Tournaments and piece rates revisited: a theoretical and experimental study of output-dependent prize tournaments

Abstract

Tournaments represent an increasingly important component of organizational compensation systems. While prior research focused on fixed-prize tournaments where the prize to be awarded is set in advance, we introduce ‘output-dependent prizes’ where the tournament prize is endogenously determined by agents’ output—it is high when the output is high and low when the output is low. We show that tournaments with output-dependent prizes outperform fixed-prize tournaments and piece rates. A multi-agent experiment supports the theoretical result.

Appendix

1.1 Derivation

In this section we derive the equilibrium effort level of the agents, starting with their expected payoff, as given in Eq. 3:

$$\begin{aligned} E u_i=\omega (x_i+\textstyle \frac{\varepsilon }{2})+ \frac{1}{\varepsilon }\displaystyle \int \limits _{0}^{\varepsilon } h(x_i,x_j,\varepsilon _j)\,d\varepsilon _j -\frac{\gamma }{2}\,x_i^2\,, \end{aligned}$$

(3 revisited)

with

$$\begin{aligned} h(x_i,x_j,\varepsilon _j)= \left\{ \begin{array}{l@{\quad }l} 0 &{} \text{ if } \quad x_i\le x_j+\varepsilon _j-\varepsilon \,,\\ \frac{1}{\varepsilon }\int \limits _{0}^{\varepsilon } [\alpha +\beta (x_i+\varepsilon _i + x_j+ \varepsilon _j)]\,d\varepsilon _i &{} \text{ if } \quad x_i\ge x_j+\varepsilon _j\,,\\ \frac{1}{\varepsilon }\int \limits _{x_j+\varepsilon _j-x_i}^{\varepsilon } [\alpha +\beta (x_i+\varepsilon _i + x_j+ \varepsilon _j)]\,d\varepsilon _i &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

Expressing the definite integrals in the definition of the function $h(x_i,x_j,\varepsilon _j)$ we get

$$\begin{aligned} h(x_i,x_j,\varepsilon _j)= \left\{ \begin{array}{l@{\quad }l} 0 &{} \text{ if } \quad x_i \le x_j+\varepsilon _j-\varepsilon \,,\\ \alpha +\beta (x_i+\frac{\varepsilon }{2} + x_j+ \varepsilon _j)] &{}\text{ if } \quad x_i\ge x_j+\varepsilon _j\,,\\ \frac{1}{2\varepsilon }(x_i+\varepsilon -x_j-\varepsilon _j)[2\alpha +\beta (x_i+\varepsilon +3x_j+3\varepsilon _j)] &{}\text{ otherwise. } \end{array} \right. \end{aligned}$$

Substituting $h(x_i,x_j,\varepsilon _j)$ in Eq. (3) and differentiating the expected profit with respect to $x_i$ we get

$$\begin{aligned} \frac{\partial E u_i}{\partial x_i}= \left\{ \begin{array}{l@{\quad }l} \omega - \gamma x_i &{} \text{ if } \quad x_i\le x_j+\varepsilon _j-\varepsilon \,,\\ \omega +\beta - \gamma x_i &{}\text{ if } \quad x_i\ge x_j+\varepsilon _j\,,\\ \frac{1}{2\varepsilon }[2\omega \varepsilon +2\alpha +\beta (2x_i+2x_j+3\varepsilon )-2\varepsilon \gamma x_i] &{}\text{ otherwise, } \end{array} \right. \end{aligned}$$

and the best reply of i to j, assuming a symmetric equilibrium, is

$$\begin{aligned} x_i=\frac{\beta }{\varepsilon \gamma -\beta }x_j+\frac{2\omega \varepsilon +2\alpha +3\varepsilon \beta }{2(\varepsilon \gamma -\beta )}. \end{aligned}$$

When $\frac{\beta }{\varepsilon \gamma -\beta }\ge 1$ (rewritten as $\beta \ge \frac{\gamma \varepsilon }{2}$) i’s best reply to any $x_j$ is $x_i>x_j$. From the symmetry between i and j it follows that when $\beta \ge \frac{\gamma \varepsilon }{2}$ both agents invest the maximal effort. When $\beta < \frac{\gamma \varepsilon }{2}$, and again considering the symmetry between i and j, the unique equilibrium effort $\hat{x}$ (in the sense of mutually best replies) must satisfy the first order condition

$$\begin{aligned} 2\omega \varepsilon +2\alpha +\beta (4\hat{x}+3\varepsilon )-2\varepsilon \gamma \hat{x} =0, \end{aligned}$$

resulting in

$$\begin{aligned} \hat{x}=\frac{2\alpha +\varepsilon (3\beta +2\omega )}{2\gamma \varepsilon -4\beta }\,\quad \text{ for } i\in \{1,2\}. \end{aligned}$$

(4 revisited)

1.2 Instructions

1.2.1 The situation

This experiment consists of multiple rounds. Before the first round, we will randomly assign you to one of two possible roles, namely the A-role and the P-role, which you will keep throughout the entire experiment. There will be groups of one P-participant and six A-participants that stay together over 10 rounds ($=$1 phase). In each round, the six A-participants in a group will be split up randomly in three pairs. Thus, each A-participant faces the same P-participant in all the 10 rounds of one phase, but is very likely to be paired with a different A-participant in each round.

1.2.2 The decision process

At the beginning of each phase, the P-participant determines a reward scheme for his/her group. The components of these reward schemes are explained below. After that, and knowing the reward scheme, the A-participants choose their action: each of the two A-participants in a pair independently chooses a number between 0 and 30.

Suppose that one A-participant chooses x and the other $\hat{x}$. These choices are linked to costs of $\frac{1}{2}x^2$ and $\frac{1}{2}\hat{x}^2$, respectively. The choice of x is linked to an output of $y=x+\varepsilon$, and the choice of $\hat{x}$ is linked to an output of $\hat{y}=\hat{x}+\hat{\varepsilon }$. $\varepsilon$ and $\hat{\varepsilon }$ are independently and evenly distributed random variables in the intervals $0\le \varepsilon \le 40$ and $0\le \hat{\varepsilon }\le 40$. In other words, any possible value of $\varepsilon$ and $\hat{\varepsilon }$ is equally likely to occur, and both random variables are drawn independently from each other.

If the output of the A-participant who chose x is larger or equal to the output of the A-participant who chose $\hat{x}$, i.e., $y\ge \hat{y}$, the A-participant who chose x earns $cy+a+b(y+\hat{y})-\frac{x^2}{2}$, and the A-participant who chose $\hat{x}$ only earns $cy- \frac{\hat{x}^2}{2}$. In other words, only the A-participant whose output is not smaller than the output of the other A-participant in the pair, receives the extra payment $a+b(y+\hat{y})$.

The first part of the extra payment, a, does not depend on the total output $y + \hat{y}$ of a pair, while the second part of the payment, $b(y + \hat{y})$, increases linearly with the total output $y+\hat{y}$ - if and when b is larger than zero.

The payment of cy and $c\hat{y}$ is independent of whether $y\ge \hat{y}$. Thus, when c is larger than zero, the payment of cy and $c\hat{y}$ ensures that A-participants receive a payment that increases in their individual output.

The P-participant earns a constant amount of 20 ECU for each unit of output, minus the payments to the six A-participants. For each pair of A-participants, the P-participant earns $(20-c-b)(y+\hat{y})-a$.

The P-participant is not free to choose any possible reward system (a, b, c), but must choose one of the following 15 rewards systems, where the first, second and third numbers in each cell stand for a, b, and c:

The experiment ends after three phases (30 rounds). Your payment is the sum of all your earnings in all periods. This sum is converted to Euro with the following conversion rate: 1 Euro $=$ 4000 ECU.

You will be paid privately in cash at the end of the experiment.

See Figs. 5, 6, and 7.

Fig. 5

Mean group efforts—all observations. Each of the 48 (16 groups $\times$ 3 phases) plots describes average efforts for a specific 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 effort axis, going from 0 (bottom) to 30 (top). The horizontal line in each plot marks the equilibrium effort of 20. Each plot is labeled with information regarding the group, phase, and the contract that was in effect. The group number (1–16) is prefixed by ‘G’; the phase number (1–3) by ‘S’; and the three numbers separated by dashes pertain to the $\alpha$, $\beta$, and $\omega$ components of the contract that was chosen by the principal for the phase. For example, the top left plot is labeled ‘G1 P1 0-6-5’. This means that the data pertains to average efforts of group number one during the first phase, and that the principal chose $\alpha =0$, $\beta =6$, and $\omega =5$

Fig. 6

Mean efforts of agents as a function of the level of each contract component and of the theoretical principal payoff, with Tobit regression lines. Each dot represents the average efforts of members of a single group in one phase

Fig. 7

Frequency of contract choices as a function of the level of each contract component and of the theoretical principal payoff, with linear regression lines. Each small dot represents one of the 15 available contracts. Larger dots indicate that multiple contracts share the same frequency and horizontal-axis value