Fatter or Fitter? On Rewarding and Training in a Contest

Competition between heterogeneous participants often leads to low effort provision in contests. We consider a principal who can divide her fixed budget between skill-enhancing training and the contest prize. Training can reduce heterogeneity, which increases effort. However, allocating some of the budget to training also reduces the contest prize, which makes effort fall. We set up an incomplete-information contest with heterogeneous players and show how this trade-off is related to the size of the budget when the principal maximizes expected effort. A selection problem can also arise in this framework in which there is a cost associated with a contest win by the inferior player. This gives the principal a larger incentive to train the expected laggard, reducing the size of the prize on offer.


Introduction
How should an employer get the most out of her work force? Similarly, how should a research council get the most out of researchers? The standard answer in many such contexts is: set up a contest with a prize to the winner -like a promotion or a research grant. But, as many contest organizers have observed, contests do not incentivize well when there are big di¤erences among the contestants at the outset. 1 So the modi…ed answer is: set up a contest, and seek to level the playing …eld among the contestants. 2 However, what if levelling the playing …eld is costly? In such cases, the contest organizer might have to trade o¤ the prize to the winner with spending resources on training the contestants so that they are both better equipped to put in e¤ort in the contest and more interested in doing so.
We address the question of how to …nd the best balance between prize and training in a setting where a principal organizing a contest has a …xed budget that she can split between a prize, which will incentivize the contestants to put in more e¤ort, and skill-improving training, which will make the e¤ort put in by a contestant more productive. When there are ex-ante di¤erences in the contestants' skills, there is also a question of who to train.
In our model, there are two contestants who compete in an all-pay auction, meaning that the winner is the contestant with the higher e¤ort. A contestant's ex-ante skills are not known by the other contestant, nor by the principal. But everybody knows the probability distribution that each contestant's skill is drawn from. The ex-ante leader is the one with skills drawn from the better distribution, while the other one is called the ex-ante laggard.
When the principal aims at maximizing the contestants'total expected e¤orts, it turns out that the exact nature of her decision on how to split her budget between prize and training depends on how large the budget is. With a mediumsized budget, the principal spends resources on training the laggard exactly so that any ex-ante di¤erences are evened out, with the rest of the budget being spent on the prize. When the budget is small, the budget is optimally split between the prize and training of the ex-ante laggard, but such that the ex-ante di¤erence is not totally evened out; and if the budget is very small, there will be no training and the whole budget is spent on the prize. When the budget is large, there is room for training both contestants in such a way that the ex-ante di¤erence is …rst evened out, and then the expected abilities of both contestants are increased symmetrically, while still having funds for a prize.
We also discuss the case where the principal cares about having the right winner of the contest. Since skills are uncertain ex-ante, there is a chance that the winner is not the ex-post more e¢ cient contestant. In order to take care of this problem, the principal should aim at minimizing the probability of erroneous selection. Interestingly, the size of the prize plays no role in this problem, so 1 See Baik (1994) for an early theoretical study. Empirical studies on the discouragement e¤ect of asymmetries are done by Sunde (2009) on tennis tournaments and Brown (2011) and Franke (2012) on golf tournaments. See also surveys by Konrad (2009), Mealem andNitzan (2016), andChowdury, et al. (2020). 2 See, e.g., the survey by Chowdury, et al. (2020).
the only remaining issue is how to split the training part of the budget between the two contestants. We show that the probability of erroneous selection is not monotonic in the amount of training given. In addition to simply minimizing the probability of selecting the less e¢ cient contestant, we also consider the case in which the principal cares about the expected cost of erroneous selection (as measured by the di¤erence in ability between the winner and the more e¢ cient loser). Maximizing a weighted combination of expected e¤ort minus expected selection cost, we show that the principal is more likely to o¤er training to the laggard for lower budgets, the less weight is placed on e¤ort.
Our analysis provides insights for many situations where a principal is faced with heterogeneity among her contestants. We will brie ‡y mention two such situations that are of particular importance.
For a country's research and technology policy, it is important to allocate public funds to researchers in the most e¢ cient way possible. There has been a concern, at least since Merton (1968), that there is too much heterogeneity among research groups and that this leads to a lack of interest among researchers in taking part in competitions for funds from the research council. A question arises whether funds should be given directly to universities and other research institutions, in such a way that laggards become more able to compete for further funding, or should be deployed through grant competitions organized by the research council. As our analysis shows, the role for the research council is larger the smaller is the total budget allocated to research funding. As this budget increases, more of it should be allocated to leveling the playing …eld among the country's research institutions.
Another equally important topic is how …rms should allocate their humanresource funds. In this research area, there are discussions whether one should focus on selecting the best workers for promotions to more prestigious and/or interesting jobs or on training the whole workforce in order to make them better able to compete for the promotions; see, e.g., Farrell andHakstian (2001), Cron, et al. (2005), and Beck-Krala (2020). Training has been given special attention in sales force management, and many companies use substantial resources on this activity (see, for example, Krishnamoorthy, et al., 2005). 3 Among others, Beck-Krala (2020) notes a tendency in human resource management to consider the "total reward" of an employee, encompassing …nancial and non-…nancial, direct and indirect rewards. Training is one costly dimension of the total reward o¤ered by a …rm. Our analysis shows that incentivizing the workforce through promotion contests and the like should be the main focus when the …rm's funds are limited, whereas a richer …rm can focus more on training the workforce in order to rectifying biases and in this way make the whole workforce interested in putting in e¤orts to obtain positions within the …rm. 4 Our paper builds on earlier discussions of all-pay auctions where players have private information about their valuations, such as Amann and Leininger (1996) and Clark and Riis (2001). In particular, Clark and Riis (2001) is close to our basic framework, since they posit two players where one has its skill drawn from a more advantageous distribution than the other, so that they, ex-ante, are leader and laggard. See also Seel (2014), where the private information is one-sided, in that one player's valuation is known by both players. This paper is related to the discussion of whether and how to rectify ex-ante biases among contestants. See, in particular, Li and Yu (2012), Kirkegaard (2012), and Franke, et al. (2018) for discussions on how to increase total e¤ort by rectifying these biases. 5 Our present analysis di¤ers from the previous work in insisting that favouring a contestant is costly and will, in the face of a …xed total budget for the contest designer, imply a lower prize for the contest winner. In fact, we …nd that, when the budget is small, it is not optimal for the contest designer to completely rectify the ex-ante biases, since doing so would leave a too small prize to the winner. 6 A number of studies discuss contest design when the designer has other concerns than simply maximizing total expected e¤orts. 7 Tsoulouhas, et al. (2007) discuss a designer facing one group of contestants with known abilities and another group with unknown abilities and …nd that it may be optimal to favour the former group. Cohen, et al. (2008) discuss a designer who has an interest in maximizing the highest expected e¤ort of a player and …nd that ex-ante biases are preferable. In Seel and Wasser (2014), the designer has concerns for both total expected e¤orts and the highest expected e¤ort, and again having a bias is optimal. Pérez-Castrillo and Wettstein (2016) study a designer who wants to select the high-type contestant and …nd conditions such that, also in this case, having an ex-ante bias is preferable. Our analysis, in contrast, shows that, for a su¢ ciently large budget, it is optimal for the designer to level the playing …eld irrespective of how much weight she puts on selecting the high-type contestant, while for smaller budgets, a high such weight means a higher concern for training the laggard in order to reduce the ex-ante bias.
The paper relates to studies of pre-contest investments. For example, Konrad high compensation in the present job. See, for example, Lazear and Oyer (2012, Sec. 5). Here, we disregard such direct bene…ts from skills, instead focusing on the principal's need for balancing spending on training and compensation. 5 Also other instruments have been suggested to increase e¤orts in asymmetric contests: Che and Gale (1998) discuss putting limits on contestants'e¤orts; Mealem and Nitzan (2016) discuss a¤ecting the contestants' contest success functions and win valuations; Sisak (2009) discusses changing the prize structure; Nilssen (2020, 2021) discuss how to split the prize fund between early and late prizes in order to counter the e¤ect of ex-ante di¤erences among contestants. See more on this in the survey by Chowdhury, et al. (2020). 6 Other research also reports that a total levelling of the playing …eld would not be optimal in games among heterogeneous contestants. For example, Ryvkin (2013) discusses how the shapes of the players' e¤ort cost functions may make it optimal to leave some heterogeneity; while Pastine and Pastine (2012) …nd that rectifying biases may result in a higher performance gap in a political-campaign game. 7 Also here, see Chowdhury, et al. (2020) for a more detailed discussion.
(2002) and Fu and Lu (2009) discuss contestants' incentives to invest in own productivity before a contest. In contrast, we discuss the principal's incentives for such pre-contest investments. Clark and Nilssen (2013) analyze contestants' incentives to put in extra e¤ort in the …rst round of a two-round competition where there is complete information and learning by doing. They discuss how the contest designer can split her prize budget across the two rounds in order to get the right balance between …rst-round learning and second-round e¤orts. This is related to the present discussion of pre-contest training versus prize award; the framework of the present paper is quite di¤erent, however, since heterogeneous players compete under incomplete information, and the ex post e¤ect of training is not deterministic. The paper is organized as follows. Section 2 outlines the basic contest played between heterogeneous participants. Section 3 considers how an e¤ort maximizing principal will divide her budget between training and the contest prize. Section 4 focuses on the selection problem in which a low-ability contestant can win the contest; the trade-o¤ between the prize and training is considered here for a principal that maximizes a weighted sum of the expected contest e¤ort and the expected cost of erroneous selection. Section 5 concludes. All proofs are to be found in the Appendix.

The contest
Two risk neutral players compete for a prize of size v by exerting irreversible e¤orts x i 0, i = 1; 2. The cost of e¤ort to player i is given by x i a i , where a i is an ability parameter that is private information to that player. It is commonly known that player 1 draws ability from a uniform distribution on [h; H], and player 2 from a uniform distribution on [l; L]. We make the following assumption: . Part (i) of the assumption implies that the players'distributions are identical up to a location shift. Part (ii) means that player 1, without loss of generality, is expected to be the more able player ex ante, with H L. It also implies that the ability distributions are overlapping, with L > h, which again implies that D > H L; and that L > D, since l = L D > 0. Ex post it can hence be the case that player 2 is actually more able, even though player 1 is expected to have the higher ability ex ante. Part (iii) is a regularity assumption. It is not a very strong assumption to make. Necessarily, H D > 1, since h = H D > 0. Suppose, moreover, that L approaches h, which would mean that h l would approach D. With L l = D and l > 0, this would imply h > D, or, since h = H D, H D > 2, which is stricter than the assumption we make here.
The player with the larger e¤ort wins the prize with certainty, with ties broken randomly, as depicted by the following contest success function giving the probability that player 1 wins the prize: At the contest stage, player i knows his own ability but not the ability of the opponent. The expected payo¤s of type a i can be written as Let the e¤ort function x i (a i ) of player i be a mapping from the player's ability to his chosen e¤ort; suppose it is continuous and strictly increasing (except possibly at zero), which implies that there exists an inverse g i ( Since abilities are uniformly distributed, we can write expected payo¤s for the two players as Using arguments explained in Clark and Riis (2001), we can state the following result, the proof of which is in the Appendix.
Proposition 1 The unique pure-strategy Bayesian Nash equilibrium is given by the equilibrium e¤ort functions x i (a i ), i = 1; 2, where x 2 (a 2 ) = 0, for a 2 2 l; Lh H ; Whilst almost all player-1 types have positive e¤ort, some low player-2 types (a 2 2 l; Lh H ) do not …nd it worthwhile to compete. Note that the two players'equilibrium e¤ort functions have the same support, Note also that, when the players draw their valuations from the same uniform distribution (i.e., when L = H), the equilibrium e¤ort functions are Figure 1 gives an illustration of the equilibrium in Proposition 1, showing that the equilibrium e¤ort function of the ex-ante less able player 2 lies over that of player 1. The superior opponent uses his expected edge to slack o¤ and save on e¤ort cost. This means that a player-2 type of inferior ability can beat a more able player-1 type. When the players have drawn the same ability a 1 = a 2 = a, which of course can only happen if a > h, it is easy to verify from (3) and (4) that x 2 (a) > x 1 (a) when H > L.

FIGURE 1 ABOUT HERE
The ex-ante total expected e¤orts (i.e., before the draws are made) are where we use the substitution h = H D.
Note that the ratio of the expected e¤orts has a simple form: Thus, even if, for a given a, player 2 has the higher e¤ort, the ex-ante expected e¤ort is higher for player 1. Moreover, in the case of symmetry, when L = H, the expression in (7) reduces to Player 1 wins the contest with certainty if player 2 draws a type in the interval l; Lh H , since 2 then has zero e¤ort in equilibrium; player 1 also wins if x 1 (a 1 ) > x 2 (a 2 ), which by Proposition 1 occurs for a 2 < L H a 1 . Hence, the probability that player 1 of type a 1 wins is 1 D a 1 L H l ; taking the expectation of this over all player-1 types gives the ex-ante probability that player 1 wins the contest in equilibrium as 8 where the inequality follows from L H. Even though the ex-ante more able player 1 is expected to have more e¤ort, this does not cost him more, since his unit cost of e¤ort is likely to be smaller. In fact, the expected ex-ante costs of e¤ort of the two players are identical: The ex-ante expected payo¤s to the players can be found as where the expected payo¤ to player 1 is higher, since he has the higher win probability in equilibrium and the players have the same expected cost of e¤ort. Player 2 must achieve a non-negative pro…t in equilibrium, i.e. (D; H) 0. It is easily veri…ed that (0; H) = 0, and that it is positive otherwise so that the expectedly weakest player has a non-negative payo¤ in equilibrium as required.

Training to maximize e¤ort
Suppose the principal aims at maximizing the total ex-ante expected e¤orts of the contestant. She has available a …xed budget B, which can be divided between giving the contest prize v and investing in the abilities of the players with s 1 0 and s 2 0, respectively. Budget balance requires B = v + s 1 + s 2 . The development of ability at the training stage is modelled as an upward shift in the ability interval of the receiving player, keeping the length of the interval constant at D. With expenditure s i , the ability improvement is simply s i ; following expenditures of s 1 and s 2 on the two players, the ability interval of player 1 becomes [h + s 1 ; H + s 1 ], while player 2 has [l + s 2 ; L + s 2 ].
At the beginning of the game, the principal announces a triple (v; s 1 ; s 2 ) that satis…es budget balance. If either of the training amounts is positive, then training takes place. Then draws are made from the modi…ed ability distributions. After this the contest is played over the prize v.
The equilibrium is based on the premise that i) the ability intervals overlap, and ii) that the ability intervals are of the same length with that of player 1 being above that of the rival. To ensure that the ability intervals overlap after any training is carried out requires that L + s 2 > h + s 1 = H D + s 1 ; after training, player 1 retains the higher ability interval as long as H + s 1 > L + s 2 . 9 Training levels that ful…ll both of these requirements hence satisfy H L D < s 2 s 1 < H L: To facilitate comparative-statics analysis when the lower and upper bounds of the interval are changed, it is convenient to rewrite the equilibrium e¤ort functions in (3) and (4) using h = H D; l = L D, since D is constant. We have We can now analyze the e¤ect that increasing the expected ability of one of the players will have on the equilibrium e¤ort functions. Suppose …rst that the support of the distribution for the laggard is moved up (i.e., l and L increase). The e¤ect that this has on the equilibrium e¤ort functions is drawn in Figure 2 for a shift from L to L 0 > L, where the new equilibrium e¤orts are given by x 0 i (a i ). 10 FIGURE 2 ABOUT HERE From this, it is apparent that the e¤ects on the e¤ort functions are monotonic; all player-1 abilities increase their e¤orts, since the rival is now expected to be more able than before. The laggard responds to the expected increase in ability by providing less e¤ort. On the other hand, the high player-2 types will have e¤ort above the previous maximum level. The common upper support of both players increases to x 0 . Figure 3 depicts the e¤ects of increasing the expected ability of the leader, i.e., increasing (h; H) to (h 00 ; H 00 ). 11 FIGURE 3 ABOUT HERE Whilst the response of the receiving player 1 is to lower e¤ort for all ability levels, except at the top of the distribution, the response of player 2 is to decrease e¤ort for low ability levels and increase it for high ones. There are also fewer player 2 types that have positive e¤ort when the opponent becomes more superior in expectation.
The principal knows that player 1 is the expectedly more able; since she does not know the actual draws made by the rivals, the principal does not know which of the players is most able ex post. As illustrated in Figures 2 and 3, increasing the expected ability of the laggard causes the e¤ort function of the leader to shift upwards and that of the laggard to shift downward; increasing the expected ability of the leader reduces the e¤orts of that player and of low laggard types, but increases the e¤ort of higher-ability laggards.
Dividing the budget between the prize and training for one of the players is not a straightforward problem as demonstrated above. The problem becomes more complex when both can receive training. However, as it turns out, the principal will not support the ex-ante leader with any training in our model, except if the budget is large, so that the optimum is to split the budget between a prize to the contest winner and training of the ex-ante laggard. In particular, we have: 10 The support moves upwards so that L 0 l 0 = D. 11 H 00 h 00 = D.
Proposition 2 A principal with a budget of B will split the budget on prize and training as follows: (i) An insu¢ cient budget, i.e., one where will lead to no training and v = B.
(ii) If the budget is small, i.e., if then the principal spends s 2 on training the ex-ante laggard and the rest, B s 2 , on the prize, where then the principal …rst spends training on the ex-ante laggard until the two players have equal expected skills, and thereafter spends equal amount of training on both players so that they continue to have equal expected skills. Total spending on training is S = s 1 + s 2 , while the rest of the budget, B S, is spent on the prize, where Consider …rst part (i), which indicates the case in which training is completely sacri…ced in order to give a contest prize as large as possible. When the principal is resource constrained in this way, training the laggard has a positive e¤ect on total e¤ort ceteris paribus, but this directly reduces the contest prize, reducing e¤ort. The second e¤ect outweighs the …rst, and no training is given.
Part (ii) covers the case when the budget is larger, but not enough that it pays to make the players symmetric. Now total e¤orts get a boost from the laggard being trained, at the same time as there is a downward pressure on total e¤orts as the prize becomes lower and lower. The amount of the budget used on training balances these two e¤ects, …nding an internal division of the budget. Increasing the budget further, as in part (iii), allows the laggard to be trained until the contestants are equal in expected ability, putting the remainder of the budget into the prize fund. Finally, in part (iv), the budget is so large that the initial laggard can be trained so that he catches up the expected leader, and then both players can be made more e¢ cient. This occurs until the marginal e¤ect of spending one unit of the budget on training is equal to the marginal e¤ect of giving that unit as a prize.
The relationship between the size of the budget and the total expected e¤orts is then straightforward to determine as . The second and fourth parts of this function are increasing and convex in B. For very small and for intermediate budget sizes, the …rst and third parts indicate that extra budget is completely given to the prize, increasing expected e¤ort linearly. This relationship between B and X is illustrated in Figure 4. 12 To give an idea of the e¤ect of training on expected e¤ort, the line X 0 indicates expected e¤ort without training, i.e. v = B; s 1 = s 2 = 0. 13 FIGURE 4 ABOUT HERE For low budget levels, all funds are spent on the contest prize, and expected e¤orts are a …xed proportion of this, as indicated by (7). When the budget reaches L(H+L) H+2L , it is possible to do better than this by training the laggard. Figure 2 shows that the leader will increase e¤ort for all ability types but that the laggard will reduce e¤ort, apart from the high types that are created by the training. Initially, as the budget increases beyond L(H+L) H+2L , the net e¤ect is positive and large enough to outweigh the fact that resources are taken away from the contest prize, which reduces e¤ort. As further resources are used on training player 2, the players become more and more alike in expected ability; this levelling of the playing …eld increases e¤ort. If the principal has a total budget of 5 3 H L, then training is given until the players are symmetric; hence, H L is used on training player 2, and 2H 3 is the contest prize. An increase in the budget from this point will 12 Figure 4 is generic, but H = 2, D = 1, and L = 1:25 are used as parameter values here. 13 Here, X 0 = L(3H 2D)(H+L) 6H 2 B, as in the …rst line of the expression for X . optimally be put in its entirety into the prize fund; however, when the budget becomes large enough (3H L 4 3 D), some resources are allocated to training both players, keeping them symmetric and increasing their ability in the contest. This leads to an increase in expected e¤ort that is larger than would be obtained simply by granting a larger prize.

FIGURE 5 ABOUT HERE
In dividing the budget between the contest prize and training player 2, the principal imparts several e¤ects on the expected payo¤s of the players. In Figure 5, the ex-ante expected payo¤s of the players when training is not given are represented by E 0 1 and E 0 2 . Since each extra unit of budget gives an equivalent increase in the contest prize, these are straight lines and player 1 -who is expected to be most able -has the larger expected payo¤. For low budget levels (up to L(H+L) H+2L ) there is no training and these basis payo¤s are achieved by both players. When player 2 is trained, he expects to draw a lower cost of participating in the contest. As illustrated by Figure 2, this causes player 1 to exert more e¤ort than before. This increases player 1's probability of winning, but it also increases its e¤ort costs, so that player 1's expected payo¤ falls for budgets in L(H+L) H+2L < B 5 3 H L -even when the principal also increases the contest prize in this region. As a response to training, most player-2 types slack o¤, which reduces the probability of this player winning. But less e¤ort, exerted at a lower cost, and an increasing contest prize mitigates this. Figure 5 shows that the expected payo¤ of player 2 increases as the budget grows, even though it is lower than it would have been with no training. Note that the expected payo¤ to player 2 is convex in this interval. As the budget increases, more and more training is given, and player 2 reduces e¤ort, but the marginal e¤ect is smaller the more his ability interval is shifted upwards. Then the fact that e¤ort becomes less and less expensive makes the payo¤ function slope up steeply, leading to a larger expected payo¤ than without training. In the interval 5 3 H L < B 3H L 4 3 D, the ability intervals of both players are identical, and the ex ante expected payo¤ of both players is the same. It increases linearly in the budget since every extra dollar is given to the prize. When the budget is large, the principal decides to train both players symmetrically, so that only a fraction of the budget increase is given to the contest prize; hence the slope of the expected payo¤ function is reduced. It can be shown that the basis expected payo¤ of player 2 (E 0 2 ) is larger than the symmetric payo¤ with training for a su¢ ciently large budget. 14 still end up losing because of the laggard's higher e¤orts. This ex-post selection problem -the problem of erroneous selection -is particularly important in settings such as promotion contests and competitions for research grants, where the winner goes on to perform tasks whose qualities may depend on the winner's skills. In this section, we therefore amend the principal's decision problem to incorporate a concern for erroneous selection. We do this by …rst studying a principal whose sole aim is to minimize the problem of erroneous selection and then use this analysis to study the general problem of a principal with an interest in both high total expected e¤orts and low expected costs of erroneous selection.
It is not possible for player 1 to win when player 2 is more able, thus we have no instance of a type-2 error. Player 1 wins when a 1 > H L a 2 and is more able in all such cases. We can calculate the probability of the principal making a type-1 error ex-post, i.e., the probability that contestant 2 wins when contestant 1 has the higher ex-post ability: Note that the size of the prize v does not a¤ect . Moreover, asymmetry (i.e., H > L) always leads to a positive probability of the contest selecting the player with the lower ability. 15 The calculation of is demonstrated in Figure 6. In all these cases, we have that a 1 > a 2 , so that the more able is selected as winner. This is also the case for area C, where player 2 wins and is more able. The areas marked by b and b 0 indicate combinations in which player 2 wins but is less able. The …rst element in (15) represents area b, while the second one is b 0 . When player 2 receives training, L increases and the line H L a 2 moves closer to the 45-degree line. This in itself reduces the areas b and b 0 . However, the square of feasible ability combinations shifts rightward in Figure 6, removing some lowability player-2 types (who mostly lose to better player-1 types) and introducing some higher-ability player-2 types who can beat better opponents. Hence, the overall e¤ect of training the laggard on the probability of erroneous selection is generally non-monotonic. In fact, we can state the following result. (ii) There exists an b H such that @ @H > 0 for H 2 L; b H and @ @H < 0 for H 2 b H; L + D .
We see from part (i) of Proposition 3 that, when H is su¢ ciently large, training player 2 by increasing L can actually increase the probability of erroneous selection for low enough levels of L. In this range, training player 2 does not contribute to the contest picking the high-ability player. For higher values of L, on the other hand, training reduces the probability of picking the wrong winner. We also see that, for low values of H, training player 2 reduces the probability of picking the low-ability player as winner.
From part (ii), we see that training the ex-ante leader by increasing H will increase the probability of erroneous selection in most cases. The exception is when H is large, in which case further increases will lead to this probability falling, since the superior player 1 will win in most cases. The exact expression for b H is given in the proof of Proposition 3 in the Appendix.
Although we could think of minimizing as a way to deal with the selection problem, it is even better to let the principal put more weight on the type-1 error the bigger the di¤erence between the contestants' ex-post abilities is -we can think of this as minimizing the expected cost of erroneous selection, Consider ability a 0 1 in Figure 6. This ability type of player 1 loses to inferior player 2 types in the interval a 2 2 [ L H a 0 1 ; L], on the line segment . The cost of losing to each opponent in this interval is a 1 a 2 > 0. Summing the expected cost over the interval gives 1 D R L a 1 L H (a 1 a 2 ) da 2 as the expected cost of erroneous selection associated with player 1 type a 0 1 . Summing over all player 1 types between H and L gives the expected cost of wrong selection associated with area b in Figure  6. This is the …rst part of the bracketed expression in (17). The second element in this equation sums up the costs associated with erroneous selection in area b 0 .
A principal solely concerned with the ex-post selection problem will seek to minimize . Note again that the prize v plays no role in this problem. The expected cost of erroneous selection is 0 at L = H; while > 0 for H > L, since the square bracket in (18) is positive for L > (H D) 3 H 2 , which holds. 16 Contrary to the probability of erroneous selection, the expected cost is strictly monotonic in L and H, as shown in the following Lemma.
Given Lemma 1, we have the following: Proposition 4 A principal who is solely concerned with minimizing the expected cost of erroneous selection will split the budget as follows.
(i) If 0 < B H L, then v equals a small amount, while the rest of the budget is spent on training to get as close as possible to symmetry.
(ii) If B > H L, then s 1 = 0; s 2 = H L, so that symmetry is obtained, and v = B H + L 0.
Since the amount of the prize does not a¤ect the cost of erroneous selection, it is only needed in order to actually induce e¤orts in the contest. In part (i) of Proposition 4, the budget is not su¢ cient to achieve full symmetry between the players and hence a cost of wrong selection of zero; in this case, a small prize is given to ensure that e¤orts are positive so that the contest can work as a selection mechanism. The rest of the budget is used to get as close to symmetry as possible. Only when the budget is large enough to achieve full symmetry (Proposition 4 , part (ii)) does the budget a¤ect the size of the contest prize. In this case, the principal trains the laggard to full symmetry, and uses the residual budget as a prize. 17 Consider next a principal who balances her concern for total expected e¤orts and that of the expected costs of erroneous selection. In particular, let her objective function be where k 2 [0; 1] is the weight put on total expected e¤orts. The cases of (1) and (0) are discussed above, with results presented in Propositions 2 and 4, respectively. There is a clear trade-o¤ that balances the two parts of the objective function, since giving more prize increases contest e¤ort, but leaves less for training, so that the cost of erroneous selection increases. Note from Proposition 2 that, when the budget is exactly B = 5 3 H L, the principal optimally trains the laggard until the contestants are expectedly of equal skill, and hence there will be no selection cost. This means that, for B = 5 3 H L, the principal sets s 2 = H L; v = B s 2 = 2 3 H, and this is independent of k. For budgets below this, the weight k will a¤ect the division between the contest prize and the training given. Again, it is optimal to only train the laggard (s 1 = 0; s 2 > 0), and we can show the following result: Proposition 5 Let k 2 [0; 1] be the weight the principal puts on total expected e¤orts. Let, for each k, T (k) = t (k) ; 5 3 H L denote a range of the non-negative real line such that, if the principal's budget B 2 T (k), then the principal's decision to train the laggard is an interior solution s 2 (k) 2 (0; H L), so that the laggard receives some training, but not enough to capture the skill level of the leader. Then, in equilibrium, ds 2 dk < 0, and dt dk 0, with dt dk > 0 whenever t (k) > 0.
As shown in Proposition 2, the optimal budget division to maximize expected e¤ort involves some training and some contest prize, except for very low budgets. Since the expected cost of erroneous selection is independent of the contest prize, lowering k from 1 gives the principal an extra incentive to train the laggard, and this incentives becomes stronger as k falls. Hence, the budget at which training starts (t (k)) is lower, the lower is k except possibly for cases where t (k) = 0. Furthermore, the amount of training given when the solution is interior will be increased, the more weight is given to preventing erroneous selection. Proposition 5 is illustrated in Figure 7. 18 FIGURE 7 ABOUT HERE In Figure 7, S(B; k = 1) is the total amount spent on training when k = 1, and this is increasing in the budget for B 2 t (1) ; 5 3 H L and constant at S = H L for B 2 [ 5 3 H L; 3H L 4 3 D]; in both cases, only player 2 receives training. Increases in the budget above 3H L 4 3 D are divided between training both contestants and adding to the prize, according to Proposition 2, so that S (B; k = 1) = s 1 + s 2 . Decreasing the weight k to expected e¤ort in the objective function increases training of the laggard for all interior solutions, and training starts at lower budgets. After the budget reaches 5 3 H L, there is no problem of erroneous selection, since the laggard has been trained su¢ ciently to have the same expected ability as the opponent, and the principal uses any budget increases to increase expected e¤ort.

Conclusion
A contest is an often-used mechanism for eliciting e¤ort. When contestants di¤er in ability or cost of e¤ort, the incentive to provide e¤ort is dampened, and many suggestions have been made as to how an e¤ort-maximizing principal may level the playing …eld. Remedies such as giving a head start or handicap, or a bias in favour of one player, or requiring threshold levels of e¤ort to obtain a prize are usually costless to the principal. In many real world situations, however, the principal implements a policy to redress the imbalance that has to be paid for from an existing and …xed budget. A sales manager can invest in training her employees for example, leaving a lower bonus to be granted to the "seller of the month". In human resource management more generally, weight has recently been attached to "total rewards" so that funds spent on an employer are spread over salary, bonuses and expenses to skill enhancement for instance.
We have considered the incentives of a principal to invest in skill-enhancing training that directly reduces the contest prize. Using a model with private information in which the ability distributions of the players overlap, we have shown how an e¤ort-maximizing principal can divide her funds to increase e¤ort through e¢ ciency gains, even when this reduces the contest prize. The potential for realizing e¢ ciency gains depends upon the size of the budget. If it is too small, then no training will be given at all, and all funds are channelled to the prize. Avoiding choosing the ex-post inferior player as winner gives the principal an extra incentive to train the ex-ante laggard, however; even small budgets may yield training if the cost of erroneous selection is given su¢ cient weight in the objective function of the principal.
Our analysis gives insight into the e¤ect on the players'expected payo¤ when the principal divides her budget between the contest prize and skill-enhancement. Several e¤ects are at work. An incremental budget unit can be used to enhance the contest prize, increasing the players'payo¤s proportionately. However, using some of the increment to train the expected laggard makes the players more similar and this draws their e¤orts closer together; the leader increases e¤ort whilst the laggard uses the e¢ ciency gain to slack o¤. This increases the probability that the ex ante leader wins the contest prize, but the fact that the contest prize is increased only in part, and the cost of exerting extra e¤ort, make his payo¤ fall. The lower cost of e¤ort and a lower e¤ort level mean that the laggard has a lower probability of succeeding, but the overall e¤ect is positive. However, it is not certain that the laggard gains a larger expected payo¤ than he would have achieved without training; this is due to the fact that training is used to increase e¤ort, without taking account of the players'payo¤s.
Two dimensions of asymmetry drive our results; the players draw ability from di¤erent intervals, and the realization is private information. In order to draw out clear conclusions we have made everything else equal, including the e¤ect and cost of training the di¤erent players. It may well be the case, however, that it is more e¤ective to train one of the players in the contest task; the expected leader may take training easier, and hence cost less to train, or it might be relatively cheap to help the expected laggard reach a higher skill level as he starts out low. Introducing these modi…cations introduces a third dimension of asymmetry and -whilst realistic -does not give clear cut results. A di¤erent base model is probably needed in order to look at these issues; one could for example dispense with the assumption of incomplete information, and look at the interplay between the known ability di¤erence and relative training cost. We leave these issues as directions for further research.

A Appendix
A.1 Proof of Proposition 1 Some properties of the equilibrium outlay functions x 1 (a 1 ) and x 2 (a 2 ) are standard (see Clark and Riis, 2001). Among these are that the e¤ort functions have a common upper support: x 1 (H) = x 2 (L) = x. For player 1, x 1 (h) = 0 and x 1 (a 1 ) > 0 for a 1 > h. For player 2, x 2 (e a 2 ) = 0 for a 2 2 [l; e a 2 ], implying an equilibrium e¤ort of zero for these types.
The …rst-order conditions for maximizing (1) and (2) are: where g 0 i (:) denotes the …rst derivative. Substituting a i = g i (x i ) into the …rstorder conditions gives a system of two di¤erential equations: Summing (A1) and (A2) yields with general solution The constant of integration, K, is determined by setting g 1 (x) = H, g 2 (x) = L into (A3): so that (A3) becomes This can then be used to substitute for g 2 (x) in the …rst-order condition in (A1): (A5) has a unique solution up to a constant of integration C: We use g 1 (0) = h to recover the constant: Thus, (A6) can be written We can use g 1 (x) = H in (A7) to …nd x: so that we can state (A7) as and g 2 (x) can be recovered from (A4) as Using g i (x) = a i and inverting (A8) and (A9) give (3) and (4) in the Proposition.

A.2 Proof of Proposition 2
Suppose the principal considers the maximization of e¤ort in two stages. At the …rst stage, she sets the contest prize v 2 [0; B] and then, at the second stage, divides up the rest of the budget B v. Working backwards, we …rst look at the problem of the principal when there is S = B v of the budget available for training, so that S = s 1 + s 2 . We initially make the assumption that H L S; so that, even if the whole training budget goes to the laggard, he is at best ex ante symmetric to the original leader. Substituting s 2 = S s 1 into (7) gives the following maximization problem for the principal: where, for now, v is treated as a constant. The maximand is decreasing in s 1 under our assumption that H D > 4 3 . It follows that, in optimum, no training will be given to 1, and the whole training budget will be given to 2: s 1 = 0; and s 2 = S. Inserting for v = B S, this means that total expected e¤ort is The solution can be found as (13), which is positive for B > L(H+L) H+2L . When B L(H+L) H+2L , total e¤ort is falling in S, making it optimal to devote the whole budget to the prize as in part (i), i.e. v = B. Furthermore, (13) satis…es our condition in (A10) only if the second inequality in (12) holds: B 5 3 H L. Otherwise, it is optimal to make the players identical through training and thereafter continue training the identical players in order to solve the following problem.
We now have H = L, and H L of the budget already being spent on player 2. So the maximization problem would be, from (8) where Z is the amount spent on training each of the two contestants after they have been equalized. The optimal additional amount of training can be determined as Z = 1 12 (3B 9H + 3L + 4D), which is positive for B > 3H L 4 3 D. Inserting Z into the expression for total training, S = H L + 2Z , we get (14) in part (iv).
When B 3H L 4 3 D, the principal will not train the players once symmetry is reached, since Z 0; hence, in part (iii), S = H L, and the prize is B H +L.

2H
> L: Recalling that L > max fH D; Dg, we have to check whether the interval

A.5 Proof of Proposition 5
Letting S denote training given to the laggard, the principal chooses S to maximize The …rst-order condition for an interior maximum is k @X @S (1 k) @ @S = 0; with second-order condition @ 2 @S 2 < 0: Total di¤erentiation in (A11) with respect to S and k gives @X @S + @ @S dk + @ 2 @S 2 ds = 0; which, using (A11), can be written as where the sign of the numerator is positive from Lemma 1, since sign @ @S = sign @ @L , and the denominator is negative from (A12). To see that dt dk > 0 when t (k) > 0, consider how S responds to a change in the budget in the range S 2 (0; H L). From (A11), we have since @X @B@S = (3H 2D)(H+2L+2S) 6H 2 > 0 and (A12) holds. Now, …x a B = t (k) such that S (t (k) ; k) = 0. Compare this with some k 0 < k, which, since dS dk < 0, implies S (t (k) ; k 0 ) > S (t (k) ; k) = 0. Since S is increasing in B, by (A13), it follows that S (t (k 0 ) ; k 0 ) = 0 for t (k 0 ) < t (k), and hence dt dk > 0.