowment (300 tokens) and the same strategy space (contributions between 0 and 600). Cross matching these two group types yields two symmetric contests and one asymmetric contest, for a total of three experimental treatments: Symmetric-homogeneous, Symmetric-heterogeneous, and Asymmetric.

The Experiment was conducted at LabSi (University of Siena) using z-Tree (Fischbacher Reference Fischbacher2007). A total of 258 participants were recruited for 15 sessions (13 with 18 participants and 2 with 12 participants each) resulting in 15 independent observations in symmetric homogeneous contests, 14 in symmetric heterogeneous contests, and 14 in asymmetric contests. At the beginning of each session, written instructions were handed out to participants and read aloud by the experimenter. After that, and before the start of the experiment, participants had to correctly answer a set of control questions to ensure correct understanding of the incentives and structure of the game. At the end of each period, players received detailed information about the effort, contributions, and earnings of their group members (sorted in descending order of efforts). They were further informed about the total contribution of their own group and the opponent group, as well as the outcome of the contest (see Appendix B in Electronic Supplementary Material for a screenshot of the feedback screen). At the end of the experiment, participants were paid their earnings in cash. Sessions lasted between 50 and 60 min, and participants earned on average around €8.50.

Table 1 summarizes our experimental design as well as the theoretical predictions for each of the three conditions. Note that the predictions for parochial altruism (players try to maximize the payoff difference between both groups) slightly differ from those derived in Sect. 2, because effort levels were capped at each individual’s endowment (w =100), while in Sect. 2 there was no such constraint. While this cap does not change the prediction that in heterogeneous groups only the high-ability player should exert any effort, predicted group efforts in asymmetric contests are no longer the same (for both homogeneous and heterogeneous groups).

Table 1 Summary of treatments and equilibrium predictions

Treatments # Participants [Contests]	Effort [Group contributions] (winning probability)
	Selfish		Joint payoff maximization		Payoff difference
	Homogeneous group(s)	Heterogeneous group(s)	Homogeneous group(s)	Heterogeneous group(s)	Homogeneous group(s)	Heterogeneous group(s)
Symmetric homogeneous N =90 [15]	${\sum }_{i=1}^3{e}_{i,j}$ = 25	–	${\sum }_{i=1}^3{e}_{i,j}$ = 75	–	${\sum }_{i=1}^3{e}_{i,j}$ = 150	–
	[ $X_j=50$ ]	–	[ $X_j=150$ ]	–	[ $X_j=300$ ]	–
	( $p_j=0.5$ )	–	( $p_j=0.5$ )	–	( $p_j=0.5$ )	–
Symmetric heterogeneous N =84 [14]	–	$e_{low,j}=0$	–	$e_{low,j}=0$	–	$e_{low,j}=0$
		$e_{medium,j}=0$		$e_{medium,j}=0$		$e_{medium,j}=0$
		$e_{high,j}=25$		$e_{high,j}=75$		$e_{high,j}={100}^{\ast }$
	–	[ $X_j=75$ ]	–	[ $X_j=225$ ]	–	[ $X_j=300$ ]a
	–	( $p_j=0.5$ )	–	( $p_j=0.5$ )	–	( $p_j=0.5$ )
Asymmetric N =84 [14]	${\sum }_{i=1}^3{e}_{i,j}$ = 24	$e_{low,j}=0$	${\sum }_{i=1}^3{e}_{i,j}$ = 72	$e_{low,j}=0$	${\sum }_{i=1}^3{e}_{i,j}$ = 150a	$e_{low,j}=0$
		$e_{medium,j}=0$		$e_{medium,j}=0$		$e_{medium,j}=0$
		$e_{high,j}=24$		$e_{high,j}=72$		$e_{high,j}=100$ a
	[ $X_j=48$ ]	[ $X_j=72$ ]	[ $X_j=144$ ]	[ $X_j=216$ ]	[ $X_j=300$ ]a	[ $X_j=300$ ]a
	( $p_j=0.4$ )	( $p_j=0.6$ )	( $p_j=0.4$ )	( $p_j=0.6$ )	( $p_j=0.5$ )a	( $p_j=0.5$ )a

Homogenous groups consist of three players who all have a medium ability of ${\alpha }_{i,j}=2.$ .Heterogeneous groups consist of three players with abilities ${\alpha }_{low,j}=1$ , ${\alpha }_{medium,j}=2$ , and ${\alpha }_{high,j}=3$

^aThese predictions are altered by the fact that in our experiment players’ endowment was capped at w = 100

5 Equity norm model

The results of our experiment constitute a challenge for the theoretical predictions reviewed in Sect. 2 (see Table 1). While the joint payoff maximization model does better than the standard and the parochial altruism models in predicting the aggregate intensity of competition, all models fail to predict the comparative statics that we observe, particularly because they all predict very different behavior for top, medium, and low ability players in heterogeneous groups, and do not anticipate that competition will be more intense in asymmetric contests, particularly for homogeneous groups.

In this section, we discuss the predictions of an equity norm model that can reconcile our results with a theoretical rationale. The model is based on a variant of the inequity aversion model developed by Fehr and Schmidt (Reference Fehr and Schmidt1999), and on a more recent social norm compliance model by Fehr and Schurtenberger (Reference Fehr and Schurtenberger2018). These models have been used to explain empirical regularities in a variety of games such as cooperative behavior in social dilemma games, giving behavior in dictator games, and reward and punishment behavior in trust and ultimatum games (see e.g., Sobel Reference Sobel2005; Cooper and Kagel Reference Cooper and Kagel2016, for overviews), but, so far, have not been applied to group contests. Yet, our results, as well as evidence from previous experimental studies, indicate that other-regarding concerns may affect behavior in this setting as well. Specifically, as highlighted here as well as by, e.g., Abbink et al. (Reference Abbink, Brandts, Herrmann and Orzen2010), in group contests individuals condition their effort choices on those of other group members, even when efforts are perfect substitutes and overall effort levels are above the standard equilibrium prediction. Such coordination behavior is consistent with the notion that individuals have a preference for (partially) matching their peers’ efforts.Footnote ¹⁵

To capture this, the main assumption of our equity norm model is that individuals derive a disutility from investing less than the average of the other group members (which we interpret as a signal of normative behavior). Assuming risk neutrality, the expected utility of player i in group j can then be written as follows:

$U_{i,j}\left(x_{i,j},X_1,X_2\right)={\pi }_i-{\tau }_i·\text{max}\left({\overline{E}}_{-i,j}-e_{i,j},0\right)$

The first part of player i’s utility function is simply given by her expected monetary payoff, ${\pi }_i$ (see Sect. 2). The second term is the nonpecuniary part of the utility function, which depends on player i’s concern for equity, ${\tau }_i$ , and the difference between the others’ average effort and player i’s effort ( ${\overline{E}}_{-i,j}-e_{i,j}$ ). The equity parameter ${\tau }_i\ge 0$ captures player i’s marginal disutility from exerting one unit of effort less than the other group members. This disutility can be interpreted as, e.g., guilt from contributing less than others (as in Fehr and Schmidt Reference Fehr and Schmidt1999), or the psychological cost of deviating from the norm (as in Fehr and Schurtenberger Reference Fehr and Schurtenberger2018). Intuitively, the larger ${\tau }_i$ , the larger is i’s disutility from contributing less than her peers, and, hence, the larger her incentive to match the effort level of her group members. When ${\tau }_i=0$ there is no disutility from contributing less than the others; in this case players care only about their own monetary payoff.

While the empirical validity of this model is appealing, it is not straightforward to make behavioral predictions based on it. Even in much simpler settings than ours, the model above (or variants thereof) typically leads to a multiplicity of equilibria (see e.g. Fehr and Schmidt Reference Fehr and Schmidt1999). The context of group contests, as considered here, is no exception; The equity component increases the complexity of the best-response functions, generating a multiplicity of equilibria in both symmetric and asymmetric contests, making it hard to provide a straightforward analytical solution. We can, however, calculate numerical solutions. Table 4 shows the outcome of such a numerical exercise, making separate predictions for symmetric and asymmetric contests, as well as for homogeneous and heterogeneous groups. For our calculations, we use two different values for the equity parameter, one low ( $\tau$ = 0.2) and one high ( $\tau$ = 0.7).Footnote ¹⁶

Table 4 Possible levels of individual efforts depending on the degree of equity concern $\tau$

	$\tau =0.2$		$\tau =0.7$
	Homogeneous group(s)	Heterogeneous group(s)	Homogeneous group(s)	Heterogeneous group(s)
Symmetric contests	$e_{i,j}\in \left(9,10\right)$	$e_{low,j}=0$ $e_{med,j}=0$ $e_{high,j}=25$	$e_{i,j}\in \left(9,27\right)$	$e_{low,j}=13$ $e_{med,j}=13$ $e_{high,j}=13$
	$X_j\in \left(54,60\right)$	$X_j=75$	$X_j=\left(54,162\right)$	$X_j=78$
	$p_j=0.5$	$p_j=0.5$	$p_j=0.5$	$p_j=0.5$
Asymmetric contests	$e_{i,j}\in \left(8,10\right)$	$e_{low,j}=0$	$e_{i,j}\in \left(8,24\right)$	$e_{low,j}\in \left(11,13\right)$
		$e_{med,j}=0$		$e_{med,j}\in \left(11,13\right)$
		$e_{high,j}=\left(24,25\right)$		$e_{high,j}\in \left(11,13\right)$
	$X_j\in \left(48,60\right)$	$X_j\in \left(72,75\right)$	$X_j\in \left(54,144\right)$	$X_j\in [66,78$ ]
	$p_j\in \left(0.40,0.44\right)$	$p_j\in \left(0.56,0.60\right)$	$p_j\in \left(0.38,0.69\right)$	$p_j\in \left(0.31,0.62\right)$

We start by discussing the effects that equity concerns have on the distribution of labor within groups. As shown in Table 4, when all group members have the same ability (homogeneous groups), even a relatively low degree of equity concerns is sufficient to guarantee uniform efforts by all group members (recall that in the standard case ( $\tau$ = 0) any combination of efforts that leads to the unique aggregate equilibrium effort is an equilibrium). The reason is that a negative deviation from others’ average effort now creates a psychological cost due to equity concerns. When all group members have the same marginal material incentive to exert effort, this additional psychological cost is sufficient to solve the free rider problem within groups, as players prefer to match the effort of their peers in order to avoid the inequity costs.

When group members differ in their ability (heterogeneous groups), in contrast, the prediction for a low degree of the equity concerns ( $\tau$ = 0.2) is the same as in the standard case: only the group member with the highest ability exerts any effort, while the others free ride. If the concern for equity is sufficiently strong ( $\tau$ = 0.7), however, the model predicts uniform efforts even in heterogeneous groups, despite the players’ different abilities. The reason is that the different material incentives to provide effort (based on the players’ marginal productivity) are counterbalanced by the disutility that is generated when group members exert different levels of effort. But the latter effect has to be strong enough for players to match each other’s effort despite their different abilities.

We now turn to the question of how equity concerns affect the overall level of effort at the aggregate level. As can be seen in Table 4, the equity norm model predicts a vast range of equilibria, especially for high levels of $\tau$ and in particular with regard to the effort of homogeneous groups. For example, in symmetric contests between two homogeneous groups, for $\tau =0.7$ any (symmetric) effort profile from 9 to 27 is an equilibrium in pure strategies of the stage game. The intuition here is simple but challenging. For any of these symmetric effort profiles, the increased probability of winning associated with a unilateral increase in effort is not worth its cost, and a unilateral decrease in effort is penalized by both a lower probability of victory and the inequity aversion disutility. The multiple equilibria predicted for homogeneous groups simply acknowledge that this logic applies for very different effort levels, from a lower bound close to the standard Nash prediction (9), up to three times that amount (27). A similar large range is predicted for homogeneous groups in asymmetric contests, where individual equilibrium efforts range from 8 to 24.

For heterogeneous groups, the model makes a unique aggregate effort prediction when the contest is symmetric, but not when it is asymmetric. In the latter case, multiple equilibria exist as well, as the range of possible contest expenditures by the homogeneous groups is vast. In Fig. A3 in Electronic Supplementary Material in Appendix A we plot the reaction functions of homogeneous and heterogeneous groups for $\tau =0.2$ (panel (a)) and $\tau =0.7$ (panel (b)).

Given the multiplicity of equilibria, without further assumptions it is a priori unclear on which equilibrium participants will coordinate on, leaving us with a complex equilibrium selection problem. To identify which equilibria may be more prominent than other, in the following we consider the well-established refinement criteria of Pareto dominance. That is, while (by definition) in all equilibria no individual has an incentive to deviate unilaterally, we assume that a group would prefer to jointly deviate to the equilibrium that yields the highest payoff.

Applying this refinement criterion to our predictions above, we find that for symmetric contests the Pareto criterion drives players towards lower effort levels. The reason is that, because in symmetric contests both groups always exert the same effort, in equilibrium the winning probability is always equal to 50%. As a result, higher efforts only lead to higher costs but not to higher expected benefits, and, therefore, payoffs are highest when efforts are lowest. In asymmetric contests, in contrast, the same logic no longer applies, as efforts by homogeneous and heterogeneous groups may differ. Here, the maximum joint payoff for homogeneous groups is reached for effort levels of either 22 (when each member of the heterogeneous group contributes 13 or 11) or 23 (when the individual efforts of the opponent is 12). Therefore, this scenario allows for the homogeneous group to be more aggressive than their heterogeneous opponent, and have a higher probability of winning than what is predicted by the standard model.Footnote ¹⁷