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In this section, a second advantage of splitting leagues will be discussed.

Although splitting leads to a negative competition effect, combining both leagues via nesting may yield additional intertemporal incentives. Here nest-ing means that the first-period outcomes of the two n/2-player leagues deter-mine the composition of future tournaments. For example, the principal can choose the best player of each league to compete in a major league after the first period whereas the other (n/2)− 1 players of each league are relegated to a minor league. I will show that the additional incentives generated by nesting dominate the negative competition effect. In order to discuss nested tournaments, the static model of Section 2 has to be replaced by a dynamic one. For this reason, I consider a dynamic model without discounting that lasts 2 periods. In each period, there is a tournament that guarantees each agent his zero reservation utility by setting appropriate tournament prizes.

If the principal does not want to split the n-player league (no-splitting

case), his expected profits will be

2π^∗(n)⁽⁶⁾= (n− 1)

2c . (22)

In the splitting case, however, the principal has the opportunity to in-troduce a promotion-and-relegation rule within a nested-tournament setting.

In the first period, there are two leagues, each with n/2 agents. Again, call these two leagues A and B, and let wL denote the winner prize of league L (L = A, B).¹⁵ Suppose that after the first period the principal promotes x (1 ≤ x ≤ ⁿ2 − 1) agents from each league to a high-prize or major league, and chooses ⁿ₂ − x agents from each league that are relegated to a low-prize or minor league. For the second period, let wM A denote the winner prize in the major league and wM I (≤ w^{M A}) the winner prize in the minor league.¹⁶ I assume that the principal chooses the first-period winner and by random x− 1 of the ⁿ₂ − 1 losers of each league L to play in the major league. The remaining ⁿ₂ − x losers of each league L are relegated to the minor league.

To sum up, in the first period we have only the two leagues A and B each consisting of n/2 players, and in the second period there only exist a major league with 2x players and a minor league with n− 2x players.

The random-selection part of the promotion-and-relegation rule is, of course, a simplification of rules that are used in practice. However, to calcu-late ex ante the probability of belonging to the x most (ⁿ₂−x least) successful agents of each league, we must compute the respective order statistics and

15Again, the principal optimally chooses (n/2) − 1 zero loser prizes in each league.

16Again, it is optimal (and feasible) for the principal to choose zero loser prizes.

these calculations would not be tractable in the given setting.¹⁷ A further justification for the random-selection rule can be seen in the fact that for the first period a symmetric equilibrium in which each agent chooses the same ef-fort will be the most plausible one because of the assumption of homogeneous agents. But then ex post the agents that are relegated and promoted must indeed be chosen by random.¹⁸ However, the most relevant justification for the random-selection rule is given by the main purpose of this section: I will show that there exist promotion-and-relegation rules that lead to splitting dominating no-splitting. As can be seen below, a random-selection rule is one of these rules. Hence, if a more exact discrimination among the (n/2)− 1 losers would lead to an even better result for the splitting case, the main result of this section would only be reinforced.

Contrary to the no-splitting case, the two periods have to be strictly distinguished when splitting leagues. In the first period, promotion and relegation can generate additional incentives, which is not possible in the second period. The game is now solved by backward induction. The solution for the second period is already given by Eqs. (3) and (4) when substituting

17In the introduction, I have mentioned that the logit-form model can be endogenously derived by assuming exponentially distributed noise; see, e.g., Loury (1979). Let si

be the score of agent i (i = 1, ..., n), which is distributed with cdf prob{si ≤ s} = Fi(s) = 1 − exp{−eⁱs} over [0, ∞] with eⁱ as agent i’s effort. Let the sis be stochas-tically independent and assume that the agent with the lowest score si is declared the winner (e.g., the agent with the least faults). Then agent i’s probability of winning is R_∞

0

hexp{−sPn/2

j6=iej}eⁱexp{−eⁱs}i

ds = ei/(ei+Pn/2

j6=iej), which yields the well-known logit-form contest success function. To see the intractability of calculating the exact or-der statistics for the relegation problem we simply have to look at the cdf of the lowest order statistic: This function will become relevant if, for example, only the least success-ful agent of each league is relegated. In this case, the relegation probability is given by prob{s_i> max[s_j|j 6= i]}=R_∞

0

hQ

j6=i[1 − exp{−ejs}] eiexp{−eis}i ds.

18Of course, assuming a random rule in advance leads to different incentives compared to a random rule which is derived endogenously.

for the respective number of contestants and the respective notation. We

for the major league, and

e^∗_{M I} = are either promoted to the major league or relegated to the minor league.

The expected utility of member i choosing effort eiL can be written as

EU_iL^∗ = eiL The first-order condition for the optimal effort choice yields

wL+^n−2x_n−2 (EU_{M A}^∗ − EUM I^∗ )

Analogously, we find for member k 6= i, j that the left-hand side of Eq. (25) must equal ekL/³P

j6=kejL

´

. Hence, we have a symmetric equilibrium in

which each agent chooses

e^∗_L=

r2 (n− 2) w^L

cn² +2 (n− 2x)

cn² (EU_{M A}^∗ − EUM I^∗ ) (L = A, B) . (26) Comparing Eq. (26) with the equilibrium effort in a static n/2-player tour-nament, i.e. e^∗(ⁿ₂) ⁽³⁾=

q2(n−2)wL

cn² , shows that nesting tournaments indeed creates additional incentives: The first term under the square root in (26) is identical with the one in the expression for e^∗(ⁿ₂), but there is an additional term, which is positive. This term is monotonically increasing in the spread between expected utility in the major and the minor league, EU_{M A}^∗ − EUM I^∗ , which can be interpreted as the ”option value” of being promoted instead of relegated after the first period. The higher this value, the higher the incen-tives for each agent in the first period. In addition, the second term under the square root in (26) is monotonically decreasing in the number of promo-tions, x.¹⁹ This finding is also intuitively plausible: The larger the number of promotions, the higher the probability that an agent is promoted by luck and, therefore, the lower the additional incentives. Hence, if all agents are promoted to the major league after the first period (i.e. if x = ⁿ₂), then the additional incentives from nesting will be zero. However, note that the overall impact of x on incentives also depends on the effect of x on e^∗_{M A} and e^∗_{M I}.²⁰

At the beginning of the first period, the principal has to decide about x,

19Note that _∂x^∂ (EU_{M A}^∗ − EUM I^∗ ) = −¹₄w_{M A}^x+1_x3 − wM In−2x+2 (n−2x)³ < 0.

20By substituting (26) into the agents’ objective function we obtain EU_iL^∗ = wLn+2 n² +

(2x−1)n+2x

n² EU_{M A}^∗ + (n + 1)^n−2x_n2 EU_{M I}^∗ , which is strictly positive since x ≤ ⁿ2.

wA, wB, wM I and wM A. He wants to maximize his objective function

π = n

2e^∗_A+n

2e^∗_B+ 2xe^∗_{M A}+ (n− 2x) e^∗M I − (w^A+ wB+ wM I+ wM A) (27)

=

r(n− 2) w^A

2c + (n− 2x)

2c (EU_{M A}^∗ − EUM I^∗ ) +

r(n− 2) w^B

2c + (n− 2x)

2c (EU_{M A}^∗ − EUM I^∗ ) +

r(2x− 1) w^{M A}

c +

r(n− 2x − 1) w^{M I}

c − (w^A+ wB+ wM I + wM A).

We obtain the following result:

Proposition 5 If tournaments can be nested, the principal will always prefer splitting to no-splitting.

Proof. See Appendix.

The proposition shows that — in the given setting — the principal is always able to design nested tournaments with relegations and promotions so that the additional incentives dominate the negative competition effect of split-ting leagues. Interessplit-tingly, this result holds although the principal faces an additional limited-liability problem when nesting tournaments: The proof in the Appendix (see Eq. (A3)) shows that optimal first-period prizes are given by

w^∗_L= (n− 2)

8c − (n− 2x)

(n− 2) (EU_{M A}^∗ − EUM I^∗ ) (L = A, B) .

Hence, if the prize spread between major and minor league and, therefore, the ”option value” EU_{M A}^∗ − EUM I^∗ becomes very large, optimal first-period prizes may become negative to prevent excessive incentives in leagues A and B. However, this is not allowed because of the limited-liability assumption.

As can be seen in the proof of Proposition 5, splitting always dominates no-splitting despite this additional restriction.

Note that Proposition 5 applies to a wide range of sport events with nested tournaments which exhibit a certain knockout rule: In the first round, there are two groups of players, A and B, with identical prizes w_A^∗ and w^∗_B. In the second round, the best x agents of each group compete for a high prize w^∗_{M A}, whereas the remaining n − 2x players only compete for a low prize w^∗M I. However, the results of Proposition 5 do not apply to regular leagues in sports which have a constant number of players. In the proof of Proposition 5, it is assumed that the number of relegations differs from the number of promotions (x6= n/4). Hence, in the second period the two leagues will be of different size. But perhaps Proposition 5 can be applied to scenarios where the size of leagues is altered in time. Such events sometimes happen even in leagues with a constant number of players due to commercial considerations or statutory changes. In such cases, the results of Proposition 5 give some hints on the resulting incentive effects.

Applications of Proposition 5 can also be found in internal labor markets of hierarchical firms. In particular, job-promotion tournaments are often nested: Only those agents that have been successful in the past are promoted to compete with other successful agents on a higher rank of the hierarchy.

For example, we can think of two divisions, A and B, and only the best x employees of each division are promoted to a higher rank where the 2x employees then compete for the high prize w_{M A}^∗ .

5 Conclusion

In this paper, two arguments in favor of splitting leagues have been discussed.

First, splitting leagues may be beneficial for the principal in the presence of possible collusion between the agents. Splitting leads to additional incen-tives and has a kind of insurance effect for the principal given that stability of collusion is stochastically independent between leagues. However, splitting also suffers from a competitive disadvantage, which becomes even worse in connection with the collusion problem. If stable collusion between leagues is rather unlikely, splitting becomes dominant compared to no-splitting. Sec-ond, splitting yields additional incentives when introducing the possibility of promotion and relegation between nested tournaments. These additional incentives will always dominate the negative competition effect from splitting leagues in the given setting.

Of course, the analysis above is somewhat restrictive since a special type of tournament — logit-form tournament with endogenous prizes — and a spe-cial type of cost function — quadratic costs — have been discussed. Perhaps other settings would lead to different results. Unfortunately, by assuming a general convex cost function instead of quadratic costs explicit solutions cannot be derived any longer. However, the analysis above only wants to em-phasize that there are situations in which splitting leagues can be profitable for the principal despite homogeneous agents and a negative competition effect.

Appendix

Proof of Proposition 3:

Using δ = 0 for the superordinate tournament, γ for the splitting case and αγ for the no-splitting case, (20) can be written as

γ² The cut-off ¯α(γ, n) will be feasible, if it is positive but smaller than one.

¯

which is always satisfied: The left-hand side is positive, because

qn−2+√ right-hand side is negative, since

1 +

is true, becausep_n

2+√

for n = 4. Furthermore, ¯α(γ, n) < 1 can be rewritten as

Now (20) has to be rewritten as µ2β

pn− 2 +√ (24)), the first-order conditions for wA and wB yield

w^∗_L= (n− 2) The first-order conditions for wM A and wM I are

(2x + 1) (n− 2x)

Substituting for wA and wB according to (A3) leads to

w_{M A}^∗ = 4 (n− 2)²(2x− 1) x⁴ (2x (2x− 1) (n − 1) − n)²c and

w^∗_{M I} = (n− 2)²(n− 2x − 1) (n − 2x)² 4 ((n− 2x) (n − 1) + 1)²c .

Inserting the four optimal prizes into the principal’s objective function gives

π = n− 2

4c + (n− 2) (2x − 1) x²

c (2x (2x− 1) (n − 1) − n) +(n− 2x − 1) (n − 2) (n − 2x) 4c ((n− 2x) (n − 1) + 1) . Comparing this expression with Eq. (22) shows that the principal will prefer splitting to no-splitting if and only if

n− 2

4c + (n− 2) (2x − 1) x²

c (2x (2x− 1) (n − 1) − n) +(n− 2x − 1) (n − 2) (n − 2x)

4c ((n− 2x) (n − 1) + 1) > (n− 1) 2c

⇔ ∆(x) := 8n (2n − 3) x³− 8n¡

n²− 2¢

x²+ 2n¡

3n²− 7n + 3¢

x + n²(2n− 1) > 0.

(A4)

The first derivative²¹

∆⁰(x) = 24n (2n− 3) x²− 16n¡

n²− 2¢

x + 2n¡

3n²− 7n + 3¢

21Note that for simplicity we abstract from the fact that x has to be a positive integer.

shows that the function ∆(x) has two relative extrema:

x^∗₁ = 1 24 (2n− 3)

³

8n²− 16 + 4√

43 + 53n²+ 4n⁴− 18n³− 81n´ and x^∗₂ = 1

24 (2n− 3)

³

8n²− 16 − 4√

43 + 53n²+ 4n⁴− 18n³ − 81n´ .

As ∆⁰(x) describes a parabola open to the top, x^∗₁ corresponds to a relative minimum and x^∗₂to a relative maximum. Note that x^∗₂ ∈£

0,ⁿ₂¤

for n≥ 4, and that x^∗₂ leads to positive prizes w^∗_L in (A3). Hence, x^∗₂ is a feasible solution.

Since ∆(0) = n²(2n− 1), ∆(ⁿ2) =−n³ + 2n², and

∆(x^∗₂) = n2Ψ³² − 2n²Λ− 1215n + 452 27 (2n− 3)²

(where Ψ = 43+53n²+4n⁴−18n³−81n and Λ = −642+8n⁴+51n²+243n− 54n³) with ∆(x^∗₂) > ∆(0) > 0, the solution x^∗₂ describes a global maximum which always satisfies condition (A4).

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