Tournaments and Piece Rates Revisited:
A Theoretical and Experimental Study of Premium Incentives
Werner Guth Rene Levnsky Kerstin Pully Ori Weiselz
June 22, 2010
Abstract
Tournaments represent an increasingly important component of organizational compensation systems. While prior research focused on xed-prize tournaments, i.e., on tournaments where the prize or prize sum to be awarded is set in advance, we introduce a new type of tournament into the literature: premium incentives. While premium incentives, just like xed-prize tournaments, are based on relative performance, the prize to be awarded is not set in advance but is a function of the rm's success: the prize is high if the rm is successful and low if it is not successful. Relying on a simple model of cost minimization, we are able to show that premium incentives outperform xed-prize tournaments as well as piece rates. Our theoretical result is qualitatively conrmed by a controlled laboratory experiment and has important practical implications for the design of organizational incentive systems.
JEL Classication: C72, C91, J33
Keywords: Tournaments, Incentives, Economic experiments
Max Planck Institute for Economics, Kahlaische Strasse 10, 07745 Jena, Germany. yEberhard Karls Universitat Tubingen, Faculty of Economics and Business Administration, Nauklerstrasse
47, 72074 Tubingen, Germany. zThe Hebrew University, Center for the Study of Rationality, Giv'at Ram, Jerusalem 91904, Israel.
1
1 Introduction and Motivation
As documented by Orrison et al. (2004) or Bothner et al. (2007), tournament incentives have developed into an increasingly important component of compensation systems; they are `pervasive in organizations' (Casas-Arce and Martnez-Jerez, 2009). Unlike piece rates, which are awarded according to absolute performance, tournament incentives are awarded according to relative performance. While the most prominent examples of tournament incentives in the organizational practice are standard promotion tournaments, any organizational incentive system awarding a (typically predetermined) prize to a predened number of top performing employees actually qualies as a tournament compensation system. Unlike Ryvkin and Ortmann (2008), we focus on the incentive properties of tournament compensation systems and not on their predictive power. Starting with Lazear and Rosen
(1981), the incentive properties of tournament compensation systems have repeatedly been analyzed in the literature (e.g., O'Keee et al., 1984; McLaughlin, 1988). However, the literature has so far principally focused on `xed-prize' tournament incentives in the sense that the prizes to be awarded are set in advance such that their actual size is not in uenced by employee performance or rm success. A prominent exception to this are
Japanese bonus tournaments (so-called J-tournaments; see Krakel, 2003), where the size of the prize to be awarded to a single contestant depends on his/her relative performance
(see, e.g., Endo, 1984). But as the size of the total wage sum or bonus bill is xed in advance (e.g., Kanemoto and MacLeod, 1984), even in the J-tournament, the prize sum in a given year is predetermined and independent of the rm's success in that year.
As long as rm performance can be easily assessed in advance, a system of predened tournament prizes that have to be paid out even if the company does badly may not pose a severe problem. However, if rm performance cannot be assessed in advance (e.g., because the rm nds itself in an uncertain economic environment), a predetermined tournament prize sum may well exceed what the rm can actually aord to pay. Accordingly, in the
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organizational practice, one should expect that the tournament prize sum depends on rm performance (as is, e.g., the case in any system of gain sharing). Besides the lack of ability to adequately forecast rm protability, there is one further argument that speaks in favor of variable instead of predetermined tournament prizes: while xed-prize tournaments carry the risk of horizontal collusion between the contestants, premium incentives clearly do not. If contestants decide for collusive behavior in a variable prize sum tournament, this will reduce the tournament prize and hence the attractiveness of collusive behavior.
In our study, we depart from the existing literature by studying tournaments that award a variable prize whose size is based on rm performance and refer to this incentive scheme as `premium incentives.' Just as in a typical tournament, the premium incentive is only awarded to the top performer(s), but in contrast to the xed prize in a typical tournament, its size is not predetermined but depends on rm performance. In order to study the comparative advantages of premium incentives, we allow for employee compensation to be composed of (a) a piece rate based on absolute performance, (b) a predetermined tournament prize awarded on the basis of relative performance (xed-prize tournament incentive), and (c) a variable tournament prize awarded on the basis of relative performance, the size of which depends on rm success (variable-prize tournament incentive or
`premium incentive').
Theoretically, we rely on a simple cost-minimization approach in search of the optimal combination of the three incentive types studied. Our analysis shows that premium incentives are more cost-eective than piece rates and xed-prize tournament incentives, the two types of incentives that have typically been studied in the literature so far.
We also test our theoretical implications by confronting them with data from a controlled laboratory experiment.1 Our data qualitatively supports the theoretical propositions:
1For reasons of data availability, empirical studies on tournaments often rely either on laboratory evidence (see, e.g., Green and Stokey, 1983; Bull et al., 1987; Orrison et al., 2004) or on data from sports
(e.g., Ehrernberg and Bognanno, 1990; Becker and Huselid, 1992; Bothner et al., 2007; Kaplan and Garstka,
2001).
3 despite agents not choosing the theoretically predicted eort level, premium incentives not only theoretically, but also empirically prove to be the prot maximizing type of incentive
(as compared to the conventional alternatives), and accordingly, most principals decide in favor of premium incentives when designing an incentive system for their subordinates.
In sum, our results suggest to foster the use of variable-prize tournaments or `premium incentives' in the organizational practice as well as to encourage future research on the subject. The remainder of this paper is organized as follows. In Section 2, we specify the model and derive its theoretical result. Section 3 presents the experimental design. The data is analyzed and discussed in Section 4. Section 5 concludes.
2 Theoretical Analysis
Assume two competing agents, 1 and 2; who may represent individual employees or teams in the same rm. Both agents i = 1; 2 must choose an eort level xi  x with x  0 :
Each eort xi generates output yi = xi + "i subject to some noise term "i 2 ["; "] with
" < " ; x+"  0 ; and density '(
) with all probability mass at interval ["; "] : According to such an iid-case, the noise levels "1 and "2 are stochastically independent and identically distributed and ensure the nonnegativity of the agents' output. With ci(xi) denoting the e ort costs of the agents, the payos of i = 1; 2 with competitor j 6= i can be dened as ui(xi; xj ; "i; "j) =
8< :
!yi ci(xi) if yi = xi + "i  yj = xj + "j
!yi +  + (xi + "i + xj + "j) c(xi) otherwise,
(1)
where ! 2 R+ is a piece rate,  2 R+ is a xed premium, and  2 R+ determines how sensitively the premium (xi +"i +xj +"j) depends on rm performance (represented by the sum of the two agents' output levels xi + "i + xj + "j ). After agents independently choose their eort levels, x1 and x2, and nature has determined "1 and "2 (according to
4
the density '(
)), the ranking of the individual (observable) output levels y1 = x1+"1 and y2 = x2 + "2 determines which agent receives  + (y1 + y2):
To test the model experimentally we restrict ourselves to a specic form of '(
) and ci(
) :
In particular, we assume the noise terms "i to be uniformly distributed2 on [0; "] and eort costs ci(xi) =

2x2i to be quadratic with > 0 for i = 1; 2:
Finally, in our analysis we assume that the participants encounter the tournament repeatedly.
Therefore, it makes sense to rely on common(ly known) risk neutrality, and the corresponding expected payos of agents i 2 f1; 2g are
Eui = !(xi + "
2 ) + 1
"
Z"
0
h(xi; xj ; "j) d"j

2 x2i ; where h(xi; xj ; "j) =
8>>>>><
>>>>>:
0 if xi  xj + "j " ;
1
"
R"
0
[ + (xi + "i + xj + "j )] d"i if xi  xj + "j ;
1
"
R"
xj+"jxi
[ + (xi + "i + xj + "j )] d"i otherwise.
For  
"
2 there is no nite best reply xi to xj : For the case  <
"
2 the unique equilibrium e orts (in the sense of mutually best replies) are xi =
2 + "(3 + 2!)
2
" 4 for i 2 f1; 2g: (2)
Since the principal can implement a three-dimensional incentive scheme (; ; !) , the natural problem of cost minimization arises, i.e., nding the cheapest incentive scheme
2Some tournament models (e.g., Lazear and Rosen, 1981) rely on normally distributed noise "i for the sake of mathematical convenience. This violation of economic nonnegativity constraints is easily sustainable in theory. However, it is dicult to test such models experimentally without deception.
5
that yields a given positive (expected) output y = y1 + y2 : Formally, this is equivalent to nding a combination (; ; !) that minimizes the linear costs of the principal
C;;!(y) =  + ( + !)y; with y = 2+"(3+2!)

"2 + " :
Considering the linearity of the problem, it suces to compare the costs of the three
`corner' incentive schemes:
(; ; !) =
8>>><
>>>:


"
2 ; 0; 0


0;
"
2+3" ; 0


0; 0;

2

; where  = y " : The corresponding costs are
C =

"
2
; C =
( + ")
"
2 + 3"
; and C! =

( + ")
2
:
One can easily see that for all nonnegative , ", and
, it holds that C  C  C! : Thus, from the principal's point of view, the premium incentive  is superior to both the piece rate ! and the xed-prize tournament  :
Proposition 1 The premium incentive  is more cost-eective than the xed prize  which, in turn, is more cost-eective than the piece rate ! :
The result is by no means obvious. When individual output yi = xi + "i is assumed to be readily observable and agents are assumed to be risk neutral, one would typically expect piece rate incentives only, which would avoid strategic interaction of agents. Our analysis shows, however, that an employer can gain by implementing tournament competition, and most preferably by using premium incentives { even in those cases where there is no potential for economizing on measurement costs (output is readily observable) and where there is no benet to eliminating common shocks (risk neutral agents).
6
While our theoretical analysis neglects the potential of collusive behavior by agents (collusion is most likely for  incentives as agents can share  even when investing very little e ort) as well as the fact that competition may endanger feelings of corporate identity
(which would seem to be least endangered by ! as agents do not strategically interact at all), both eects may play a role in the experiment.
3 Experimental Design
The experiment was run at the computer laboratory of the Max Planck Institute of Economics with 112 participants, mostly undergraduates of the University of Jena, enrolled in dierent elds. Each of the four computerized experimental sessions (28 participants per session) lasted about 100 minutes. Earnings, including a show-up fee of €2.50, ranged from €4.60 to €17.44. Upon arrival, each participant was seated in a visually isolated cubicle.
Detailed written and oral instructions (to establish common knowledge) explained the rules and payos of the game and were followed by a control questionnaire. After the experiment, participants were paid individually and left the laboratory separately. In each session the 28 participants were randomly partitioned into four 7-person groups. In each group, one participant was assigned the role of `principal' and 6 were assigned to be `agents'. The 7-person groups remained constant throughout the experiment, and this was made known to the participants. However, they did not know which of the other participants was in their group. Each session was divided into three phases with 10 rounds each. Each of the three phases began with principals selecting one of 15 available combinations of the three incentives ; ; and ! (displayed in Table 1). This choice set the stage for the interactions of `their' three pairs of agents in the following ten rounds (phase). The principal's payo from each agent-pair-interaction was up(x1; x2; "1; "2) = (20!)(x1+
"1+x2+"2) ; based on the interpretation that principals can sell whatever `their' agents produce at a constant price of 20 per unit and that they must reward the agents according
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Contract Equilibrium Empirical ! A-eort P-prot A-prot Frequency A-eort P-prot
0 8 0 20 960 320 16 18.08 910
200 6 0 20 920 340 7 18.95 877
400 4 0 20 880 360 5 19.20 872
600 2 0 20 840 380 1 16.27 784
800 0 0 20 800 400 1 16.05 656
0 6 5 20 720 440 6 18.16 690
200 4 5 20 680 460 4 21.44 696
400 2 5 20 640 480 1 19.18 606
600 0 5 20 600 500 0 { {
0 4 10 20 480 560 3 21.66 500
200 2 10 20 440 580 3 21.80 489
400 0 10 20 400 600 0 { {
0 2 15 20 240 680 1 23.35 273
200 0 15 20 200 700 0 { {
0 0 20 20 0 800 0 { {
Table 1: Available contracts { equilibrium predictions and empirical results to the chosen contract.
After learning which incentive scheme (, , !) had been implemented by the principal, each agent was randomly paired, in each round, with one of the other ve agents in the same group. Agents were not told with whom they were randomly paired. Agent i 2 f1; 2g could choose the eort level xi 2 [0; 30]; knowing that the random variable "i takes values
"i 2 [0; 40] according to the constant density having all probability mass at interval [0; 40] and that both cost functions ci(xi); i 2 f1; 2g are given by x2i
2 ; i.e., = 1.
After each round, principals were informed about the production (yi) of each agent, the joint revenue (20(y1 + y2)), their cost ((! + )(yi + y2) + ), and their prot (up). This information remained on the principal's screen, and information from the next round was appended to it. Thus, after each phase of 10 successive rounds, the principal had information about all tournaments (three pairs of agents in ten rounds). Additionally, after
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each of the three phases, principals received feedback information on average production, revenues, costs, and prots across all thirty tournaments that took place in the phase.
To capture the fact that the organizational structure of a rm is rarely re-designed, and that such changes, when they are made, are mostly made in the light of much experience with a status quo structure, the principals in the experiment could change the incentive scheme only twice. In both cases they completed thirty tournaments before they were allowed to re-structure the incentive scheme.
After each round, agents were informed about both production levels (y1, y2) and the , and ! components of their earnings before they were randomly rematched with another agent of the same group except for the last round of the phase when they knew that the principal could change the incentive scheme.
All fteen available contracts yield the same individual equilibrium eort of twenty. If both agents play optimally, i.e., both choose twenty, they suer when the principal switches to a superior contract from her point of view. In view of such con icting interests agents may be inclined to collude rather than to compete. Obviously, the strongest incentive for collusion by agents is oered by the pure -scheme: since the -component does not depend at all on output, agents can collect  even when eort levels are very low. Such collusion, however, is rather unlikely, since agents are randomly rematched with one out of ve possible partners in each period. Similarly, one may ask whether a group consisting of one principal and six agents, rematched to three work teams in each period, might develop something like a `corporate identity' and aim at group eciency. Such potential eciency seeking, however, is unproblematic since equilibrium eorts are also ecient due to rent dissipation (Tullock, 1980).3
3In the sense of symmetric eorts maximizing the rm's expected surplus p(y1 + y2)

2 (x21
+ x22
) :
9
4 Results
4.1 Agents' Choice of Eort
The rst part of our analysis examines agents' eorts whose dynamics of group averages are graphically illustrated in Figure 1 of the Appendix. Since the equilibrium eort is always twenty, regardless of the contract installed by the principal, we rst check whether agents' eorts are indeed identical across all chosen contracts. For this we check whether the contract, characterized by the principals' equilibrium prot, or by the level 4 of each contract component (; ; !) associated with it, is a good predictor of the eort invested by agents. We use Tobit regressions, taking into account that only observations across groups are independent, with the agents' eorts as a dependent censored variable.5
The result is that agents systematically deviate from their equilibrium eort of twenty; when we use the principals' theoretical prot as a predictor, the coecient is 0:00744
(Z = 4:05; p < 0:0005). The small value of the coecient is somewhat misleading and results from the dierence in scales between the theoretical payos (0 to 960) and the e ort level intervals agents could choose from (0 to 30). The interpretation of the coe- cient is straightforward; the more protable a contract is for the principal, the lower the e ort invested by agents. Specically, an increase of 100 points in the principal's theoretical payo results in a decrease of 0.744 in the agents' eort. Considering the 960-point di erence between the minimal and maximal theoretical payos, the eect on the agents'
4For the purpose of the this and the following analyses we use the level of each contract component rather than the absolute value. For example, while the possible values of the  component are 0, 200, 400,
600, and 800, the variable included in the analyses has corresponding possible values of 0, 1, 2, 3, and 4.
The same holds for the  and ! components.
5Each of these Tobit regressions uses only one explanatory variable (the principal's equilibrium prot, the level of the  component, the level of the  component, or the level of the ! component). It is not possible to include all of these as explanatory variables in the same regression model because they are not really distinct from each other; the level of each contract component can be determined by the other two, and it follows that the principals' equilibrium prot can also be determined by the levels of any pair of components. 10 e orts could be substantial.
When we use the level of each of the contract components, rather than the principal's theoretical prot, as explanatory variables, additional Tobit regressions reveal that agents react dierently to each component. Figure 2 in the Appendix, which should be selfexplanatory, visualizes the dependency of mean eort separately on , , ! and the equilibrium prot of the principal, depending on contract choice. The coecient on the level of the ! component is 1.78 (Z = 3:83; p < 0:0005), indicating a rather strong and positive relation between the level of ! and the agents' eorts. For the  component the coecient is 0:78 and only marginally signicant (Z = 1:57; p = 0:116), indicating that agents possibly exert less eort the higher the level of the  component. Most strikingly, agents were not sensitive at all to the  component of the contracts (coecient:
0:18; Z = 0:30; p = 0:763), the main incentive component in nearly all the tournament literature. Given the negative relation between the principal's equilibrium prot and the agents' e ort, principals' (empirical) prots are clearly higher than the equilibrium prot for the relatively inferior (from the principals' point of view) contracts and lower for the relatively superior contracts. Possibly, such a pattern could lead to a situation, where theoretically superior contracts are empirically inferior (and vice versa). However, this is not the case; using principals' theoretical prot to predict their actual prot in a linear regression (taking into account that only observations across groups are independent) reveals a very strong and positive relation. The coecient of the theoretical prot is 0.67
(t = 5:47; p < 0:0005), indicating that an increase of 1 point in the theoretical prot was accompanied by an increase of 0.67 points in the actual prot. The data in Table 1 also makes it clear that the principal's theoretical and actual prots are closely linked and that, despite the pattern of agents' deviations from their equilibrium eort of twenty, principals enjoyed higher prots when they chose the theoretically superior contracts.
Agents' eorts may, of course, adjust during the ten rounds of each phase. However, including the period number as an explanatory variable along with the principals' equilibrium
11
prot in a Tobit regression yields an insignicant coecient of 0:10 (Z = 0:94; p =
0:345). To exclude eects of experience from previous phases we ran the same regression for rst-phase decisions only, which also yielded an insignicant result (coecient:
0:014; Z = 0:07; p = 0:940), leading us to conclude that eort choices do not reveal any systematic dynamics during phases with constant contracts.
Result 1 Agents systematically deviate from the equilibrium eort of twenty; the better a contract is for the principal (in equilibrium), the less eort agents choose to invest.
Eorts are positively related with !, negatively with , and are not related at all with .
However, agents' actual eorts do not question the theoretical ranking of contracts from the principal's point of view. Eort choices during all ten-period phases are rather stable.
4.2 Principals' Choice of Incentive Scheme
The second part of our analysis examines the contract choice by principals. In Figure 3 in the Appendix, we visualize separately for , , !, and the principal's equilibrium prot how the frequency of contract choices depends on each of them. Since the contracts are clearly ranked in terms of the equilibrium prot of the principal, and especially since the empirical prots closely preserve this ranking, we checked if principals indeed chose contracts that were more protable to them, namely, contracts with a high  (and ) component and a low ! component. The data in Table 1 makes it clear that this is mostly the case. Both the contracts' theoretical and empirical prots are highly correlated with the frequency with which they were chosen (r = 0:61; p = 0:0161; r = 0:60; p = 0:049; respectively). Principals display a very strong tendency to choose contracts with high  levels (r = 0:90; p < 0:0001), and a weaker tendency to rely on contracts with low ! levels
(r = 0:50; p < 0:0594). The correlation between the level of the  component and the contracts' frequency is negative and marginally signicant (r = 0:4056; p < 0:1337).
Do principals change their contract choices in a systematic way during the experiment?
Although principals have only two opportunities to adapt the contract { once after the rst
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Equilibrium prot Empirical prot
Comparison S p S p
Phase one { phase two -14.5 0.2622 -5 0.8209
Phase one { phase three -6.0 0.6338 -11 0.5966
Phase two { phase three 10.5 0.4033 5 0.8209
Table 2: Principals' contract design dynamics. S { Wilcoxon signed rank sum test statistic; p { signicance level phase and once after the second { they do receive a great deal of feedback information
(thirty tournaments per phase), making it reasonable to expect that they will choose more favorable contracts as the experiment progresses, e.g., due to the `law of eect,' as propagated by reinforcement learning.
To check for systematic changes in principals' contract choices across phases, we compared for each principal her theoretical prots of the contracts chosen in the rst and second phase. Similarly, we compared `rst phase { third phase' and `second phase { third phase.'
We conducted equivalent comparisons for the empirical prots. Wilcoxon signed rank sum tests, however, do not reveal any systematic dierences (Table 2).
It should be noted that the lack of noticeable dynamics in principals' contract choices may, at least partly, be due to the fact that in many cases principals already started out by relying heavily on premium incentives () and that any contract dynamics therefore would have meant adopting worse contract schemes.
Result 2 Principals chose the superior contracts. They were primarily sensitive to the  component of their contract choices.
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5 Discussion and Conclusions
Tournaments are often used by rms and organizations to inspire their agents' performance and supplement the usual reward schemes for employees, like salaries or piece rates. The general reward scheme and the specic rules of such tournaments do not only inspire higher eorts but also have many quite diverse side eects, e.g., crowding out of intrinsic motivation (e.g., Frey and Jegen, 2001) or fostering of collusive behavior { both of which we excluded from our theoretical and experimental analysis.
Nonetheless, we derived an important and { to the best of our knowledge { new result: tournament compensation systems, especially if they come in the form of premium incentives, outperform piece rates even in cases where individual output is readily observable and agents are risk neutral. In the organizational practice the comparative advantage of premium incentives as opposed to piece rates is further supported by the former's ability to economize on monitoring and measurement costs (a tournament requires only ordinal information on individual performance) and to reduce agents' risk exposure by eliminating common shocks. While the latter two arguments apply to traditional xed-prize tournaments and premium incentives alike, premium incentives have the additional advantage that they bear less potential for the collusion of agents since collusion will result in lower rm performance and hence lower the tournament prize. However, when compared to traditional xed-prize tournaments, premium incentives have the disadvantage of additionally exposing agents to the risks inherent in the production process: not only is the probability of winning the tournament aected by the amount of idiosyncratic risk but also the size of what is at stake. While in real-life tournaments with risk averse agents this e ect may overcompensate the cost advantage of premium incentives such that traditional xed-prize tournaments are superior, our laboratory ndings for the (usually risk averse) student participants do not suggest that this is the case.
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! "# $ %& ' ( ) * + , -%. / 0 12%3 -%# 4 4
Figure 1: Mean group eorts { all observations. Each of the 48 (16 groups  3 phases) plots describes average eorts for a specic group in one (10-round) phase. The horizontal axis in each plot is the `round' axis, going from 1 (left) to 10 (right), and the vertical axis is the eort axis, going from 0 (bottom) to 30 (top). The horizontal line in each plot marks the equilibrium eort of 20. Each plot is labeled with information regarding the group, phase, and the contract that was in eect. The group number (1-16) is prexed by `G', the phase number (1-3) by `S'; the 3 numbers separated by dashes pertain to the , , and ! components of the contract that was chosen by the principal for the phase. For example, the top left graph is labeled `G1 P1 0-6-5'. This means that the data pertains to average eorts of group number one during the rst phase, and that the principal chose  = 0,  = 6, and ! = 5.
Appendix
17
Figure 2: Mean eorts of agents as a function of the level of each contract component and of the theoretical principal payo. Each dot represents the average eorts of members of a single group in one phase.
Figure 3: Frequency of contract choices as a function of the level of each contract component and of the theoretical principal payo. Each small dot represents one of the 15 available contracts. Larger dots indicate that multiple contracts share the same frequency and horizontal-axis value.
18