3.1 Experimental design and comparative static predictions

Study 1 examined performance on the ball-catching task under piece-rate incentives. Each subject worked on the same ball-catching task for 36 periods. Each period lasted 60 s.Footnote ⁴ In each period one combination of prize-per-catch (either 10 or 20 tokens) and cost-per-click (0, 5 or 10 tokens) was used, giving six treatments that are varied within subjects (see Table 1). We chose a within-subject design to be able to observe reactions to changes in incentives at an individual level. The first 6 periods, one period of each treatment in random order, served as practice periods for participants to familiarize themselves with the task. Token earnings from these periods were not converted to cash. The following 30 periods, five periods of each treatment in completely randomized order (i.e., unblocked and randomized), were paid out for real. In all, 64 subjects participated in the experiment with average earnings of £13.80 for a session lasting about 1 h.Footnote ⁵

Table 1 Within-subject treatments in study 1

Treatment no.	1	2	3	4	5	6
Prize per catch (P)	10	10	10	20	20	20
Cost per click (C)	0	5	10	0	5	10

Given a particular piece-rate incentive, how often would subjects click? Basic piece-rate theory assumes that subjects trade off the costs and benefits of clicking in order to maximize expected utility. Assume that utility is increasing in the financial rewards, which are given by $PQ-Ce$ , where $Q$ is the number of catches and $e$ is the number of clicks, and assume the relationship between $Q$ and $e$ is given by $q=f\left(e,\epsilon \right)$ , where $f$ is a production function with $f^{{}^{\prime }}>0$ and $f^{{}^{\prime \prime }}<0$ , and $\epsilon$ is a random shock uncorrelated with the number of clicks. Given these assumptions the expected utility maximizing number of clicks satisfies:

(1) $e^{\ast }=f^{\prime }\left(\frac{C}{P}\right).$

This analysis posits a stochastic production function linking individual catches and clicks, and so an individual’s optimal number of clicks may vary from trial to trial as the marginal product of a click varies from trial to trial. This may reflect variability in the exact pattern of falling balls from trial to trial. We also recognize that the marginal product function might vary systematically across individuals. To make predictions at the aggregate level, we will estimate the production function (in Sect. 3.3) allowing for individual specific random effects and then use this estimate, evaluated at the mean of the random effects, along with our incentive parameters to predict the average optimal number of clicks. Before we proceed to this estimation, we discuss some features of the optimal number of clicks and how they relate to our experimental design.

The first feature to note is that the optimal number of clicks is homogeneous of degree zero in $C$ and $P$ . That is, a proportionate change in both input and output prices leaves the optimal number of clicks unchanged. This feature reflects the assumption that there are no other unobserved inputs or outputs associated with working on the task that generate cognitive or psychological costs or benefits. In fact we can think of two plausible types of unobservable inputs/outputs. First, output may be a function of cognitive effort as well as the number of clicks. For example, output may depend not just on how many clicks a subject makes, but also on how intelligently a subject uses her clicks. If the production function is given by $f\left(e,\kappa ,\epsilon \right)$ , where $\kappa$ represents cognitive effort, then $e^{\ast }$ will reflect a trade-off between $e$ and $\kappa$ . If all input and output prices were varied in proportion (including the “price” of $\kappa$ ), the optimal number of clicks would be unaffected. However, a proportionate change in just C and P would affect $e^{\ast }$ . If $e$ and $\kappa$ are substitute inputs then a proportionate increase in $C$ and $P$ will result in a decrease in $e^{\ast }$ as the subject substitutes more expensive clicking with more careful thinking. Second, subjects may enjoy additional psychological benefits from catching balls. For example, suppose that in addition to the pecuniary costs and benefits there is a non-monetary benefit from a catch, and suppose this psychological benefit is worth B money-units per catch. Again, proportionate changes in P, C and B would leave optimal effort unchanged, but a change in just P and C would not. Maximization of $(P+B)Q-Ce$ implies that a proportionate increase in $C$ and $P$ (holding B constant) will decrease $e^{\ast }$ .

Our experimental treatments allow us to test whether unobservable costs/benefits matter compared with induced effort costs in the ball-catching task. Our design includes two treatments that vary $C$ and $P$ while keeping the ratio $\frac{C}{P}$ constant (treatments 2 and 6 in Table 1). In the absence of unobserved costs/benefits, the distribution of clicks should be the same in these two treatments. The presence of unobserved costs/benefits could instead lead to systematic differences. Note that with this design the prediction that optimal clicking is homogeneous of degree zero in C and P can be tested without the need to derive the underlying production function, $f$ , since all that is needed is a comparison of the distributions of clicks between these two treatments.

A second feature of the optimal number of clicks is that, for positive costs of clicking, the optimal number of clicks decreases with the cost-prize ratio. Our design includes four further treatments that vary this ratio. Comparisons between treatments with different cost-prize ratios allow simple tests of the comparative static predictions of piece-rate theory, again without the need to estimate the production function. The variation in incentives across treatments serves an additional purpose: it allows us to recover a more accurate estimate of the underlying production function over a wide range of clicks.

A final feature of the optimal solution worth noting is that when the cost-per-click is zero optimal clicking is independent of P. In this case, since clicking is costless the individual’s payoff increases in the number of catches, and so regardless of the prize level the individual should simply catch as many balls as possible. Again, if there are psychological costs/benefits associated with the task this feature will not hold. Indeed, one could use the ball-catching task without material costs of clicking, basing comparative static predictions (e.g. that the number of catches will increase as the prize per catch increases) on psychological costs of effort. However, like in many existing real effort tasks, the ball-catching task without material costs might exhibit a “ceiling effect”, that is unresponsiveness of the number of clicks to varying prize incentives.Footnote ⁶ For this reason our design includes two treatments where the material cost of clicking is zero (treatments 1 and 4 in Table 1). These allow us to test whether there is a ceiling effect in the ball-catching task without induced clicking costs.

3.2 Comparative statics results

Figure 2 shows the distributions of clicks for each treatment, pooling over all subjects and periods. Clear differences between panels show that clicking behavior varies across incentive treatments. We begin by examining how these differences relate to the comparative static predictions based on optimal clicking (1).Footnote ⁷

Fig. 2 Distributions and kernel density distributions of the number of clicks in Study 1

Consider first the comparison between treatments 2 (P = 10, C = 5) and 6 (P = 20, C = 10). These treatments vary the financial stakes without altering the cost/prize ratio. The basic piece-rate theory prediction is that this will not have a systematic effect on clicking. As discussed in Sect. 3.1 however, unobserved psychological costs/benefits associated with the task will lead to systematic differences between the distributions of clicks in the two treatments. We find that the distributions of clicks are very similar, with average clicks of 18.6 under low-stakes and 18.4 under high stakes. Using a subject’s average clicks per treatment as the unit of observation, a Wilcoxon matched-pairs signed-ranks test (p = 0.880) finds no significant difference between treatments 2 and 6. Thus, we cannot reject the hypothesis that the average number of clicks is invariant to scaling up the financial stakes.

Next we ask whether variation in the cost-prize ratio affects clicking as predicted. Will increasing the cost-per-click, holding the prize-per-catch constant, reduce the number of clicks? And will the number of clicks depend on the prize level for a given clicking cost? First, we compare the top three panels of Fig. 2, where the prize is always 10. We observe a clear shift of the distribution of the number of clicks when moving across treatments with lowest to highest induced clicking costs. The average number of clicks falls from 58.7 to 18.6 to 8.8 as the cost-per-click increases from 0 to 5 to 10. Friedman tests for detecting systematic differences in matched subjects’ observations, using a subject’s average clicks per treatment as the unit of observations, show that the differences across treatments are highly significant (p < 0.001). A similar pattern is observed in the bottom three panels, where the prize is always 20, and again the differences are highly significant (p < 0.001).

Next, we perform two vertical comparisons between treatments 2 and 5 and between treatments 3 and 6. Holding the clicking costs constant, we find that a higher prize leads to higher number of clicks in both comparisons (Wilcoxon matched-pairs signed-ranks test: p < 0.001).

Finally, a comparison between treatments 1 and 4 offers an examination of whether a ceiling effect, observed in many real effort tasks, is present in the ball-catching task. In these treatments the cost-per-click is zero, but the prize-per-catch is 10 in treatment 1 and 20 in treatment 4. If there is no “real” psychological cost/benefit associated with working on the task, subjects should simply catch as many balls as possible and we should observe the same distribution of the number of clicks in these two treatments, thus exemplifying the typical ceiling effect. Comparing the distributions of clicks across the zero-cost treatments illustrated in Fig. 2 suggest that distributions are very similar. Average clicks are 57.8 in the low prize treatment and 58.7 in the high prize treatment. The closeness of average clicking between treatments 1 and 4 is statistically supported by a Wilcoxon matched-pairs signed-ranks test (p = 0.215), again using a subject’s average clicks per treatment as the unit of observation. The sharp contrast between the strong prize effect in treatments with induced effort costs and the absence of a prize effect in the zero-cost treatments illustrates that the ceiling effect can be avoided by incorporating financial costs in the ball-catching task.Footnote ⁸

In sum, as stated in the following result, we find that the comparative static predictions of basic piece-rate theory are borne out in the experimental data.

Result 1: The main comparative statics predictions are supported:

1. (1) Varying the financial stakes without altering the cost//prize ratio does not affect clicking behavior.

2. (2) Increasing the cost-per-click while keeping the prize-per-catch constant reduces the number of clicks; increasing the prize-per-catch while keeping the cost-per-click constant increases the number of clicks.

3. (3) When the cost-per-click is zero, the value of the prize-per-catch does not affect clicking behavior (ceiling effects).

Our next goal is to derive point predictions about the number of clicks in the various treatments and to compare them to the data. To be able to do so, we next estimate the production function, which we will then use to derive the point predictions.

3.3 The production function

Our empirical strategy for estimating the production function is to first specify a functional form by fitting a flexible functional form to the catches-clicks data using the full sample. Next, we estimate the production function, allowing for persistent as well as transitory unobserved individual effects and fixed period effects. We then test whether the production function is stable across periods and invariant to varying prize levels. We will also examine the stability of the production function across experimental sessions.

Figure 3 shows the observed catches-clicks data from all treatments along with a fitted production function based on a fractional polynomial regression.Footnote ⁹ The fitted production function has a clear concave shape, indicating a diminishing marginal rate of return to clicks. After a point, the production function is decreasing, indicating that there is a “technological ceiling” beyond which more clicking may actually lead to lower production levels. Observations in the decreasing range are predominantly from the treatments with a zero cost of clicking. As one of the main advantages of using the ball-catching task is precisely that clicking can be made costly, the decreasing part of the production function should be of little concern, since with positive clicking costs the number of clicks will be within the range where the empirical production function is concave and increasing.

Fig. 3 The relation between clicks and catches and the estimated production function. Note the first entry in (*, *) denotes the prize per catch and the second the cost per click. The fitted production functional form is given by $Q=9.507+5.568e^{0.5}-0.003e^2$ , where $Q$ denotes the number of catches and $e$ the number of clicks. The estimates of coefficients are from a fractional polynomial regression

Using the functional form suggested by the fractional polynomial regression, we move on to estimate the following random coefficients panel data model:

$Catche{s}_{i,r}={\beta }_0+{\beta }_{1}Click{s}_{i,r}^{0.5}+{\beta }_{2}Click{s}_{i,r}^2+\left({\delta }_r+{\omega }_i+u_{i,r}\right)Click{s}_{i,r}^{0.5},\left(2\right)$

where $Catche{s}_{i,r}$ and $Click{s}_{i,r}$ are respectively the number of catches and the number of clicks of subject $i$ in period $r$ . Period dummies ${\delta }_r$ (with the first period providing the omitted category), an individual random effect ${\omega }_i$ with mean zero and variance ${\sigma }_{\omega }^2$ , and a random error $u_{i,r}$ with mean zero and variance ${\sigma }_u^2$ are all assumed to be multiplicative with $Click{s}_{i,r}^{0.5}$ . Our specification of multiplicative heterogeneity and heteroskedasticity allows both persistent and transitory individual differences in the marginal product function, which could also vary across periods. The model thus predicts heterogeneity in clicking both across and within subjects.Footnote ¹⁰ All equations are estimated using maximum likelihood and estimates are reported in Table 2.Footnote ¹¹

Table 2 Panel data regressions for model (2) in study 1

Dep. var. catches	Coefficient estimates (SE)
	(1) full sample	(2) Prize = 10	(3) Prize = 20
Intercept	10.107*** (0.230)	10.477*** (0.308)	9.405*** (0.423)
Clicks0.5	5.495*** (0.132)	5.402*** (0.216)	5.660*** (0.171)
Clicks2	−0.003*** (0.000)	−0.003*** (0.001)	−0.003*** (0.000)
${\sigma }_{\omega }$	0.366*** (0.038)	0.352*** (0.045)	0.384*** (0.042)
σ u	0.796*** (0.013)	0.870*** (0.021)	0.694*** (0.016)
N	1905	946	959

All period dummies are included and insignificant except for period 2 using the full sample. *** p < 0.01

Columns (1), (2) and (3) in Table 2 reports the coefficient estimates for the full sample, the sub-sample with the prize of 10 and the sub-sample with the prize of 20 respectively. Note the similarity between the estimates of the parameters of the production function in all equations. The fitted production functions for the two sub-samples with different prizes are shown in Fig. B1 in the online supplementary materials: the two production functions almost coincide. Furthermore, we find that both persistent and transitory unobserved individual effects are statistically significant, and that the transitory unobservables account for more of the variation in clicking than the persistent individual differences.

To formally test whether the production function is invariant to different prize levels, we proceed to estimate an augmented model by adding interactions of the intercept, covariates $Click{s}^{0.5}$ and $Click{s}^2$ with a binary variable indicating whether the prize is 10–20. We then perform a likelihood ratio test of the null hypothesis that the coefficients on the interaction terms are all zero. We cannot reject the null hypothesis, indicating that the production function is stable across prize levels ( ${\chi }^2\left(3\right)$ = 4.70, p = 0.195).

To test the stability of the production function across experimental sessions, we estimate an augmented model by adding interactions of the intercept, $Click{s}^{0.5}$ and $Click{s}^2$ with a session dummy. We cannot reject the null hypothesis that the production function is invariant across sessions ( ${\chi }^2\left(3\right)$ = 2.60, p = 0.458). In fact the fitted production functions are barely distinguishable.Footnote ¹² We summarize these findings in the following result.

Result 2: The estimated production function, that is, the relationship between catches and clicks, is increasing in clicks and concave. The production function is stable across different prize levels as well as across different experimental sessions.

3.4 Comparing the predicted and actual number of clicks

With the estimated production function from model (2) and treatment parameters, we are ready to see how quantitative predictions on clicking perform.

Table 3 compares the predicted number of clicks that is derived from Eq. (1) given the estimated production function reported in column (1) of Table 2 and the cost-prize parameters, with the actual number of clicks for every treatment.Footnote ¹³ We find that average actual clicks are very similar to the predicted number of clicks in treatments 1, 2, 4 and 6 and near to, but statistically significantly different from, predicted clicks in treatments 3 and 5 (subjects seem to have over-clicked in treatments 3 and under-clicked in treatment 5).Footnote ¹⁴ Thus, overall, not only did they change their clicking behavior in the predicted direction when incentives changed, but also for given incentives their clicking was close, on average, to the profit maximizing level. The results are surprising given that subjects cannot know the production function a priori and therefore are in no position to calculate the optimal level of clicking. Nonetheless, on average, they behaved as if they knew the underlying structural parameters and responded to them optimally. These findings are summarized in our next result.

Table 3 Comparisons between the predicted number of clicks and the actual number of clicks

Treatment no.	1	2	3	4	5	6
Prize per catch (P)	10	10	10	20	20	20
Cost per click (C)	0	5	10	0	5	10
Predicted clicks	57.4	19.5	6.9	57.4	34.5	19.5
Av. actual clicks (SD)	57.8 (12.2)	18.6 (9.44)	8.8 (5.02)	58.7 (12.5)	30.9 (15.8)	18.4 (9.80)
p-value	0.723	0.367	0.000	0.276	0.040	0.294

p values are based on two-tailed one-sample t-tests using a subject’s average clicks per treatment as the unit of observation when testing against the predicted clicks

Result 3: The average number of clicks is close to the point prediction in all treatments but deviates statistically significantly from point predictions in treatments 3 and 5.

Figure 4 shows the predicted clicks and the distribution of actual clicks by combining categories whenever the treatments have the same predicted clicks. The distribution of clicks is approximately centered on the predicted clicks in each case, but shows variability in clicking for any given C/P ratio. As noted earlier, if the marginal product of clicking is subject to individual-specific and idiosyncratic shocks variability in clicking is to be expected.

Fig. 4 Distributions and kernel density distributions of the actual number of clicks and the predicted clicks. Note the vertical line in each panel represents the predicted number of clicks

In the next section, we provide further tests for the suitability of the ball-catching task by investigating its performance in well-known experimental settings that hitherto have typically used induced-value designs. This will be a further opportunity to see whether the ball-catching task produces behavior that is consistent with equilibrium comparative static or point predictions.

4.1 Team production

The understanding of free-riding incentives in team production is at the heart of contract theory and organizational economics (Holmstrom Reference Holmstrom1982). A standard experimental framework for studying team production is the voluntary contribution mechanism in which the socially desirable outcome is in conflict with individual free-riding incentives (see a recent survey in Chaudhuri (Reference Chaudhuri2011) in the context of public goods).

Our team production experiment was run over three sessions. One session included a team production (TP) treatment, in which four team members worked on the ball-catching task independently over 10 periods. The same four subjects played as a team for the entire 10 periods. For each ball caught, the subject contributed 20 tokens to team production while he/she had to bear the cost of clicking, with a cost per click of 5 tokens. At the end of each period, total team production was equally shared among the four team members. Each member’s earnings were determined by the share of the production net of the individual cost of clicking. Note that an individual’s marginal benefit from another catch is 5 tokens, whereas the marginal benefit accruing to the entire group is 20 tokens. The other two sessions included control treatments where individuals play 10 periods according to a simple individual piece-rate. In the first treatment (PR20) an individual receives a prize per catch of 20 tokens and incurs a cost per click of 5 tokens. The second treatment (PR5) has a prize per catch of 5 tokens and a cost per click of 5 tokens.

The amount of clicking in PR5 gives a “selfish” benchmark for the TP treatment, while clicking behavior in PR20 gives an “efficiency” benchmark. If a subject in the TP treatment is only concerned about her own private costs and benefits from clicking and catching, and equates marginal costs to marginal private benefits, she should click the same as in PR5. On the other hand, if she is concerned about total team production and equates marginal costs to marginal social benefits, then she should provide the same clicks as in PR20. Our hypothesis is that free-riding incentives would drive clicking towards the selfish benchmark, as is observed in many similar experiments using induced values (e.g., Nalbantian and Schotter (Reference Nalbantian and Schotter1997) and many public goods experiments using voluntary contribution mechanisms).

Figure 5 displays the average numbers (± 1 SEM) of clicks in the three treatments. The two horizontal lines represent the Nash predictions on optimal clicking levels in PR20 and PR5 respectively (using the estimated production function from Study 1 to compute the optimal clicking levels).

Fig. 5 Average clicks over time in team production

The figure shows a clear declining average number of clicks over time in TP. Average clicks decrease from 30 clicks to just above 17 clicks in the last period. By comparison, average clicks in PR20 decrease from 38 to 32 and in PR5 from 16 to 8 and thus is consistent with our findings in Study 1. Subjects in TP under-provide effort, relative to the efficiency benchmark, from the very first period and steadily decrease their clicking. Even in the final period, however, average clicks exceed the extreme selfishly optimal level. This empirical result is qualitatively similar to previous findings from experiments using induced values, such as Nalbantian and Schotter’s (Reference Nalbantian and Schotter1997) revenue sharing treatment and many public goods experiments, and also from some real effort experiments on team incentives (e.g., Corgnet et al. (Reference Corgnet, Hernán-González and Rassenti2015)).

4.2 Gift exchange

The gift exchange experiment (Fehr et al. Reference Fehr, Kirchsteiger and Riedl1993) examines reciprocal behavior between subjects in the role of firms and subjects in the role of workers. The gift exchange game using induced value techniques has been a workhorse model for many experimental investigations of issues in labor economics and beyond (see Gächter and Fehr Reference Gächter, Fehr, Bolle and Lehmann-Waffenschmidt2002; Charness and Kuhn Reference Charness, Kuhn, Ashenfelter and Card2011 for surveys).

Our version of the bilateral gift exchange experiment follows Gächter and Falk (Reference Gächter and Falk2002), except that they used induced values whereas we use the ball-catching task and slightly different parameters which we deem more suitable for the present purpose. In our experiment, in each period the firm offers a wage between 0 and 1000 tokens to the matched worker who then works on the ball-catching task. Each ball caught by the matched worker adds 50 tokens to the firm’s payoff. The worker’s payoff is the wage minus the cost of clicking. To compensate for possible losses, every firm and worker received 300 tokens at the beginning of each period. We implemented the gift exchange game in two sessions, one using a treatment with stranger matching over ten periods and the other using a treatment with partner matching over ten periods.

We made two key changes to the task compared with the version used in Study 1. First, we reduced the number of balls that could be caught within 60 s from 52 to 20 by increasing the time interval between falling balls. We made this change because we wanted to reduce the influence of random shocks as much as possible. The change makes it easy for a subject to catch every ball so that reciprocal behavior by workers could be reflected in their clicks as well as in their actual outputs. Second, the cost schedule was changed to a convex function in accordance with the parameters used in most gift exchange experiments. The cost for each click is depicted in Table 4. For example, the 1st and 2nd clicks cost 5 tokens each, the 3rd click cost 6 tokens, etc., and finally the last column with No. 30 + means that the 30th and any further clicks cost 12 tokens each. If, for example, the worker makes a total of three clicks she will incur a total cost of 5 + 5 + 6 = 16 tokens.

Table 4 The cost schedule in gift exchange

No. of click	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Cost	5	5	6	6	6	7	7	7	7	8	8	8	8	8	9
No. of click	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30+
Cost	9	9	9	9	10	10	10	10	10	11	11	11	11	11	12

Based on many gift exchange experiments and in particular the results by Gächter and Falk (Reference Gächter and Falk2002) and Falk et al. (Reference Falk, Gächter and Kovacs1999) who also compared partners and strangers in gift exchange, we expect gift exchange and predict that the reciprocal pattern is stronger with partner matching where it is possible to build up a reputation between a firm and a worker. Figure 6 confirms both predictions. It shows the relationship between outputs and wages on the upper panel and the relationship between clicks and wages on the lower panel. The data suggests a clear reciprocal pattern in both treatments and an even stronger pattern in the partner treatment whether we look at outputs or clicks.

Fig. 6 Reciprocal patterns in gift exchange. The upper panel shows the relationship between outputs and wages in both treatments and the lower panel displays the relationship between clicks and wages. The relationship in the stranger matching treatment is shown in the left panels and in the partner matching treatment in the right panels. The fitted lines are estimated from non-parametric lowess regressions with the bandwidth equal to 0.8

For formal statistical tests we estimate the following random effects panel data model for the number of clicks on the wage received:

$Clic{k}_{i,r}={\beta }_0+{\beta }_{1}wag{e}_{i,r}+{\omega }_i+{\delta }_r+u_{i,r}$

where ${\omega }_i$ is an individual-specific random effect identically and independently distributed over subjects with a variance ${\sigma }_{\omega }^2$ , ${\delta }_r$ denotes a period dummy for the $r$ ^th period (with the first period providing the omitted category), and $u_{i,r}$ is a disturbance term, assumed to be identically and independently distributed over subjects and periods with a variance ${\sigma }_u^2$ .

Table 5 reports the estimates for both treatments and also for the pooled sample with an additional interaction term. Consistent with gift exchange reciprocity and the graphical evidence from Fig. 6, workers in both treatments respond to higher wages by clicking more, and the number of clicks differs systematically from zero clicks. Furthermore, the strength of reciprocity is stronger with partners than strangers as the interaction term between the wage received and the treatment dummy in the column (3) is highly significant. These results in our ball-catching gift exchange experiment are qualitatively similar to findings from induced value experiments in Falk et al. (Reference Falk, Gächter and Kovacs1999) and Gächter and Falk (Reference Gächter and Falk2002). Our results from the stranger treatment are also consistent with an early real effort gift exchange experiment by Gneezy (Reference Gneezy2002) who used a maze solving task (without induced values) to measure worker’s performance, although Gneezy’s experiment was conducted in a one-shot setting.

Table 5 Random effects regressions for worker’s clicks in gift exchange

Dep. var.: clicks	Coefficient estimates (SD)
	(1) Stranger	(2) Partner	(3) Pooled
Wage	0.003** (0.001)	0.014*** (0.003)	0.004** (0.002)
Partner			1.154 (0.797)
Wage × partner			0.014*** (0.003)
Intercept	3.279*** (0.681)	3.746*** (1.444)	2.200** (0.952)
${\sigma }_{\omega }$	1.753	3.346	2.293
σ u	2.649	4.397	3.972
Hausman test for random versus fixed effects	df = 10 p = 1.000	df = 10 p = 0.956	df = 11 p = 0.984
N	160	160	320

All period dummies are statistically insignificant. Partner is a dummy which equals 1 if the treatment is the partner matching and 0 if the stranger matching. *** p < 0.01, ** p < 0.05

4.3 Tournament

Tournament incentive schemes, such as sales competitions and job promotions, are an important example of relative performance incentives (Lazear and Rosen Reference Lazear and Rosen1981). One early laboratory experiment by Bull et al. (Reference Bull, Schotter and Weigelt1987) found that tournament incentives indeed induced average efforts in line with theoretical predictions. But the variance of behavior was much larger under tournament incentives than under piece-rate incentives. Many induced value tournament experiments have been conducted since (see Dechenaux et al. (Reference Dechenaux, Kovenock and Sheremeta2014) for a survey).

In one session we included a simultaneous tournament treatment. The 32 subjects were randomly matched into pairs in a period and each pair competed in the ball-catching task for a prize worth 1200 tokens. The winner earned 1200 tokens net of any cost of clicking, whereas the loser received 200 tokens net of any cost of clicking. The cost per click was always 5 tokens. Each player’s probability of winning followed a piecewise linear success function (Che and Gale Reference Che and Gale2000; Gill and Prowse Reference Gill and Prowse2012): prob{win} = (own output – opponent’s output + 50)/100. This procedure was repeated over 10 periods.

We use this contest success function because it allows us to make a point prediction on the expected number of clicks. This is because the specified piecewise linear success function implies that an additional catch increases the probability of winning by 1/100. Thus, the marginal benefit of clicking is equal to the prize spread between the winner prize and the loser prize, 1000, multiplied by 1/100, multiplied by the marginal product of a click. The marginal cost of clicks is 5 tokens. Once again, we utilize the estimated production function from Study 1 to compute the optimal number of clicks, which turns out to be 20 clicks. Notice that while an additional catch increases earnings by 10 tokens in treatment 2 of Study 1, here an additional catch increases expected earnings by 10 tokens.

Figure 7 displays the average clicks (± 1 SEM) across all subjects and periods. We observe quick convergence to the predicted clicking level. The variance of clicks in tournament also appears to be larger than that observed in treatment 2 of Study 1. The standard deviation of clicks is around 12 in the former and 9.4 in the latter, perhaps reflecting the stochastic nature of the relationship between catches and earnings under tournament incentives.Footnote ¹⁶ Both results are qualitatively similar to previous findings from Bull, et al. (Reference Bull, Schotter and Weigelt1987).

Fig. 7 Average clicks over time in tournament

5.1 The ball-catching task on Amazon Mechanical Turk

As a third test of the versatility of the ball-catching task, we introduce an online version. This online version is programmed in PHP and has been designed to resemble the lab version as closely as possible.Footnote ¹⁷ The purpose of this section is to show the potential (and limitations) of using the ball-catching task in online experiments, which increasingly appear to be a valuable complement to experiments in the physical laboratory.

We ran the same experiment as in Study 1 on Amazon Mechanical Turk (MTurk; see the supplementary materials for instructions).Footnote ¹⁸ In total, we recruited 95 subjects from MTurk and 74 of them finished the task. Recruitment took around 10 min. Given the unusually long duration of the task (50 min), the 78% completion rate suggests that our promised payment is sufficiently attractive to most of the workers on MTurk. The average payment, including a $3 participation fee, was around $5.90, which was well above what most MTurk tasks offered. The average age was 35 years, ranging from 20 to 66 years; and 52% were male.

Paralleling the presentation of Study 1 results, Fig. 8 summarizes the distribution and the kernel densities of the number of clicks for each treatment. In general, we find that the comparative statics results are very similar to those in Study 1. A Wilcoxon signed-ranks test using a subject’s average clicks per treatment as the unit of observations suggests that homogeneity of degree zero also holds here: the difference in clicks between the two treatments with the same C/P ratio is not systematic (p = 0.309). The same is true for the difference in clicks between the two treatments with C = 0 (p = 0.832). Similarly, when comparing treatments with the same prize, Friedman tests indicate that comparative static predictions for different costs are supported (p < 0.001 in both comparisons).

Fig. 8 Distributions and kernel density distributions of the number of clicks in study 3

We observe some notable differences between the online and the lab version. The variance of clicking in each treatment for MTurkers appears to be higher than in the lab with student subjects. Moreover, we find that the production function is not invariant to prize levels, nor is it stable across sub-samples, thus preventing us from making meaningful point predictions.Footnote ¹⁹