NBER WORKING PAPER SERIES RANK-ORDER TOURNAMENTS AS OPTIMUM LABOR CONTRACTS Edward P. Lazear Sherwin Rosen Working Paper No. 401 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge MA 02138 November 1979 We are indebted to Merton Miller for helpful comments on a first draft. The research was supported in part by the National Science Foundation. The research reported here is part of the NBER's research program in Labor Studies. Any opinions expressed are those of the authors and not those of the National Bureau of Economic Research. NBER Working Paper 401 November 1979 Rank-Order Tournaments as Optimum Labor Contracts ABSTRACT It is sometimes suggested that compensation varies across individuals much more dramatically than would be expected by looking at variations in their marginal products. This paper argues that a compensation scheme based on an individual's relative position within the firm rather than his absolute level of output will, under certain circumstances, be the preferred and natural outcome of a competitive economy. Differences in the level of output between individuals may be quite small, yet optimal "prizes" are selected in a way that induces workers to allocate their effort and investment activities efficiently. In particular, by compensating workers on the basis of their relative position in the firm, one can produce the same incentive structure for riskneutral workers that the optimal and efficient pie~e rate produces. It might be less costly however, to observe relative position than to measure the level of each worker's output directly. This results in the payment of prizes, wages which for some workers greatly exceeds their presumed marginal products. When risk aversion is introduced, the prize salary structure no longer duplicates the allocation of resources induced by the optimal piece rate. For activities which have a high degree of inherent riskiness, payment based on relative position will dominate. Finally, when workers are allowed to be heterogeneous, an important result is obtained. Competitive contests which pay workers on the basis of their relative position will not, in general, sort workers in a way which yields an efficient allocation of resources. In particular, low quality workers will attempt to contaminate a firm comprised of high quality workers, even in the absence of production complimentarities. This suggests that high quality firms will use non-price techniques to allocate jobs to appli- cants. Edward P. Lazear Sherwin Rosen Department of Economics University of Chicago 1126 E. 59th Street Chicago, II 60637 312/753-4525 312/753-4503 I. INTRODUCTION It is a familiar proposition in labor economics that under competitive conditions workers are paid the value of their marginal products. In order for this statement to be meaningful, it is apparent (see Alchian and Demsetz, 1972) that some mechanism exist for ascertaining and monitoring productivity. If inexpensive reliable monitors are available, the optimal compensation method is a periodic wage that pays on the basis of observed input. However, when the firm cannot monitor input as closely as the worker can, compensation based on input that the firm actually observes invites shirking by workers and may be inefficient. In these cases, the situation can be improved if rewards are based on output, so long as that is more easily observed by both parties. It is true that compensation geared to , input is generally superior to that based on output when monitoring costs are negligible, because workers must bear more risk in an incentive contract than in a wage contract, creating gains from trade if the firm is less riskaverse than are its employees. Nevertheless, when monitoring is not free, the gain in efficiency from output or incentive payments may outweigh the loss of utility due to additional risk-bearing of workers. A wide variety of incentive payment schemes are used in practice and one of them, simple piece rates, has been extensively analyzed (e.g., see Cheung (1969), Stiglitz (1975), Mirrlees (1976)). In this paper we consider another type of incentive payment, namely contest and prizes, that has not been analyzed very much, yet which seem to be prevalent, either implicitly or explicitly, in many labor contracts. The main difference between prizes and other types of incentive compensation is that in a 2 contest earnings depend on rank order among a group of workers, whereas piece rates typically are paid on the basis of individual performance. 1 The prototype contest is the tennis match, where winning and losing prizes are fixed in advance, independently of each player's performance in that particular contest. The analysis of alternative compensation schemes is related to the problem of moral hazard and providing appropriate incentives for eliciting effort and investment when information is costly and monitoring costs are asymmetric. Each type of compensation method is characterized by certain parameters, such as the piece rate, the prize structure and so forth. For each scheme we seek the values of the parameters that maximize utility of workers, subject to a zero profit constraint for competitive firms. For familiar reasons, the scheme that actually emerges in competitive markets achieves the unconditional maximum utility of workers among the set of conditional maximums. Two dimensions of incentives need to be distinguished: one is investment or skill acquisition prior to the time a work activity is entered and the other is the effort expended, after skills have been acquired, in a given work situation or play of the game. In this work we concentrate on the first and ignore the second. The emphasis on prior investment lends itself most readily to an interpretation of earnings prospects over the whole life cycle, or lifetime rather than annual earnings. Section II demonstrates that two-player tournaments achieve the same allocation of resources as piece rates when workers are risk-neutral. Therefore, choice between the two forms of payment depends on costs of assessing rank rather than individual performance. These issues are important for the structure of executive pay. Section III extends these 3 results to N-player tournaments; sequential tournaments with eliminations, which give rise to skewed realized reward structures; and to problems where rank rather than total output is valued by consumers. Section IV shows the very surprising result that tournaments can be the social optimum contract when workers are risk-averse. This is applied to a problem originally formulated by Friedman (1951) on the relation between skewness in the overall earnings distribution and workers' preferences for lotteries. Section V discusses problems of adverse selection in tournaments in the presence of population heterogeneity and asymmetric information, and the economic structure of handicapping systems when these differences are known to everyone. II. PIECE RATES AND TOURNAMENTS WITH RISK-NEUTRALITY A. Piece Rates Consider the simplest linear production structure in which worker j produces (lifetime) output qj according to where ~ is the worker's precommitted choice of investment in the activity (a measure of skill) and € is a random or luck component drawn out of a known distribution with 2E(€.) = o. J Production requires only labor and production risk is completely diversifiable for firms, so entrepreneurs act as expected value maximizers, or as if they were risk-neutral. Notice that this production structure, which is maintained throughout, specifies that a worker's product is a random variable, but that the mean of the probability 4 distribution can be affected by the worker's own actions. The incentive mechanism is designed to elicit the optimum value of ~, or to choose the "correct" distribution. Analysis of the case where workers can also affect variance is left for some other occasion. The piece rate solution is very simple to analyze in this case. Let the piece rate be r. Ignoring discounting, the worker's net income is rq - ~(~), where C(jJ) is the cost of producing skill level jJ, with C' and C" > O.. Risk-neutral workers choose jJ to maximize which, for given r, implies C'(jJ) =r: investment equates marginal cost and marginal return. Let the firm sell the output on a competitive market at price Then expected profit is E(Vq - rq) = (V - r)jJ so entry and competition for workers bids up the piece rate to r = V. This implies, in conjunction with the worker's investment criteria, that (2) C'(jJ) = V Therefore the marginal cost of investment equals its social return, the 4standard result that piece rates are efficient. B•. Rank-Order Tournaments Now consider a two-player tournament in which the rules of the game specify a fixed prize WI to the winner and a fixed prize W 2 to the loser. Production of each player, j and k, follows (1), with €:.,i=j,k ~ 5 independently and identically distributed, and E(E:.) = 0 1. and Th~ winner is determined by the largest drawing of q. The contest is rank-order because the margin of winning does not affect earnings. Contestants precommit their investment strategy knowing the rules of the game and the prizes and do not communicate or collude during the investment period. We seek the competitive equilibrium prize structure (W 1 ,W Z ) . . Consider the contestant's problem, assuming that both have the same costs of investment C(~), so that their behavior is identical. A contestant's expected utility (wealth) is P(Wl - C(~» + (1 - P)(WZ - C(~» (3) = PW I + (1 - P)W Z - C(~) where P is the probability of winning. The probability that j wins is P = prob(qj > qk) = prob(~j - ~ > E: k - E: j ) (4) - prob(~j - ~k > ~) = G(~j - ~) where ~ =E: k - E: j • ~ ~ g(~), G(o) = zcrZ (because E: and E: E: j k is the cdf of g(~), E(~) = 0 and are i.i.d.). Each player chooses ~i to maximize (3), which requires, assuming interior solutions, (WI - WZ) ap c' (~ ) = 0---a~. i 1. (5) i = j,k (WI - WZ) aZp _ C"(~.) < 0 Z 1. a~i We adopt the Nash-Cournot assumptions that each player optimizes against the optimum investment of his opponent.(or he plays against the market over which he has no influence). This is perhaps justified 6 because investment is precommitted, each player is a small part of the market and does not know the identity of his opponent at the time investment decisions are made. Thus j takes ~ as given in determining his investment and conversely for k. It then follows from (4) that, for player j which upon substitution into (5) yields j's reaction function Player k's reversed. reaction function looks the same except with ~j and The symmetry of (6) implies that when the Nash solution exists ~j = ~k and P = G(O) =~. However, it is not necessarily true that there is a solution because with arbitrary density functions the objective function (3) may not be concave in the relevant range. 5 It is possible to show that a solution exists provided that cr2 is sufficiently large, i.e., contests e: are feasible only when the variance of chance is large enough. This result accords with intuition and is in the spirit of the old saying that a (sufficient) difference of opinion is necessary for a horse race. Technical details are slightly tedious and therefore are relegated to an appendix. Existence of an equilibrium is assumed in all that follows. Substituting ~j = ~k at the Nash equilibrium equation (6) reduces to (7) i = j ,k so each player's investment depends on the spread between winning and losing prizes. The levels of the prizes only influence the decision to enter the 7 game and the players' expected rent if entry is profitable. The condition for entry is that maximum expected utility be nonnegative. With P = ~ equation (3) becomes which simply says that expected winnings at the best investment strategy must be at least as large as opportunity cost if workers are to enter the tournament. The risk-neutral firm is more passive. Its actual gross receipts are (qj + qk)'V and its ~osts are the total prize money WI + WZ• Competition for labor bids up the purse to the point where expected total receipts equal costs, or WI + Wz = (~j + ~k) ·V. the zero profit condition reduces to But since ~ = ~ = ~ j k in equilibrium, Therefore the expected value of product equals the expected prize in equilibrium. Substituting (8) into the worker's utility function (3) with P = ~ in equilibrium, the worker's expected utility at the optimum investment strategy is (9) V~ - C(~). The equilibrium prize structure selects WI and W 2 to maximize (9) or (10) i = 1,2 and the marginal cost of investment equals its social marginal return, v = C'(~), in the tournament as well as the piece rate. So competitive tournaments, like piece rates, are efficient and both result in exactly the same allocation of resources. 8 To summarize the analysis, the problem for the firm is to choose a prize structure (W 1 ,W 2 ) that maximizes profits. The decision by individuals to invest in skill or effort (~) depends upon the spread between winning and losing. As the spread increases, the incentive to devote additional resources to improving one's probability of winning increases. The firm would like to increase the spread and thereby induce higher productivity "play" (which increases the firm's revenue). However, as the spread increases contestants invest more, but their costs are increased as well. The latter is what limits the spread. A firm offering too large a spread induces excessive investment. A competing firm can attract all the workers by·decreasing the spread, because it lowers workers' costs by more than it lowers expected earnings and therefore raises expected utility. If the marginal cost of skill acquisition is increasing, there is a unique equilibrium spread between the prizes that maximizes expected utility. Some further manipulation of the equilibrium conditions yields an interesting interpretation in terms of the theory of agency (see Ross (1972), Becker and Stigler (1974) and Harris and Raviv (1978». Solve (7) and (8) for WI and W2 to obtain WI = V~ + C'(~)/2g(O) = V~ + V/2g(O) (11) Wz =V~ - C'(~)/2g(O) = V~ - V/2g(O) The second equality follows from V = C'(~). Now think of the term C'(~)/2g(O) or V/Zg(O) in (11) as an entrance fee or bond that is p~ted by each player. The winning and losing prizes payoff the marginal 9 value product plus or minus the entrance fee. That is, the players receive their marginal product combined with a fair winner-take-all gamble over the total entrance fees or bonds. The appropriate social investment incentives are given by each contestant's attempt to win the gamble. This contrasts with the main agency result, where the bond is returned to each worker after a satisfactory performance has been observed. There the incentive mechanism works through the employee's attempts to work hard enough to recoup his own bond. Here it works through the attempts to win the contest. Let us conclude this part of the discussion with some comparative statics, all of.which follow from the marginal conditions (10) and the worker's investment decision (7). These two imply also that We first see what happens when the distribution of luck is changed. From (7), all that matters about the random variables is the value of the distribution of ~ at ~ = 0, or g(O). It is clear, however, from the definition ~= Ek - E. that a reduction in the variance of luck 0 2 J E concentrates the pdf of ~ around zero so that g(O) increases. For example, if E is normal then g(O) = 1/21 'IT02 and ag(O)/aOE < O. Yet E condition (10) shows that the optimal investment is independent of the higher moments of the distribution of chance for risk-neutral workers: marginal cost equals marginal return irrespective of g(~) so long as E(~) =O. Therefore a reduction in luck does not change the quality of the contestants. It does reduce the spread (WI - W 2 ), from (12). Equation (11) shows also that WI falls and W 2 rises, i.e., the total entrance fees decline and the size of the gamble falls. The reason is 10 that a given incremental investment buys each player a smaller incremental probability when luck is more important. The stakes must therefore increase to give contestants the proper marginal incentives to invest. This implies that among risk-neutral workers the optimal prizes are closer together in occupations that are inherently less riskY~ Note that paying by the piece gives a similar result. There are only two other exogenous factors in this problem: parameters of the cost function, C(]..I), and V. From condit.ion (12) the spread is independent of costs for a given value of V. However, anything that increases marginal costs must reduce investment since V = C'(]..l). From (11), this condition implies no reduction in the entrance fee, which remains at V/Zg(O), but a reduction in the "certain income" V']..I. Both prizes are reduced by the same amount because the lower total product cannot support the same total prize money.7 An increase in V with given costs raises investment since its value has increased, from (10). The incentive to invest more is signalled by an increase in the spread, from (lZ). Greater total productivity also supports a larger total purse WI + WZ' Equation (11) implies that W 1 must increase, but W z need not rise. The entrance fee and magnitude of the gamble are incr.,ased in any case, which is another way of saying that investment incentives rise. Finally, there is an important practical implication of these results. While it remains true from the zero profit condition (8) that ex ante expected wages equal expected marginal product in a tournament, the actual realized earnings definitely do not equal marginal productivity in either an ex ante or ex post sense. Consider ex ante first. Since ]..lj = ]..Ik = ']..I, expected marginal products are equal. So long as W 1 > W z' which it must 11 to induce any investment, the payment that j receives never equals the payment that k receives. It is impossible that the prize is equal to ex ante marginal product, because ex ante marginal products are equal. It is equally obvious that wages are not equal to ex post marginal products. Actual marginal product for j is Vqj rather than V~j. But qj is a random variable, the value of which is not known until after the game is played~ while Wi and W 2 · are fixed in advance. Only under the rarest coincidence would Wi = vqj and W 2 = Vqk. Therefore, wages do not equal ex post marginal products. C. Comparisons All compensation schemes can be viewed as transforms of the distribution of productivity into the distribution of earnings. The piece rate of Section II.A is a linear transform and apart from a change in the mean and a change in scale, the earnings distribution absolutely follows the distribution of output. The tournament in Section II.B is an entirely different animal. It is a distinctly nonlinear transformation that converts the continuous distribution of productivity into a discrete binomial distribution of earnings. In spite of this difference, we have shown that a competitive prize-tournament structure duplicates the allocation of resources achieved by a piece rate structure, when workers are risk-neutral. This is clear from examination of conditions (2) and (10). In both cases the fundamental marginal condition is the same: V = ct(~) so investment is the same. The reason for this is that risk-neutral workers only care about the first moment of the earnings distribution and in both cases that is given by V~ - C(~), where ~ is determined by (2) or (10). Even Ii though the higher moments differ, those differences do not affect expected ut~lity or wealth. In Section IV we analyze cases where workers are riskaverse. There, preferences for higher order moments in addition to the mean serve to break the tie between the two schemes on the grounds of tastes alone. Nevertheless, we believe that even in the risk-neutral setting there are important factors that influence the choice between the two. Chief among them are possible differences in costs of information. If rank is more easily observed than each individual's level of output, tournaments dominate piece rates. On the other hand, occupations in which an individual's output level is easily observed would have no particular preference for a prize scheme. For example, salesmen whose output is easily observed are paid by piece rates, whereas many business executives, whose output is much more difficult to observe, engage in contests. Consider the salary structure for executives. It appears as though the salary of, say, the vice-president of a particular corporation is substantially below that of the president of that same corporation. Yet presidents are often chosen from the ranks of vice-presidents. On the day that a given individual is promoted from vice-president to president, his salary may triple. It is difficult to argue that his skills have tripled in that one-day period, presenting difficulties for standard theory, where supply factors should keep wages in those two occupations approximately equal. It is not a puzzle, however, when interpreted in the context of a prize. If the president of a corporation is viewed as the winner of a match and as such receives the higher prize, WI' then that wage payment is settled upon not only because it reflects his current productivity as 13 president, but rather because it induces that individual and all other individuals to perform appropriately when they are in more junior positions. This interpretation suggests that presidents of large corporations do not necessarily earn high wages because they are more productive as presidents, but because this particular type of payment 'structure makes them more productive over their entire working life. A contest provides the proper incentives for skill acquisition prior to coming into the position~ III. RISK-NEUTRALITY: SOME EXTENSIONS This section extends the analysis to N players and also discusses some aspects of sequential contests. A. Several Contestants It is easy to show that the formal equivalence between piece rates and tournaments is not an artifact of two-player games, when all players are risk-neutral and have identical costs. players and prizes, W. for i th place. ~ worker's expected utility is N (13) EPiW i - C(~) Let there be N The probability of i th place is P. , ~ N so a Since the players are identical, the Nash solution, if it exists, implies Pi = l/N for all i and for all players. Therefore (13) becomes, in equilibrium, (14) (EW.)/N - C(~) ~ 14 However, for zero expected profit, the total purse must be exactly supported N by expected revenues. Expected revenue is E(6Vq.) = NV~, so competitive ~ equilibrium requires 6W. = NV~. ~ Substituting into (14), each worker acts to maximize V~ - C(~). That is, ~ is chosen to satisfy V = C'(~) just as in the two-player case. Therefore a tournament is efficient independent of the number of players. There is a curious feature of the N-player case that only first and last place prizes are uniquely determined in competitive equilibrium. This is sufficiently nonobvious that it is worth discussing, but requires a few details and is therefore relegated to appendix 2. This indeterminacy is a feature of the risk neutral case and vanishes when risk aversion is intro- duced. B. Sequential Contests This section considers a few aspects of sequential games. Before doing so, it is necessary to clarify a possible point of confusion between what might be labeled repetitive and sequential contests. 1. Repetitive games The point at issue concerns the proper interpretation of the stochastic term E in the output technology q = jl + E. As stated at the outset, we think of as investment and of q as lifetime output. Therefore E must be interpreted as "lifetime luck"; or alternatively as a drawing out of a known distribution of life-persistent person effects or 15 ability whose realization is unknown to all agents at the time investment decisions and contracts are drawn up. On this interpretation, € is revealed only very slowly, and strictly speaking in the .formal model above, at the end of the lifetime. Clearly, there is only a single period or onetime tournament in a whole lifetime, though the "period" is a long one to be sure. In repetitive play one might think of a series of, say, annual contests. This stands in the same relation to a single lifetime contest as a compound lottery does to a simple lottery. Let annual output be written Suppose now that €t is i.i.d.--qt = Jl + €t' where t is a year index. 2 2 E(€t) = 0, E(€t) = cr and E(€t€s) = 0 for t +s . In this case has the interpretation of "pure luck." In this case also, however, annual risk is diversifiable by each worker over his lifetime, e.g., by using a saving account, since a good outcome in a given year is likely to be offset by an equally poor outcome in some other year. The point is that with sufficient repetition and independent error all risk would be diversified away, for basically the same reasons that the sample standard deviation of the mean of a distribution shrinks to zero as the sample size increases. Incottle could be made constant over the lifetime and the tournament structure would unravel. Consequently, in order for a tournament to make sense there must be some risk that is nondiversifiable by the worker and that is revealed only 9 relatively slowly. Put in another way, € is the remaining element of chance after all independent components have been diversified through repetition. The simplest error structure consistent with this requirement is a variance-component specification €it = 0i + nit' where i refers to persons and t to years. Clearly, we have nothing to say about risk that the worker can diversify himself. 16 2. Sequential games: information and skew The main interest of sequential contests is their possible use in gaining information about the undiversifiable chance component in lifetime productivity. The analogy with sequential statistical analysis is suggestive of how € (or 0 in the variance-component) comes to be revealed. We cannot do justice to this complicated problem here and the following brief ff . 10 comments must su ~ce. Contests with eliminations give rise to a skewed income distribution. Consider sequential two-player games starting with N players. A winner is selected through a series of paired contests, "quarter-finals," "semifinals," and "finals." The first round consists of N/2 two-person matches: N/2 players lose, earning W 21 . N/2 win, receiving WII plus the opportunity to advance to the next round. In the second round N/4 are losers and N/4 are winners. So N/4 end up with W II + W 22 and N/4 get W II + W I2 plus the opportunity to advance to the next round. The final distribution of income has N/2 with W 21 ; N/4 with W ll + W 22 ; Z N/8 with W il + W I2 + W 23 , etc., and the winner, with L W as income, t=1 It where Z = in N/in 2. This distribution has positive skew. A plausible story can be told which yields sequential contests and skew. Returning to the point that rank may be less costly to determine than individual outputs, suppose further that it is cheaper to determine relative position in a two-person game than in a multi-person game. Let 0i be an unobserved ability component for player i, with neither contestants nor firms taking this into account when selecting an optimal ~ because it is unobserved at the time of investment. Then where n is pure luck and t th refers to the t 17 contest. If we want to select the players with the largest ~ + ° it also pays to have only winners play winners, since Therefore a sequential elimination tournament may be a cost-efficient way of selecting the best person.ll Information about prior outcomes influences wage rates and prizes in sequential games. To illustrate, let production take place in two periods and consider a two-player contest played in period 1 only. The winner receives (W 11 ,W I2 ) in periods 1 and 2 respectively and the loser gets (W Z1 ,W Z2 )' Ignoring discounting for simplicity, lifetime income is WI = W 11 + W 12 for the winner and W z = W Z1 + W 22 for the loser. Each player chooses ~ before the game is played to maximize expected wealth PW 1 + (1 - P)W 2 C(~) and the Nash solution is identical to Section II.B: C'(~) = g(O) (WI - W 2 ). The budget constraint is only slightly altered to WI + W z = 4V~, so in competitive equilibrium WI and W z are basically the same as before, though twice as large because production takes place over two periods instead of one. 12 If workers are not bound to the same firm over their lifetime then competition would bid up the second period wage of the winner to the expected value of the second or largest order statistic: W12 :I E(Vq j2 1j wins) = V~ + VE(oj Iqj 1 > qkl) , where qil is first-period production and qi2 is second-period production of player i. The reason is that a competing firm, knowing a game had been played, could infer the second-period conditional expectation of the winner's product simply by knowing his identity. The same logic implies W Z2 = E(vqjZlj loses) = V~ + E(ojlqjl < qkl). If the firm organizing the contest attempted to 18 ensure that the winner stayed by paying more than E(VQj2 !j wins), it would have to pay less than the conditional expectation to the loser. Competition for losers would make the firm unactuarial in period 2 and invite bankruptcy. With W 12 and W 22 fixed by their conditional expectation, W ll and W 21 are determined to be consistent with the competitive game equilibrium comditions on gross wealth WI and W 2 , the firm is actuarially balanced period by period and it makes no difference if the players depart or not in period 2. One can even imagine the existence of firms whose major activity is running contests among young workers (period i) that provide sorting and information services for other firms for older workers (period 2), minor leagues in a sense. This is essentially financed by the workers themselves in the manner of the entrance fees discussed in Section II. This places restrictions on the shape of the "age-earnings" profile, but there remains an element of indeterminacy to the prize or wage in each period, though not to wea.lth, and W 13 2' C. Rank-Order Objective Functions It is of some interest to note that the fundamental solution to the tournament prize structure survives a broad class of alternative specifications of the revenue function. Consider a horse race. The criterion v . (Uj + Uk) assumes that gate receipts are proportional to the total or average speed in the race. It may be that spectators also care about rank as well as overall speed. Let Q i be the expected value of the speed of 19 the i th place horse (an order statistic). If spectator's willingness to pay depends on the expected quality of the winning horse, revenue is instead of So long as win and place prizes so gate receipts also dependis the cdf ofthenof and W 2 are the stakes, contestants still behave exactly as in Section II: ~ is the same for both and depends on the spread. If F(e) is the cdf F 2 (Q -~)1 on the spread through its influence on the distribution of the winner's expected speed. The competitive prize structure still maximizes contestants' expected utility but the zero profit constraint is somewhat altered from the above. However, only the details differ: The equilibrium spread solves the maximum problem and the prize levels satisfy the constraint. Similar considerations apply if revenue depends on Q 1 and Q 2 so that spectators care about expected speeds of both ranks and on the closeness of the contest. Notice that since the revenue function always involves ~,there always exists an alternative piece rate scheme: For example, each horse could be paid in proportion to its speed. Therefore virtually the same issues of comparison arise as those considered above: Even if revenue depends on rank the optimum compensation method need not depend on rank. As an historical note, one of the few papers we have been able to find on the prize structure in tournaments is by Sir Francis Galton (1902).15 Galton considered a race with n-contestants where a fixed sum was to be divided among the first two places. With prizes W 1 and W 2 Galton proposed the following division on strictly a priori grounds 20 That is, the ratio of first to second prize stands in the same relation as th~ corresponding expected rank-order differences over third place. This criterion is roughly related to expected relative outcomes or productivity. Galton's work is an important paper in statistics but is less interesting to economists because it does not take account of a contestant's incentive to run fast. His contest is a random draw of a sample of size n from a fixed population. By contrast, the crucial aspect of our work is in allowing the prize structure to change the population distributions through incentives to invest. The population distributions are therefore endogenous in our model but exogenous in his model. Nonetheless, Galton went on to show the remarkable result that (Ql - Q3)/(Q2 - Q3) is approximately 3.0 independent of n when the parent distribution is normal. Therefore his criterion leads to a highly skewed prize structure. This result would perhaps be less surprising today, given what is known about the characteristic skew of extreme value distributions, yet it does roughly concur with the prize structures commonly observed in sports activities such as tennis and horse racing. Skew is not necessarily an implication of a single n-player contest in our model, though it is not inconsistent with it. We believe, however, that skewed prizes in each play of a repetitive game structure may be a strong implication of the incentive to elicit effort in each round and that dimension has been ignored here. 21 IV. OPTIMAL PRIZE STRUCTURE WITH RISK-AVERSION In the previous section, we obtained the surprising result that contests and piece-rate compensation schemes yield identical and efficient allocations of resources. In this section, we allow for risk-aversion of contestants and show that the choice of compensation scheme is no longer indeterminate. Even when costs of ascertaining output are zero, the piece-rate scheme tends to dominate for distributions of € with low variance, while prizes tend to dominate for high variance distributions. The reason is roughly as follows: In a piece-rate scheme, players are paid a fixed sum (to be determined optimally) plus some proportion of their output. The higher is the variance in chance, €j' the higher will tend to be the variance in income (although this can be offset by reducing the variable and increasing the fixed part of the payment). A prize structure truncates the extremes of the distribution by converting it to a binomial. However, the necessity of a positive spread in the prize structure implies that there must be some variance in income. For small 2 a , € it appears as though the variance introduced by the requisite spread exceeds that which results from paying by the piece. But as 2 a gets large, higher expected € utility is produced by fixing maximum gain and loss in advance with prizes than by taking one's chance and allowing €j to affect wages strongly. The fixed component would have to be so high relative to the variable piece component in the piece rate that investment incentives are too small and prizes dominate. 22 A. Optimum Linear Piece Rate 16 The piece-rate scheme analyzed pays workers a fixed amount, I, plus rq, where r is piece rate per unit of output. The problem for the firm J is to pick an r, I vector that maximizes workers' expected utility (15) max{E(U) = maxfU(y)8(y)dy} l,r where (16) y:: I + rq - C(1l ) - I + rll + r8 - C(ll) and 8(y) is the pdf of y. Since V~ is expected revenue from a worker and I + r~ is expected wage payments, the zero profit market constraint is (17) V~ := I + r~ . The worker's problem is to choose ~ to maximize expected utility, given r· and r. If € '" f (€) , the worker's problem is Max E(U) := fU(I + r~ + r€ - C(~»f(€)d€ • ~ The first-order condition is aEa~U) = f[U'(y)](r - C'(~»f(€)d€ := 0 , which conveniently factors so that (18) r = C'(~) . Condition (18) is identical to the risk-neutral case, and it should be. Investing in ~ gives a certain return of r; the error, €, is 23 independent of investment effort, ~, and as a result e does not affect Solving (,1'0 for I and substituting into u.6) the optimum contract maximizes with respect to r, where ~ = ~(r) satisfies u.8). The marginal condition is d~ !U'(·)[(V - C'(~» dr + e]f(e)de = 0 or in more compact notation [v - C'(~)] d~ EU' + EeU' = 0 . dr Solving ~9) yields the optimal r. Then the optimal I is found from (17) • Condition (19) does not lend itself to ready interpretation. However, using Taylor series approximations to the utility function, the optimum is approximated by (20) (21) h f b 1 . k . 17 t e measure 0 a so ute r1S -averS10n.where _ -U" s = U' increases (see (20» in V and decreases in s, C" and 2 cr e Investment because 24 all these changes imply corresponding changes in the marginal piece rate r which influences investment through condition (18). The same changes in V, sand C" have corresponding effects on the variance of income (see (21» but an increase in 2(J €: can actually reduce variance if 2(J €: is large because it reduces r and increases I. B. Optimum Prize Structure The worker's expected utility in a two-player game is (~2) E(U) = P[U(W1 - C(p*»] + (1- P)[U(W2 - C(p*»] where * denotes the outcome of the contest rather than the piece-rate scheme. The optimum prize structure is the solution to (23) max P U(W 1 - C(p*) p* + (1 - P) U(W 2 - C(p*) } subject to the product market constraint Vp* = PW + (1 - P)W1 2 In maximizing (27) the worker sets oE(U) = 0 oll* . Since cost functions are the same and and €k are i.i.d. the Nash solution implies Pj =Pk and P = ~ as before, and the worker's investment criterion simplifies to (25 ) C'(].l*) =2[U(I) - U(2)]g(0) U' (1) + U' (2) where U(.) = U(W - C(ll*» • and = U' (W - C(ll*»,' • = 1,2.18 • Equation (25) implies 25 and the optimum contract (W1,W Z ) maximizes (27) 1 1 EU* =2" U(W 1 - C(lJ*)) + 2" U(WZ - C(lJ*)) . subject to (24), with P =~, and (26). Increasing marginal cost of investment and risk-aversion (U" < 0) guarantees a unique maximum to (27) when a Nash solution exists. Again, the marginal conditions are not very revealing and second-order approximations are required. Assuming a normal density for €, these are z • (Jy* = where = W - C(Il*) Z if if statics of (28) and (29) are similar and need not be repeated. and 19 cov(€j'€k) =O. The comparative to the piece rate (20) and (21) and 26 C. Comparisons Compare (21) to (29). This difference is rr (=3.14159) in the denominator and numerator of ~9). This illustrates the proposition piece rates produce an income with a lower variance 2 At high values of cr, the prize has a lower variance than e than stated at the outset: 2 .;; - 1 (j < ----"--- e: sC"(rr _ /;) the prize. For low values of 2 cr , e: specifically when the piece rate. This, of course, does not imply that piece rates dominate when 2 1;- 1 That would be true only if investment were thecr < e: sC" ('IT - /;) same, but generally fl :f: fl*, so expected utility must be examined to determine which scheme is best. Since it is difficult to "nest" the two schemes analytically we have constructed two examples which demonstrate the proposition that piece rates dominate for low values of and initially let the utility function be while prizes dominate for and v = 1,2 e. 'V N(O,cr ), J e -U" /U' = 1.soU = -e-y 2 cr , e 2 = fl /2We assume that C(fl) Z 20 cr • e high values of Equations (25) , (27) and (28) are solved for WI' Wz' fl* and E(U*). Then equations (19) , (17) and (15 ) are solved for r, I, fl and E(U). The computations were carried for various values of 2 and the resultsout cr , e: 21 are given in Table 1: 27 Table 1 Constant Absolute Risk-Aversion 2 )l* E(U) - E(U*)(J )l -f.. .5 .667 1.03 .275 1.5 .. 40 .678 .088 2.5 .285 .567 .029 3.0 .25 .532 .012 3.5 .222 .505 -.0008 6.5 .133 .413 -.041 8.0 .117 .395 -.051 The same was done for a utility function of the quadratic form, U =_y2 + 1000y. Table 2 contains those results: Table 2 Quadratic Utility Function 2 )l* E(U) - E(U*)(J )l -f.. .01 .999 .999 .021 1 .999 .993 2.12 100 .909 .614 92 150 .869 .514 61 200 .833 .443 -46.2 Both tables tell the same story. When (J2. 11·~s sma p~ece rates e: 2 dominate, but when (J goes above some level a prize yields the largest e: expected utility. With risk-neutrality we showed that there is always a unique prize structure that duplicates the resource allocation of the optimal piecerate wage structure: There, each compensation scheme yields the same expected utility, and choice between them is on the basis of different monitoring and observation costs. In this section, risk-aversion supplies 28 another criterion for choice. Because the prize structure that duplicates the investment of a piece rate structure has a different variance (and distribution in general), workers generally prefer one to the other: The expected utility of the optimal prize scheme does not, in general, equal the expected utility of the optimal piece rate structure. In the examples in Tables 1 and 2, piece rates dominate when luck is relatively unimportant (o~ small) and prizes dominate when luck is more important (cr~ ,large). This suggests perhaps, that high level executives, whose decisions may have a large effect on output will have salaries which more closely parallel prizes. Production workers, on the other hand, for whom tend to be compensated according to a 0 2 E: . t 22p~ece ra e. is likely to be low, will An additional consideration affects the choice between piece rates and tournaments. Let the worker be risk-averse and let cr2 be sufficiently small E: so that the piece rate optimum yields a higher expected utility. If there is measurement error associated with gauging output, then the prize structure may well dominate. Suppbse that the output estimator for worker i is vit is random error, and Pt is common measurement error which is matchspecific but affects both workers similarly.23 For example, j and k may have the same supervisor who always undervalues output. Or a mechanical counting device might run too fast or too slow. This type of measurement error affects the actual income variance in a piece rate scheme, but if compensation is organized by prizes, the common measurement error drops out: A A qjt qkt = qjt - qkt + V jt - vkt . 29 The sign of is unaffected by Contest-specific measurement error adds no noise to the tournament, but does contribute variance to piece rate income. Therefore, if structure may dominate even if 2 cr p is sufficiently large, the prize t is small. D. Skewed Earnings Distribution with Risk-Aversion In the risk-neutral world, we obtained a positively skewed distribution of income by ,introducing contests with eliminations. Skewed overall income distributions fallout of the analysis with risk-aversion quite easily. Suppose that there are two occupations, A and B, and that all individuals are alike. Let jls output in A or B be given by m m =ll.+e:. J J m =: A,B Assume for simplicity that costs of investment are the same in both let it be the case that as the result of the 2 > cr B. Further, E: a piece rate 2 is inherently riskier, i.e., cr A E: risk differences, Aoccupations, but that Those in B end up with income that reflects the distribution of scheme is optimal for B, whereas the prize scheme is optimal for A. B E: • If that is normally distributed, then their incomes are normally distributed with mean either WI B I + r].l 2 2 and variance r cr B. Those in A receive incomes of E: Since A is inherently riskier than B, the expected utility associated with A is lower than that for B at any common V. Therefore, relative supplies of workers must adjust to increase the value of services in A until expected utilities are equal across occupations. This implies that the mean income in A must exceed the mean in B to compensate for the additional risk. The distribution of income for both groups is shown in Figure 2: because 30 The overall distribution is the sum of a normal and a binomial, weighted by the number of individuals in each occupation. It is positively skewed _A _B Y > Y . This result is reminiscent of a paper by Friedman (1951) on social arrangements producing income distributions that satisfy workers' risk preferences. It demonstrated how a Friedman-Savage utility function leads to a two-class income distribution: Those people who are risk-averse are assigned to relatively safe occupations, where income follows productivity; whereas those with preferences for gambles buy lottery tickets in very risky occupations where a small number win very large prizes. The overall distribution is the sum of the two and exhibits skewness in the right tail. While Friedman's work is a remarkable precursor to more recent treatments, it is flawed by inadequate attention to the problem of moral hazard or incentive. When the role of incentives is modeled more completely we 31 obtain a similar result with only risk-aversion, since the optimum contract is a tournament if the underlying distribution of outcomes has a large variance. V. HETEROGENEOUS CONTESTANTS Workers are not sprinkled randomly across firms, but rather seem to be sorted by ability levels. One explanation for this has to do with complementarities in production. But even in the absence of such complementarities, sorting may be an integral part of optimal labor contract arrangements. Therefore, this section analyzes tournament structures in the risk-neutral case when investment costs differ among persons. Two types of persons are assumed, a's and b's- , with marginal costs of the a's being smaller than that of the b's: C~(~) ~. We are led to the unalterable conclusion that a pure price system cannot sustain an efficient competitive equilibrium in the presence of population heterogeneity and private information. This does not say that price competition cannot separate markets, but it does imply that if markets are separated on the basis of price incentives alone then they must be inefficient. It is easy to produce an example of such an equilibrum. The a's want to differentiate themselves from the b's because they are potentially more productive. Consider the following two contests. In one the prize structure is set up as in Section II, equations (11) and (12). In the other there is a much larger spread and a smaller losing prize. Following the development of the first part of this section, assume that only b's are attracted to the first type of contest and only a's to the second. The situation is shown in Figure 4. The curve labeled R1(~) is the expected payoff from entering the first type of contest at investment ~ and is identical to ~(~) in Figure 3. R2(~) is the expected payoff to entering a contest with greater spread and smaller losing prize. It is therefore steeper than R (~) a in Figure 3 and has a smaller intercept than In distinction to Figure 3 where R (~) a everywhere dominated for larger values. The overall return function is therefore the upper envelope of these two curves. As illustrated, the b's find it optimal to invest ~~, which is efficient for them; but the a's invest , ~a' 38 ., ; , ; I,. :/ /,' / ", ,, , :. II I ------.- If, (A.) / J / which exceeds J.l*.a These values satisfy zero profit because R (J.l*) =Vj.I*1 b b and That is, in order to differentiate themselves the a's must overinvest and engage in a gamble with larger variance (cf. Aker10ff (1978) for a similar kind of result in other contexts). This example bears a family resemblance to the work of Rothchild and Stiglitz (1977), Wilson (1978), and Riley (1975) on adverse selection in inSurance markets. We .know from that work (and it is obvious from construction of Figure 4) that a separating equilibrium need not exist; but difficulties remain even when it does: (1) The equilibrium in Figure 4 may be less efficient than a nonseparating equilibrium with mixed play. (2) This type of separation may not be sustainable against a broader set of strategies. To elaborate (this point also applies to the insurance 39 problem), suppose the leagues and prizes are set up as in Figure 4 with players "signing" into a league prior to committing investment. The mere act of signing into league 1 or 2 therefore reveals their type. But once types are revealed all those labeled as a's can gain from trade: There are post-signing incentives to set up yet another league that satisfies conditions (11) and (12) for that yields higher rents for the a's than league 2 and no game in league 2 is ever played. If that happens no b signs up in league 1, knowing that signing in the "dummy" league 2 labels him an a and ultimately entitles participation in what appears to be a dominant game, as in Figure 3. This behavior, akin to a time inconsistency, unravels the two-league structure and separation is not achieved. The practical resolution of these difficulties, which has its obvious counterpart in the structure of real-life tournaments, is the use of nonprice rationing and certification to sort people into the appropriate games, based on past performances. Similarly, firms use nonprice factors in allocating jobs among applicants. Presumably the market solves a complicated experimental design problem, perhaps with a hierarchical structure, that elicits this information in an efficient manner. The issues are similar to those sketched in Section III.B. Upon reflection, it is not terribly paradoxical that price information alone does not allocate resources efficiently in these circumstances and that nonprice rationing can be more effective. After all, only a fool or a person with tastes for random dissipation of his wealth would examine only the price of a transaction when there is a nontrivial probability of misrepresentation of the other terms. That nonprice factors are ubiquitous in labor markets is recognized in the theory of equalizing differences. For practical examples of nonprice rationing one need go no 40 further than the problem of allocating academic economists to departments. Yet there appears to be a special problem with tournament structures that should be pointed out. Strictly speaking, in the formal model above there is no social value of sorting inherent in the technology, since total output is the sum of independent outputs of all the workers. The tournament introduces a socially contrived dependence through strategic factors involved with attempting to improve one's chance of winning, for which relative output is crucial. The independence of production implies that sorting is no problem in the piece rate solution when workers are riskneutral: It makes no difference who works with whom because rewards are based on independent individual performance?5 Of course, there would be productive value of sorting if the objective function were not summable, e.g., V(qj,qk) or the order statistic example in Section III.C instead of V· (qj + qk). Nevertheless, the above differences may suggest that stratification and sorting present greater difficulties for tournaments than for individual performance-geared incentive schemes. Still, tournaments may be the socially efficient arrangement if rank is easier to observe than is individual performance. B.Handicap Systems This section moves to the opposite extreme of the previous discussion and assumes that the identities of each type of player are known to everyone. Gambles involving N players of known talent are said to be fair if each player has win probability of liN. When the quality of play is drawn out of fixed distributions that differ among players, fairness is achieved by handicaps that equalize the upper liN quantiles of the various 41 distributions; e.g., with N = 2 the weaker player is given "points" to equalize the medians. It is surprising that this criterion of fairness fails to hold true in competitive markets where prizes affect probability distributions and the gambles are productive: The market clearing handicap implies less than full equalization so that the better player always has a competitive edge. Consider again two types a and b now known to everyone. Prize structures in a - a and b - b tournaments satisfying (11) and (12) are efficient, but those conditions are not optimal in mixed a - b play. Denote the socially optimal levels of investment by lJ* and lJ~ , their a difference by LllJ, and the prizes in a mixed league by WI and W 2 , Let h be the handicap awarded to b with LllJ ~ h ~ O. Then the Nash solution in the a - b tournament satisfies C' (lJ ) a a (37) (The second condition in (37) follows from symmetry of g(~).) Since independence of outcomes implies no social preference for alternative structures when revenues are additive, the efficient investment criterion is v = C'(lJ*) = C' (u:k) a a b . b ' independent of pairings, Therefore from (37) the optimum spread in a mixed match must be (38) LlW = V/g(LllJ - h) to induce proper investments by both contestants. The spread is larger in mixed than pure contests unless a gives b the full handicap 42 h = ~~ - ~~. Otherwise, the appropriate spread is a decreasing function of h. - - -and W z must also satisfy the zero profit constraint WI + Wz = v . (~~ + ~~) independent of h since the spread is always adjusted to induce investments ~~ and ~~. The gain to an a from playing a b with handicap h rather than another a with no handicap is the difference in expected prizes since costs are the same in all matches C (~*): a a Y (h)a where P = G(A~ - h) is the probability that a wins the mixed match. The corresponding expression for b is The zero profit constraints in a - a, a - band b - b imply that ya(h) + yb(h) = 0 for all admissible h; so the gain of playing mixed matches to a is completely offset by the loss to b and vice versa. If Ca(~) is not greatly different from Cb(~) then A~ = ~: - ~~ is small and P ~ t+ g(A~ - h). This approximation and the zero profit constraint reduce (39) to (41) Y (h) ~ -VA~r:i _ g(O) (l _ J!.)l a l: g(A~ - h) A~J The expression for yb(h) is the same except for the absence of the minus sign in front of VA~. y~(h) = -yb(h) < 0 so the gain to a decreases in h and the gain to l increases in h. Moreover, Y (0) = Vt.].l [g(O) - 1] > 0 and Y (t.].l) = -Vt.].l/2. Therefore there exists a g(t.].l) 2 a a unique h* satisfying Ya(h*) = yb(h*) = 0 and 0 < h* < t.].l. If the actual handicap is less than h* , a's prefer to play in mixed contests rather than with their own type while b's prefer to play with ~'s only. The opposite is true if h > h*. Since the gain to the one is the loss to the other, side payments and guarantees could induce b's to play against a's in the first case and a's to play against b's in the second. However, side payments are unnecessary when h = h*, for then no one cares who they are matched against. Therefore h* is the competitive market clearing handicap and the condition 0 < h* < t.].l implies that it is less than full. The a's are given a compe~itive edge in equilibrium, - 1 1 p ~ 2 + g(t.].l - h*) > 2 because they contribute ~ to total output in mixed matches than the b's do. This same result holds if € has a a different variance than €b' but may be sensitive to the assumption of statistical independence and output additivity. VI. SUMMARY AND CONCLUSIONS This paper proposes an alternative to compensation in proportion to marginal product. Under certain conditions, the new scheme yields ~~ allocation of resources identical to that generated by the efficient piece rate. By compensating workers on the basis of their relative position in the firm, one can produce the same incentive structure for risk neutral workers that the optimal piece rate produces. It might be less costly, however, to observe relative position than to measure the level of each worker's output directly. This would result in paying salaries which resemble "prizes"; wages which for some workers greatly exceed their presumed marginal products. 44 When risk aversion is introduced~the prize salary structure no longer duplicates the allocation of resources induced by the optimal piece rate. For activities which have a high degree of inherent riskiness~ contests will tend to dominate. For sufficiently small levels of inherent riskiness~ the competitive piece rate tends to dominate. Given risk aversion~ a positively skewed overall distribution of income is the natural outcome of the mixing of the distribution of income for those paid prizes with that for those paid piece rates. Finally, we allow workers to be heterogeneous. This complication adds an important result: Competitive contests will not, in general, sort workers in a way which yields an efficient allocation of resources. In particular, low quality workers will attempt to contaminate the firm comprised of high quality workers, even if there are no complementarities in production. This contamination results in a general breakdown of the efficient solution if low quality workers are not prevented from entering. This is, therefore, one rationalization for the use by high quality firms of initial credentials when allocating jobs to applicants. 45 FOOTNOTES 1. This statement needs qualification. For example, annual bonuses may depend on group measures such as total profits of the firm, but not on rank order of workers within each labor category. 2. Virtually all the results of this paper hold true if the error structure is multiplicative rather than additive, but the exposition is slightly less convenient in the case. 3. Throughout the analysis, we assume that the worker has control only over ~. A more general specification would allow him to affect the variance of E. Although this will not alter the solution in the risk neutral case, risk averse workers might sacrifice ~ for lower variance of E. This problem, of inducing the worker to avoid cautious strategies, is one which we do not analyze here. 4. A nonlinear piece rate schedule r(q) provides the correct marginal incentives so long as r'(q) =V at equilibrium. It is clear that a one-parameter linear piece rate is the competitive solution in this case, since nonlinearities or two-part tariffs can only transfer inframarginal rents to employers, thus violating the zero profit condition. A one-parameter piece rate is definitely nonoptimal in the presence of risk aversion. See below. 5. Since a~P =g(~j -~) and g(.) is a pdf, a2P/a~~ =g'(~j - ~k) j may be positive and fulfillment of second-order conditions in (5) may imply sharp breaks~in the reaction function. See the appendix for elaboration. 46 6. With normal errors (12) and (8) imply The ratio of the spread to the mean prize is proportional to the coefficient of variation of output. This formula also applies to N an N-player game with W N replacing W 2 in the numerator and rWi/N in the denominator. 7. Of course, in a full analysis any change in costs of all participants would change industry supply and therefore alter V, but those qualifications are straightforward. 8. Lazear (1979, 1979a) uses this notion in a deterministic but dYnamic setting. By paying an individual more than his marginal product later in life, and less than marginal product earlier in life, one induces optimal effort and investment behavior throughout the worker's lifetime. 9. Of course, the risk is diversifiable across workers and therefore by firms. One might also think that risks could therefore be pooled among groups of workers through, say, sharing agreements, but that is false because of moral hazard. A worker would never agree to share prizes since doing so (completely) results in ~ =0 so that E(qj + qk) =O. As the result, firms offering tournaments' or piece-rates in the pure sense yield higher expected utility than the sharing arrangement. 10. There is some discussion of contests in the statistics literature for the method of paired comparisons. Samples from different populations are compared pairwise and the object is to choose the one with the largest mean. If all pairs are compared, the experimental design is similar to a round-robin tournament. An alternative design is a 47 knockout tournament with single or double elimination, which does not generate as much information as the round-robin but which requires fewer samples and is cheaper. These problems are discussed by David (1963) and Gibbons, Olkin and Sobel (1977), but are not helpful here because they assume fixed populations, whereas the essence of our problem is to choose an experimental design and payoff mechanism that induces players to choose the correct distributions to draw from. A complete treatment would add the additional complexity of allowing investment strategies to change as new information becomes available. 11. With the current technology and risk neutral workers, there is no advantage to being able to distinguish the best from second-best, etc. The motivation for this analysis comes from considering other payoff structures where level is important in nonadditive ways. A multiplicative technology, where revenue equals Vf(qjqk), for example, implies that sorting of workers is important. Similarly, a payoff which depends on the level of the highest output individual may also require sorting. This latter case is discussed below in another context. 12. Strictly speaking, the production variables in the two-period problem should be rescaled to make the outcomes identical with Section II, but that is a detail. 13. See Lazear (1979) for additional details. 14. Economies of joint consumption, in some activities such as performance, may imply very large rewards to a small number of people. To a potential entrant the game might look like a lottery based on rank order and it may appear as if consumers value rank perse. See Rosen (1979) for elaborat ion. 48 15. We are indebted to George Stigler for this reference. There is a well-known paper by Mosteller (1951) on how many games must be played before one is fairly certain that the best team emerges victorious, as well as the literature on paired comparisons referred to above, but these are not directly relevant. 16. The following is similar to a problem analyzed by Stiglitz (1915). A linear piece rate structure is a simplification. A more general structure would allow for nonlinear piece rates (see Mirrlees, 1916). A still more general model allows for the compensation scheme to be a random variable. Prescott and Townsend (1919) consider this in the 11. context of a selection problem. Furthermore, r::: V/(l + sC"cr2) e: r = V and I = ° in the case of risk ·aversion, s = 0, as asserted in Section II. All these approximations use first-order expansions for terms in U'(.) and second-order expansions for terms in U(·). The same is true of the approximations below for the tournament. 18. With N players and N prizes (WI the first place prize, W2 second N place, etc.) equation (25) becomes C,(~) =k[U(l) - U(N)]/EU'(T), where k is a constant. As in Section III.A, the numerator contains terms only in first and last place prizes. But now the denominator contains terms in all the other prizes as well, removing the source of indeterminacy of the entire prize structure that occurred in the risk-neutral case. 19· Furthermore, C,(~*)::: g(O)(W l - W 2 ), so the spread is still crucial for investment incentives, as in the risk-neutral case. 49 20. The following contract contains the prize and piece rates of Section IV as special cases. Denote gross income by Y: 11 + r l qj if qj > qk Y. =J 12 + r 2 q j if qj < qk The piece rate of Section IV is (Nl) with the additional constraints and The prize is (Nl) with imposed. It is not difficult to write down competitive equilibrium conditions for contract (Nl) but they defy interpretation. Following the general theorem that less constraints are better, (Nl) must be superior to either piece rates or prizes in the case of risk ·aversion. In the riskneutral case all three are tied. 21. We have not verified that a Nash solution exists at each value of cr~ in the contest, but since existence is less likely for smaller values of cr 2 this cannot affect the conclusions drawn from these examples. €: 22. In this context, no distinction is drawn between time rates and piece rates. To the extent that time rates are dependent upon output level over some time period, they fall into the piece rate category. 23. Pt must be match-specific and not firm-specific. A firm-specific error that applied to all matches in that firm would lead to systematic bias. If the firm survives, an efficient market would come to learn of the bias and that would be fully adjusted in the piece rates or prizes it offered, in the nature of an equalizing difference. 24. Expand G(~ - ~~) ~ around zero up to third order, the cubic being necessary becaus.e g'(O) = 0 since g(F;) is synnnetric around O. 50 Substitute into (36) to obtain This difference exceeds zero for ]..1 > 0 since ]..1* > ]..1b* and g"( 0) < o.- a 25. The qualification of risk neutrality is important. There is definitely a sorting problem if workers are risk averse (see Stiglitz, 1975). 51 REFERENCES Alehian , A.A. and H. Demsetz. "Production, Information Costs and Economic Organization," American Economic Review. LXII, No.5 (pp. 777-795) December, 1972. Becker, G. and G. Stigler. "Law Ecforcement, Malfeasance and Compensation of Enforcers," Journal of Legal Studies. (pp. 1-18) 1974. Cheung, Steven. The Theory of Share Tenacy: With Special Application to Asian Agriculture and the First Phase of Taiwan Land Reform. Chicago: University of Chicago, 1969. David, H.A. The Method of Paired Comparisons. London: Charles Griffin, 1973. Friedman, M. "Choice, Chance and the Personal Distribution of Inc9me," Journal of Political Economy. Vol. LXI, No.4 (pp. 277-90) August, 1953. Galton, Francis. "The Most Suitable Proportion Between the Values of First and Second Prozes," Biometrika. Vol. 1 (pp. 385-390) London: Cambridge University Press, 1901-1902. Gibbons, J.D., I. Olkin and M. Sobel. Selecting and Ordering Populations: A New Statistical Methodology. John Wiley and Sous, New York: 1977. Harris, M. and A. Reviv. "Some Results on Inceptive Contract with Application to Education and Employment, Health Insurance and Law Enforcement," American Economic Review. Vol. 68, No.1 (pp. 20-30) 1978. Lazear, Edward. "Why Is There Mandatory Retirement," Journal of Political Economy. December, 1979. "Agency, Earnings Profiles, Productivity and Layoffs," unpublished draft. Chicago: University of Chicago, April, 1979, (1979a). Mirrlees, J.A. "The Optimal Structure of Incentives and Authority Within an Organization," Bell Journal of Economics. 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Vol. 6, No.2 (Pl" 552-579) Autumn, 1975. Wilson, Charles A. "A Model of Insurance Markets with Incomplete Information," Journal of Economic Theory. Vol. 16 (2) (Pl'. 167-207) December, 1977. A-I APPENDIX 1. Nash Solution (AI) (A2) J's reaction function satisfies (5) and (6) (w W)g'( ) - C"(".) < 01 - 2 llj - llk "J g(llj - llk) is a symmetric function with mode at O. The Nash solution, if there is one, implies llj :0 llk or C'(ll.) J and is unique. Differentiate (AI) to obtain Since g is symmetric the reaction function is positively sloped for llj > llk and negatively sloped for llj < llk' Assume jj = ll. = llk is an J equilibrium satisfying (AI) . The difficulty is that llj = jj is a consistent reply to II = ii, but may not be the best reply. Figure Al shows a case k where it is the best reply and Figure A2 shows a case where it isn't. Reaction functions for both j and k corresponding to each of these cases are shown in Figures A3 and A4. Notice that the reaction functions are discontinuous: Inflection points in g(llj - llk) imply there always exists a value of llk at which there are two equally good replies by j satisfying (AI), as is implicit in the construction of Figures Al and A2. It is difficult to completely characterize all conditions that ensure equilibrium because they do not involve marginal conditions. It is clear from Figures Al and A2, however, that the variance in s cannot be too small for equilibrium (as is much smaller in Figure A2 than in Figure AI). A-2 ~ J '1 A2 "u,. J i I'j, 't'~4..t..,."rJ +a..7\C+,cm ?~ .)I, I"e4.cT,Ot\ -'----- .J.U.llc...T'OY\ APPENDIX 2 For a given prize structure WI' W 2 , W 3 and opponents' investments is chooses ll. ] to maximize (13). The first-order condition ap ap ap (1\3) W _1 + W _2 + W _3 - C' (ll .) = 0 1 all. 2 all 0 3 all]o ] ] ] As in Section II, we must evaluate the marginal effects of investment on the probabilities. For example, j wins if or and Define ~k = £k - £j and ~t =£t - £j and denote the joint density of (~k'~t) by ¢(~k'~t)· The ~'s have zero means, E(~k) = E(~t) = 0, but nonzero covariance, E(~k ~t) = 20; +0, because of the common component £j in both ~k and ~t. Define the skill or investment differences between j and his opponents by and Then the probability ,of win may be written Similarly, the probability of show is The probability of place is P2 = 1 - PI - P 3 . Finally, since the probabil~ ities add up to unity, (A5) A2 The Nash solution, if it exists, implies ~j = ~k = ~t and P = P = P = 1/3 for each player. Therefore Sk = Sn = 0 in equilibrium. 1 Z 3 "We now show that at equilibrium aPl/a~j = -aP3/a~j so that ap Z/dllj = 0 in order to satisfy (4.5·). Refer to Figure Itt, The ellipses show the probability contours of ·(i;;kfi;;t). This joint pdf is symmetric because E: .', E: k and E: t are Ll.d. But Sk = St = 0 in equilibrium, so J PI is the integral of over the third quadrant, P3 the integral over the first quadrant and P z the inte-gral over the second and fourth quadrants. Positive correlation between ~t and ~k makes all these probabilities equal to 1/3. Suppose now that j makes an incremental investment, given ~k and ~t' that increases Sk and St from 0 to 65 k and 65 t . Then PI rises by the integral of (~k'~t) over the two shaded strips. By the same token, P 3 falls by integral over the two unshaded strips. But since (~k'~t) is symmetric, these integrals are ~ ~ A3 equal and opposite in sign and P2 is unchanged. Therefore aP2/a~j = 0 and the optimum condition (A3) reduces to ~) Finally) elementary manipulations.of (A4) imply which is just the marginal density of ~k evaluated at ~k = O. Substituting this into (A6) gives i = j,k)i which is virtually identical to equation (7) of Section II. Note two things: First) W z does not appear in ~6). This is the source of the indeterminacy. Any W z suffices so long as 1 V~ = 3 [WI + W z + W 3 ] ~ C(~j). Second) again only the win-show spread is important in the determination of investment. The intuition is clear. A risk-neutral player views games as equal so long as they have the same expected value. Thus) increasing W z and reducing WI and W 3 but leaving (WI - W 3 ) unchanged has no effect on the expected value of the game. And because it leaves the spread between WI and W 3 unchanged) it does not change the value of ~. It is) therefore) like a lump-sum tax coupled with offsetting lump-sum subsidies) and does not affect behavior. As mentioned in the text, the indeterminacy vanishes when risk-aversion is introduced.