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In order to provide a clear theoretical framework for the incentive models presented in sections three and four, first, it is necessary to review the existing literature on this topic. This section begins describing the main models on Tournament theory. Then, it outlines the most relevant problems attributed to these reward schemes and the results of the main empirical research done in the most recent literature. Finally, this section reviews other literature related to this paper.
The baseline of the Tournament theory came up with the paper "Rank-Order Tournaments as Optimum Labor Contracts" by Lazear & Rosen (1981). They compare (in efficiency terms) the classical piece rate compensation  with a tournament based on relative performance, in a simple two workers-one period model.
In this paper, it is assumed that the output of a worker (q) is given by his/her effort (ï) plus a random variable (ï¥), which symbolizes luck [qi = ïi + ï¥i], and that exerting effort is costly for the worker, which is given by the cost function C(ï) (where C'(ï),C''(ï) > 0)  .
When workers are risk-neutral, piece rates system is efficient in competitive equilibrium. Therefore, workers' expected pay is the value of their effort. The incentive mechanism of the rank-order tournament is slightly more complicated. Here prizes W1 and W2 (where W1 > W2) are assigned to the winner and looser, respectively (the winner is the worker who produced a higher output level (q) ex-post). In contests with competitive prize structures, workers' decision on effort investment (ï) affects both, the probability of winning and the prize. In order to find the optimal prize spread the firm should set, Lazear & Rosen (1981) derive the Nash-Cournot symmetric reaction function of each worker:
(W1 - W2)dP/dïi - C'(ïi) = 0 
Then, they show that when players are homogeneous (same ability meaning same cost of effort) the optimal prize spread in equilibrium induces workers to exert the same effort level than implementing piece rate compensation, V = C'(ï)  . This happens because the firm sets the optimal prize spread taking into account that the marginal cost of effort is increasing in ï, which implies that there is a unique equilibrium prize spread maximizing workers expected utility.
Furthermore, Lazear & Rosen (1981) consider the case of Risk-Averse workers and they show that the outcome depends on the specific form of workers' utility function. Although usually Piece Rates dominate Tournaments in effort investment and expected income, in some cases of decreasing absolute risk aversion, Tournaments could dominate Piece Rates  .
They also deal with the case of heterogeneous contestants, but in a very simple way. They conclude that asymmetric information on workers' abilities turns into Adverse Selection, because low ability contestants will prefer to enter in higher prize contests and therefore they will tend to jeopardize high ability contestants. This results in inefficient mixed contests  . However, if there is perfect information, an optimal Handicap System could be introduced to induce workers to exert the efficient level of effort  .
In this first approximation towards Tournament theory, Lazear & Rosen show that contest may provide appropriate incentives for workers. Moreover, when workers face some common uncertainty, tournament-based rewards could provide the appropriate insurance scheme as well.
"Economic Contests: Comparative Reward Schemes" by O'Keeffe, Viscosi & Zeckhauser (OVZ) (1984) is another relevant paper in Tournament theory. They are concerned with the design of optimal contests for different situations. Thus, they analyze the sustainability of the incentive mechanisms of tournaments under different conditions as fairness/unfairness, endogenous monitoring accuracy and heterogeneity in abilities. To do so, OVZ introduce some variations to the basic tournament model due to Lazear & Rosen and evaluate if the Nash solutions show sustainable Global incentives in each case.
They begin analyzing the simple case with homogeneous workers in a fair contest, but considering the case of an indivisible prize (a job promotion for example). In this case, the valuation that workers have of the prize may not be optimal (either too high or too low prize spread), and so, it may induce an inefficient effort investment. While Lazear & Rosen (1981) take monitoring precision as exogenous  , OVZ (1984) assume that the employer can (at least partially) manage it. Therefore, a suboptimal prize spread could be compensated by increasing/decreasing monitoring accuracy in order to induce an efficient investment in effort. Thus, for a too high prize gap employer could reduce monitoring precision, while for a too small prize gap he/she could increase it.
Moreover, OVZ (1984) take the case of heterogeneous contestants, as described by Lazear & Rosen (1981), and develop it to include employers' discretion in monitoring precision. Then, they analyze this case within different informational situations:
If ex-ante all workers are identical (none knows anyone's' ability), a high prize spread and accurate monitoring could provide global incentives and select the most able worker.
If there is perfect information (everyone knows each other's ability), there may exist an optimal handicap mechanism to induce efficient effort (Lazear & Rosen, 1981). However, in the case of an indivisible prize it may be harder to set this handicap tournament.
If each worker only knows his/her own ability and the distribution of abilities, then precise monitoring necessarily results in an inefficient outcome. But reducing the precision of monitoring could allow for an efficient outcome for a certain type of ability distributions  .
In similar fashion, Nalebuff & Stiglitz (1983) look for a general overview of compensation schemes by introducing some variations to the basic rank-order tournament model. For example, they describe contests where payment depends on relative performance. Here, tournaments' rank-order nature disappears, as this compensation scheme combines piece rate system and tournament by setting pay spread according to workers' output difference. As McLaughlin (1988)  points out, one potential problem with this system is that workers may have incentives to collude and exert low levels of effort.
Apart from this, Nalebuff & Stiglitz (1983) deal with the N-contestants and single (indivisible) prize tournaments, in order to evaluate how does the increase in the number of players affect workers' incentives  .
Green & Stokey (1983) also analyze the case of N-contestants, but within multiple prize tournaments, providing some insights of the optimal compensations schemes under risk aversion.
Problems with Tournaments
Some potential problems with tournament-based compensation schemes were pointed out early in the literature. Dye (1984) points out some potential problems with tournament-based compensation schemes. The most relevant for our study is the possibility of an ex-ante agreement among contestants to exert the minimum effort level for entry. Besides, Dye (1984) refers to the nature of output, which could be multidimensional, as a source of difficulties in determining the winner of a contest.
Another example is given by Nalebuff & Stiglitz (1983), who conclude, "In competitive systems, there are no incentives for cooperation"  . This could be interpreted as the failure of tournaments in providing incentives for a better inter-worker cooperation.
"Pay Equality and Industrial Politics" by Lazear (1989) is an attempt towards a deeper analysis on the non-cooperative properties of tournaments. He studies the adequacy of pay compression and the interaction of personalities in tournaments. Contests induce workers to exert effort through competition; however, if pay spread is too high uncooperative behaviour may arise, causing inefficiencies.
Lazear (1989) analyzes how does the trade-off between the incentives of competition and the loose generated by sabotage affect the optimal prize structure. Although he admits that the optimal degree of pay compression depends on each job type's factors (include examples), the paper presents a simple two-worker tournament model where sabotage is possible.
Sabotage is defined as any (costly) action that one worker takes to adversely affect the output of another worker  . Thus, sabotage is a major source of inefficiency because it reduces aggregate output and workers' utility (it is assumed to be costly). This is reflected in workers cost function, which here depends on effort and sabotage:
C(ï,ï±) where Cï(ï,ï±), Cï±(ï,ï±) > 0 and Cïï±(ï,ï±) > 0
Firstly, Lazear (1989) analyzes the case for symmetric workers, that is, when both workers have the same marginal cost of effort and sabotage. He derives the first order conditions for workers' equilibrium reaction function, which reveals that as pay spread increases sabotage does so too. On the other hand, if pay spread decreases so does workers' effort. Therefore, when taking a decision on prize spread, there is a trade-off between sabotage and effort. The symmetric players' analysis concludes that the possibility of sabotage lowers the expected output and that optimal pay spread should depend on how hard workers could damage each other's output by sabotaging (that is the marginal cost of sabotage).
Finally, Lazear (1989) also considers the case of heterogeneous workers, that is, when workers differ in their marginal cost of sabotage. When there is perfect information, firms can match worker depending on their type. Alternatively, firms can design handicap prize spreads, so that, pay varies with the identity of the winner (lower if the winner has lower marginal cost of sabotage). However, when workers ability to sabotage is not verifiable ex-ante, Lazear (1989) shows that they do not self-sort. That is, workers with low marginal cost of sabotage (Hawks) prefer to compete against rivals with higher marginal cost of sabotage (Doves)  . It happens because, in this way, they increase their probability of winning the contest. This generates an adverse selection problem for employers. Besides, in mixed tournaments, it is more likely that a Hawk wins, due to his/her better ability for sabotage. This fact can damage firms when the prize is a promotion, as they might be promoting the most able saboteur instead of the most able worker in terms of effort.
Apart from Lazear (1989), the non-cooperative effects of competitive contests have been widely discussed in the literature regarding tournaments  . However, it is important to highlight the paper "Selection Tournaments, Sabotage and Participation" by Münster (2004). He does a further development on how the possibility of sabotage may create inefficiencies in tournaments. Although this paper is closely related to psychological economics, it is valuable to account some of the negative effects that sabotage appears to have in promotion tournaments.
Münster (2004) analyzes the case of perfect information among at least three contestants. In opposition to Lazear (1989), in this model, contestants could undermine one specific rival, which has important implications for workers' decision on effort and sabotage.
This paper focuses on identifying how contestants sabotage efforts may affect the selection property of tournaments. On one hand, Münster (2004) shows that sabotage equalizes the probabilities of winning among workers, because the workers with higher ability would be undermined most. On the other hand, it is likely that the most able workers  do not participate in the tournament, as the prevision of receiving too much sabotage may deter them form entering. Therefore, Münster (2004) shows that under perfect information and having more than two players, the selection property of tournaments might be lost due to sabotage.
Empirical research on tournaments
Many authors have conducted empirical researches in order to test tournament models within different contexts. For example, Main, O'Reilly & Wade (1993) researched on corporate executive's pay; Bognano (2001), Audas, Barmby & Treble (2004) tests tournament model within corporate hierarchies; and Ehrenbergh & Bognanno (1990), Szymanski (2003) analyze tournament theory on sporting contests.
However, for the purpose of this paper, it deserves to mention the paper by Drago & Garvey (1998), where they test how promotion incentives affect workers' individual and collective efforts within a corporation. Their empirical research supports Lazear's (1989) main conclusions by showing that when prize spread increases (promotion incentives are strong), workers focus on their individual performance and allocate fewer resources to unassigned collective effort  .
Other related literature
Apart from the tournament models described above, Drago & Turnbull (1991) develop an interesting analysis that is related to this paper. They compare two different reward schemes: a competitive tournament and a non-competitive quota scheme. In the former, workers compete for a single prize/promotion, which is awarded to the best performer. In the latter, the promotion of a worker depends only on his/her absolute performance  .
In a model with two homogenous non-risk loving workers and a risk-neutral firm, Drago & Turnbull (1991) assume that production technology enables helping efforts. In other words, workers have to decide on own effort, which increases own output, and on helping effort, which increases co-workers output. Of course, helping effort could only happen when it is expected to be reciprocal. Therefore, this paper evaluates each rewarding scheme's (competitive and non-competitive) efficiency under three alternative worker behaviours: (1) Cournot behaviour  , (2) possibility for agreement on mutual help effort and (3) possibility for agreement on both, help and own effort (collusion).
Drago & Turnbull (1991) show that in a symmetric equilibrium, regardless of which behaviour is assumed for workers, there are no helping efforts if the reward scheme is a competitive tournament. Furthermore, if workers are allowed to bargain over helping and own efforts (case 3), the equilibrium own effort is zero. That is, it is expected that workers will collude on shirking. On the other hand, under the non-competitive scheme, helping efforts will not occur only if Cournot behaviour is assumed. If workers are allowed to bargain over helping effort this will happen as long as it is efficient (if helping efforts are corresponded and add significant value to workers output).
Drago & Turnbull (1991) conclude that competition based tournaments do not induce cooperation among workers. Besides, workers may collude on zero effort (or the minimum stated by the firm) if they are allowed to bargain over it (and workers are homogeneous in abilities). On the contrary, non-competitive reward schemes might foster cooperation in the workplace if it is efficient to do so.
However, Drago & Turnbull's analysis does not take into account the case of heterogeneous workers. Moreover, in their model, the only incentive to help other workers is to receive help back from them in exchange. The nature of cooperation stems from the desire to increase own output, not from increasing aggregate output. Furthermore, when the prize is in form of a promotion, the selection effects of the non-competitive compensation scheme might be sub-optimal. In other words, this compensation scheme could promote more/less workers than what it would be optimal. This does not happen in the competitive scheme, as the number of promotions is fixed in advance.