Strategic mismatching and competing teams

Journal of Economic Behavior & Organization Vol. 50 (2003) 355–372

Strategic mismatching and competing teams Matthias Kräkel∗ Department of Economics, BWL II, University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany Received 26 September 2000; accepted 28 February 2001

Abstract Informational problems are the traditional reason for mismatching and resulting efficiency losses. This paper discusses the strategic role of mismatching, where players voluntarily form inefficient teams or forego the formation of efficient teams, respectively. Strategic mismatching arises (1) when players can realize a competitive advantage (e.g. harming other competitors), (2) or when players want to choose the organizational form—team or self-employment—that minimizes work incentives, which can be rational in certain tournament situations. In this context, negative externalities and free-riding may be beneficial for a team. Informational problems are not necessarily the cause for mismatching, but may improve match efficiency by mitigating the problem of strategic mismatching. © 2003 Elsevier Science B.V. All rights reserved. JEL classification: J41; J44; M21 Keywords: Externalities; Free-rider effect; Mismatch; Teams; Tournament

1. Introduction Two aspects of team production are widely discussed in the literature. First, team formation will be collectively rational for individuals if positive externalities or complementarities are generated (Alchian and Demsetz, 1972; Milgrom and Roberts, 1995). Secondly, if individual contributions to collective output cannot be enforced by contracts, the problem of free-riding in teams will arise (Holmström, 1982; McAfee and McMillan, 1991). These two aspects are usually discussed separately. In this paper, the aspects of externalities and free-riding in teams are combined. Since positive externalities favor and free-riding discourages team formation, we can suspect that teams will (will not) form if the first (second) effect dominates the other. Especially, it seems to be plausible that in the absence of positive externalities and, moreover, in case of negative externalities teams will not form. However, the following results will show that even ∗ Tel.: +49-228-73-9211; fax: +49-228-73-9210. E-mail address: [email protected] (M. Kräkel).

0167-2681/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 2 ) 0 0 0 2 7 - 6

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in this situation individuals may decide to form a team. Borrowing from labor economics, teams without positive externalities can be called mismatches (Jovanovic, 1984; Mortensen, 1988). In labor economics, inefficient combinations of employers and employees are called mismatches, which are usually caused by informational problems. The following results will show that mismatches may even arise without any informational problem. These mismatches can be called strategic mismatches. In general, strategic mismatching is given when individuals voluntarily form inefficient teams or forego the formation of efficient teams, respectively. This paper focuses on the question under which circumstances strategic mismatching will occur. The model discussed in this paper considers two incumbent players who one after another can offer a team contract (i.e. an equal sharing arrangement) to a new entrant into the market. If no team is formed, the two incumbents and the entrant will compete as single players against each other in a tournament at the final stage of the game. If one of the incumbents forms a team with the entrant there will be tournament competition between a two-player team (consisting of the one incumbent and the entrant) and a one-player team (consisting of the other incumbent). This model is best reflected by the market for professional services (e.g. the market for lawyers) where a new entrant (e.g. a new lawyer) can either work as a self-employed or form a partnership (e.g. a law firm) with one of the incumbents. In general, we can distinguish three possible types of strategic mismatches: • An incumbent and the entrant form a team, although the other incumbent would generate larger positive externalities when forming a team with the entrant, i.e. the wrong team arises (mismatch I). • An incumbent and the entrant form a team in spite of negative externalities, i.e. an inefficient team forms (mismatch II). • There is no team formation between an incumbent and the entrant despite positive externalities, i.e. an efficient team does not form (mismatch III). Some interesting results can be derived for this model. For example, when luck plays a dominant role in the tournament, the marginal costs of effort are large, and the winner prize is small, strategic mismatching can be rational where players choose the organizational form— team (mismatch II) or self-employment (mismatch III)—that minimizes the incentives to exert effort. In this context, negative externalities and free-riding can be beneficial for the team to reduce work incentives. We can also demonstrate by example that informational problems are not necessarily the cause for mismatching, but may improve match efficiency by alleviating the problem of strategic mismatching. There are parallels between the problem of strategic mismatching discussed in this paper and the theory of auctions with externalities (Jéhiel et al., 1996; Caillaud and Jéhiel, 1998). In principle, an auction can be characterized as an allocation mechanism that results in a match between the buyer and the seller of a good. Auctions with externalities consider cases where the utility of an unsuccessful bidder is affected by someone else having won the auction. Moreover, in case of identity-dependent externalities an unsuccessful bidder’s utility depends on which individual has won the auction. In such auctions, negative externalities lead to additional incentives for submitting a high bid not because the object being auctioned is very valuable but because the bidder wants to avoid the negative externalities. In the model discussed in this paper, two incumbents “bid” for a new entrant

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by offering a team contract (i.e. a partnership contract). Negative externalities can arise due to the fact that the incumbent who does not form a team with the entrant has to compete against the team consisting of the other incumbent and the entrant in the subsequent tournament game, and this team may be a strong competitor because of cost advantages. The parallels between bidding in auctions to avoid negative externalities and the result of Proposition 2 are quite obvious. Here, the first incumbent will offer a side payment to the entrant if he agrees to form a team. In this situation, team formation is not primarily motivated by cost advantages. In particular, the first incumbent wants to prevent a team consisting of the other incumbent and the entrant. In addition, there are also parallels between strategic non-participation in auctions with negative externalities (Jéhiel and Moldovanu, 1996) and the results of Proposition 1. On the first stage of the auction game, the potential winner decides not to participate to avoid winning the auction and paying a huge price on the subsequent bidding stage. Proposition 1 is based on a similar kind of self-commitment. Here, players strategically choose a certain organizational form on the first stages of the game to bind themselves to exert only little effort on the subsequent tournament stage. The paper is organized as follows. Section 2 describes the model and solves the tournament competition at the final stage of the sequential game. Section 3 contains the results of the paper. The last section concludes.

2. The model In the following model, three risk neutral players compete for a benefit B in a regionally separated market. Let B, for example, be a highly profitable order which is gained by the player with the best performance. The three players are the two incumbents I1 and I2 and the new entrant E into the market. For example, we can think of I1 and I2 as two local lawyers, and E as a law student from the regional university who has just passed his final exams. Furthermore, we can assume that the best performer in this local market for lawyers will become the exclusively legal adviser of the largest corporation in this region. This will lead to the benefit B for the most successful competitor. In this situation, the question arises whether E will either form a professional partnership—a team—with one of the incumbents or become self-employed. To discuss this question formally it is assumed that I1 and I2 can sequentially offer a partnership contract to the new entrant E. Fig. 1 describes the timing structure of the game. In t1 , incumbent I1 can offer a partnership contract to the entrant E. If I1 offers a contract and E accepts, the game will continue in t5 , where the team consisting of I1 and E competes against I2 in a simple tournament. The winner of the tournament—the team{I1 , E} or I2 — receives the benefit B (B > 0), whereas the loser gets nothing. If I1 does not offer a contract to E or E rejects I1 ’s offer in t2 , the incumbent I2 has to decide about a contract offer. If I2 offers a team contract and E agrees to it, the team {I2 , E} will compete against I1 in t5 . If I2 chooses not to offer a contract or E rejects I2 ’s offer in t4 , there will be a standard tournament between the three single players I1 , I2 , and E in t5 . Before I start to discuss the possible mismatches that can arise during t1 –t4 , further details about the tournament subgame in t5 have to be described.

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Fig. 1. Timing of the sequential moves.

There are three possible states in t5 : s1 , s2 and s3 . State s1 describes the situation where no team {I1 , E} or {I2 , E}, respectively, has been formed. In state s2 the incumbent I2 and the entrant E have formed a team {I2 , E}. State s3 denotes the case where the other incumbent I1 and the entrant E have decided to work together as team {I1 , E}. For the rest of this section, these three states will be dealt with in turn. In the first state, s1 , the three players are independent competitors. The performance of each player i (i = I1 , I2 , E) can be described by a simple linear function qi = ei + εi where ei denotes i’s effort and εi a random or luck component.1 In the example sketched at the beginning of this section, εi would represent the corporation’s measurement error when choosing the best performer as the exclusively legal adviser. The εi s are assumed to be independently and identically distributed (i.i.d. assumption) according to a cumulative distribution function F (ε) with density f (ε). For simplicity, we assume that the εi s are uniformly distributed over the interval [0, ε¯ ] with F (ε) = ε/¯ε and f (ε) = 1/¯ε .2 The player with the highest qi will be the winner of the tournament and receives the benefit B. Player i’s disutility of effort (in monetary terms) is described by the cost function c(ei ) = (k/2)ei2 (k > 0). Each competitor wants to maximize his expected utility which is identical with his expected net income: EUi (s1 ) = B · Pr{i wins} − 21 ke2i .

(1)

The analysis is restricted to symmetric equilibria in the tournament subgame, where each ∗ .3 Thus, the winning player chooses the same amount of effort e∗ = eI∗1 = eI∗2 = eE 1

Most of the assumptions follow the standard model by Lazear and Rosen (1981). In the following, we have to calculate with order statistics to derive the equilibrium efforts for the tournament subgame. It is well-known that calculating with order statistics implies some difficulties. Therefore, the simplifying assumption of uniformly distributed luck is used in this model. 3 This restrictive assumption is not unusual in the tournament literature; see, e.g. Nalebuff and Stiglitz (1983, pp. 26–27), Lazear (1989), p. 565. There may be asymmetric equilibria, too. But intuitively it is reasonable to think of symmetric equilibria, because the three players have identical characteristics. 2

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probability Pr{i wins} can be written as: Pr{iwins} = Pr{qi > q(2) } = Pr{ei +εi > e∗ +ε(2) } = Pr{X < ei −e∗ } = G(ei −e∗ ), where both q(2) and ε(2) denote the highest of two order statistics, respectively, and X := ε(2) − εi with cumulative distribution function G(x) and density g(x). Therefore, Eq. (1) can be rewritten as: EUi (s1 ) = B · G(ei − e∗ ) − 21 ke2i .

(2)

EUi (s1 ) is assumed to be strictly concave in ei so that the first-order condition describes an absolute maximum. Using the concrete form for g(·) from Appendix A we see that strict concavity is guaranteed by k ε¯ 2 > 2B,

(3)

which is assumed to hold throughout the paper. This condition also implies that the expected utilities of all players in each state s1 , s2 , and s3 are strictly positive in equilibrium. Moreover, condition (3) implies strict concavity of the expected utility functions of all players in s2 and s3 , too. The first-order condition ∂EUi = Bg(ei − e∗ ) − kei = Bg(0) − kei = 0 ∂ei gives the Nash equilibrium effort ei∗ = e∗ =

Bg(0) B = , k k ε¯

i = I1 , I2 , E,

(4)

where the last expression follows from the assumption of uniformly distributed luck. Eq. (4) shows that the Nash equilibrium effort is increasing in the benefit B, decreasing in the cost parameter k and decreasing in the influence of luck ε¯ , which is all intuitively reasonable. By substituting (4) into (2) we obtain4 EU∗i (s1 ) =

B B2 , − 3 2k ε¯ 2

i = I1 , I2 , E.

(5)

In the second state, s2 , the team {I2 , E} competes against I1 in the tournament.5 It is assumed that I2 and E perform according to a linear team production function q(eI2 , eE ) = eI2 + eE + εI2 . Here, εI2 describes the same stochastic luck as in state s1 . The externalities from team production are reflected by the cost function of each team member.6 c(ei ) =

k 2 e , 2γI2 i

γI2 > 0, i = I2 , E.

(6)

Note that G(0) = 1/3. Drago et al. (1996) discuss a tournament between teams, where the tournament prizes are shared equally among the team members, but not in connection with matching problems. 6 Externalities are not modeled by using a non-separable production function (Alchian and Demsetz, 1972). This allows for a better comparison of the three states. 4

5

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The new cost parameter γI2 can be either greater or lower than 1/2, where γI2 > 1/2 indicates positive externalities and γI2 < 1/2 negative externalities between the team members, respectively. To see this, note that a first cost advantage from team formation already arises from the fact that the individual cost functions are convex and, therefore, team members minimize costs by sharing the total team effort equally: A given effort level e¯ causes 2(k/[2γI2 ])(e/2) ¯ 2 = (k/[4γI2 ])e¯2 effort costs for the team {I2 , E} and (k/2)e¯2 effort costs for the individual I1 . Thus, team formation leads to lower costs when γI2 > 1/2. Since there are no team effects on the production or performance side, match efficiency can be solely defined via the cost function. A team {I2 , E} will be called efficient (inefficient), if γI2 > 1/2 (γI2 < 1/2), i.e. if there exist positive (negative) externalities within the team.7 Furthermore, we assume that if the team wins the benefit B will be shared equally among I2 and E.8 Thus, I2 ’s and E’s expected utility in state s2 is given by EUi (s2 ) =

B k 2 e , Pr{team{I2 , E} wins} − 2 2γI2 i

i = I2 , E.

(7)

Incumbent I1 remains alone. It is assumed that he has the same performance function and the same cost function as in state s1 . Therefore, his expected utility is EUI1 (s2 ) = BPr{I1 wins} − 21 ke2I1 .

(8)

Comparing Eqs. (7) and (8) we see that a trade-off has to be taken into account by I2 and E when they decide about forming a team. Positive externalities can make the formation of a team attractive, because I2 and E would realize a cost advantage. On the other hand, I2 and E will only receive half of the benefit B if they win against I1 . This effect reduces the team members’ incentives to exert effort and is well known as free-rider effect. The winning probability Pr{team{I2 , E} wins} can be written as: Pr{team{I2 , E} wins} = Pr{eI2 + eE + εI2 > eI1 + εI1 } = Pr{Y < eI2 + eE − eI1 } = H (eI2 + eE − eI1 ) where Y := εI1 − εI2 has a cumulative distribution function H (y) and a density h(y). Now, Eqs. (7) and (8)can be written as: EUi (s2 ) =

B k 2 e , H (eI2 + eE − eI1 ) − 2 2γI2 i

i = I2 , E,

k EUI1 (s2 ) = B[1 − H (eI2 + eE − eI1 )] − eI21 . 2

(9)

(10)

An efficient (inefficient) team {I1 , E} can be defined analogously using γI1 . Of course, such an equal sharing arrangement is not always optimal. It is a simplifying assumption, which can be motivated by the fact that individual contributions to the joint team performance are often non-contractible. Therefore, the team members agree on an equal sharing contract. In addition, equal sharing is often observed in real partnership contracts; see, e.g. Newhouse (1973), p. 39; Getzen (1984), p. 209. 7 8

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From the first-order conditions we obtain the following reaction functions:9 B k ei = 0, h(eI2 + eE − eI1 ) − 2 γI2 Bh(eI2 + eE − eI1 ) − keI1 = 0.

i = I2 , E,

(11) (12)

Eq. (11) shows that in equilibrium the team members I2 and E will exert the same amount of effort, i.e. eI2 = eE . Combining this result with (11) and (12) yields the following condition: γI eE = eI2 = 2 eI1 . (13) 2 Eq. (13) shows that whether the team {I2 , E} or the individual I1 exerts more effort in equilibrium depends on the relation between positive externalities and the free-rider effect. Total team effort amounts to eE + eI2 = γI2 eI1 . Thus, if γI2 > 1(< 1) the positive externalities (free-rider effect) will be dominant. If γI2 = 1 the two effects will cancel each other. To sum up, we have to differentiate between three situations: When 0 < γI2 < 1/2, there are two negative effects from team formation—negative externalities and free-riding. When 1/2 < γI2 < 1, there are positive externalities from team formation, but the free-rider effect is still dominant. When γI2 > 1, the positive externalities dominate the free-rider effect.10 To derive the equilibrium efforts we have to substitute the concrete form of the density h(y) into (11) and (12). The results of Appendix A show that the convolution h(y) of two uniform (or rectangular) densities is a triangular density function, which is symmetric around zero. Thus, we have to discuss two different cases. First, it is possible that eI2 + eE ≤ eI1 . In that case, we have to use the left-hand part of the triangular density h(y) for solving (11) and (12). From (13) we know that this scenario will be relevant, if and only if γI2 ≤ 1, i.e. if the free-rider effect is dominant. Secondly, the opposite relation eI2 + eE > eI1 can hold. In this case, the right-hand part of the density h(y) becomes relevant and we have a dominant influence of positive externalities, i.e. γI2 > 1. After some calculations, the “negative scenario” eI2 + eE ≤ eI1 yields the following expressions for the equilibrium efforts: γI B ε¯ B ε¯ ∗ = 2 and eI∗1 = . (14) eI∗2 = eE 2 B − BγI2 + k ε¯ 2 B − BγI2 + k ε¯ 2 Relation (14) emphasizes the trade-off between the positive externalities and the free-rider effect (in analogy to (13)). In addition, we see that the strategic interaction between the team {I2 , E} and I1 in the tournament results in a spillover of the positive externalities. An ∗ as well as the denominator of e∗ and increasing γI2 reduces the denominator of eI∗2 = eE I1 results in increasing efforts of all players. However, the positive externalities are larger for the two team members, because an increasing γI2 additionally increases the numerator of 9 For the derivation of h(y) see Appendix A. Condition (3) implies that the functions EU (s ) (j = I , I , E) j 2 1 2 are strictly concave in ej . 10 There is a third effect which can be called the competition effect. Without team formation, three players compete against each other in the tournament, whereas there are only two competitors when I2 and E form a team. However, this competition effect is not an original one. The effect directly follows from the interplay of the positive externalities and the free-rider effect, which can be seen from g(x) and h(y) in Appendix A.

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∗ . Substituting (14) into (9)−(10) and using the concrete form of the distribution eI∗2 = eE function H (y) (see Appendix A) gives the expected utilities in equilibrium:11

EU∗− i (γI2 ; s2 ) =

1 2k ε¯ 2 − BγI2 , Bk¯ε 2 8 (B − BγI2 + k ε¯ 2 )2

EU∗− I1 (γI2 ; s2 ) = B

i = I2 , E,

(15)

B 2 (1 − γI2 )2 + Bk¯ε 2 ((3/2) − 2γI2 ) + (1/2)k 2 ε¯ 4 . (B − BγI2 + k ε¯ 2 )2

(16)

Similar calculations can be made for the “positive scenario”. Now we obtain ∗ = eI∗2 = eE

γI2 B ε¯ , 2 BγI2 − B + k ε¯ 2

eI∗1 =

B ε¯ , BγI2 − B + k ε¯ 2

(17)

as equilibrium efforts and EU∗+ i (γI2 ; s2 ) =

1 4B 2 (1 − γI2 )2 + Bk¯ε 2 (7γI2 − 8) + 2k 2 ε¯ 4 , B 8 (BγI2 − B + k ε¯ 2 )2

EU∗+ I1 (γI2 ; s2 ) =

k ε¯ 2 − B 1 Bk¯ε 2 , 2 (BγI2 − B + k ε¯ 2 )2

i = I2 , E,

(18)

(19)

as expected utilities in equilibrium.12 The third state, s3 , considers the case where I1 and E form a team {I1 , E} in t2 which competes against I2 in the tournament in t5 . The earlier considerations concerning state s2 analogously hold for s3 . We obtain the equilibrium efforts and expected utilities by substituting s3 for s2 , and interchanging the indices I2 and I1 in (14)−(19). In the following, we will look for subgame perfect equilibria in the game described by Fig. 1 that lead to strategic mismatching between I1 , I2 and E. In principle, three types of mismatching can be distinguished − the wrong team forms, an inefficient team forms, or an efficient team does not form. To reduce complexity, the following analysis is restricted to mismatches that can arise in connection with player I1 . Then, there are three possible types of mismatch equilibria in the game: • Mismatch I equilibrium: I1 and E form a team, although the team {I2 , E} would generate larger positive externalities, i.e. γI2 > γI1 . Without I1 ’s offer in t1 there would be an efficient match between I2 and E. • Mismatch II equilibrium: I1 and E form a team in spite of negative externalities, i.e. γI1 < 1/2. Without I1 ’s offer in t1 there would be efficient matching, where all players remain self-employed. • Mismatch III equilibrium: I1 and E do not form a team despite positive externalities, i.e. γI1 > 1/2. The players remain self-employed which is inefficient. Mismatch I deals with the classical problem of mismatching where the wrong players decide to form a team. Mismatch II considers the case in which the players form a team when there should be no team because of efficiency reasons. Mismatch III considers the opposite 11 12

Here, “−” indicates that the “negative scenario” is considered. Here, “+” indicates the “positive scenario”.

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case where players forego an efficient match. In the following section, the existence of these three types of mismatch equilibria will be discussed. 3. Results Information problems are the traditional reason for mismatching in economics. In the beginning of this section, information problems are completely excluded. Thus, we can concentrate on purely strategic mismatching, because all the mismatches left must be voluntary. We assume that the cost parameters γI1 and γI2 are common knowledge in the whole game. By that means, each player is able to correctly calculate the consequences of externalities and the free-rider effect and therefore the possible outcomes in the states s1 , s2 and s3 in t5 . Under these assumptions, we obtain the following results:13 Proposition 1. If all the players have complete cost information, the following will hold: (i) There does not exist a mismatch I equilibrium. (ii) There is the possibility of a mismatch II equilibrium. (iii) If B is sufficiently small, k is sufficiently large and ε¯ is sufficiently large there will be the possibility of a mismatch III equilibrium. Proof. See Appendix B.

䊐

The result of Proposition 1(i) seems to be plausible. In the case of no cost uncertainty E knows that forming a team with I2 is better for him than forming a team with I1 because of γI2 > γI1 . In t2 player E anticipates that he will receive a contract offer from I2 in t3 . Therefore, he would never accept the less attractive offer from player I1 in t2 . Hence, without any information problem a mismatch I equilibrium cannot exist. The results of Proposition 1(ii and iii) seem to be puzzling. There is the possibility that I1 and E voluntarily form a team in spite of the free-rider effect and negative externalities (mismatch II), i.e. there are no positive externalities, which can outweigh the disadvantages of the free-rider effect. Thus, we would expect the non-existence of a mismatch II equlibrium as a plausible result. Such expectations are supported by the fact that with Cournot competition instead of tournament competition in t5 a mismatch II will not arise in a situation with no cost uncertainty (Kräkel, 2001). On the other hand, it is also possible that I1 and E do not want to form an efficient team (mismatch III). As the proof of Proposition 1(iii) shows, this becomes possible even when the positive externalities dominate the free-rider effect. The intuition that lies behind these two results becomes clear when first looking at Proposition 1(iii). Here, I1 and E may forego to form an efficient team if ε¯ and k are sufficiently large, and B is sufficiently small. First, this parameter condition means that luck plays a dominant role for the outcome of the tournament as ε¯ is required to be large which implies a large variance of the stochastic luck component εi .14 Then it is rational for each player to 13 Here, “possibility” indicates that the values for the exogeneous parameters γ and γ must lie in the ranges I1 I2 given by the earlier definitions of the three types of mismatch equilibria. 14 A large ε ¯ means that the εi s are distributed over a wide range [0, ε¯ ]. Note that ε¯ = 1/g(0). For the interpretation of 1/g(0) as a measure of luck in the symmetric tournament equilibrium (here, in state s1 ) see Lazear (1995), p. 29.

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exert only minimal effort, because in this situation effort is not decisive for the tournament outcome but generates costs c(ei ). Secondly, the parameter condition requires that the cost parameter k is large. This would strengthen the players’ incentives to exert only little effort. At last, the benefit B has to be small. This would also result in minimal work incentives, because the winner prize B is the driving force for exerting effort in tournaments. Altogether, a large ε¯ , a large k, and a small B imply that it is rational to withhold effort, because winning the tournament is not very attractive and effort does not have any real influence on the outcome of the tournament but generates considerable costs. In this situation, it is optimal for I1 and E to choose the organizational form—team or self-employment—that minimizes effort incentives and, thereby, their effort costs. Here, the formation of a team would lead to high efforts due to positive externalities. Thus, I1 and E prefer to remain self-employed. In this state s1 , all players will realize large expected utilities EU∗i (s1 ) (i = I1 , I2 , E) if B is small and k and ε¯ are large. Eq. (5) shows that EU∗i (s1 ) decreases in B from a certain level on and increases in k ε¯ 2 . This can be explained by the fact that in the symmetric equilibrium of situation s1 , each player’s probability of winning is always 1/3, irrespective of the players’ effort levels. Since effort costs c(ei ) rise in ei , the expected utility EU∗i (s1 ) becomes large for small values of B and large values of k ε¯ 2 , which lead to low effort levels ei according to (4). The result of Proposition 1(ii) can be explained in an analogous way. Again, let k and ε¯ be sufficiently large and B sufficiently small so that it is rational for I1 and E to choose the organizational form which minimizes work incentives. But now in the case of γI1 < 1/2 it can be attractive for I1 and E to form a team, which leads to free-riding and negative externalities, because both effects increase the incentives to exert minimal effort. In such a situation, I1 and E bind themselves to reduce effort and the disutility of effort by forming a team.15 In other words, I1 and E form a team not in spite of but because of the free-rider effect and negative externalities. Unfortunately, there are countervailing effects that make this result less obvious than the previous result of Proposition 1(iii). Especially, there is a trade-off concerning the effort level when forming a team, whereas in the equilibrium of state s1 (i.e. all players are self-employed) expected utility is strictly decreasing in effort because of the given winning probability due to symmetry. In the case of team formation, however, Eq. (9) shows that withholding effort will lead to low effort costs, but also to a low winning probability. Here, I1 and E are already less likely to win the tournament than I2 due to the “negative scenario” (i.e. γI1 < 1/2). The trade-off will make team formation no longer attractive as an organizational form to minimize efforts, if k ε¯ 2 (B) becomes too large (small). Formally, the proof of Proposition 1(ii) shows that (for sufficiently small ∗ B and sufficiently large k ε¯ 2 ) the term EU∗− I1 (γI1 ; s3 ) − EUI1 (s1 ) decreases with increas∗ ing k ε¯ 2 , for example. There exist large values for k ε¯ 2 so that EU∗− I1 (γI1 ; s3 ) < EUI1 (s1 ). In addition to the trade-off, condition (B.4) in Appendix B shows that there is a lower bound for γI1 (i.e. γI2 < γI1 < 1/2). Therefore, when choosing a team to withhold effort, γI1 cannot become too small to induce low effort incentives. To sum up, Proposition 1 shows that strategic mismatching can arise when players want to choose the organizational 15 From Section 2 we know that the effort of I (and E) is (B/(k ε ¯ )) when not forming a team (state s1 ) and 1 (γI1 /2)(B ε¯ )/(B − BγI1 + k ε¯ 2 ) when forming a team (state s3 ). As long as γI1 is small enough, i.e. the negative externalities are large enough, there will be less effort if I1 and E form a team.

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form—team or self-employment—that leads to minimal work incentives which can be rational in certain tournament situations. For simplicity, the analysis has been restricted to team contracts in form of equal sharing arrangements. In this setting, a mismatch I equilibrium does not exist. However, an interesting result can be derived when allowing entrance fees, which can be charged by I1 , I2 , or E when forming a team. Proposition 2. If there is no cost uncertainty but entrance fees are allowed in team contracts, a mismatch I equilibrium will become possible. Proof. See Appendix B.

䊐

The proof of Proposition 2 shows that a mismatch I equilibrium will become possible for purely strategic reasons if I1 is allowed to offer an entrance fee in addition to the equal sharing agreement. Then, for a sufficiently high entrance fee E will accept I1 ’s offer in spite of larger efficiency gains from a team contract with I2 (i.e. γI2 > γI1 ). The crucial point in this sort of strategic mismatching is the fact that I2 has no chance to prevent E from accepting I1 ’s offer despite γI2 > γI1 . Player I2 could promise to add an entrance fee to his equal sharing offer, too, but such a promise could not be credible. At t3 of the sequential game, I2 would never offer an entrance fee in addition to the standard equal sharing contract, because at this point of time E cannot threat I2 to form a team with I1 . Player E anticipates this in t2 when he decides about accepting or rejecting I1 ’s offer.16 The proof of Proposition 2 also points out, which parameter constellations make a mismatch I equilibrium (with entrance fee) possible. For example, γI2 has to be high enough for a mismatch I. A high γI2 means that offering a team contract to E would be profitable for I2 . In addition, a high γI2 results in large competitive disadvantages for I1 when I2 and E form a team in this situation. This effect is indicated by Eq. (19), which shows that EU∗+ I1 (γI2 ; s2 ) is decreasing in γI2 . Therefore, if γI2 is sufficiently high player I1 will have strong incentives to prevent a team {I2 , E}. Furthermore, the proof of Proposition 2 shows that γI1 must also be sufficiently high. A high γI1 guarantees that there are considerable efficiency gains for I1 and E when forming a team. This induces E to accept I1 ’s equal sharing offer and, in addition, allows I1 to pay an entrance fee that is sufficiently high for E to forego an efficient match with I2 .17 To sum up, the Propositions 1 and 2 have demonstrated that the three types of mismatches are possible for strategic reasons in sequential team contracting, although all players are completely informed about the cost parameters γI1 and γI2 . 16 Note that there are parallels to deadline threats in bargaining. If we drop the assumption of equal sharing contracts, I2 will have a strong bargaining position due to the deadline threat. He would make E a take-it-or-leave-it offer so that E is just indifferent between forming a team with I2 or remaining alone. With the entrance fee this strength becomes a weakness, because now I2 cannot credibly promise E to offer him more than his reservation value. 17 Some anecdotal evidence for the result of Proposition 2 can be found in professional team sport. On 31 July 1993, an interesting transaction was sealed within the Major League Baseball (MLB). On that day, the Toronto Blue Jays acquired Rickey Henderson, “the best lead-off hitter in the history of baseball” (O’Malley and O’Malley, 1994, p. 11). This deal seems to be puzzling, because the Blue Jays needed a pitcher and not a hitter at that time. However, the acquisition of Henderson was quite rational for the Blue Jays since it successfully prevented that the Yankees or other competitors were able to acquire Henderson for strengthening their teams.

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At last, it can be shown that information problems, which are the traditional reason for mismatching, may alleviate the problem of strategic mismatching. In the following example, it is assumed that there is symmetric uncertainty about the cost parameters γI1 and γI2 . The two incumbents and the entrant only have two probability distributions γˆI1 ∼ FI1 (·) and γˆI2 ∼ FI2 (·) over the interval [ 21 , 1] for the true cost parameters γI1 and γI2 as common knowledge. In addition, let γˆI1 and γˆI2 be unbiased estimators of γI1 and γI2 , i.e. E[γˆI1 ] = γI1 and E[γˆI2 ] = γI2 .18 Of course, neither of the three players knows that γˆI1 and γˆI2 are unbiased, because otherwise there would be no real cost uncertainty. It is assumed that players cannot offer entrance fees, and that all uncertainty regarding γI1 and γI2 is resolved in t5 before the tournament starts, i.e. the formation of a team immediately reveals its external effects. Hence, we can still use the results of Section 2. For this simple example, it can be shown that a mismatch III is less likely (i.e. holds for a smaller parameter space) compared to the situation with no cost uncertainty.19 In this scenario, with symmetric cost uncertainty a mismatch III, where players forego to form an efficient team, is still possible. But now the players overestimate the benefits of teamwork and this informational bias works against strategic mismatching. Thus, informational problems are not necessarily the cause for mismatching, but may improve match efficiency by mitigating the incentives for strategic mismatching.

4. Conclusions This paper contains the following results: • Strategic mismatching can be beneficial to realize a competitive advantage (e.g. harming other competitors), if this advantage dominates the disadvantages from mismatching. • Strategic mismatching can arise when players want to minimize work incentives in situations in which effort only plays a subordinate role in tournament competition (caused by the dominant influence of luck) but the marginal costs of effort are high. In this context, the formation of teams can be explained even in situations where team formation usually seems to be irrational because of free-riding and negative externalities. • Informational problems are the traditional reason for mismatching. Thus, we might expect that adding cost uncertainty to the analysis will result in more mismatching. The example considered in Section 3 shows that this is not necessarily true. Informational problems may work in the opposite direction than strategic mismatching. • Comparing the results of this paper and the results derived in Kräkel (2001), it becomes clear that the kind of competition at the final stage of the game can play a major role in strategic mismatching. Kräkel (2001) shows that a mismatch II equilibrium (i.e. forming an inefficient team) cannot be possible when considering Cournot competition at the final stage. This result seems to be intuitively plausible because both the free-rider effect and negative externalities work in the same direction and make a mismatch II unprofitable. In this paper, tournament competition takes place at the final stage. However, this kind 18 19

Here, E[·] denotes the expectation operator. This result can be easily proved by using Jensen’s inequality.

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of competition makes a mismatch II equilibrium possible, especially because of its luck component. Altogether, the aspect of strategic mismatching seems to be quite interesting and may also apply to other cases than the sketched partnership example. Further applications may include (strategic) alliances between firms, or enticing away managers from competing firms (e.g. a firm may raid a competing firm by acquiring some of its managers despite their large amount of firm-specific human capital solely to harm the competing firm). Another application may be an architectural contest where architects—as single competitors or teams— compete for a given winner prize. Nevertheless, it would be fruitful to generalize the model discussed in this paper. For example, the sequential matching process can be criticized.20 In some situations, it may hold that players have a fixed agenda about the sequence of contract bargaining. In other situations, however, it would be more realistic to assume that the entrant has to choose between different contract offers simultaneously. Replacing the assumption of sequential matching by assuming simultaneous contracting would affect the results concerning mismatch I, especially the result of Proposition 2. But it would not affect the mismatches II and III in general, which consider whether a team forms or not (e.g. we can think of two identical incumbents so that the kind of matching process does not really matter). Acknowledgements I would like to thank Silke Becker, John Conlisk, Markus Irngartinger, Stephanie Rosenkranz, Dirk Sliwka, Gunter Steiner and an anonymous referee for helpful comments. Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. Appendix A The functions g(x), h(y) and H (y) can be derived in analogy to Kräkel (2000, pp. 398– 401). Here, we obtain    ε¯ +x 2ε(2) (¯ε + x)2     dε , if − ε ¯ ≤ x < 0 , if − ε¯ ≤ x < 0  0  (2) 3 ε¯ ε¯ 3 g(x) = = 2  2ε      xε¯ (2) dε(2) ,  1 − x , if 0 ≤ x ≤ ε¯ . if 0 ≤ x ≤ ε¯ 3 3 ε¯ ε¯ ε¯ and  ε¯ + y   if − ε¯ ≤ y < 0  2 , ε¯ h(y) =  ε¯ − y   , if 0 ≤ y ≤ ε¯ . ε¯ 2 20 However, the matching models in labor economics also assume sequential instead of simultaneous matching (Jovanovic, 1984; Mortensen, 1988).

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Integrating h(y) and noting that H (−¯ε ) = 0, H (0) = (1/2) and H (¯ε ) = 1 yields  1 y2 y   + + , if − ε¯ ≤ y < 0 2 ε ¯ 2 2¯ ε H (y) = 2   y − y + 1 , if 0 ≤ y ≤ ε¯ . ε¯ 2 2¯ε 2 Appendix B Proof of Proposition 1. The intuition of result (i) is rather obvious and given in the discussion subsequently to Proposition 1. A formal proof is therefore omitted here. Next, result (ii) is considered. There are three conditions for a mismatch II: γI1 , γI2 < 21 ,

(B.1)

∗ EU∗− i (γI2 ; s2 ) < EUi (s1 ),

i = I2 , E,

(B.2)

∗ EU∗− j (γI1 ; s3 ) > EUj (s1 ),

j = I1 , E.

(B.3)

Relations (B.1) and (B.2) define a mismatch II. Condition γI1 < 1/2 means that the team {I1 , E} is an inefficient match. Together with γI2 < 1/2 this implies that not forming a team would be efficient. Inequality (B.2) guarantees that I2 and E do not want to form a team {I2 , E}. Therefore, without I1 ’s contract offer there would be efficient matching (i.e. state s1 ), where all players remain self-employed. Condition (B.3) ensures that both I1 and E prefer competing as a team to competing as independent individuals in the tournament. ∗− Relations (B.2) and (B.3) together imply γI2 < γI1 , because EU∗− I2 (γI2 ; s2 ), EUE (γI2 ; s2 ), ∗− ∗− EUI1 (γI1 ; s3 ) and EUE (γI1 ; s3 ) are all identical functions of γI2 or γI1 , respectively, which are increasing in γI2 or γI1 . Thus, (B.1) can be rewritten as: (B.1 )

γI2 < γI1 < 21 .

Substituting the concrete expressions for the expected utilities into (B.2) and (B.3) we see that the conditions (B.1 )–(B.3) are equivalent to Ψ (γI2 ) < 0,

Ψ (γI1 ) > 0,

γI2 < γI1 < 21 ,

(B.4)

with Ψ (γ ) :=

Λ(γ ) (B − Bγ + k ε¯ 2 )2 k(¯ε 2 /B)

and Λ(γ ) := γ 2



1 3 2B

(B.5)

   13 2 4 − 13 k ε¯ 2 B 2 + γ 24 k ε¯ B − 13 k ε¯ 2 B 2 − B 3

1 3 6 − 12 k ε¯ + 23 k ε¯ 2 B 2 − 16 k 2 ε¯ 4 B + 21 B 3 .

(B.6)

Note that the denominator of Ψ (γ ) in (B.5) is positive so that (B.4) is equivalent to Λ(γI2 ) < 0,

Λ(γI1 ) > 0,

γI2 < γI1 <

1 . 2

(B.7)

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Since according to the concavity condition (3) the inequality k ε¯ 2 > 2B holds, the function Λ(γ ) describes a concave parabola. This parabola has the following two roots: √ −24B 2 + 13k 2 ε¯ 4 − 8Bk¯ε 2 + 144B 2 k 2 ε¯ 4 + 105k 4 ε¯ 8 − 240k 3 ε¯ 6 B γ¯ = (B.8) −2(12B − 8k ε¯ 2 )B √ = −24B 2 + 13k 2 ε¯ 4 − 8Bk¯ε 2 − 144B 2 k 2 ε¯ 4 + 105k 4 ε¯ 8 − 240k 3 ε¯ 6 B γ = . (B.9) −2(12B − 8k ε¯ 2 )B The concavity condition (3) implies that both the denominator of γ¯ and the denominator of = = γ are positive. In addition, γ < γ¯ . Thus, there are parameter values so that (B.7) will hold if and only if =

0 < γ < 21 .

(B.10)

Substituting (B.9) into (B.10) and rearranging yields:21  12B 2 < 144B 2 k 2 ε¯ 4 +105k 4 ε¯ 8 −240k 3 ε¯ 6 B−13k 2 ε¯ 4 +24B 2 +16Bk¯ε 2 < 8Bk¯ε 2 . (B.11) The condition 12B 2 < 8Bk¯ε 2 is always met because of the concavity condition (3). In general, a lot of feasible parameter constellations meet the double inequality (B.11) (e.g. k ε¯ 2 = 4 and B = 1.5). Relation (B.11) shows that there are upper and lower bounds for both k ε¯ 2 and B. For example, the second part of the double inequality (B.11) will hold as long as k ε¯ 2 is sufficiently large for a given B. But by inspection of the first part of (B.11), we see that there is an upper bound for k ε¯ 2 . Intuitively, this upper bound is due to the fact that −NB2 ∂ ∗− ∗ {EU (γ ; s ) − EU (s )} = I 3 1 1 I I 1 1 ∂k ε¯ 2 8(BγI1 − B − k ε¯ 2 )3 k 2 ε¯ 4

(B.12)

with N = k 2 ε¯ 4 [B(11γI1 − 12) + γI1 (BγI1 − 3k ε¯ 2 )] − 12k ε¯ 2 B 2 (1 − γI1 )2 + 4B 3 (γI31 − 1) + 12B 3 γI1 (1 − γI1 ),

(B.13)

becomes negative for sufficiently large (small) values of k ε¯ 2 (B) because of γI1 < 1/2 and the concavity condition (3) (note that the last expression in the numerator term N is the only positive one). Thus, if k ε¯ 2 becomes too large a mismatch II equilibrium cannot be attractive ∗ for I1 any longer, i.e. EU∗− I1 (γI1 ; s3 ) < EUI1 (s1 ). At last, result (iii) has to be proved. This result requires the following three conditions to be met: γI1 , γI2 > 21 , EU∗i (γI1 ; s3 ) < EU∗i (s1 ), 21

(B.14) i = I1 , E,

Note that 144B 2 k 2 ε¯ 4 + 105k 4 ε¯ 8 − 240k 3 ε¯ 6 B = 4k 2 ε¯ 4 (6B − 5k ε¯ 2 )2 + 5k 4 ε¯ 8 is always positive.

(B.15)

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with EU∗i (γI1 ; s3 )

=

EU∗− i (γI1 ; s3 ),

if

1 2

EU∗+ i (γI1 ; s3 ),

if

1 < γI1

EU∗j (γI2 ; s2 ) < EU∗j (s1 ),

< γI1 ≤ 1

j = I2 , E,

(B.16)

with EU∗j (γI2 ; s2 )

=

EU∗− j (γI2 ; s2 ),

if

1 2

EU∗+ j (γI2 ; s2 ),

if

1 < γI2 .

< γI2 ≤ 1

Inequality (B.14) means that team formation—especially the team {I1 , E}—would be efficient. Together with condition (B.15) we have a mismatch III equilibrium, where I1 and E forego to form an efficient team. Relation (B.16) ensures that all players remain self-employed, which is inefficient. Let, for simplicity, γI1 = γI2 so that (B.16) can be neglected and we can concentrate on player I1 .22 Now, we have to differentiate between two cases. (1) When γI1 > 1, the “positive scenario” with EU∗+ i (γI1 ; s3 ) (i = I1 , E) has to be considered. Using the concrete expressions for the expected utilities we see that the conditions (B.14) and (B.15) are equivalent to Ω(γI1 ) < 0,

γI1 > 1,

(B.17)

where    Ω(γ ) := γ 2 B 3 + 13 k ε¯ 2 B 2 + γ 43 k ε¯ 2 B 2 +

5 2 4 12 k ε¯ B

− 2B 3



+ 13 k 2 ε¯ 4 B − 53 k ε¯ 2 B 2 − 16 k 3 ε¯ 6 + B 3 . The convex parabola described by (B.18) has two roots: √ 24B 2 − 5k 2 ε¯ 4 − 16Bk¯ε 2 + 57k 4 ε¯ 8 + 192k 3 ε¯ 6 B + 144k 2 ε¯ 4 B 2  γ = 8B(k ε¯ 2 + 3B) √ 24B 2 − 5k 2 ε¯ 4 − 16Bk¯ε 2 − 57k 4 ε¯ 8 + 192k 3 ε¯ 6 B + 144k 2 ε¯ 4 B 2  . γ = 8B(k ε¯ 2 + 3B)

(B.18)

(B.19)

(B.20)

Therefore, the minimum of the parabola Ω(γ ) lies below the horizontal axis. In this case, (B.17) means that the right-hand root of Ω(γ ) must be greater than 1, i.e. γ  > 1. This condition can be simplified to  57k 2 ε¯ 4 +192Bk¯ε 2 +144B 2 > 5k ε¯ 2 +24B ⇔ 27B 2 + 3Bk¯ε 2 < 2k 2 ε¯ 4 , (B.21) 22 I only plays a passive role in a mismatch III equilibrium. γ 2 I1 = γI2 implies that a team {I2 , E} will also not form and the inefficient outcome will result. All the players remain self-employed. Note, that the assumption γI1 = γI2 is not decisive for the result of Proposition 1(iii), which only gives a sufficient condition for the possibility of a mismatch III equilibrium.

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which will hold if B is sufficiently small and k and ε¯ are sufficiently large. (2) When (1/2) < γI1 ≤ 1, we have the “negative scenario” with EU∗− i (γI1 ; s3 ) (i = I1 , E). In this case, (B.14) and (B.15) will be equivalent to 1 < γI1 ≤ 1, 2

Λ(γI1 ) < 0,

(B.22) =

where Λ(γ ) is the concave parabola defined in (B.6) with the left-hand root γ and the right-hand root γ¯ (see (B.8) and (B.9)). There are different ways to check whether (B.22) = holds. For example, if γ > 1, the condition Λ(γI1 ) < 0 will hold for all γI1 $((1/2), 1]. The = inequality γ > 1 can be rearranged to  (B.23) k ε¯ 2 (13k ε¯ 2 − 24B) > 144B 2 k 2 ε¯ 4 + 105k 4 ε¯ 8 − 240k 3 ε¯ 6 B. Since the right-hand side of (B.23) is positive, the left-hand side has to be positive, too. (B.23) can be simplified to k ε¯ 2 > 29 B,

or

2k ε¯ 2 < 3B.

(B.24)

Comparing (B.24) and the concavity condition (3), we see that 2k ε¯ 2 < 3B cannot hold. Thus, k ε¯ 2 > (9/2)B must hold, which means that B has to be sufficiently small, and k and ε¯ have to be sufficiently large.23 䊐 Proof of Proposition 2. Consider the case where I1 offers E an entrance fee η > 0 in addition to the equal sharing arrangement to make him sign the team contract. Proposition 2 only claims that there is the possibility of a mismatch I equilibrium. Therefore, it suffices to show that a mismatch I may arise in certain situations. For this reason, the analysis is restricted to the case where γI1 , γI2 > 1, i.e. the “positive scenario” holds. Then, the conditions for a mismatch I are as follows: γI2 > γI1 > 1, ∗ EU∗+ i (γI2 ; s2 ) > EUi (s1 ),

(B.25) i = I2 , E,

(B.26)

∗+ EU∗+ I1 (γI1 ; s3 ) − η > EUI1 (γI2 ; s2 ).

(B.27)

∗+ EU∗+ E (γI1 ; s3 ) + η ≥ EUE (γI2 ; s2 ).

(B.28)

Conditions (B.25) and (B.26) define a mismatch I. First, a team {I2 , E} would be better than a team {I1 , E} from an efficiency viewpoint (i.e. γI2 > γI1 ); (B.25) also implies that the team {I2 , E} would be efficient (i.e. γI2 > 1/2). Secondly, (B.26) ensures that I2 and E prefer competing as a team {I2 , E} in the tournament to competing as single players. Therefore, without the interference of player I1 efficient matching would arise. Condition (B.27) guarantees that I1 offers a contract to E. Inequality (B.28) ensures that E will accept this offer. In this situation, I1 has all the bargaining power, because in t1 he makes a take-it-or-leave-it offer to E, whereas E has EU∗+ E (γI2 ; s2 ) as reservation value and can 23

Note that the left-hand side of (B.23) is always positive for k ε¯ 2 > (9/2)B.

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only choose between accepting or rejecting the offer in t2 . Therefore, I1 chooses η to make E just indifferent between his offer and forming a team with I2 , i.e. η = EU∗+ E (γI2 ; s2 ) −EU∗+ (γ ; s ). After substituting for the expected utilities in (B.26)−(B.28) and using I1 3 E ∗+ η = EU∗+ (γ ; s ) −EU (γ ; s ) we obtain the following three conditions for a mismatch I2 2 I1 3 E E I equilibrium with entrance fee: γI2 > γI1 > 1,

(B.29)

B 4B 2 (1 − γI2 )2 + Bk¯ε 2 (7γI2 − 8) + 2k 2 ε¯ 4 B B2 > , − 8 3 (BγI2 − B + k ε¯ 2 )2 2k ε¯ 2

(B.30)

2[4B 2 (1 − γI1 )2 + Bk¯ε 2 (7γI1 − 8) + 2k 2 ε¯ 4 ](BγI2 − B + k ε¯ 2 )2 −4k ε¯ 2 (k ε¯ 2 − B)(BγI1 − B + k ε¯ 2 )2 − [4B 2 (1 − γI2 )2 +Bk¯ε 2 (7γI2 − 8)+2k 2 ε¯ 4 ] ×(BγI1 − B + k ε¯ 2 )2 > 0

(B.31)

These conditions hold for a lot of feasible parameter constellations which can easily be checked by a numerical example (let, e.g. k = 2, ε¯ = 2, B = 1, γI2 = 5, γI1 = 4.5). 䊐 References Alchian, A.A., Demsetz, H., 1972. Production, information costs and economic organization. American Economic Review 62, 77–795. Caillaud, B., Jéhiel, P., 1998. Collusion in auctions with externalities. Rand Journal of Economics 29, 680–702. Drago, R., Garvey, G.T., Turnbull, G.K., 1996. A collective tournament. Economics Letters 50, 223–227. Getzen, T.E., 1984. A “brand name firm” theory of medical group practice. Journal of Industrial Economics 33, 199–215. Holmström, B., 1982. Moral hazard in teams. Bell Journal of Economics 13, 324–340. Jéhiel, P., Moldovanu, B., 1996. Strategic nonparticipation. Rand Journal of Economics 27, 84–98. Jéhiel, P., Moldovanu, B., Stacchetti E., 1996. How (not) to sell nuclear weapons. American Economic Review 86, 814–829. Jovanovic, B., 1984. Matching, turnover and unemployment. Journal of Political Economy 92, 108–122. Kräkel, M., 2000. Relative deprivation in rank–order tournaments. Labour Economics 7, 385–407. Kräkel, M., 2001. Strategic mismatches in sequential contracting: the case of professional partnerships. Jahrbuch für Wirtschaftswissenschaften 52, 25–39. Lazear, E.P., 1989. Pay equality and industrial politics. Journal of Political Economy 97, 561–580. Lazear, E.P., 1995, Personnel Economics. Cambridge University Press, Cambridge, MA. Lazear, E.P., Rosen, S., 1981. Rank-order tournaments as optimum labor contracts. Journal of Political Economy 89, 841–864. McAfee, R.P., McMillan, J., 1991. Optimal contracts for teams. International Economic Review 32, 561–577. Milgrom, P.R., Roberts, J., 1995. Complementarities and fit: strategy, structure and organizational change in manufacturing. Journal of Accounting and Economics 19, 179–208. Mortensen, D.T., 1988. Wages, separations and job tenure: on-the-job specific training or matching? Journal of Labor Economics 6, 445–471. Nalebuff, B.J., Stiglitz, J.E., 1983. Prizes and incentives: toward a general theory of compensation and competition. Bell Journal of Economics 14, 21–43. Newhouse, J.P., 1973. The economics of group practice. Journal of Human Resources 8, 37–56. O’Malley, M., O’Malley, S., 1994. Game day: the Blue Jays at sky dome, Toronto.