DISCUSSION PAPER SERIES IN ECONOMICS AND MANAGEMENT Competitive Careers as a Way to Mediocracy Matthias Kräkel Discussion Paper No. 10-14 GERMAN ECONOMIC ASSOCIATION OF BUSINESS ADMINISTRATION – GEABA Competitive Careers as a Way to Mediocracy∗ Matthias Kräkel† Abstract We show that in competitive careers based on individual performance the least productive individuals may have the highest probabilities to be promoted to top positions. These individuals have the lowest fall-back positions and, hence, the highest incentives to succeed in career contests. This detrimental incentive eﬀect exists irrespective of whether eﬀort and talent are substitutes or complements in the underlying contest-success function. However, in case of complements the incentive eﬀect may be outweighed by a productivity eﬀect that favors high eﬀort choices by the more talented individuals. Switching from wages-attached-to-jobs to pay-for-performance will work against mediocracy if applied to top jobs, but may be detrimental at lower career levels. The mediocracy problem will be aggravated if highability individuals decide to sandbag on lower career levels in order to avoid strong opponents at higher levels. Key Words: career competition; contest; mediocracy; sandbagging. JEL Classification: D72; J44; J45; M51 ∗ I would like to thank Oliver Gürtler, Daniel Müller, Petra Nieken, Anja Schöttner, the participants of the economics workshop of the Leibniz University of Hannover, in particular Hendrik Hakenes, Vilen Lipatov and Georgios Katsenos, and the participants of the annual meeting of the "Unternehmenstheoretischer und -politischer Ausschuss" of the German Economic Association (Verein für Socialpolitik), particularly Jürgen Eichberger, Dieter Pfaﬀ, Andreas Pfingsten, Kerstin Pull, Ulrike Stefani and Siegfried Trautmann, for helpful comments. Financial support by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR 15, is gratefully acknowledged. † University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228 733914, fax: +49 228 739210, e-mail: [email protected] 1 "mediocracy = A society in which people with little (if any) talent and skill are dominant and highly influential." 1 (dictionary) Introduction Career systems that are not based on individual performance but on criteria like seniority may end up in a situation where people with average or less than average talent are assigned to key positions. At the level of society, the direct consequence would be the emergence of a mediocracy. At first sight, one might suspect that competitive career systems with performance-based job assignment should lead to a significantly better outcome. However, in our paper we show that competitive career systems have an inherent tendency to promote the least productive individuals, thus leading to mediocracy. The intuition for this result comes from the fact that more productive people have better fall-back positions than less productive ones when failing in the competition for top positions. Hence, highly productive people have only moderate incentives to win the competition for top jobs, whereas individuals with low productivity have strong incentives to avoid their rather unattractive fall-back positions. We use a contest model to analyze competitive careers of heterogeneous individuals. At the beginning, all individuals have the chance to reach the top position with the highest possible career income.1 However, during the career we have losers and winners, where winners still compete for the top job but losers strive 1 Another motive for career competition is typically an increase in status. See Frank (1985) on the role of status as motivation in individual careers. Moldovanu, Sela and Shi (2007) discuss competition for status in a formal contest model. 2 towards less important positions in a so-called consolation match.2 Winning such consolation match defines the fall-back position of an individual. The fact that more productive individuals have better fall-back positions than less productive individuals does not depend on the underlying contest-success technology: In any kind of consolation contest, a more productive player can either achieve the same winning probability at less eﬀort costs or a higher winning probability at the same eﬀort costs compared to a less productive opponent. Hence, the expected utility of an individual when participating in a consolation contest will always be positively correlated with his productivity. As natural consequence, the individuals with the lowest productivities have the highest incentives to climb the ladder in order to avoid their respective fall-back positions.3 In the basic career model, we analyze a setting with eﬀort and talent (or productivity) being substitutes in the contest-success function of an individual. Our results show that near the top position — individuals only have to pass one hurdle for being assigned to the top job — the individuals with the worst fall-back positions will be most likely to win the contest for the top position, due to the incentive eﬀect mentioned above. If individuals have to pass more than one hurdle to reach the top, it will be crucial whether productivity has a higher impact on the fall-back position or the future career and how dissipative the competition is on diﬀerent career levels. For example, if consolation contests are suﬃciently dissipative (i.e., expected utility from participating in these contests is rather low), 2 See, for example, Uehara (2009) on consolation contests in a Japanese corporation. For a theoretical analysis of consolation contests see Kiyotaki (2004). 3 Anecdotal evidence for such kind of incentive eﬀect can be seen in the careers of Hollywood’s most successful actresses and actors. Many of them either had very unattractive jobs before they became superstars or orriginated from rather poor families (see, among many others, the CVs of Sandra Bullock, Harrison Ford, Megan Fox and Hillary Swank). Starting from the hypothesis that talent is uniformly distributed over all social classes of a society, the fact that so many superstars in Hollywood have really low fall-back positions can most naturally be explained by very strong incentives induced by the diﬀerence of superstar income and individual fall-back position. Note that, of course, this possible anecdotal evidence is not based on low ability but on low outside options or reservation utilities. 3 fall-back positions will be quite unattractive, but most unattractive for the least productive individuals. In this situation, the less productive individuals are most likely to reach the top. In the discussion, we consider a career model where talent and eﬀort are complements in the contest-success function. Again, there is a detrimental incentive eﬀect that makes winning of the top position by the least productive individuals most likely. However, now an additional eﬀect works into the opposite direction: The larger an individual’s productivity the more eﬀective will be the exertion of eﬀort to win the contest. For this reason, a highly talented individual may prefer to spend high eﬀort despite an attractive fall-back position. If this productivity eﬀect is dominated by the incentive eﬀect, society will still tend to mediocracy. Otherwise, the most talented people will assert themselves in the career competition and be assigned to the top position. Besides the application to society in general, our model also oﬀers insights for competitive careers in a more concrete situation. For example, it can be applied to career competition between politicians, who first struggle for being elected as local leader of their political party. Thereafter, they may further compete for becoming national party leader, governor or president of a certain state. As another example, consider the case of internal careers within corporate hierarchies. These hierarchies are often organized as internal labor markets with clearly structured career paths.4 Here, a manager first wants to be promoted to become division head. In further career steps, the manager may be elected to the board of directors or even compete for the position as chief executive oﬃcer. Moreover, there are many associations that have local oﬃces and a national or even international headquarter (e.g., worker and employer associations). In these organizations, individual careers typical proceed as follows: Individuals that suc4 See, for example, Baker, Gibbs and Holmström (1994). 4 cessfully managed their jobs in the local establishment get the chance of becoming chief of the local oﬃce. Thereafter, they may be promoted to the national or international headquarter where they can succeed in further career steps up to the top of the organization. There are also career ladders that include moves from one organization to another. Consider, for example, the European Central Bank (ECB). Often, individuals employed by the ECB come from national central banks where they have climbed the ladder before. In all these examples, the individuals that do not succeed to the top may win the consolation contest and become head of the local establishment or climb an inferior ladder in the headquarter. In the applications mentioned before, those wage policies would work against the problem of mediocracy that ensure highly talented individuals a significantly higher career income at the top position than less talented candidates. In other words, if wages are not attached to jobs but depend on the productivity of the respective individual, the top position should be quite attractive for the most talented candidates. For example, top positions could be combined with performancebased incentive schemes so that highly productive individuals have suﬃciently large expected incomes when reaching the top. By this wage policy, the detrimental incentive eﬀect stated above could be turned into a beneficial one. This possible solution by using pay for performance at the top is addressed in the discussion part of the paper. However, our results also point out that pay for performance at lower career levels may aggravate the mediocracy problem. Furthermore, there are situations where highly talented individuals can either compete against other strong opponents for the top positions or play against a significantly weaker field for less attractive jobs. If the pay (and status) gap between the top positions and the less attractive ones is not too large and the competition against other strong opponents really exhausting it may be optimal for talented individuals not to strive towards the top positions. This eﬀect in 5 competitive careers would further explain the deficit of top-class candidates for top positions. Note that this sandbagging strategy may be optimal for either high-potential individual so that they face a coordination problem: If there are two highly talented players and both prefer to sandbag, then they will end up in a consolation contest with strong competition. Of course, such failed coordination would exacerbate the mediocracy problem. We discuss the sandbagging issue in the final subsection of the discussion. Our paper is related to the literature on contests and rank-order tournaments. There are two widely used classes of contest-success functions — the ratio-form and the diﬀerence-form contest-success function. The former one was introduced by Tullock (1980) for the discussion of rent seeking, and generalized and axiomatized by Skaperdas (1996). The latter one is used by rank-order tournament models in labor economics that discuss job promotions and worker careers. Most of these papers build on Lazear and Rosen (1981) and Nalebuﬀ and Stiglitz (1983). A dynamic tournament with subsequent career steps is analyzed by Rosen (1986).5 The basic model and most of the subsequent parts are based on Skaperdas’ (1996) ratio-form contest-success function. The discussion of the sandbagging problem in Subsection 4.3 uses a diﬀerence-form model to highlight the competitive eﬀects in a clearly laid out setting. There are also parallels to the public economics literature on bad politicians. This literature oﬀers an explanation for why political leaders and presidents often have rather low qualifications or are of low quality. Contrary to our paper, they do not consider a contest framework but follow either the political-agency or the citizen-candidate approach (see, among many others, Caselli and Morelli (2004), Messner and Polborn (2004), Mattozzi and Merlo (2007)). In the model 5 There are also contest papers that discuss the diﬀerence-form contest-success function in a more general context; see Dixit (1987), Hirshleifer (1989), Baik (1998) and Che and Gale (2000). Konrad (2009) oﬀers a comprehensive overview of the most frequently used contestsuccess functions, applications of contest models and the most important results in contest theory. 6 by Caselli and Morelli (2004), bad and dishonest candidates have rather low opportunity costs and extract more money from their political positions. These two eﬀects make them more likely run for oﬃce. Messner and Polborn (2004) consider a similar approach, but in their setting political candidates care for both their salaries and the impact of the politician’s ability on the quality of the political oﬃce. Mattozzi and Merlo (2007) show that, due to competition for highly talented individuals between the political sector and the lobbying sector, a political party may prefer not to recruit the best candidates although they would accept a contract oﬀer. However, all these papers do not address the individuals’ incentives and endogenous investments or activities during their careers, which take center stage in our model. Finally, the part on sandbagging is related to the ratchet eﬀect in dynamic principal-agent models (Baron and Besanko 1984; Freixas, Guesnerie and Tirole 1985; Laﬀont and Tirole 1988): Agents might prefer to withhold eﬀort in the first period in order to influence the incentive contract for the second period. High eﬀort and, hence, good performance in the first period may make the principal adjust the performance standard upwards for the next period. In order to meet the new high standard the agent has to exert significantly more eﬀort. Since he anticipates the adjustment of the standard, the agent optimally chooses low eﬀort in the beginning of the contractual relationship. The sandbagging problem diﬀers from the ratchet eﬀect as the latter one does not lead to a coordination problem between several agents. The paper is organized as follows. We start with the analysis of talent and effort being substitutes in the contest-success function. Section 2 deals with the case where individuals are near the top and only have to pass one hurdle to reach the highest possible position. Section 3 considers a two-hurdle career model with productivity influencing both an individual’s fall-back position and his future career. 7 Section 4 discusses the main findings. Subsection 4.1 oﬀers a robustness check. Here, we analyze whether the mediocracy result also holds under the assumption of talent and eﬀort being complements in the contest-success function. In Subsection 4.2, pay for performance at top positions is analyzed as a possible solution to the mediocracy problem. Subsection 4.3 addresses sandbagging of high-ability players, which may exacerbate the mediocracy problem. Section 5 concludes. 2 One-Hurdle Career 2.1 Basic Model We consider a career game between ≥ 3 risk neutral individuals (e.g., workers of the same cohort entering a firm in a certain period). The structure of this game is sketched in Figure 1. [Figure 1] The players start by simultaneously choosing productive activities (e.g., eﬀorts) ≥ 0 ( = 1 ) to become the winner of the major career contest. This winner receives a high career income . The − 1 other players enter a consolation match where they compete for less attractive positions. The winner of this consolation contest earns , whereas the − 2 losers get with 0 .6 We assume that the three career incomes are suﬃciently large so that players choose positive eﬀorts in equilibrium. In both, the major contest for and the consolation contest each player has to bear the costs of his activity (e.g., disutility of eﬀort). For simplicity, when player exerts activity level let his costs be equal to . Moreover, in both 6 Note that in our model career incomes are exogenously given, but as we can see below the general eﬀects still exist in a situation where a career designer endogenously chooses incomes. 8 contests, players face the same contest-success function based on the one suggested by Skaperdas (1996):7 If player chooses and the other players 6= choose activity levels , then ’s probability of winning is given by = ( − ) P ( − ) + 6= ( − ) (1) with (·) as monotonically increasing impact function that is concave and strictly positive.8 Hence, each player increases his winning probability by exerting more eﬀort, but the other contestants’ eﬀorts as well as luck or measurement error also influence the outcome of the contest. The parameter 0 indicates player ’s productivity.9 The lower the value of the more productive will be the player. If, for example, all players choose identical activity levels, the most productive player will have the highest winning probability, whereas the player with the lowest productivity is least likely to win the contest. The scaling parameter 0 is identical for all contestants. As can be seen from the equilibrium outcomes below, measures how dissipative the contest is (i.e., the larger the smaller will be an individual player’s expected utility from participating). We assume that productivity parameters 1 are common knowledge. This assumption can be justified for at least two reasons. First, players typically observe each others’ qualifications, which are positively correlated with productivity.10 Second, note that the common-knowledge assumption is only introduced 7 This assumption can be motivated by the fact that both contests take place between the same individuals excepting the first-round winner, who is missing in the consolation round. 8 As the player’s activity is assumed to be productive, ( − ) may describe a worker’s or a politician’s output. 9 Hence, eﬀort and ability/productivity are substitutes in the players’ impact function. The case of complements will be discussed in Subsection 4.1. 10 On the one hand, following human capital theory based on Becker (1962), investment in a player’s skills rises his productivity, which is certified by a formal qualification. On the other hand, according to Spence (1973), high qualifications can be a credible signal for corresponding high ability. 9 for the contestants.11 These players often observe each other when working close together at the same organization every day. For example, the two arguments hold for a situation where workers compete for job promotion in internal labor markets and for a situation where politicians compete for becoming the leader of their political party. Although we do not explicitly specify a welfare function, we assume that welfare will be highest if the most productive player (i.e., the one with parameter value min{ 1 }) passes the decisive hurdle by winning the major contest and is assigned to the most important position, which is associated with career income . 2.2 Consolation Contest We start the analysis by solving the contest game of those − 1 players that have failed in the major contest and, therefore, enter the consolation stage. Here, the winner earns career income but the − 2 losers end up with the lower income . Player chooses activity to maximize expected utility g ( ) = + ( − ) ( − ) P − ( − ) + 6= ( − ) Since this objective function is strictly concave and, by assumption, career incomes are suﬃciently large to prevent a corner solution where neither player chooses positive eﬀort, the equilibrium is described by the − 1 first-order conditions 11 P ( − ) 0 ( − ) 6= ( − ) = 1 h i2 P ( − ) + 6= ( − ) We do not have a contest designer, who shares the same information as the contestants. Note that our results will not change when introducing an uninformed designer, who endogenously chooses optimal tournament prizes but cannot observe individual productivities. 10 which can be rearranged to − 1 h i i2 = P −1 0 ( − ) ( − ) + 6= ( − ) =1 ( − ) hP with player denoting an arbitrary opponent of player . In analogy, the first-order condition of player 6= can be written as − 1 h i hP i2 = P −1 0 ( − ) ( − ) + ( − ) 6= =1 ( − ) Combining the right-hand sides of both equations yields P P ( − ) + 6= ( − ) ( − ) + 6= ( − ) = 0 ( − ) 0 ( − ) As both sides describe the same monotonically increasing function of − and − , respectively, we have − = − ∀ ( ) Hence, the higher the productivity of a player the lower will be his equilibrium eﬀort. Intuitively, a highly productive player prefers to save eﬀort costs by choosing a low activity level since activities and productivity parameters are substitutes in the impact function (·). By inserting − = − ∀ ( ) in player ’s first-order condition, the equilibrium activity level ∗ is described by ¶ µ ( − 2) ( − ) 1 ( − 2) ( − ) ∗ + ⇔ = = 0 ∗ ( − ) ( − 1)2 (∗ − ) ( − 1)2 with (·) denoting the monotonically increasing inverse function of 0 . Equilibrium activity increases in the size of the career income diﬀerence − , because 11 any contestant earns at least in the consolation round. Furthermore, ∗ is nonincreasing and for 3 strictly decreasing in the number of contestants. This eﬀect can be labeled discouragement eﬀect: Each player exerts less eﬀort when the number of opponents increases since his relative impact on the outcome of the contest becomes smaller. g ( ). Finally, we insert equilibrium activities in the objective function Thus, in equilibrium player ’s expected utility is given by ¶ µ ∗ ( − 2) ( − ) − ∗ g − − ( ) = + −1 ( − 1)2 Now, we can see why measures the contest’s degree of dissipation. Note that, in equilibrium, a player’s winning probability is always 1 ( − 1), irrespective of the value of . However, a player’s equilibrium activity and, therefore, his eﬀort costs rise in . Consequently, the higher , the lower will be each player’s expected utility from participating in the consolation match. Nevertheless, a player strictly g ∗ (∗ ) decreases in ). benefits from higher productivity (i.e., 2.3 Major Contest A player will earn the highest career income , if he passes the hurdle by winning the major contest. In case of losing, he will be relegated to the consolation contest g ∗ (∗ ). Hence, player ’s objective function in associated with expected utility the major contest can be written as ( − ) P ( − ) + 6= ( − ) Ã ! ∗ − ) ( g (∗ ) 1 − P + − ( − ) + 6= ( − ) d ( ) = 12 ³ ´ ∗ ∗ g − ( ) · ( − ) g ∗ (∗ ) + P − = ( − ) + 6= ( − ) In analogy to the consolation contest, the first-order conditions of two arbitrary players and are given by and g ∗ (∗ ) 1 − P = 0 P ( − ) [ ( − ) + 6= ( − )] [ =1 ( − )]2 (2) ∗ g (∗ ) − 1 P = 0 P ( − ) [ ( − ) + 6= ( − )] [ =1 ( − )]2 (3) g ∗ (∗ ) g ∗ (∗ ) and Hence, if , then we have 1 h i P 0 ( − ) ( − ) + 6= ( − ) 1 h i⇔ P 0 ( − ) ( − ) + 6= ( − ) P P ( − ) + 6= ( − ) ( − ) + 6= ( − ) 0 ( − ) 0 ( − ) P which implies − − since [ () + 6= ( − )] 0 () is a monotonically increasing function of . Let (1) (2) · · · () denote the order of the players’ productivity parameters (i.e., player (1) is the most productive one), ∗(1) ∗(2) ∗() the respective equilibrium eﬀorts and ∗(1) ∗(2) ∗() the winning probabilities in the major contest. Then we obtain the following result: Proposition 1 The players’ winning probabilities in the major contest satisfy ∗(1) ∗(2) ∗() and the corresponding equilibrium eﬀorts ∗(1) ∗(2) ∗() . 13 Proof. The first part immediately follows from the fact that implies P − − and that = ( − ) [ ( − ) + 6= ( − )] is strictly increasing in − . The second part follows from − − ⇔ − − ( − ) 0. Proposition 1 shows that the more productive a player, the less likely he will win the major contest and the less eﬀort he will choose. The intuition for the first result comes from the players’ diﬀerent fall-back positions in the major contest. If a player has a large productivity (i.e., a small ), then he will also be a strong player in the consolation match, which will guarantee him a large expected utility g ∗ (∗ ) as a kind of fall-back position. This fact reduces his incentives in the major contest so that we have a tendency to mediocracy where key positions are filled by less productive individuals, leading to welfare losses. In other words, participation in the consolation match is rather unattractive for the less productive players so that they have very strong incentives to win the major contest. Note that although we assume career incomes to be exogenously given, our results of Proposition 1 as well as the following results will qualitatively hold under endogenous incomes or prizes that are optimally chosen by a contest designer before the competition starts. The career incomes or contest prizes of course influence the levels of all players’ equilibrium eﬀorts. However, they neither have an impact on the players’ eﬀort diﬀerences − in the consolation match nor an impact on the ranking between − and − in the major contest. As the same holds for the findings in the following sections our results are robust with respect of exogeneity/endogeneity of the contest prizes. The result on eﬀort ranking ∗(1) ∗(2) ∗() stems from the intuition before together with the fact that eﬀort and productivity are substitutes in the impact function. Hence, even if all players had identical fall-back positions, the order ∗(1) ∗(2) ∗( ) would not change. Welfare was solely defined via 14 the career decision based on the outcome of the major contest. However, if the activity levels ∗ ( = 1 ), which are productive by assumption, were also important for welfare considerations, we might have a second source for welfare losses since the most productive individuals choose the lowest activity levels. 3 Two-Hurdle Career In this section, we consider the case where a player has to pass two career hurdles to reach the top. This situation is described by Figure 2. [Figure 2] First, a player must assert himself in the organizational unit that he belongs to (e.g., a division or aﬃliate of a corporation, a certain political party). Second, when being successful in becoming head of the organizational unit (e.g., division head or party leader), the player enters a higher-order contest to reach the top position of his career path (e.g., CEO of the corporation or president of a state). As crucial diﬀerence to the one-hurdle career, now a player has to succeed two times before reaching the top and a player’s productivity has both an influence on his fall-back position and an influence on his future career. 3.1 Model Modifications There are two organizational units, and , that consist of and members, respectively, with ≥ 3 ( ∈ { }). In each unit , the members compete for becoming unit head (level-I contests). These two contests are modeled analogously to those in Subsection 2.1. Each member ∈ {1 } chooses activity level ≥ 0 (at cost ), which influences his probability of winning the 15 level-I contest for unit head , = ( − ) P ( − ) + 6= ( − ) (4) Again, indicates the productivity of the respective player, with lower values corresponding to higher productivities. The parameter 0 measures the grade of dissipation in the level-I contests.12 If player wins and becomes unit head he will enter the higher-order level-II contest where he competes against the other unit head to reach the top of his career. However, the − 1 losers are relegated to a consolation contest in unit where the winner earns career income 0 and the − 2 losers get 0. As outcome of the level-I contests, two players — the unit heads and — enter the level-II contest. Here, they compete for the unique top position, which is associated with career income . Whereas the winner of the level-II contest is assigned to this top position, the loser gets a lower career income with 0 . Career incomes are assumed to be suﬃciently large so that the players exert strictly positive eﬀorts in each contest. The contest-success function for the level-II contest is again a Skaperdas-type function like (1) and (4). The scaling parameter for measuring the degree of dissipation at level II is denoted by 0. As in Section 2, we assume that from a welfare perspective the player with the highest productivity (i.e., the one with parameter min{1 | ∈ { }}) should be assigned to the top position with career income . 12 Our results will not change qualitatively, if we define diﬀerent parameters and for the two units. 16 3.2 Level-II Contest Let and denote the productivity parameters of the two unit heads, who enter the level-II contest, and and the corresponding activity variables. In the contest, the head of unit maximizes ( − ) − Ω∈{} (Ω − Ω ) d ( ) = + ( − ) P Proceeding in the same way as in Section 2, we find that, in equilibrium, activity levels are described by − 1 i2 = ( − ) 0 ( − ) Ω∈{} (Ω − Ω ) hP = 1 ( − ) 0 ( − ) yielding ( − ) 0 ( − ) = ( − ) 0 ( − ) and, hence, − = − . Altogether, in the level-II contest the head of organizational unit optimally chooses ∗ µ − = 4 ¶ + and gets expected utility d ∗ µ ¶ − − − − ( ) = + 2 4 We can see that in equilibrium the more productive unit head exerts less eﬀort than the other head, but has the same probability of being promoted to the top position since the higher productivity completely outweighs the eﬀort deficit. Thus, it is pure luck whether the better head is assigned to the top job or not. 17 3.3 Level-I Contests Consider the contest in organizational unit . The objective function of member can be written as ´ ³ ∗ ∗ d g − · ( − ) g ∗ + P ( ) = − ( − ) + 6= ( − ) If player wins, he will earn the expected utility from participating in the leveld ∗ . In case of losing, he will enter the consolation contest of his II contest, g ∗ . Hence, his fall-back position is unit , where he receives expected utility g ∗ and the extra utility from being successful at level I by characterized by g ∗ . From the first-order condition we obtain the following description d ∗ − of player ’s equilibrium activity: g ∗ d ∗ − 1 hP i i2 = 0 ( − ) ( − ) 6= =1 ( − ) hP (5) To further characterize the equilibrium activities, we have to calculate player ’s g ∗ . Applying the results for the consolation contest of Subfall-back position section 2.2 yields g ∗ (∗ ) ¶ µ ( − 2) − − = − 1 ( − 1)2 and, therefore,13 g ∗ = + − + ( − ) d ∗ − 2 − 1 ¶ µ ¶ µ − ( − 2) − + 4 ( − 1)2 13 Recall that, by assumption, career incomes guarantee interior solutions for the equilibrium ∗ ∗ g 0. d − activities. Hence, we must have 18 Hence, if then for two arbitrary members and of unit in the level-I d ∗ − g ∗ d ∗ − g ∗ ⇔ . The first-order contest we have conditions (5) of players and can be written as " P d ∗ − g ∗ =1 #2 = 0 1 P ( − ) [ ( − ) + ( − )] 6= ( − ) and " g ∗ d ∗ − P =1 #2 = 0 ( − ) 1 P ( − ) [ ( − ) + ( − )] 6= d ∗ − g ∗ d ∗ − g ∗ then Thus, if ( − ) + P 6= ( − ) 0 ( − ) ( − ) + P 6= ( − ) 0 ( − ) which implies − − , analogously to the findings in Subsection 2.3. Since this comparison holds for any two members of organizational unit ∈ { } and since players have identical winning probabilities in the level-II contest, we have proven the following result: Proposition 2 If , then the most (least) productive player of each unit ∈ { }, i.e., the player with = min{ 1 } (with = max{ 1 }), has the lowest (highest) probability to reach the top position with career income . Proposition 2 shows that the mediocracy result of the one-hurdle career will prevail under the assumptions of the two-hurdle career if the level-I contests are more dissipative then the level-II contest. The intuition for this result is the 19 following. The higher relative to , the less attractive will be participation in the consolation match for the players. Moreover, since expected utility increases in the productivity of a player (i.e., decreases in ), the less productive a player the stronger will be his incentives to avoid a consolation match and, hence, to pass the hurdle that leads to participation in the level-II contest. A situation with level-I contests being more dissipative than the level-II contest seems quite realistic. For example, note that a single player is less visible in the large contest against − 1 opponents on level I compared to the level-II contest with two players. Therefore, it is rather diﬃcult for an individual to stand up to his opponents on level I compared to level II. In order to become winner on level I, each player has to spend a lot of eﬀort, which makes participation in the contest costly and, hence, level-I contests quite dissipative.14 4 Discussion The purpose of this section is twofold. First, we will check the robustness of the main result on mediocracy. We will analyze whether the most able individuals are still least likely to reach the top if activities and productivities are not substitutes any longer (Subsection 4.1) or if salary at the top job depends on the job holder’s ability (Subsection 4.2). Second, we will address the sandbagging problem mentioned in the introduction: High-ability individuals might prefer to sandbag in order to avoid playing against a strong field (Subsection 4.3). This eﬀect could exacerbate the mediocracy problem. 14 Note that equilibrium eﬀorts strictly increase in the dissipation parameters , and , respectively. 20 4.1 Activities and Productivities as Complements So far we have assumed that activities ( ) and productivity parameters ( ) are substitutes in the players’ impact function (·). This assumption drives part of the previous results, in particular the findings that players with higher productivities (i.e., lower values of ) choose lower eﬀorts in equilibrium. However, this paper does not focus on eﬀort choice but on the probability that the most productive player is not assigned to the top career position. In this subsection, we will check the robustness of the finding that the correlation between productivity and promotion probability of a player may be strictly negative. For this purpose, we reconsider the one-hurdle career model and assume that player ’s contest-success function is given by ˇ = ³ ´ + ³ ´ P 6= ³ ´ (6) with (·) denoting the same strictly positive, increasing and concave impact function as before. Again, the smaller the more productive will be the respective player, and for the case of identical activity levels by all players the most productive one has the highest winning probability. However, comparison of (1) and (6) shows that (besides skipping the dissipation parameter ) the contest-success functions and ˇ diﬀer significantly. Contrary to (1), now the activity variable and the productivity parameter are complements in the sense that lower values of make higher activity levels more eﬀective.15 As in Section 2, we first solve the consolation contest game and then turn g ( ) = to the major contest. In the consolation contest, player maximizes 15 Activities ³ ´ and productivities are complements in the contest-success function in the sense of 0. 2 1 21 + ( − ) · ˇ − . The first-order conditions of two arbitrary players and can be combined to ( − ) ³ ´h ³ ´ P ³ ´i ³ ´i2 = hP −1 0 + 6= =1 = (7) ³ ´h ³ ´ P ³ ´i 0 + 6= which yields the following equilibrium outcome: Lemma 1 In the consolation contest, if then (i) ˇ∗ ˇ∗ and (ii) g ∗ (∗ ). g ∗ (∗ ) Proof. Part (i) can be shown by contradiction. Suppose that ˇ∗ ≤ ˇ∗ ⇔ ∗ ∗ (8) From the first-order conditions (7) we obtain h ³ ∗´ P h ³ ∗´ P ³ ∗ ´i ³ ∗ ´i + 6= + 6= ³ ∗´ ³ ∗´ = 0 0 Since [ () + P 6= (9) ³ ∗ ´ ] 0 () is a monotonically increasing function of , (8) and (9) can only be satisfied at the same time if , a contradiction. (ii) Since player can always choose the same eﬀort level as so that he has the same g ∗ (∗ ) g ∗ (∗ ) eﬀort costs but a higher winning probability we must have in equilibrium. Lemma 1 points out that, in the consolation match, more productive players have higher winning probabilities and larger expected utilities than less productive players. Result (ii) is also important for the major contest, where players compete 22 for the top position with income . Due to the positive correlation between productivity and expected utility, more productive players have better fall-back g ∗ (∗ ) in the major contest, leading to less incentives. This eﬀect also positions works in the models with substitutes (Sections 2 and 3) and will be called incentive eﬀect. d ( ) = g ∗ (∗ )+( − g ∗ (∗ ))· In the major contest, player maximizes ˇ − . The first-order conditions of two players and , g ∗ (∗ ) − ³ ´i hP ³ ´i2 = ³ ´ h ³ ´ P 0 + 6= g ∗ (∗ ) − ³ ´i and hP ³ ´i2 = ³ ´ h ³ ´ P 0 + 6= (10) g ∗ (∗ ) imply g ∗ (∗ ) together with h ³ ´ P h ³ ´ P ³ ´i ³ ´i + 6= + 6= ³ ´ ³ ´ 0 0 The inequality shows that solutions of type (11) are always possible. Such outcomes coincide with the findings above where more productive players are less likely to obtain the top career position. However, the parameters and in the numerators of the two sides in (11) indicate that we cannot rule out solutions with if is suﬃciently small and suﬃciently large. This eﬀect can be labeled productivity eﬀect. Hence, if activities and productivity parameters are complements in the impact function, we will have two eﬀects that work into opposite directions. Coming back to our question regarding eﬃcient assignment at the top we obtain the following result: Proposition 3 Consider the major contest for the top position with income 23 and let . If g ∗ (∗ ) − ∗ g (∗ ) − then ˇ∗ ˇ∗ . (12) Proof. Combining the first-order conditions (10) yields ³ ´ + P 0 6= ³ ´ ³ ´ i ³ ´ P ³ ´ h ∗ ∗ g − ( ) + 6= ³ ´ i = h 0 g ∗ (∗ ) − If (12) is satisfied, we will have and, thus, ³ ´ , + P 0 6= ³ ´ ³ ´ ³ ´ + P 0 6= ³ ´ ³ ´ which implies ˇ∗ ˇ∗ for the winning probabilities in equilib- rium according to (6). From Lemma 1(ii) we know that both sides of inequality (12) are larger than one so that there exist parameter constellations for which (12) holds and others for which (12) does not hold. Condition (12) points out the two opposing eﬀects. While the right-hand side describes the incentive eﬀect, the left-hand side characterizes the productivity effect. If the incentive eﬀect dominates the productivity eﬀect, more productive players will have lower winning probabilities in the major contest than less productive ones. In this case, the mediocracy result under substitutes qualitatively still holds for activities and productivities being complements. 4.2 Individual Salaries So far, we have assumed that all players have the same career incomes as contest prizes. This assumption is realistic for those cases where wages are attached to 24 jobs. Such wage policy can be often observed in politics and (public) bureaucracies. Here, we have a clear bundle of tasks that is assigned to a certain job. These tasks determine the job holder’s qualification as well as his salary. Moreover, wages that are attached to jobs are one of the key assumptions within the concept of internal labor markets.16 Finally, if workers’ performance signals are unverifiable, tying wages to jobs is necessary for a firm to use job-promotion tournaments as credible incentive schemes.17 However, there are also jobs with verifiable performance signals, allowing pay for performance. In these cases, a more able player has a higher expected career income at a certain position than a less able one. In this subsection, we will discuss whether such individual salaries for the same job will change the main findings on competitive careers and mediocracy. As in the previous subsection, we reconsider the one-hurdle career model of Section 2. The only modification of this model is the introduction of individualized career incomes or contest prizes in this subsection. Hence, player has career incomes , and with ( = ) if player is more able than player (i.e., if ). First, we can analyze the eﬀect of individual career incomes on players’ expected g ( ) ( = 1 ). utilities from participating in the consolation contest, Since, by assumption, individual career incomes increase in the players’ abilities, participation in the consolation contest will become more attractive for high-ability players than for low-ability ones. This eﬀect would strengthen the mediocracy result of Section 2. However, since players’ equilibrium eﬀorts react to individualized career incomes, which influences both eﬀort costs and winning probabilities, we 16 See Doeringer and Piore (1971), Williamson, Wachter and Harris (1975). See Malcomson (1984, 1986). Without the self-commitment property of wages-attached to jobs, the employer would always promote the worker with the lowest promised salary for the vacant job in order to save labor costs. Since such opportunistic behavior is anticipated by the workers, job-promotion tournaments can only create incentives if salaries are linked with positions. 17 25 have to do a comparative-static analysis of the equilibrium eﬀorts with respect to career incomes. Let, for simplicity, = 3 so that two players remain in the consolation contest, say players and . Moreover, let player be more able than player . Player maximizes g ( ) = + ∆ ( − ) − ( − ) + ( − ) g ( ) = +∆ ( − ) − ( − ) + ( − ) with ∆ := ( − ) Analogously, the objective function of player reads as with ∆ := ( − ) Obviously, a player’s expected utility increases in the respective base income or which would strengthen the mediocracy result because of . Eﬀort costs as well as winning probabilities depend on the players’ income spreads ∆ and ∆ . From the first-order conditions for the equilibrium eﬀorts ∗ and ∗ we obtain the following comparative static results (see Appendix A): ∗ 0 ∆ ∗ T 0 ∆ ∗ T 0 ∆ ∗ 0 ∆ if ∗ − T ∗ − if ∗ − T ∗ − The results show that a player’s equilibrium eﬀort will increase if his income spread becomes larger. The reaction to the opponent’s income spread depends on the initial situation. If, for example, the more able player is leading in the initial situation in the sense of ∗ − ∗ − , then a larger income spread of his opponent, ∆ , implies a further increase of ∗ . Intuitively, higher incentives for player (via an increase of ∆ ) brings him back into the race so that the contest 26 becomes more intense. Technically, we obtain ∗ ∆ 0. As a consequence, competition becomes more balanced which implies a higher eﬀort by player as well. In general, it is not clear whether higher absolute career incomes for more able players are accompanied by higher income spreads. However, the comparative static results show what happens in this case. Suppose we switch from a situation with wages-attached-to-jobs and ∆ = ∆ = ∆ to individual career incomes, leading to ∆ ∆ = ∆ . This switch additionally motivates the more able player and discourages the less able one; thus, ∗ goes up and ∗ goes down so that both player ’s winning probability and his eﬀort costs increase. The total eﬀect unambiguously leads to a higher expected utility of player :18 Starting from ∗ − = ∗ − and ∆ = ∆ in the wages-attached-to-jobs situation and marginally increasing ∆ only leads to a reaction by player — he will increase his eﬀort whereas player will not change his eﬀort since initially ∗ − = ∗ − . Therefore, winning probability and eﬀort costs of increase. Note that this reaction g ∗ (∗ ), because player optimally reacts to an increase yields a higher value of in and and not changing ∗ at all would already lead to a higher expected utility due to higher career incomes. The next marginal increase of ∆ results into reactions by both players and since now ∗ − ∗ − . Player is discouraged and reduces ∗ whereas increases ∗ . Player ’s reaction benefits g ∗ (∗ ) since ’s winning probability increases. Player ’s reaction must increase as well since it would not be rational otherwise. A third marginal increase of ∆ has the same implications as the marginal increase before and so on. Altogether, the introduction of individualized career incomes that lead to and ∆ 18 As an example, consider the case of a linear impact function ( − ) = − . For this case, we can compute explicit solutions for the players’ equilibrium eﬀorts. We obtain ∗ − = ∆ ∆2 [∆ + ∆ ]2 and ∗ − = ∆2 ∆ [∆ + ∆ ]2 . Inserting into ’s ∗ ∆3 g (∗ ) = + objective function gives 2 − , which is strictly increasing in [∆ +∆ ] and ∆ . 27 g ∗ (∗ ) and g ∗ (∗ ), thus strengthening ∆ further increase the gap between the mediocracy result. In a second step, we can analyze the impact of individualized career incomes on players’ behavior in the major contest. Here, individual incomes and with have straightforward consequences. We obtain first-order conditions similar to (2) and (3). As only diﬀerence now the left-hand sides are given by g ∗ (∗ )][P ( − )]2 and [ − g ∗ (∗ )][P ( − )]2 , [ − =1 =1 respectively. Career incomes additionally motivate the more able player, thus working against the mediocracy result. If individual career incomes at the ∗ ∗ g (∗ ) − g (∗ ), the mediocracy problem top are so large that − will be even reversed. To sum up, whereas individual career incomes along the players’ career paths may aggravate the mediocracy problem, individual career incomes at the top position dependent on the job holder’s ability or performance unambiguously work against mediocracy. This finding clearly speaks against wages being attached to jobs at the top of bureaucracies and governments. For example, it may be worthwhile thinking about pay for performance at top positions in politics that depends on voter satisfaction being evaluated in regular time intervals. 4.3 Sandbagging In this subsection, we consider another eﬀect which may further exacerbate the mediocracy problem: If a high-ability player expects to compete against other highability individuals when entering the major contest for the top job, he may prefer not to participate in that contest. In particular, he could sandbag at the preceding career stage to avoid promotion to the major contest. Such sandbagging has three advantages. First, the high-ability player saves eﬀort costs at the preceding career stage. Second, he avoids strong opponents and, hence, a relatively low winning 28 probability in the major contest. Third, he avoids a rather homogeneous field in the major contest, which would lead to strong competition and high eﬀort costs. If the income at the top job does not diﬀer too much from the income at less important jobs, sandbagging can be a rational strategy. However, since each of the high-ability players may prefer to sandbag, these players face a non-trivial coordination problem when they choose eﬀorts at the preceding career stage. To analyze the sandbagging problem, we consider a stylized career model with two rounds and two contests in each round. The two contests in the first round have heterogeneous players: In each contest, a more able player with talent is matched with a less able one of talent ∈ [0 ). In the second round, the two winners of the first-period contests enter the major contest. The winner of this contest gets the top job with income , whereas the loser gets nothing. The two losers of the first-period contests are assigned to a consolation contest for less important positions. The winner of the consolation contest receives income ∈ (0 ), whereas the loser gets nothing. In order to highlight the diﬀerent strategic eﬀects that may lead to sandbagging, in this subsection we switch from the ratio-form contest-success function to the diﬀerence-form one.19 We assume that player ’s winning probability when competing against player is now given by ˆ = ( ( ) + − ( ) − ) with and denoting the players’ eﬀort choices, whereas and describe the players’ exogenous talents. (·) denotes the same positive, monotonically increasing and concave impact function as in the basic model and (·) a cumulative 19 For the diﬀerence-form contest-success function see Dixit (1987), Hirshleifer (1989), Baik (1998), Che and Gale (2000), and the literature on rank-order tournaments in labor economics, based on Lazear and Rosen (1981) and Nalebuﬀ and Stiglitz (1983). 29 distribution function with density (·) := 0 (·) that is symmetric around its mean at zero (e.g., a normal distribution with mean zero).20 Player ’s winning probability can be written as ˆ = 1 − ( ( ) + − ( ) − ) = ( ( ) + − ( ) − ) We assume that the functions (·) and (·) are well-behaved so that equilibria in pure strategies exist.21 We can solve the two-stage game by backward induction, beginning with the major contest and the consolation contest in round two. The equilibrium outcomes of these two contests are anticipated by the players in the two heterogeneous contests in round one when deciding on eﬀorts and, hence, on whether to sandbag or not. Let ∆ := − , and let (·) denote the inverse function of 1 0 (·), which is monotonically increasing since (·) is concave. Furthermore, call, for example, ¯ denote the -player’s winning the two round-one contests and ; then let probability in contest from the perspective of the two players in contest . In other words, when the two heterogeneous players choose their eﬀorts in one of the round-one contests they know that the high-ability player of the other contest will ¯ The solution of the game has the following properties: win with probability . Proposition 4 In the round-one contests, either both -players are more likely to win or in one contest the -player has a higher winning probability and in the 20 For example, in the rank-order tournament models considered in labor economics, contestants and have performance signals = ( )+ + ( = ) with and as idiosyncratic noise terms (e.g., denoting measurement errors) that are identically and independently distributed (see, e.g., Lazear and Rosen 1981). The winning probability of player is given by prob{ ( )+ + ( )+ + } = ( ( ) + − ( ) − ) with (·) as cumulative distribution function of the composed random variable − . If, for example, and are normally (uniformly) distributed, then the symmetric convolution (·) := 0 (·) is normal (triangular) with zero mean. 21 As is well-known from the literature on diﬀerence-form contests, equilibria in pure strategies will only exist if uncertainty about the outcome of the contest is suﬃciently large (i.e., the density (·) is suﬃciently flat and | 0 (·)| suﬃciently small); see, for example, Lazear and Rosen (1981), p. 845, fn. 2. 30 other contest the -player. In the latter case, the condition ¶ ¶ µ µ 1 1 ¯− ( − ) 2 {[ ( (0)) − ( (∆))] (∆) − 2 2 (13) + [ ( (0)) − ( (∆))]} must be satisfied. Proof. See Appendix B. The proposition shows that it is impossible that both -players are more likely to win in round one. Intuitively, if a -player anticipates that in the other contest the less able player has a higher winning probability than the more able one, for two reasons he prefers to enter the major contest and, thus, to win the first round: first, he wants to avoid being matched with the other -player in the next round; second, in the major contest he can win and in the consolation contest only . However, it is possible that the players face a coordination problem with at least two coexisting equilibria. In one equilibrium, the more able player has a higher winning probability in the first round-one contest and the less able player is more likely to win the second round-one contest. In the other equilibrium, we have the opposite constellation with the less able player being more likely to win the first round-one contest and the more able player the second round-one contest. For such coordination problem to exist, it is necessary that condition (13) is satisfied. The two sides of (13) highlight two eﬀects that favor a coordination problem (given ¯ 1 ): According to the left-hand side of (13), − must be suﬃciently 2 small.22 The smaller − the less attractive will be the major contest relative to the consolation contest. For − → 0, every player is completely indiﬀerent between the two contests concerning the winner prizes. However, both types of 22 Note that for − → 0 the condition is clearly satisfied. 31 players have a major interest not to be matched with the same player type in the second round, which is pointed out by the right-hand side of condition (13): From the proof of the proposition we know that player chooses eﬀort ( · ( − )) ( = ) when competing against player in the second round. Hence, if two players of the same type are matched in one contest, their marginal winning probability (0) ( (∆) = (−∆)) will be quite large since (·) is single-peaked at zero, implying high eﬀorts and, therefore, high eﬀort costs ( · (0)). The expression 2{[ ( (0)) − ( (∆))] + [ ( (0)) − ( (∆))]} describes the players’ saved eﬀort costs in case of successful coordination. To sum up, if condition (13) is satisfied the players may primarily be interested to coordinate their matches in the second round. In such situation, one -player prefers sandbagging in round one to compete against a -player in the consolation contest. Of course, if the coordination of the players fails they may end up in the constellation where both -players sandbag and enter the consolation contest in the second round — thus leading to mediocracy. 5 Conclusion At first sight, one might expect that career contests perfectly correspond to the well-known phrase "survival of the fittest". According to Darwin, there should be a natural selection among heterogeneous individuals so that the best suited ones will win the competition for reproduction. Relating to career competition, the most talented or most productive players should win and be promoted to the top positions within structured career paths. However, in our paper we show that, contrary to the Darwinian view, the least productive players may have the highest probability of winning career competition. The intuition comes from the fact that the phrase "survival of the fittest" implicitly assumes that all individuals choose 32 the same activity level. Maybe, in biology this crucial assumption holds. In an economic context, however, equilibrium behavior of heterogeneous players usually diﬀers. Since the least productive players have the lowest fall-back positions, these individuals are strongly motivated to win career competition and, thereby, to avoid their unattractive fall-back options. In this sense, we have a natural tendency that the least productive players succeed. It is important to emphasize that under identical activity levels our model would replicate the Darwinian outcome: The individuals with the highest productivities would most likely win the career contest. In our paper, we used a game-theoretic perspective to show how the detrimental eﬀect of fall-back positions may lead to adverse career outcomes for society. Switching to a contract-theoretic view would not qualitatively alter our results. Allowing endogenously chosen, optimal career incomes would probably lead to a change in the levels of equilibrium activities. However, as mentioned in Section 2 the ranking between the players’ activities and winning probabilities would not change. Introducing reservation utilities for the players may even reinforce our findings. If the most productive players have also the highest reservation utilities, these players may prefer their outside options and decide not to participate in the consolation contest. If the fall-back positions are now determined by the players’ diﬀerent reservation utilities, we will have the same natural tendency that the most productive players have the lowest incentives to succeed in a given career contest. All these considerations are based on the assumption that the contest designer cannot use a mechanism to reveal the players’ types and then choose type-dependent contest prizes to adjust individual incentives. However, as has been emphasized by Malcomson (1984, 1986), identical prizes for all contestants are important if individuals only have unverifiable but observable performance signals. In that case, under diﬀerent prizes the principal would ex-post always claim that the player with the lowest winner prize has performed best, thereby saving labor costs. Since 33 the players can anticipate such opportunistic behavior, contest incentives would break down if prizes diﬀer. According to the Peter Principle, individuals are promoted as long as they reach their level of incompetence. This observation was made by Peter and Hull (1969). The outcome of the Peter Principle is based on two rules — currently good performance is rewarded by job promotion and demotions are not possible. In the subsequent economic work, the Peter Principle has been used as a synonym for the misallocation of managers at (high) hierarchy levels. For example, Prendergast (1992) explains such misallocation by the personnel policy of hiding good talents, whereas the explanation of Lazear (2004) is based on temporary luck. In the light of our model, misallocation at higher hierarchy levels can be explained by a detrimental incentive eﬀect that gives less talented individuals strong incentives to assert themselves in competitive careers. 34 6 6.1 Appendix Appendix A: Comparative Statics to Individual Salaries The two first-order conditions lead to the following set of implicit functions: ∆ 0 (∗ − ) (∗ − ) −1 = 0 [ (∗ − ) + (∗ − )]2 ∆ 0 (∗ − ) (∗ − ) − 1 = 0 2 := [ (∗ − ) + (∗ − )]2 1 := Let ¯ ¯ ¯ || = ¯¯ ¯ 1 ∗ 1 ∗ 2 ∗ 2 ∗ ¯ ¯ ¯ ¯ ¯ ¯ denote the Jacobian determinant, which is strictly positive since 1 ∗ 1 ∗ 2 ∗ 2 ∗ 00 (∗ − ) Ψ − 2 [ 0 (∗ − )]2 0 Ψ3 (∗ − ) − (∗ − ) = ∆ 0 (∗ − ) 0 (∗ − ) Ψ3 (∗ − ) − (∗ − ) = ∆ 0 (∗ − ) 0 (∗ − ) Ψ3 00 (∗ − ) Ψ − 2 [ 0 (∗ − )]2 = ∆ (∗ − ) 0 Ψ3 = ∆ (∗ − ) with Ψ := [ (∗ − ) + (∗ − )]. Comparative statics yield ∗ ∆ ¯ ¯ ¯ 1 1 ¯ ¯ − 1 ¯ ∆ ∗ ¯¯ = || ¯¯ − 2 ∗2 ¯¯ ∆ ¯ 0 ∗ − ∗ − ( ) ( ) ¯ 1 ¯¯ − Ψ2 = ¯ || ¯ 0 ∆ 0 (∗ − ) 0 (∗ − )[ (∗ − )− (∗ − )] Ψ3 2 ∆ (∗ − ) 00 (∗ − )Ψ−2[ 0 (∗ − )] Ψ3 35 ¯ ¯ ¯ ¯0 ¯ ¯ ∗ ∆ ¯ ¯ 1 1 ¯¯ − ∆ = || ¯¯ − 2 ∆ 1 ∗ 2 ∗ ¯ ¯ ¯ ¯ ¯ ¯ 0 ¯= 1 ¯ ¯ || ¯ 2 ¯ ¯ − ∆ 1 ∗ 2 ∗ ¯ ¯ ¯ ¯ ¯ ¯ (∗ − ) − (∗ − ) ∆ 0 (∗ − ) [ 0 (∗ − )]2 (∗ − ) = || Ψ5 T 0 if ∗ − T ∗ − Analogous results can be computed for ∗ ∆ and ∗ ∆ . 6.2 Appendix B: Proof of Proposition 4 In the second round, two players compete in the major contest for , whereas the two other players compete in the consolation contest for . In the major contest, player maximizes · ( ( ) + − ( ) − ) − and player · [1 − ( ( ) + − ( ) − )] − The two first-order conditions together lead to · ( ( ) + − ( ) − ) = 0 1 1 = 0 ( ) ( ) ¡ ¢ Hence, if an equilibrium ∗ ∗ in pure strategies exists,23 it must be symmetric: ∗ = ∗ = ∗ with ∗ = ( · ( − )). Inserting in player ’s objective function yields = ( − ) − ( ( − )) as ’s expected utility from competing against in the major contest. In analogy, we obtain = ( − ) − ( ( − )) as player ’s expected utility from participating in 23 Recall that we assume well-behaved functions that guarantee existence. 36 the consolation contest. In the first round, we have two heterogeneous contests, each one between a -player and a -player. In each contest, the -player chooses to maximize £ ¡ ¢ ¤ ¯ · ¯ · + 1− ( ( ) + ∆ − ( )) £ ¡ ¢ ¤ ¯ · ¯ · + 1− − + [1 − ( ( ) + ∆ − ( ))] whereas the -player decides on to maximize £ ¡ ¢ ¤ ¯ · ¯ · + 1− [1 − ( ( ) + ∆ − ( ))] ¤ £ ¡ ¢ ¯ · + 1 − ¯ · − ( ( ) + ∆ − ( )) As first-order conditions we obtain £ ¡ ¢ ¡ ¢ ¡ ¢¤ ¯ · ¯ · − − + 1− = ( ( ) + ∆ − ( )) 1 0 ( ) and £ ¡ ¢ ¡ ¢ ¡ ¢¤ ¯ · ¯ · − − + 1 − = ( ( ) + ∆ − ( )) 0 1 ( ) Using the symmetry of the density function (·) (implying (0) = 12 , (∆) = (−∆) and (−∆) = 1 − (∆)) the expected utilities can be computed as follows: − ( (0)) 2 = (∆) − ( (∆)) = = = [1 − (∆)] − ( (∆)) 37 with = . The winning probability of the high-ability player, ( ( )+∆− ( )), shows that for equal eﬀort levels of both players the -player has always a higher winning probability because of his "lead" ∆. Hence, if the -player is more likely to win the contest, we must have that . From the two first-order conditions we can see that this condition is equivalent to ¡ ¢ ¡ ¢ ¡ ¢ ¯ · ¯ · − − + 1− ¡ ¢ ¡ ¢ ¡ ¢ ¯ · ¯ · − − + 1 − ⇔ µ ¶ 1 (∆) − ( − ) 2 £ ¤ ¯ − 1 ([ ( (0)) − ( (∆))] + [ ( (0)) − ( (∆))]) + 2 the inequality given in the condition. Note that the left-hand side is strictly positive since (∆) 1 . 2 ¯ If 1 , 2 the inequality cannot be satisfied since the right-hand side is negative due to (0) (∆). In words, if in the other contest the -player is more likely to win than the -player, the same cannot be true in the given contest, because here the -player already leads by ∆ and, in addition, chooses more eﬀort than his opponent . Thus, it is impossible that in both contests the two -players have higher winning probabilities. However, if ¯ 1 2 and − is suﬃciently large so that the inequality is violated then ¯ in both contests the -players are more likely to win. Finally, if 1 2 and − → 0 then the inequality is clearly satisfied. 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