4.4 Solution and Main Results
4.4.4 Example with dierent optimal structures
In this section, we provide a simple but rich enough example to illustrate dierent optimal struc-tures.
Suppose that the output function has the following form:
Π(.) =X
n
(λnX
i
xi,n), λ >0.
Costs are quadratic:
C(x) =x2
Thus, the designer cares about the average level of eorts at dierent stages. The parameter λdetermines the weights she attaches to dierent stages.
Then, we can apply Proposition 4.2 and get the following optimal prize structure:
∆Wn = m2 m−1
λ2n
2 (1−κλ2), n6=N,
∆WN = m2 m−1
λ2N 2 . Now we consider several cases for the parameterλ:
1. λ < 1 (Figure 4.1). In this case, the designer values the later stages less than the earlier stages. The optimal prize structure is increasing (1−κλ2 >0) and concave (λ2n decreases with larger values ofn, and hence, ∆Wn falls)6.
Figure 4.1: Concave prize shape
6At the last stage, the prize structure is not necessary concave.
2. λ= 1 (Figure 4.2). All stages are equally important. The optimal prize structure is linearly increasing with a jump in the nal (∆Wn = const > 0, n 6= N). This result is the case of Rosen (1986), where the designer maximizes the same average level of eort during the tournament.
Figure 4.2: Linear prize shape
3. 1 < λ <
q1
κ (Figure 4.3). The designer values the later stages more, but not drastically.
The optimal prize structure is increasing (1−κλ2 >0) and convex (λ2nincreases with larger values onn, and hence,∆Wnincreases).
Figure 4.3: Convex prize shape
4. λ= q1
κ (Figure 4.4). The optimal prize structure is winner-take-all (all prize spreads equal to zero ∆Wn = 0, except the rst and the last stages). Further, there are several papers where the winner-take-all structure turns out to be optimal in other settings (Krishna and Morgan, 1998, Moldovanu and Sela, 2001).
5. λ >
q1
κ (Figure 4.5). The designer values the later stages drastically more than the earlier
Figure 4.4: Winner-take-all
of the tournament. Therefore, the optimal prize structure is decreasing, with a large nal prize being awarded to the winner. This is an example of a "trap structure". When the designer values each subsequent stage much more than the previous one, her valuation of the nal is so high that she tries to make the gap between the prize for the winner and prizes for the other nalists as high as possible. Thus, using negative prize dierences - and, hence, negative prizes - the designer puts agents in a situation where they are punished more if they go closer to the nal and lose there. At the later stages, stakes become extremely large, which enforces very high levels of eorts, as is needed by the principal. Though the prizes become more negative and agents who survive longer obtain smaller prizes at all stages except the nal, the value from surviving until the later stages increases because the agent gets closer to the nal prize.
Figure 4.5: Trap
4.4.5 (Non-)Monotonicity of prize structures
In the previous discussion, we have shown that optimal prize structures may vary a lot. In other words, the shape of a prize structure may not be only convex or concave but even non-monotone or decreasing. Here, we consider the case of a separable output function and directly address the question of monotonicity, not in terms of the ecient level of eorts but in terms of the output function.
We assume here that the output function is separable with respect to dierent stages, that is, Π(x1(Ω), ...,xN(Ω)) =P
n
Πn(xn(Ω)).7 To begin with, suppose that ∂Πn∂x(x,...,x)n ≥ ∂Πn+1∂x(x,...,x)
n+1
for any x, which means that eorts at earlier stages are more important for the designer than at later stages. Then, the equilibrium level of eorts falls during the contest and a(m−1)γm2 (C(x∗n)− κC(x∗n+1)) ≥08. Hence, all prize spreads are non-negative and the prize structure is increasing.
However, in real-life situations, this assumption usually does not hold. Thus, we need to consider a more plausible case where ∂Π∂xn(xn∗n) < ∂Πn+1(x
∗ n+1)
∂xn+1 ,that is, eorts at later stages are more important than at the earlier ones. This is a natural assumption in many real-life situations such as corporate tournaments and sports tournaments. For example, in application to a rm, this would mean that the activities of workers at higher levels of corporate hierarchies are more important than the activities of those at lower ones. In a sport tournament, the assumption means that the performance of contestants in last rounds is valued more than in the early ones. This case is not only the most reasonable but also the most interesting one: the prize structure is not necessarily monotone here.
The main result here is that if the valuations of eort do not increase too much from each stage to the next stage, and, simultaneously, the output function is concave enough, then the optimal prize structure is always non-decreasing. The inverted conditions together serve as sucient conditions for "trap" structures. The exact statement is the following9:
Proposition 4.3. 1. If ∂Πn+1∂xn+1(x,...,x) ≤ 1
κ
∂Πn(x,...,x)
∂xn and x∂Πn∂x(x,...,x)
n is decreasing for allx and n, then the optimal prize structure is increasing at all stages. If the inequality is strict and x∂Πn∂x(x,...,x)
n is strictly decreasing, then the optimal prize structure is strictly increasing.
2. If at some stagen, the opposite holds, that is, ∂Πn+1∂xn+1(x,...,x) ≥ 1
κ
∂Πn(x,...,x)
∂xn and x∂Πn∂x(x,...,x)
n is
7In many real-life applications, this is a reasonable assumption. For example, it is natural to assume that for sports events the revenues from selling tickets on semi-nal matches do not depend on the teams' eorts in quarter-nals.
8As ∂Π∂xni,n(.) = C0(x∗n), a decrease of the derivative of the output function would lead to a decrease of the equilibrium level of eorts.
9For simplicity of notations we skip index because all agents are treated in the same way.
increasing for all x,then the optimal prize is decreasing at stagen. If the inequality is strict and x∂Πn∂x(x,...,x)
n is strictly increasing then the optimal prize is also strictly decreasing at this stage. Hence, the optimal prize structure would be non-monotone10.
Proof. See Appendix 4.A.
However, these conditions are only sucient, not necessary conditions.11 The most interesting nding is in the second part of this proposition, which implies that if the designer values some stage suciently higher than the previous one, the prize at this stage must be lower than the prize at the previous stage. The intuition here is similar to the intuition in the 5th case of the example. Agents react to the prize spreads at all later stages. By decreasing prize at some stage, the designer is able to increase eorts applied at that stage.