Competition and Fatigue
?May 28, 2021
Vera Angelovaa, Thomas Giebeb,∗, Radosveta Ivanova-Stenzela a
Technische Universit¨at Berlin, Faculty of Economics and Management, Str. des 17. Juni 135, 10623 Berlin, Germany,
vera.angelova@tu-berlin.de, ivanova-stenzel@tu-berlin.de b
Linnaeus University, Department of Economics and Statistics, 351 95 V¨axj¨o, Sweden, thomas.giebe@lnu.se
Abstract
We study how subjects deal with fatigue in a sequence of tournaments that are linked through fatigue spillovers. Our contribution is threefold. First, we develop a model that allows us to predict the con-sequences of varying the severity of competition as well as the ease of recovery over time. Second, we test how fatigue spillovers affect subjects’ effort provision. Third, as we employ both, a chosen-effort and a real-effort task, we contribute to the methodological question of the consistency of insights obtained from both paradigms. Our experimental results suggest that subjects have difficulties in dealing with fatigue within a dynamic competitive environment. The model predicts strategic resting before and after a tournament with higher incentives. While an increase in incentives in one tournament does lead to higher effort in that tournament, we do not observe the expected strategic resting before and after that tournament. As a consequence, the increase in incentives does not yield the expected higher total effort. Keywords: incentives; fatigue; recovery; tournament; theory; experiment;
JEL codes: C72, C91, D91, J22, L2, M52
?
We thank Paul Schweinzer in particular and are grateful for discussions with Florian Englmaier, Dorothea K¨ubler,
Martin Siegel, participants at EEA ESEM in Manchester 2019, ESA 2018 in Berlin, Contests: Theory and Evidence 2018 in Norwich, The Lisbon Meetings in Game Theory and Applications 2018, the Annual Meeting of Swedish Economists 2018, the
UCSD-Rady Visitors Conference on Incentives and Behavior Change in Modica 2017, the department seminars at Universit´e
Laval, Linnaeus University, and LUISS University Rome. Financial support by Deutsche Forschungsgemeinschaft through CRC TRR 190 “Rationality and Competition” as well as support by the Berlin Centre for Consumer Policies (BCCP) are gratefully acknowledged. The experiments reported in this paper have been approved by the management of the Experimental
Laboratory at Technische Universit¨at Berlin.
∗
1. Introduction
Many work environments are characterized by the simultaneous presence of fatigue and competition. There is growing evidence that competitive pressure may cause stress and fatigue with potentially negative consequences. Fatigue impairs cognitive and physical performance and leads to inefficient work outcomes, inferior decision making, but also sick-leave, burnout or an early exit from the work force (Kant et al., 2003). In this paper, we study how fatigue affects behavior in dynamic competitive environments.
Designing incentive schemes for dynamic competitive environments requires understanding of how incentives and variations in incentives over time affect performance. It has been established that an increase in monetary incentives positively affects effort provision in the short run (Jenkins Jr et al., 1998). Over time, however, individuals often respond with strategic effort allocation, which can have unintended consequences (Asch, 1990, Mikl´os-Thal and Ullrich, 2015). Indeed, a few studies have found that an increase in incentives in dynamic environments might not result in an increase in performance, even though there is a clear behavioral response (Goette and Huffman, 2006, Angelova et al., 2018).
A natural explanation for why performance in dynamic settings might not respond to changes in incentives as expected is the presence of fatigue and the need for recovery. Fatigue is an established empirical phenomenon. Recent empirical work shows that a large part of the working population accu-mulates fatigue during the work week, and (partially) recovers over the weekend (˚Akerstedt et al., 2018). There is a small number of contributions that explicitly focus on effort choice in the presence of fatigue and recovery (Dragone, 2009, Baucells and Zhao, 2019). However, far too little is known about the role of fatigue and recovery in competitive work environments. Our study contributes to filling this gap.
Our approach is twofold. First, we propose a model in which work periods are linked through fatigue spillovers and investigate how a variation of the severity of competition and the ease of recovery affect effort provision.1 Second, in order to test how fatigue spillovers influence subjects’ effort provision, we conduct two experiments: a chosen-effort and a real-effort experiment. In the chosen-effort experiment, subjects’ effort choice implies clear monetary cost. In the real-effort experiment, subjects experience fatigue while working on a physical task. Both approaches have their justification, as effort allocation under fatigue is a decision problem with both cognitive and physical aspects. The chosen-effort task focusses subjects’ attention on the forward-looking planning or strategic side of the problem, while the real-effort task enables the physical and cognitive experience of fatigue during a longer time span in which subjects can instantaneously adjust their effort.
The starting point for our theoretical and experimental analysis is the rank-order tournament model of Lazear and Rosen (1981).2 We augment it in two dimensions. First, we allow for a dynamic competitive environment, i.e., a sequence of tournaments. A sequence of tournaments can be interpreted as consecutive work periods. These periods can be working days, weeks, or months, and employees (partially) recover
1
Our modelling approach follows the logic of the “effort recovery model” by Meijman and Mulder (1998) that is widely used in the psychology literature and has been validated in various studies (e.g. recently for breaks during the workday by Hunter and Wu (2016)). Broadly speaking, in this model, finite resources are depleted during effort exertion and have to be restored through sufficient recovery. In the presence of (qualitatively and quantitatively) sufficient breaks, the baseline resources are recovered, otherwise fatigue will accumulate, negatively affecting well-being and performance.
2
This (static) model has been widely used in the literature on management, organization, personnel economics, and experimental economics, e.g., Prendergast (1999), Connelly et al. (2014). See, e.g., Dechenaux et al. (2015) for an overview of standard tournament models and corresponding experimental studies.
from fatigue over night, on weekends, or during holidays.3 Second, we introduce fatigue and recovery through a non-time-separable cost function, in which marginal effort cost increases in previous periods’ effort, and decreases with resting. The model allows us to vary the quality of recovery between work periods.
One inherent feature of dynamic competitive environments is fluctuations in incentives over time. These can be natural, or deliberately introduced, as well as objective or perceived.4 In our model, we
allow for variations in incentives in some period, in order to analyze spillover effects to the work periods “before” and “after”. A short-term increase in incentives in one period may justify higher effort in that period, but also strategic resting before and after, as fatigue requires a sensible effort allocation over time. We provide a theoretical analysis of the model for an arbitrary number of agents and tournament periods. We show that a sequence of three tournaments is sufficient to generate the key comparative statics effects regarding incentives and fatigue. The benchmark symmetric equilibrium for this setup exhibits a V-shaped effort profile under constant incentives over time and an inverse V-shape if monetary incentives in the middle tournament are sufficiently higher than in the tournaments before and after. Higher(-powered) incentives in the middle tournament imply higher effort in that tournament. This is combined with strategic resting in the tournaments before and after. Overall, total effort, i.e., the sum of efforts in all tournaments, increases. For a given incentive scheme, if recovery is made harder, i.e., if fatigue becomes more severe, agents respond with lower effort in all tournaments resulting in a decrease in total effort.
In the chosen-effort experiment, subjects state an effort profile, taking into account the monetary incentive scheme and the non-time-separable effects of fatigue. We employ two tournament incentive schemes, the first with constant incentives over time, and the second with higher-powered incentives in the middle tournament. Corresponding to the theory, in the second incentive scheme, subjects choose higher effort in the middle tournament, and the observed average effort profiles exhibit the expected inverse-V shape. However, we do not observe the predicted strategic resting before and after the middle tournament. In the case of constant incentives over time, the pattern of behavior is not that clear although a substantial number of subjects’ effort profiles feature the predicted V-shape. Total effort does not positively respond to higher-powered incentives in the presence of fatigue. When we completely remove fatigue, total effort does increase as predicted. This suggests that dealing with the strategic implications of fatigue in complex environments is challenging. A variation of subjects’ ability to recover between tournaments reveals that, as predicted, subjects choose lower effort in all tournaments if there is less opportunity to recover between tournaments.
3
For example, consider competition for promotion. In each work period, employees choose effort in order to contribute to being positively evaluated by their supervisors, relative to their co-workers. Monetary incentives in the form of tournament prizes can be understood as contributions (“points”) towards positive performance evaluation and a favorable promotion decision in the (distant) future.
4
Promotions are often understood as prizes in a competition between workers that takes place over a longer period (Lazear and Rosen, 1981). During that time, it is natural that work periods differ in their relevance towards promotion. For instance, an employee might have an important presentation on Tuesday, and regular office days for the rest of the week. Then the employee’s performance on Tuesday will be more visible or count more towards promotion, or might be perceived as being more important than regular working days. As an example for deliberate variations in incentives, consider sales contests. The business literature recommends varying the severity of competition over time by introducing short-term sales contests on top of existing incentive schemes (Roberge, 2015).
In the real-effort experiment, subjects’ task is to move sliders to a certain position on the screen within a given amount of time. Thus, they experience physical and/or mental exertion. As in the chosen-effort experiment, we implement two different incentive schemes, featuring either constant incentives over time or higher incentives in the middle tournament. We also vary the length of the break between tournaments. Observed behavior is comparable to that of the chosen-effort experiment, though subjects’ response to the presence of fatigue spillovers is even more heterogeneous and, on average, not as pronounced. Similar to the chosen-effort experiment, subjects react to the increase in incentives in the middle tournament, but do not exhibit ample strategic resting before and after that tournament. The short-term increase in incentives does not lead to the expected increase in total performance. These results corroborate that subjects have difficulties in dealing with fatigue within a dynamic competitive environment and support the view that ensuring sufficient resting both before and after times of higher incentives should be seen as a managerial task, as one cannot rely on individuals to sufficiently respond to dynamic fatigue spillovers. In the chosen-effort task, subjects state their effort level being aware of the monetary cost of fatigue. In the real-effort task, they experience fatigue while continuously providing physical effort. Comparing the results obtained from both tasks, the findings point in a similar direction, but the tasks are not fully substitutable in our setup.
The paper proceeds as follows. The next section reviews the related literature. Section 3 presents the theoretical model. Section 4 discusses the design, hypotheses and results of the chosen-effort experiment, while section 5 covers the real-effort experiment. In section 6, we compare the main results obtained in the two different experimental environments. Finally, section 7 provides concluding remarks. The Supplementary Material contains proofs, additional data analysis, as well as instructions and further details on the lab experiments.
2. Related Literature
The economics literature has established that people respond to financial incentives (see, e.g., the meta study Jenkins Jr et al., 1998). Individuals are also able to strategically allocate effort over time as a reaction to dynamic incentives. Asch (1990) shows that Navy recruiters allocate their recruitment effort over time in response to the incentive scheme. Stiroh (2007) reports that NBA players’ performance goes down significantly after signing multi-year contracts. Fehr and Goette (2007) conduct a field experiment with Swiss bicycle messengers who face an anticipated but transitory increase in incentives. This wage increase results in an overall increase in labor supply, where, interestingly, the number of hours worked rises but the effort per hour falls. They show that a neoclassical model with non-separable utility as well as loss aversion are able to organize their empirical results. In particular, their model in which last period’s effort raises this period’s marginal disutility of effort could be interpreted as a model of fatigue. Goette and Huffman (2006) measure performance of bicycle messengers who initially face a lower piece rate, which is then increased once and forever. Although messengers initially react to the piece rate increase, their performance drops in the course of the day, such that their total performance does not differ between the two piece rates. Angelova et al. (2018) study a one-time increase in incentives in a sequence of identically incentivized tournaments in a real effort lab experiment. Performance is measured as the number of correctly positioned sliders on a computer screen. This study reports a behavioral response to the short-term increase in incentives with zero net effect on total performance. For tournament settings,
there is sufficient empirical evidence showing that participants respond to a change in the prize spreads (i.e., the differences between available prizes), e.g., Becker and Huselid (1992) on NASCAR racing data, and Harbring and L¨unser (2008) for experimental evidence.
Ryvkin (2011) presents a race-like contest model with binary effort choice, and tests it experimentally. In the model, a player gets an ‘asymmetry advantage’ in the form of a higher winning probability in the next stage contest only if the player chooses the low effort while the rival chooses the high effort. This modelling approach combines both a coordination and a fatigue problem. This is because on one hand, choosing higher effort than the other player results in a competitive disadvantage in the future (fatigue), but this effect does not arise if both players choose the same, high or low, effort (which is the equilibrium prediction). Thus, fatigue is not a result of high effort per se, but requires asymmetric play. This does not correspond to the intuitive view of fatigue in work environments that we seek to analyze in this paper.
We mention that there is a range of contributions in the tournament literature that assumes an effort budget which can be seen as a way of modelling fatigue (e.g. Amegashie et al., 2007). This modelling approach is also related to multi-battle contests or the so-called Blotto games (e.g., Roberson, 2006). We neither model effort as a budget or stock of resources nor do we assume a hard limit to effort choice. Fatigue is induced through the convexity of the cost function and wears off over time due to resting between work periods.
There are very few theoretical contributions that study the implications of fatigue and recovery for optimal (physical or cognitive) effort choice. Paarsch and Shearer (1997) develop a theoretical model of intertemporal effort choice that includes fatigue in the production as well as the utility function. They use panel data from tree planters in British Columbia in order to identify fatigue effects in workers who are incentivized with fixed wages. Dragone (2009) introduces a continuous-time principal-agent model in which a firm commits to a wage and an agent chooses effort, where effort is modelled similar to a renewable resource. Baikenova (2016) studies the effort choice of a forward-looking decision maker. Effort in one period causes fatigue in the next (or higher utility in the learning-by-doing version of the model). In addition, the decision maker gets (dis)utility from (negative) positive differences between the current and the previous period’s payoffs. She also surveys older economics literature that is related to the study of fatigue, and links the topic to the literature on time-inseparable utility in general (e.g., on the Equity Premium Puzzle). Baucells and Zhao (2019) present a continuous-time model in which fatigue decays over time. The decision maker faces constant incentives over time in a non-competitive situation. In contrast to these contributions, in our setting agents compete with each other. Furthermore, we derive and test comparative statics predictions between different incentive profiles over time.
Tournaments are wide-spread and well-studied as incentive systems within firms. Respectively, in-centive systems in firms are often seen as tournaments, see, e.g., the seminal Lazear and Rosen (1981) or Green and Stokey (1983) on rank-order tournaments and Prendergast (1999) on incentive systems in firms. Our theoretical model is based on an n-player version of Lazear and Rosen (1981). While our model is dynamic, similar static games have recently been analyzed in a more general way by, e.g., Bastani et al. (2021) and Kirkegaard (2020).
Finally, there is related literature that covers issues like the tradeoff between the benefits of breaks and the cognitive advantages of uninterrupted work (e.g., Eden, 2020), or involuntary work interruptions (e.g., Cai et al., 2018).
3. Theoretical model and predictions for the experiments
3.1. A Tournament Model with Fatigue and Recovery
We develop and analyze a dynamic model with fatigue and recovery based on the seminal static rank-order tournament model of Lazear and Rosen (1981). All computations and proofs can be found in the Supplementary Material.
In our model, each agent i ∈ {1, . . . , n}, n ≥ 2, simultaneously chooses unobservable costly effort et,i ≥ 0 in a sequence of tournaments t ∈ {1, . . . , T }. Introducing fatigue and recovery into this dynamic
environment intuitively implies that effort choice in one tournament affects that tournament as well as future tournaments.
First, we introduce agent i’s fatigue level St,i at the start of tournament t. Fatigue is a function of
all previous tournaments’ effort choices and the fatigue parameter F ∈ (0, 1). This parameter governs the speed or extent of recovery between tournaments and describes the persistence of past effort choices. We exclude the cases of full recovery, F = 0, and no recovery, F = 1, to simplify some of the arguments. However, F can be arbitrarily close to the excluded values. For F = 0 the model collapses to a sequence of independent standard tournaments.
In the model, fatigue evolves according to the process
St,i = 0 t = 1, F · (et−1,i+ St−1,i) t > 1. (1)
Second, we embed the fatigue level St,i in a standard convex (quadratic) effort cost function,
Ct,i:= k (et,i+ St,i)2, (2)
making effort cost non-time-separable. The parameter k > 0 can be used, e.g., to select suitable parame-ters for experimental tests.
Agent i’s cost function in tournament t, Ct,i, is a function of the current effort et,i and the current
fatigue level St,i. Fatigue is thus implemented as an increase in marginal cost that is caused by effort
in all previous tournaments, ∂2Ct,i/(∂et,i∂eτ,i) > 0 for all τ < t. Recovery is governed by the fatigue
parameter F . The share F of last period’s effort as well as last period’s fatigue is carried over to the following tournament.
Note the distinction between an agent’s (accumulated) fatigue level or tiredness St,i, and the fatigue
parameter F , which describes to which extent fatigue is carried over to the future. Intuitively, F can be understood as the quality of recovery.
Figure 1 illustrates the cost function for the case of T = 3 tournaments, with effort choice (e.g., time spent working) on the horizontal, and effort cost (tiredness) on the vertical axis. Intuitively, it shows how an agent becomes more tired during working days, and (partially) recovers between working days. A low value of F = 0.1 (left panel) implies nearly full recovery between periods, whereas F = 0.5 (right panel) illustrates substantial accumulating fatigue, e.g., during the work week. Figure 1 plots equal efforts in all tournaments, regardless of optimality.
t=1 t=2 t=3 0 e10 e20 e3 effort effort cost F=0.1 t=1 t=2 t=3 0 e10 e20 e3 effort effort cost F=0.5
Figure 1: Illustration of the cost function for fatigue parameters F = 0.1, F = 0.5.
the prize Lt. The winner of tournament t is the agent with the highest output, where output is the
sum of effort et,i and an individual i.i.d. noise term t,i that is distributed according to the cdf H. This
noise term is frequently interpreted as a measurement error or a productivity shock. Agent i’s output in tournament t exceeds that of a single rival player j if
et,i+ t,i > et,j+ t,j ⇐⇒ t,j < et,i− et,j + t,i. (3)
The probability of that event is H(et,i− et,j+ t,i). In order for agent i to win tournament t, i’s output
must exceed all n − 1 rivals’ output. This event has probability
Z ∞ −∞ n Y j=1,j6=i H(et,i− et,j+ x)H0(x)dx. (4)
As in the standard model of Lazear and Rosen (1981), agents are risk-neutral and fully rational. There is no information revealed between tournaments. Thus, there is no room for learning or updating one’s strategy between tournaments, and, consequently, the backwards induction solution coincides with the solution obtained from analyzing a one-shot interaction. Therefore, our setup is analyzed as a one-shot interaction, in which each agent’s decision is represented by a T -dimensional effort vector or profile (e1,i, . . . , eT ,i).
We denote the prize spread, i.e., the additional payoff of a winner, in tournament t by Pt= Wt− Lt.
Each agent i’s problem is to maximize total expected payoff from participation in all tournaments,
max (e1,i,...,eT ,i) T X t=1 Pt Z ∞ −∞ n Y j=1,j6=i
H(et,i− et,j+ x)H0(x)dx + Lt− Ct,i
. (5)
Given the symmetric setup of the game and following the literature on rank-order tournaments, we restrict attention to symmetric pure-strategy equilibria, denoted by e∗t,i = e∗t,j =: e∗t. Furthermore, we only consider equilibria with positive efforts. In fact, a pure-strategy equilibrium with positive efforts in all tournaments ceases to exist if the fatigue parameter F is sufficiently large, combined with sufficiently
large differences in prizes between tournaments. Upper bounds for the fatigue parameter will be specified further down, in Proposition 3.
We start from the baseline case of a constant prize spread over time, P1 = · · · = PT =: P . Then
equilibrium efforts are also constant over time, except for the first tournament (in which, by definition, there is no initial fatigue, S1,i := 0) and the last tournament (which does not cause fatigue for future
tournaments). In particular, equilibrium efforts are
e∗t = P B 2k(1 − F ) t = 1, P B 2k(1 − F ) 2 t = 2, . . . , T − 1, P B 2k(1 − F + F2) t = T, (6) where B := Z ∞ −∞ (n − 1)H(x)n−2(H0(x))2dx (7)
is the (symmetric equilibrium) marginal probability of winning contest t. By definition, S1∗ = 0, while
St∗= P B
2k F (1 − F ), t ∈ {2, . . . , T } (8)
in all other periods.
Now, consider a one-time change in incentives. In the following, we demonstrate that such a change affects only three consecutive tournaments.
In particular, consider a tournament sequence with T ≥ 5 tournaments, and a one-time increase in the prize spread from P to P + δ in some tournament tδ where tδ ∈ {3, . . . , T − 2}. The restriction
T ≥ 5 makes the presentation clearer, as it avoids that the incentive change affects the tournaments t = 1 or t = T . According to (6), efforts in these two periods are different from the constant effort over the rest of the time. Denoting the tournament with higher incentives by tδ, the three affected tournaments’
equilibrium efforts are
e∗t δ−1 = B 2k(P (1 − F ) 2− δF ) (9) e∗tδ = B 2k(P (1 − F ) 2+ δ(1 + F2)), t δ∈ {3, . . . , T − 2} (10) e∗t δ+1 = B 2k(P (1 − F ) 2− δF ) (11)
while all other tournaments’ efforts remain as in (6). As can be seen, there is an effort reduction before and after tournament tδ (strategic resting), and an increase in effort in tournament tδ.
Proposition 1. Consider a sequence of T ≥ 5 tournament periods with constant prize spread over time (P := P1 = · · · = PT). Let (e∗1,i, . . . , e∗T ,i) characterize a symmetric pure-strategy equilibrium with positive
efforts. Then, ceteris paribus, increasing the prize spread in a single period tδ ∈ {3, . . . , T − 2} from P
to P + δ affects equilibrium efforts only in the three periods tδ−1, tδ, and tδ+1. In particular, compared to
the case of constant prize spreads over time,
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● δ=0, F=0.1 δ=5, F=0.1 δ=0, F=0.5 δ=5, F=0.5 1 2 3 4 5 6 7 8 9 1 2 3 tournament effort
Figure 2: Equilibrium effort in a nine tournaments model. Solid lines: uniform prize spread. Dashed lines: larger prize spread in tournament t = 5. Parameters: Uniform distribution on [−3, 3], n = 2, k = 0.5, P = 20, δ ∈ {0, 5}, F ∈ {0.1, 0.5}.
while all other periods’ efforts are unaffected;
b) the fatigue level at the start of period tδ is smaller, fatigue at the start of period tδ+1 is larger, while
there is no effect on the fatigue level in any other period.
Qualitatively similar results can be obtained for T = 3 and T = 4 with the difference that tournaments 1 or T will be affected by the prize increase δ, which results in efforts different from those in (9)–(11).
In Figure 2 we graphically illustrate equilibrium efforts in a nine tournaments model with constant prize spread (solid lines) and a one-time increase in incentives in tournament 5 (dashed lines) for two different levels of the fatigue parameter F . We clearly see the strategic resting before and after tournament 5, whereas effort is unchanged in all other tournaments (t = 1, 2, 3, 7, 8, 9). Comparing the two solid, resp. dashed lines, we see that a larger fatigue parameter results in an overall lower effort.
In other words, agents respond to the larger prize in period tδ by reducing effort in the period before,
tδ−1, in order to start period tδ at a lower fatigue level. Then effort in tournament tδ is larger in response
to the larger prize spread. This leaves agents with a higher fatigue level at the start of tournament tδ+1.
In this tournament, agents reduce effort such that, at the end of that tournament, they return to the fatigue level that is obtained under equal prize spreads over time.
3.2. Theoretical Predictions for the experiments
The results from Proposition 1 have shown that a one-time change in incentives affects only three consecutive tournaments. Thus, in order to test the major comparative statics predictions of the model, we consider a three-tournament setup, T = 3.
F2e1,i+ F e2,i. The cost function is
Ct,i= k (et,i+ St,i)2 =
k(e1i)2 t = 1, k(F e1i+ e2i)2 t = 2, k(F2e1i+ F e2i+ e3i)2 t = 3. (12)
Allowing for different prize spreads over time, the symmetric pure-strategy equilibrium with positive efforts (interior solution) is characterized by
e∗1= B 2k(P1− P2F ) (13) e∗2= B 2k P2(1 + F 2) − (P 1+ P3)F (14) e∗3= B 2k P3(1 + F 2) − P 2F , (15)
where B is defined in (7). As can be seen, if we set F = 0, tournaments are independent and effort only depends on the given tournament’s prize, as in a standard tournament. Gradually introducing a small fatigue parameter, F > 0, tournaments become interdependent but efforts remain positive. In fact, (13)– (15) are always positive if the prize spread is constant over time. This can be different, however, if prize spreads differ too much between tournaments for a given fatigue parameter F . The comparative statics statements of Propostions 2 and 3 below need to be seen in this light. They, as well as our experimental tests, are concerned with this interior solution only.5
We continue with formally stating some comparative statics results with respect to changes in incen-tives and the fatigue parameter that we test experimentally. Intuitively, larger prize spreads Pt should
imply overall higher effort, wheras an increase in the fatigue parameter F should lead to overall reduced effort. While this general intuition is confirmed in Propositions 2 and 3, we also see that the individual tournament efforts e∗t can be differently affected by these changes.
Proposition 2 shows that changing incentives in the middle tournament 2 also affects effort in the tournaments before and after.
Proposition 2. Let (e∗1, e∗2, e∗3) characterize a symmetric pure-strategy equilibrium with positive efforts. Then, ceteris paribus,
a) an increase in the prize spread Pt of any tournament t ∈ {1, 2, 3} implies higher effort in that
tournament, e∗t, as well as higher total effort, e∗1+ e∗2+ e∗3.
b) an increase in the prize spread in tournament 2 implies lower efforts in tournaments 1 and 3 (strategic resting).
Next, we consider a discrete increase in the fatigue parameter from some value F ∈ (0, 1) to F + ∆F ∈ (0, 1). As indicated above, existence of the interior solution requires that F and ∆F are sufficiently small relative to the configuration of prize spreads (allowing for different prizes between the tournaments).
Correspondingly, Proposition 3 provides upper bounds on the fatigue parameter such that effort in tour-nament t ∈ {1, 2, 3} goes down with an increase in the fatigue parameter. Total effort unambiguously goes down if recovery becomes harder, i.e., if the fatigue parameter is increased.
Proposition 3. Let (e∗1, e∗2, e∗3) characterize a symmetric pure-strategy equilibrium with positive efforts. Then, ceteris paribus, increasing the fatigue parameter from F to F + ∆F leads to
a) lower efforts in tournament 1, in tournament 2 if 2F + ∆F < P1+P3
P2 , and in tournament 3 if 2F + ∆F < P2
P3,
b) lower total effort e∗1+ e∗2+ e∗3.
In the experiments, we restrict attention to two-player (n = 2) tournaments with noise terms being uniformly distributed on support [−a, a], and the prize structure (P1, P2, P3) = (P, P + δ, P ) with δ ≥ 0.
Comparing the cases δ = 0 and δ > 0 allows us to study how the severity of competition affects effort provision including spillovers between periods. We consider only small to moderate fatigue parameters (in terms of the model) such that all efforts e∗t are predicted to decrease as recovery is made harder (i.e., F increases, see Proposition 3). Fatigue levels that typically accumulate during the work week and from which employees recover over the weekend (˚Akerstedt et al., 2018, see also Figure 1) belong to this parameter range.
These assumptions imply that (7) becomes B = 2a1, and efforts (13)–(15) simplify to
e∗1= 1 4ak (P (1 − F ) − F δ) (16) e∗2= 1 4ak P (1 − F ) 2+ δ(1 + F2) (17) e∗3= 1 4ak P (1 − F + F 2) − F δ . (18)
As can be seen in (16) and (18), positive equilibrium efforts cease to exist for sufficiently large F and δ > 0. This is intuitive: it is optimal to concentrate effort in the second tournament if incentives are larger there and there is a strong fatigue effect.6 In order to reasonably test the theoretical predictions, we chose the model parameters for our chosen-effort experiment such that predicted efforts satisfy three criteria: (i) they constitute a unique pure-strategy equilibrium (note that there is no asymmetric equilibrium under these parameters), (ii) they are positive, in order to differentiate between a dropout decision in the experiment and a zero effort decision that corresponds to the prediction, (iii) they are sufficiently different between and within our experimental treatments.7
Figure 3 plots the equilibrium effort profiles for the model parameters that have been used in the chosen-effort experiment. In the left panel, the fatigue parameter has a low value, F = 0.1, and in the right panel a high value, F = 0.5. The gray lines in both panels depict the case of flat incentives over time (δ = 0). The black lines show the shape of the effort profiles in the case of larger incentives in the middle tournament (δ = 13).
6
A sufficient condition for positive efforts is δ/P < (1 − F )/F . For the parameters used in the chosen-effort experiment, the condition δ < P is sufficient. See the Supplementary Material for details.
● ● ● ● ● ● 30 27 30.33 27.83 48.88 28.17 δ=13, F=0.1 δ=0, F=0.1 1 2 3 10 20 30 40 50 tournament effort ● ● ● ● ● ● 16.67 8.33 25 5.83 35.42 14.17 δ=13, F=0.5 δ=0, F=0.5 1 2 3 10 20 30 40 50 tournament effort
Figure 3: Equilibrium effort profiles. Left: smaller fatigue parameter F = 0.1; Right: larger fatigue parameter F = 0.5; Gray line: flat incentives (δ = 0), Black line: larger incentives in tournament 2 (δ = 13).
Let us keep the fatigue parameter F fixed in the following discussion. Start with the case δ = 0 (flat incentives over time). In any symmetric pure-strategy equilibrium with positive efforts, the effort profile is V-shaped (e∗1> e∗2 < e∗3, see both gray lines in Figure 3): Effort in tournament 3 is highest, because this effort does not imply spillovers into the future. In that sense, it is the cheapest effort and, correspondingly, the other two efforts are smaller in order to induce low fatigue at the beginning of tournament 3. Effort in tournament 1 is the second highest, because at the beginning of tournament 3, fatigue from tournament 1 has decayed to a larger extent than that of tournament 2. Finally, effort in tournament 2 is the smallest, as that effort has the strongest fatigue spillover on tournament 3.
Now compare the flat incentive scheme (δ = 0), i.e., the gray lines in the left and right panels of Figure 3, to the scheme with higher incentives in the second tournament (δ > 0), i.e., the black lines in the left and right panels of Figure 3. As a direct response to the larger prize spread, optimal effort in tournament 2 is higher. However, due to fatigue spillovers, there need to be (optimal) adjustments in the other two tournaments as well. The effort in tournament 1 needs to be reduced in order to start tournament 2 at a lower fatigue level. The effort in tournament 3 will be reduced as well, because fatigue is high at the end of tournament 2, implying high effort cost in tournament 3.
4. Experiment I: Chosen-Effort Task
4.1. Design and Hypotheses
The design of the computerized laboratory experiment follows the setup of the model. In each round, two subjects simultaneously chose effort from the interval [0, 70], for each of three tournaments.8 We implemented incentive schemes (P, P +δ, P ), with prize spread P = 20, δ ∈ {0, 13}, and fatigue parameter F ∈ {0.1, 0.5}. Table 1 gives an overview of the four treatments. “Hump” denotes the incentive scheme with δ = 13, i.e., the larger prize spread in the second tournament, whereas “Flat” refers to δ = 0, i.e., same incentives in all three tournaments. The abbreviations “L” and “H” refer to the low (F = 0.1), respectively the high value (F = 0.5) of the fatigue parameter. Subjects were randomly assigned to treatments, and each subject participated in one treatment only. To explain the cost function, we provided a graph and
8All values were denoted in a fictitious currency ECU (Experimental Currency Unit). The cash conversion rate was 20
a cost calculator on each decision screen in addition to the verbal explanation and the mathematical formula in the instructions. We verified that subjects understood the instructions with a quiz.
The random number (i.e., the noise term t,i in the model) was drawn from a uniform distribution
with support [−30, 30], and the loser prize was 30. At the end of each round, subjects were informed about their own effort choice, cost, random number and output, as well as which prize they won and their total payoff, for each of the three tournaments. There was no feedback between tournaments, in order to avoid feedback effects (e.g., discouragement). Three rounds were randomly selected and paid in cash after the experiment.
Each treatment consisted of 30 rounds. In each round, subjects were matched with the same opponent. Between rounds, subjects were randomly rematched within a matching group of 8. Per treatment, we had 6 matching groups which qualify as independent observations. Altogether, 192 subjects participated in the experiment, i.e., 48 subjects per treatment.
At the end of the experiment, we also collected data on cognitive ability (using a 5-minute Raven test), individual risk preferences (using a lottery experiment), and some demographic characteristics. The experiment was conducted with students (37% female), mostly from economics, natural sciences, or engineering at Technische Universit¨at Berlin.
Low fatigue, F = 0.1 High fatigue, F = 0.5
Flat incentives, δ = 0 FlatL FlatH
Hump-shaped incentives, δ = 13 HumpL HumpH
Table 1: Treatments in the chosen-effort experiment
Our experimental analysis focusses on the comparative statics predictions stated in Propositions 2 and 3. According to Proposition 2, increasing the prize spread in tournament 2 only, from P to P + δ, leads to higher effort in tournament 2, and lower effort in tournaments 1 and 3. The effect on total effort is unambiguously positive.
However, the magnitude of these effects depends on the fatigue parameter: with a larger fatigue parameter, strategic resting becomes more pronounced while the increase in total effort becomes weaker. In particular, for the model parameters used in the experiment (see Figure 3), the predicted difference in effort for tournament 1 (resp. 3) rises from 2.17 (resp. 2.16) to 10.84 (resp. 10.83) with the increase of the fatigue parameter F from 0.1 to 0.5. At the same time, the predicted difference in total effort goes down from 17.55 to 5.42. Obviously, F = 0.5 (right panel of Figure 3) provides the appropriate environment for the test of “strategic resting”, whereas F = 0.1 (left panel of Figure 3) is better suited to test the effect on total effort. Thus, we can use the different levels of the fatigue parameter in the experiment to test the two effects in isolation.
Hypothesis 1. (i) For any fatigue parameter F , effort in the middle tournament is higher in the Hump-treatments than in the Flat-treatments.
(ii) For F = 0.5, efforts in the tournaments before and after are lower in the Hump-treatment than in the Flat-treatment.
Hypothesis 2. For a given incentive parameter δ, effort in each tournament and, hence, total effort, is higher in the L-treatments than in the H-treatments.
4.2. Results
4.2.1. Testing the Hypotheses
● ● ● ● ● ● 31.7 29.6 31.8 29.6 39.0 28.4 1 2 3 tournament 10 20 30 40 mean effort ● FlatL ● HumpL ● ● ● ● ● ● 21.9 19.6 23.9 18.3 27.9 21.1 1 2 3 tournament 10 20 30 40 mean effort ● FlatH ● HumpH
Figure 4: Average effort by treatment with 95% confidence intervals.
Figure 4 illustrates the average observed effort choice by treatment, whereas Figure 5 depicts average effort by tournament and treatment over time together with the theoretical predictions. In the Hump-treatments, average effort in tournament 2 is always above effort in the other two tournaments, and also above effort in tournament 2 in the respective Flat-treatments. For a given fatigue parameter, effort levels in tournaments 1 and 3 in the Hump-treatments are slightly lower than those in the Flat-treatments. Regardless of the incentive scheme, efforts in all tournaments are lower in the H-treatments compared to the L-treatments. Efforts in the L-treatments, except for tournament 2 in HumpL, are close to the theoretical benchmark.9 In all treatments, we observe a downward adjustment of effort over time with behavior stabilizing in the second half of the experiment.
9The comparison of the average efforts in the three tournaments to the point predictions provides mixed evidence. In the
L-treatments, average effort differs significantly from the theoretical prediction only in tournament 2 in HumpL. In the H-treatments, behavior is never in line with the theoretical prediction except for tournament 3 in FlatH (see the Supplementary Material).
0 5 10 15 20 25 30 35 40 45 50 Effort 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Round e 1 e2 e3 e 1* e2* e3* FlatL 0 5 10 15 20 25 30 35 40 45 50 Effort 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Round e 1 e2 e3 e 1* e2* e3* HumpL 0 5 10 15 20 25 30 35 40 45 50 Effort 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Round e 1 e2 e3 e 1* e2* e3* FlatH 0 5 10 15 20 25 30 35 40 45 50 Effort 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Round e 1 e2 e3 e 1* e2* e3* HumpH
Figure 5: Average effort and equilibrium prediction by tournament over time
To test our hypotheses, we need to compare effort in a given tournament across treatments. As the effort choices for tournament 1, 2, and 3 are dependent on each other, and thus correlated, we use multivariate regression analysis. We simultaneously regress effort in the three tournaments on dummy variables for the four treatments. Based on this regression, we run post-estimation tests that pairwise compare the estimated coefficients of the (appropriate) dummy variables for each effort. The unit of observation is an average for each subject and each tournament. We report the regression results in Panel A of Table 2, and the results of the post-estimation tests in panel B. Throughout the paper, we report results from two-sided tests. Note that our hypotheses are always directed, thus one-sided testing would also be justified.
The regression results confirm that for any given fatigue parameter, (i) subjects exert significantly more effort in tournament 2 in the Hump-treatments than in the Flat-treatments (FlatL vs. HumpL, resp. FlatH vs. HumpH, p ≤ 0.0147); (ii) in tournaments 1 and 3, effort in the Hump-treatments does not significantly differ from effort in the Flat-treatments. Hence, although we observe a tendency for strategic resting, it is not statistically significant for any value of the fatigue parameter. In all tournaments, subjects provide significantly less effort when resting is made harder (FlatL vs. FlatH, resp. HumpL vs. HumpH, p ≤ 0.0175). The results do not change when we instead use seemingly unrelated regressions, non-parametric Wilcoxon-Mann-Whitney tests, or run either analysis on the second half of the experiment
only.10 Results also do not change if we include our control variables for gender, cognitive ability, risk aversion, self-reported impulsiveness, age, and the number of semesters studied.
Panel A: Multivariate regression
e1 e2 e3
Reference category FlatL FlatL FlatL
FlatH-dummy -9.77*** -10.07*** -7.88** (3.36) (3.38) (3.02) [0.004] [0.003] [0.010] HumpH-dummy -13.38*** -1.76 -10.66*** (3.36) (3.38) (3.02) [0.000] [0.604] [0.001] HumpL-dummy -2.13 9.36*** -3.42 (3.36) (3.38) (3.02) [0.527] [0.006] [0.258] Constant 31.71*** 29.64*** 31.79*** (2.37) (2.39) (2.14) [0.000] [0.000] [0.000] Observations 192 192 192
Panel B: Post-estimation tests
FlatH vs. HumpH HumpL vs. HumpH
e1 0.2828 0.0010
e2 0.0147 0.0012
e3 0.3580 0.0175
Table 2: Panel A: Multivariate regression comparing effort in each tournament across treatments. Standard erros in paren-theses, p-values in brackets (*** p<0.01, ** p<0.05, * p<0.1). Panel B: Two-sided p-values from post-estimation tests comparing the regression coefficients.
Table 3 displays the results of linear regressions comparing total effort across treatments. All specifi-cations are generalized least squares (GLS) models with random effects at the subject level and clustered standard errors at the matching group level to account for correlated decisions by the same subject and within the same matching group. Specification (1) contains treatment dummies whose coefficients can either directly be compared to the reference category FlatL, or via post-estimation tests (p-values are reported in the last two rows of Table 3). Specification (2) contains our control variables (none of which is significant) and also round as an additional control variable.
Specification (1) shows that total effort does neither significantly differ between FlatL and HumpL (p = 0.678) nor between FlatH and HumpH (p = 0.7806). However, total effort in FlatH is significantly lower than in FlatL (p = 0.001). Similarly, total effort in HumpH is significantly lower than total effort in HumpL (p = 0.0002). These results do not change when we include the controls (specification (2)).
Additionally, we compare pairwise the distributions of total effort across treatments using Kolmogorov-Smirnov tests. We do not find a significant difference between the Hump-treatments and the Flat-treatments for any given fatigue parameter (p ≥ 0.368). In contrast, distributions differ significantly for a given incentive scheme between the L- and H-treatments (p ≤ 0.002).
10
Dependent variable: (1) (2) Total effort
Reference category FlatL FlatL
Constant 93.14*** 121.54*** (6.451) (29.389) [0.000] [0.000] FlatH -27.72*** -26.19*** (8.225) (8.198) [0.001] [0.001] HumpH -25.80*** -24.32*** (7.937) (8.053) [0.001] [0.003] HumpL 3.810 5.605 (9.190) (8.777) [0.678] [0.523] Round -0.930*** (0.122) [0.000] Controls No Yes Observations 5,760 5,760 Number of subjects 192 192 βF latH = βHumpH 0.7806 0.7990 βHumpL = βHumpH 0.0002 0.0002
Table 3: GLS regressions with random effects at the subject level and clustered standard errors at the matching group level. Standard errors in parentheses, p-values in brackets (*** p<0.01, ** p<0.05, * p<0.1). Two-sided p-values of post-estimation tests (last two rows). Controls: gender, Raven score, risk aversion, self-reported impulsiveness, age, and the number of semesters.
Result 1. (i) Compared to a flat incentive scheme, stronger incentives in the middle tournament lead to higher effort in that tournament, supporting H1(i). In the tournaments before and after, there is no significant strategic resting. In the presence of fatigue, introducing higher-powered incentives in one tournament does not lead to higher total effort. Thus, we reject H1(ii) and H1(iii).
(ii) For a given incentive scheme, effort in all tournaments, as well as total effort, is lower when resting is made harder. Both results are consistent with H2.
4.2.2. Heterogeneity in Behavior: the Shape of Effort Profiles and Dropout Behavior
The shape of effort profiles. A prominent feature of our model are the predicted V-shaped effort profiles in the Flat-treatments and the inverse V-shaped effort profiles in the Hump-treatments. Figure 4 provides support for the predicted pattern of behavior on the aggregate level. In fact, we observe the expected shape of the average effort profiles almost throughout the experiment: in 27 (FlatL), 28 (FlatH), 30 (HumpL), and 29 (HumpH) out of the 30 rounds. Figure 6 illustrates behavior at the individual level. For each subject, it plots the average change in effort between tournaments 1 and 2, ∆12= effort2−effort1
(black dots), and tournaments 2 and 3, ∆23= effort3−effort2(hollow circles). Subjects are ranked by ∆12
on the horizontal axis, while the vertical axis measures the size of ∆12 and ∆23. To formally investigate
for each treatment separately. The dependent variable in the regressions is the average ∆12 and ∆23 by
subject. We report the p-values from these regressions together with the sign of the observed change and its frequency (the regressions can be found in the Supplementary Material). Results do not change if, instead of taking the average ∆12and ∆23 per individual as our unit of observation, we consider ∆12 and
∆23 for each subject in all thirty rounds and additionally control for individual random effects.
The predicted V-shape for the Flat-treatments prescribes ∆12< 0, i.e., the black dots in the upper half
of Figure 6 should be in the negative range of the vertical axis, and ∆23> 0, i.e., the hollow circles should
be in the positive range. As evident from the upper two panels of Figure 6, for the majority of subjects in both Flat-treatments, ∆12is indeed negative (71% in FlatL and 60% in FlatH) and ∆23is indeed positive
(60% in FlatL and 58% in FlatH). The decrease in effort from tournament 1 to tournament 2 is never significant (p=0.207 in FlatL and p=0.487 in FlatH), while the increase in effort from tournament 2 to 3 is significant only for FlatH, p = 0.071 (p = 0.487 in FlatL). In total, we observe the V-shape, i.e., both ∆12 < 0 and ∆23> 0, for 42% of subjects in FlatL, and 35% in FlatH. Note that this is the most
frequently observed shape in the two Flat-treatments.11
In the Hump-treatments, the prediction is reversed, i.e., the inverse V-shape implies ∆12 > 0 and
∆23 < 0. In the two lower panels of Figure 6 (Hump-treatments), the majority of dots is indeed in the
positive range of the vertical axis, i.e., ∆12> 0 for 73% in HumpL and 71% in HumpH, and most circles
are indeed in the negative range, i.e., ∆23 < 0 for 83% in HumpL and 67% in HumpH. The change
in effort from tournament 1 to tournament 2 is significantly positive in both, HumpL (p = 0.008) and HumpH (p = 0.010). The change from tournament 2 to tournament 3 is significantly negative for both, HumpL (p = 0.001) and HumpH (p = 0.051). In total, we observe the inverse-V shape for the majority of subjects’ effort profiles in both Hump-treatments, i.e., 69% of subjects in HumpL, and 54% in HumpH.
To sum up, despite substantial heterogeneity in behavior, the majority of the observed changes in effort between tournaments has the predicted sign. In addition, the V-shape in the Flat-treatments, and, respectively, the inverse-V shape in the Hump-treatments is the most frequently observed shape within each treatment. Finally, in the Hump-treatments the predicted changes in behavior are always significant. All of this suggests that the theory can rationalize the observed shapes of individual effort profiles to a large extent.
-60 -40 -20 0 20 40 60 0 4 8 12 16 20 24 28 32 36 40 44 48 Subject FlatL -60 -40 -20 0 20 40 60 0 4 8 12 16 20 24 28 32 36 40 44 48 Subject FlatH -60 -40 -20 0 20 40 60 0 4 8 12 16 20 24 28 32 36 40 44 48 Subject HumpL -60 -40 -20 0 20 40 60 0 4 8 12 16 20 24 28 32 36 40 44 48 Subject HumpH
Figure 6: Heterogeneity in behavior in the chosen-effort task: the black dots represent the average change in effort from
tournament 1 to 2 (∆12), and the hollow circles the average change in effort from tournament 2 to 3 (∆23), for each of 48
subjects in each treatment.
Dropout behavior. Recall that, given the model parameters for our experiment, we should observe strictly positive efforts in all tournaments. In fact, 73 subjects choose positive efforts in all tournaments throughout the experiment, whereas 119 subjects drop out, i.e., choose a zero effort level in at least one tournament of their 30 effort profiles. Nobody drops out in all tournaments and all rounds.12
Dropout behavior seems to be driven by the size of the fatigue parameter rather than the incentive scheme. Using Kolmogorov-Smirnov tests, we reject the equality of distributions of subjects’ dropout frequency between FlatL and FlatH (p = 0.018) and between HumpL and HumpH (p = 0.018). In contrast, we cannot reject the equality of distributions between FlatL and HumpL (p = 0.960), and between FlatH and HumpH (p = 0.997).
We investigate to what extent the observed dropout behavior, i.e., a choice of a zero effort, is respon-sible for the results related to our hypotheses. Table 4 describes the relative frequency of effort profiles containing only positive effort choices (first row), only zero effort choices (second row), and effort profiles containing either one or two zero effort choices (third row). The majority of observations is located in the first row, i.e., most effort profiles contain only positive effort choices. We repeat our analysis considering two subsamples: (1) excluding all (0,0,0) profiles, i.e., the second row, and (2) additionally excluding all profiles that consist of one or two zero efforts, i.e. the second and third rows of Table 4. We confirm our findings established in Result 1. As the number of strictly positive effort profiles significantly decreases over time (p = 0.0000 based on logit regressions with random effects on the subject level, correlating the number of strictly positive effort profiles and period), we conduct the same analysis on the second half of
12
the experiment only, i.e., when behavior has stabilized. Results do not change.13
Rounds 1-30 (16-30) 1-30 (16-30) 1-30 (16-30) 1-30 (16-30)
(e1, e2, e3) FlatL FlatH HumpL HumpH
(> 0, > 0, > 0) 77% (75%) 56% (50%) 81% (76%) 56% (49%)
(0, 0, 0) 9% (10%) 14% (19%) 13% (17%) 10%(12%)
else 14% (15%) 30% (31%) 6% (8%) 34% (39%)
Table 4: Distribution of effort profiles across treatments (rounds 1 to 30: N = 1440, rounds 16 to 30: N = 720 per treatment)
Result 2. (i) There is strong heterogeneity in behavior between the subjects. In the Flat-treatments, a substantial number of subjects’ effort profiles features a V-shape. In the Hump-treatments, average effort profiles are inverse-V-shaped.
(ii) Dropout behavior seems to be driven by the size of the fatigue parameter rather than the incentive scheme. Dropouts do not affect the results related to our hypotheses.
4.3. Robustness Check
One important finding so far is that total effort does not positively respond to higher-powered incen-tives. To check whether this and all other experimental results depend on the presence of fatigue, we conducted two treatments without fatigue, Flat0 and Hump0, as a robustness check. We also use them to check for feedback effects. These treatments are identical to those investigated so far with two exceptions: (i) there is no fatigue, i.e., F = 0, and (ii) in half of the 12 matching groups, subjects received feedback after each tournament rather than after each sequence of three tournaments.
Theoretically, these feedback differences should be irrelevant. To verify this, we run multivariate regressions for Flat0 and Hump0 separately. The dependent variable effort in tournaments 1, 2, and 3 is regressed on a feedback-dummy taking the value of one if there was feedback after each tournament. This dummy is never significant – for none of the tournaments under any incentive scheme. Therefore, we conclude that feedback does not play a role, and pool the data for the Flat0-treatments with and without feedback, and similarly for the Hump0-treatments.
Figure 7 provides the descriptives for Flat0 and Hump0. The theoretical prediction for this case follows directly if we set F = 0 in our model, corresponding to three strategically independent static tournaments. As predicted by the theory, subjects exert significantly more effort in tournament 2 in Hump0 than in Flat0 (p = 0.000, Table 5). Strategic resting is not an issue in the absence of fatigue. Indeed, there is no significant difference in observed average effort in tournaments 1 and 3 between Hump0 and Flat0 (p = 0.645, resp. p = 0.319, Table 5). Moreover, in line with the theory, the switch from the Flat to the Hump incentive scheme leads to a significant increase in total effort (p = 0.021, Table 6). Results do not change if the analysis is conducted on the second half of the experiment only or with linear random effects regressions with clustered standard errors on the matching group level that we run separately for effort in each tournament.
Result 3. In the absence of fatigue, total effort in Hump is higher than in Flat.
13
Altogether, our results suggest that it is due to the presence of fatigue that total effort is not higher in the Hump-treatments than in the Flat-treatments.
● ● ● ● ● ● 29.9 28.6 29.3 31.4 43.1 32.5 1 2 3 tournament 20 30 40 mean effort ● Flat0 ● Hump0
Figure 7: Average effort with 95% confidence intervals in the treatments without fatigue.
e1 e2 e3
Reference category Flat0 Flat0 Flat0
Hump0-dummy 1.485 14.50*** 3.249 (3.210) (3.065) (3.244) [0.645] [0.000] [0.319] Constant 29.87*** 28.65*** 29.25*** (2.270) (2.167) (2.294) [0.000] [0.000] [0.000] Observations 96 96 96
Table 5: Multivariate regression comparing effort in each tournament across treatments. Standard errors in parentheses, p-values in brackets (*** p<0.01, ** p<0.05, * p<0.1).
Dependent variable: (1) (2)
Total effort
Reference category Flat0 Flat0
Constant 87.77*** 24.98 (4.387) (25.32) [0.000] [0.324] Hump0 19.23** 19.74*** (8.352) (7.264) [0.021] [0.007] Round -0.539*** (0.201) [0.007] Controls No Yes Observations 2,880 2,880 Subjects 96 96
Table 6: GLS regressions with random effects at the subject level and clustered standard errors at the matching group level. Standard errors in parentheses, p-values in brackets (*** p<0.01, ** p<0.05, * p<0.1). Controls: gender, Raven score, risk aversion, self-reported impulsiveness, age, and the number of semesters.
5. Experiment II: Real-Effort Task
5.1. Design
The real-effort task is a modified version of the slider task developed by Gill and Prowse (2012). Subjects had to move sliders from position 0 exactly to the middle of 100 possible integer positions (i.e., 50) within a limited time, using the computer mouse only.14 This slider task has the advantage that effort cost is actual disutility of the combined experience of effort and fatigue. It therefore allows us to capture the idea of fatigue being a cognitive and physical experience. As subjects confirmed in our post-experimental questionnaire, the task is perceived as tiring and tedious work (participants mentioned, e.g., tiredness, decreasing ability to concentrate, need for a break, or that the task was physically demanding). This experiment consisted of two identical parts. In each part, subjects participated in the same sequence of three two-player tournaments, corresponding to the three-period version of our theoretical model and the chosen-effort experiment. Part one was conducted in order to allow subject to get familiar with the task and to experience the strategic environment. After the completion of part one, subjects were informed that part two would be a repetition of part one.
In each tournament, subjects had eight minutes to work on the slider task. Every two minutes a new screen with 48 sliders appeared in order to ensure that subjects could not run out of work. At the end of a tournament, a feedback screen containing the subject’s total number of correctly positioned sliders in that tournament was shown for 5 seconds, followed by a new screen reminding subjects of the incentive scheme that applied to the next tournament (15 seconds). During a tournament, subjects received live feedback on their own performance through an on-screen counter of correctly positioned sliders. As in the chosen-effort experiment, there was no feedback about other subjects’ performance, nor about earnings or who won or lost a tournament.
As in the chosen-effort experiment, we implemented two different incentive profiles (P, P + δ, P ) with P = 200, δ = 0 in the “Flat” treatments, and δ = 600 in the “Hump” treatments.15 Subjects were informed about the winner and loser prizes for all three tournaments before starting with the task.
We also vary the length of the break between tournaments, in order to manipulate the ease of re-covery. Contrary to the “ShortBreak” treatments, subjects had an additional 100 seconds to rest in the “LongBreak” treatments. The additional time ensures that subjects’ physical work on the slider task is interrupted for a longer period of time. In order to avoid influencing subject’s behavior during this time in any way, the additional 100 seconds break between tournaments was not mentioned in the instructions (i.e., instructions were identical for a given incentive scheme). Table 7 gives an overview of the treatments. Earnings were determined by randomly pairing subjects after each tournament. The subject who correctly placed more sliders in that tournament received the winner prize and the other one the loser prize. When tied, each subject received half of the sum of the winner and loser prize. We paid for all tournaments and all parts. Additionally, we paid 1 ECU=0.005 EUR per correctly positioned slider.
The real-effort experiment was also conducted at Technische Universit¨at Berlin. The data for the LongBreak treatments were collected in February 2020. For the ShortBreak treatments, we used data
14Following Gill and Prowse (2012), the computer mice were set to the lowest possible speed, the wheels were switched off,
and access to the keyboards was prevented. 15
from Angelova et al. (2018) collected in the same lab. As in the chosen-effort experiment, participants in the real-effort experiment were students (39% female), mostly from economics, natural sciences, or engineering. Altogether, we observed 241 subjects (FlatLongBreak: 38, HumpLongBreak: 46, FlatShort-Break: 79, HumpShortFlatShort-Break: 78).
Long break (120 seconds) Short break (20 seconds)
Flat incentives, δ = 0 FlatLongBreak FlatShortBreak
Hump-shaped incentives, δ = 600 HumpLongBreak HumpShortBreak
Table 7: Treatments in the real-effort experiment
5.2. Behavioral Predictions
Testing our model predictions with a real-effort experiment is challenging. First, the process of fatigue and recovery is not observable, and the levels of fatigue and the corresponding spillover effects between periods are not exactly measurable. Second, results may depend on the characteristics of the real-effort task. Third, actual effort is not observable and hence not measurable. Instead, we have to rely on subjects’ performance as a proxy for effort. Nevertheless, a real-effort experiment can provide an environment for decision making under actual fatigue. Thus, it allows a more realistic, albeit much less controlled, test of the model.
A distinct feature of our model with fatigue spillovers is the predicted V-shaped effort profile under flat incentives. This prediction is robust to the distribution of the noise term, the prize spread P , and the size of the fatigue parameter F . Hence, in a sequence of three tournaments with fatigue spillovers and flat incentives in the real-effort experiment, we expect to observe a V-shaped effort profile. A second distinct prediction of the model is an inverse V-shaped effort profile provided that the value of δ in the incentive scheme (P, P + δ, P ) is sufficiently large, given the other model parameters. In the model, values of δ ≥ P/3 are sufficient for obtaining the inverse V-shape, for all values of the fatigue parameter F . In our real-effort experiment, we used a value of δ = 3P , and therefore it seems reasonable to expect inverse V-shaped effort profiles in the Hump-treatments.
In addition to these distinct shapes, we can check how well the qualitative predictions from Propo-sitions 2 and 3 organize the data in the real-effort environment. Hence, for a given length of the break, we expect a higher effort in tournament 2 and a lower effort in tournaments 1, and 3 (strategic resting), as well as a higher total effort in the Hump-treatments than in the Flat-treatments. Furthermore, if the manipulation of the break length has any measurable effect, it should be a decrease in effort in each tournament and hence lower total effort with a shorter break between tournaments.
5.3. Results
In our analysis, a subject’s number of correctly positioned sliders in a given tournament is referred to as “performance”. A subject’s (vector of) measured performance in tournaments 1, 2, and 3 (in each part) is called “performance profile”. Note that, due to the absence of feedback between tournaments, each subject yields an independent observation.
The upper panel of Figure 8 presents the comparison between the average performance in both Long-Break treatments, whereas the lower panel plots the average performance profiles observed in the Short-Break treatments. Note that between the time the LongShort-Break- and the ShortShort-Break-treatments were
conducted, the equipment in the laboratory was changed, in particular, the resolution of the computer mice. This explains the observed substantial difference in subjects’ performance between the two data sets. In fact, in the very first tournament, the average performance in the LongBreak-Treatments differs from the average performance in the ShortBreak-Treatments by a factor of 1.83 (note that we used the same instructions in both cases).
In part one, performance increases from one tournament to the next in all treatments. There are no significant differences across treatments for a given fatigue-regime for any given tournament in this part. Subjects seem to predominantly learn in this part and they do so at a similar pace in all treatments, presumably reaching at the end of part one a comparable level of familiarity with the task and level of fatigue for a given fatigue-regime.16
In part two, average performance in tournament 2 in Hump is always above average performance in tournaments 1 and 3. It is also above average performance in tournament 2 in the respective Flat-treatments. In FlatLongBreak, average performance always increases. In FlatShortBreak, we also observe an increase from tournament 2 to 3, while average performance in tournaments 1 and 2 is similar.
In the LongBreak-treatments, the differences are insignificant for all tournaments (p ≥ 0.2234, t-test). Compared to FlatShortBreak, performance in HumpShortBreak is significantly higher in tournament 2 (p = 0.0762, t-test), whereas performance in tournaments 1 and 3 is not significantly different (p ≥ 0.3195, t-test). Using a difference-in-differences approach, we compare the performance differences between two consecutive tournaments across treatments with the same length of break. The average change in perfor-mance from tournament 2 to 3 in both Flat-treatments is positive and (marginally) significantly different from the average change in the Hump-treatments, which is negative (p = 0.1003 in the LongBreak-treatments, and p = 0.0073 in the ShortBreak-LongBreak-treatments, t-test). OLS regression of total performance on treatment dummies (see Table 8) do not reveal any significant differences between the Hump- and the Flat-treatment regardless of the break length.17 We also perform Kolmogorov-Smirnov tests to compare the distribution of total performance between the treatments. We cannot reject the null hypothesis of equality of distributions for both fatigue-regimes (p ≥ 0.382).
To sum up, the observed performance levels in part two exhibit a reaction to the higher incentives in tournament 2, i.e., higher effort in that tournament (ShortBreak-treatments) and a tendency for slacking in tournament 3 in Hump compared to Flat. However, introducing higher-powered incentives in the middle tournament does neither trigger sufficient strategic resting nor lead to higher total effort.
16
For more details of the statistical analysis, see the Supplementary Material. 17
● ● ● ● ● ● ● ● ● ● ● ● Part 1 Part 2 43.8 48.4 51.9 43.1 50.2 53.0 50.4 53.8 55.8 54.3 56.7 54.6 1 2 3 1 2 3 tournament 30 40 50 60 mean performance ● FlatLongBreak ● HumpLongBreak ● ● ● ● ● ● ● ● ● ● ● ● Part 1 Part 2 79.8 87.9 90.1 79.4 88.0 92.9 92.3 92.0 96.5 95.8 98.6 97.4 1 2 3 1 2 3 tournament 70 80 90 100 110 mean performance ● FlatShortBreak ● HumpShortBreak
Figure 8: Average performance by treatment with 95% confidence intervals.
Dependent variable: (1) (2)
Total performance in part two
Reference category FlatLongBreak FlatShortBreak
HumpLongBreak 5.727 (9.399) [0.544] HumpShortBreak 10.90 (10.12) [0.283] Constant 159.9*** 280.8*** (6.955) (7.133) [0.000] [0.000] Observations 84 157
-80 -60 -40 -20 0 20 40 60 80 100 0 4 8 12 16 20 24 28 32 36 40 44 Subject FlatLongBreak -80 -60 -40 -20 0 20 40 60 80 100 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 Subject FlatShortBreak -80 -60 -40 -20 0 20 40 60 80 100 0 4 8 12 16 20 24 28 32 36 40 44 Subject HumpLongBreak -80 -60 -40 -20 0 20 40 60 80 100 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 Subject HumpShortBreak
Figure 9: Heterogeneity in behavior in the real-effort task. The black dots represent the change in performance from
tournament 1 to 2 (∆12), and the hollow circles the change in performance from tournament 2 to 3 (∆23), for each of 48
subjects in each treatment in part two.
Next, we investigate whether the individual performance profiles feature the expected shapes. The analysis will focus on part two where subjects have become familiar with the task and have experienced the strategic environment. Figure 9 (the equivalent of Figure 6 for the chosen-effort task) plots changes in performance from one tournament to the next at the individual level, i.e., the differences ∆12 =
performance2−performance1 (black dots) and ∆23= performance3−performance2 (hollow circles). Recall
that in the Flat-treatments (Hump-treatments), the black dots should be in the negative (positive) range of the vertical axis, and the hollow circles in the positive (negative) range.
Behavior at the individual level is very heterogeneous, even more so than in the chosen-effort experi-ment. Only 34% of subjects in each, FlatLongBreak and FlatShortBreak, exhibit a negative ∆12 (upper
half of Figure 9). In fact, in FlatLongBreak, the change in performance from tournament 1 to 2 is positive and significant (p = 0.0618, t-test), while in FlatShortBreak performance in tournaments 1 and 2 is not significantly different (p = 0.8832, t-test). However, for the majority of subjects, ∆23 is positive (58%
in FlatLongBreak and 61% in FlatShortBreak) and there is a significant increase in performance from tournament 2 to 3 (p = 0.0904 for FlatLongBreak, p = 0.0163 in FlatShortBreak, t-test). Overall, we observe that 26% (28%) of performance profiles in FlatLongBreak (FlatShortBreak) feature the V-shape, i.e. ∆12< 0 and ∆23> 0.
In the Hump-treatments (lower half of Figure 9), the majority of subjects exhibit ∆12 > 0: 52% in
HumpLongBreak and 63% in HumpShortBreak. Indeed, the change in performance from tournament 1 to 2 is (marginally) significant for both treatments (p = 0.1054 for HumpLongBreak and p = 0.0015 for HumpShortBreak, t-test). For around half of the subjects, ∆23 is negative, 48% in HumpLongBreak and
