A primer on cardinal versus ordinal tournaments.

Tsoulouhas, Theofanis

I. INTRODUCTION

The tournament literature has considered primarily two relative performance evaluation schemes that are consistent with what we observe in practice. The rationale behind each of these incentive schemes is that relative performance evaluation is useful when agent performance is subject to common shocks, in which case individual performance is not a sufficient statistic for individual effort. That is, the performance of other agents can enhance the inference of the effort exerted by an individual agent. One important difference between these schemes is that under rank-order or "ordinal" tournaments the prizes are prespecified and agents compete for rank. By contrast, under two-part piece rate or "cardinal" tournaments, the total prize is fixed and agents compete for a share of the pie. (1) Each agent receives a base payment and a bonus depending on the difference of his performance from the average performance of his peers. (2) Average performance provides an informative signal about the value of common shocks inflicted on agents.

Ordinal tournaments are prevalent in contests where rank is more important than individual performance (for instance, sports contests where contestants compete for medals or fixed prizes), or when individual performance data are not readily available (for instance, promotion contests where the individual contribution of contestants cannot easily be separated from that of their co-workers or subordinates). By contrast, cardinal tournaments are popular in several occupations or industries where cardinal performance data are readily available (e.g., salesmen contracts, physician contracts with HMOs, and agricultural contracts). (3) Both ordinal and cardinal tournaments are relatively simple to design, even though they are only proxies of theoretically optimal schemes. (4)

Prior literature has contrasted the efficiency properties of either ordinal or cardinal tournaments against those of piece rates. Under piece rates, each agent is evaluated according to his absolute performance, thus, agents receive a base payment and a bonus which is based, typically linearly, on their individual output. (5) Piece rates are sometimes expressed as "fixed performance standards'' when an agent's performance is evaluated against a fixed standard instead of the average output obtained. Lazear and Rosen (1981), Green and Stokey (1983), and Nalebuff and Stiglitz (1983) have shown that ordinal tournaments are superior to piece rates when agent activities are subject to common shocks that affect performance. (6) More recently, Marinakis and Tsoulouhas (2012, 2013) investigated and verified this result for cardinal tournaments. Relative performance evaluation via tournaments constitutes a Pareto superior move. By removing common uncertainty from the responsibility of agents, and by charging a premium for this insurance, the principal increases his profit without hurting the agents. Moreover, by providing this type of insurance, tournaments enable the principal to implement higher power incentives than under piece rates.

What is still an open question in the literature above is a comparison of ordinal to cardinal tournaments. The consensus among tournament theorists is that even though ordinal tournaments are informationally wasteful (as in Holmstrom 1982) by ignoring the agents' cardinal performance data if available, they are, however, more general because they are less restrictive on the functional form of the contract (i.e., they are step functions with a linear or non-linear trend). Part of the appeal is that the founders of tournament theory focused on ordinal tournaments. Nevertheless, a significant part of the current tournament literature has analyzed cardinal tournaments instead. (7)

This paper fills in this gap in theory by developing a framework that can be used to analyze both cardinal and ordinal tournaments, as well as piece rates, and in doing so it provides a primer on tournaments and piece rates. The analysis aims at obtaining a Pareto ranking of cardinal versus ordinal tournaments. To the best of our knowledge, this is the first paper to attempt this on a theoretical level. The analysis shows that, contrary to common belief, cardinal tournaments are superior to ordinal tournaments. The rationale is that, by utilizing all the available information more efficiently, cardinal tournaments allow the principal to implement higher power incentives, which makes them superior even though they restrict the form of the contract more than ordinal tournaments.

If we believe that markets select for better contracts, the evolution of contracts for the production of broiler chickens in the United States supports our theoretical results. Initially, in the 1950s and 1960s, chicken growers were rewarded with a kind of piece rate based on the feed conversion (i.e., feed to output) ratio they achieved. Gradually, this contract was replaced by a rank order tournament (Knoeber and Thurman 1995). In the 1980s, the form of the contract changed again to a scheme in which payments were determined by the difference of a grower's feed conversion from the average of all other growers. This contract resembles the standard cardinal tournament. (8)

Even though cardinal tournaments are shown to be superior to ordinal tournaments, it is not a priori dear whether the principal should provide full or partial insurance against common shocks. Insulating agents fully may lead them to exert less effort than if the principal only provides partial insurance. There is a trade-off; the principal may potentially increase his profit by charging for the insurance against common shocks, but too much insurance can be detrimental to efforts. Based on this intuition, Tsoulouhas (2010a) generalized cardinal tournaments by examining a form of cardinal tournaments in which the weights on absolute and group average performance are not necessarily equal. Under these "hybrid" tournaments, the total prize is not necessarily fixed. In addition, one can analyze standard cardinal tournaments by setting the weights equal, and analyze piece rates by setting the weight on group average performance to zero. The analysis in Tsoulouhas (2010a) showed that hybrid cardinal tournaments are Pareto superior to standard cardinal tournaments. In particular, providing partial insurance against common uncertainty via a hybrid tournament is always better for the principal than providing full insurance against common uncertainty via a standard tournament or than providing no insurance at all against common uncertainty via piece rates. The principal can induce the agents to exert more effort by subjecting them to some common uncertainty. Agents, then, work harder in order to insure themselves against bad realizations of the common shocks.

Thus, overall and by combining the hybrid tournament case in Tsoulouhas (2010a), our analysis leads to the conclusion that hybrid cardinal tournaments are Pareto superior to standard cardinal tournaments, which are Pareto superior to ordinal tournaments, which are Pareto superior to piece rates, if there is enough common uncertainty. (9) In all, our analysis provides a theoretical justification for using cardinal instead of ordinal tournaments when cardinal performance data are readily available and agent performance is subject to common shocks.

II. MODEL

A risk-neutral principal signs a contract with n homogeneous agents. (10) Each agent i produces output according to the production function [x.sub.i] = a + [e.sub.i] + [eta] + [[epsilon].sub.i], where a is the agent's known ability, [e.sub.i] is the agent's effort, [eta] is a common shock inflicted on all agents, and [[epsilon].sub.i] is an idiosyncratic shock. We assume that both production shocks [eta] and [[epsilon].sub.i], follow independent normal distributions with zero means. Agents exert effort first then the production shocks are realized. The effort of each agent and the realizations of the production shocks are privately observed by each agent, but the output obtained is publicly observed. (11) The price of output is normalized to 1 so that the output produced by the agents is revenue to the principal. Agent preferences are represented by a CARA utility function n([w.sub.i], [e.sub.i]) = -exp(-r[w.sub.i] + 1/2 r/a [e.sup.2.sub.i), where r is the agent's coefficient of absolute risk aversion and w, is the compensation he receives from the principal. This utility function has been widely used in the literature (for instance, see Meyer and Vickers 1997). (12) We also assume that the production shocks have finite variances var ([eta]) = [[sigma].sup.2.sub.[eta]] and var ([[epsilon].sub.i]) = [[sigma].sup.2.sub.[epsilon]], [for all] i. One of the advantages of this model is that it leads to closed form solutions and its basic results conform with those obtained by Lazear and Rosen (1981) who relied on first-order Taylor approximations of utility functions. (13)

The principal compensates agents for their effort based on their outputs by using a piece rate scheme [w.sub.i] = b + [beta][x.sub.i], or a two-part piece rate or "cardinal" tournament [w.sub.i] = b + [beta] ([x.sub.i] - [bar.x]), or a rank-order or "ordinal" tournament with pre-specified prizes [w.sub.n] > [w.sub.n-1] > [w.sub.n-2] > ... > [w.sub.1]. Lazear and Rosen (1981) focused on ordinal tournaments. Both cardinal and ordinal tournaments compensate agents based on a relative performance evaluation. The piece rate scheme (or absolute performance evaluation) provides a benchmark for the analysis. Tsoulouhas (2010a) has shown that "hybrid" cardinal tournaments of the form [w.sub.i] = b + [beta][x.sub.i] - [gamma][bar.x] are dominant over standard cardinal tournaments of the form [w.sub.i] = b + [beta] [[x.sub.i] - [bar.x]). Moreover, the piece rate scheme is a special case of the hybrid cardinal tournament by setting [gamma] = 0, and the standard cardinal tournament is a special case for [beta] = [gamma]. However, in the limit, that is, for a sufficiently large number of workers, [beta] = [gamma]. (14)

We begin by briefly analyzing the benchmark case of piece rates followed by the case of standard cardinal tournaments. This part of the analysis follows directly from Marinakis and Tsoulouhas (2013) (note that Marinakis and Tsoulouhas 2012 also analyzed the corresponding case of risk-neutral agents). The case of ordinal tournaments is analyzed afterwards, and it is the novel part of the analysis.

III. THE PIECE RATE BENCHMARK

The piece rate scheme (R) is the payment scheme in which the compensation to each agent is determined by absolute performance evaluation and the compensation to the ith agent takes the form [w.sub.i] = [b.sub.R] + [[beta].sub.R][x.sub.i], where ([b.sub.R], [[beta].sub.R]) are the contractual parameters to be determined by the principal through backward induction. First, the principal calculates each agent's expected utility:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the expression in square brackets is the certainty equivalent compensation of the agent which admits a mean-variance representation. The principal then calculates the effort level that maximizes Condition (1). First-order conditions yield the following incentive compatibility condition for the agent

(2) [e.sub.iR] = a[[beta].sub.R],

so that effort depends on inherent ability and on the magnitude of the piece rate. Given that the principal is endowed with the bargaining power, the principal selects the value of the base payment, [b.sub.R], that satisfies the agent's individual rationality constraint with equality so that he receives no rents but still accepts the contract. For ease of exposition, we normalize the agent's reservation utility to -1, (15) hence, given Condition (1) the agent's individual rationality constraint implies

(3) E[U.sub.R] = -1 [??] [b.sub.R] = r ([[sigma].sup.2.sub.[eta]] + [[sigma].sup.2.sub.[epsilon]]) - a/2 [[beta].sup.2.sub.R] - a[[beta].sub.R].

Thus, by choosing the piece rate [[beta].sub.R], the principal can precisely determine the agent's effort because the agent will optimally set his effort according to Condition (2). In addition, by setting [b.sub.R] in accordance with Condition (3) the principal can induce agent participation at least cost.

Given Conditions (2) and (3), the principal maximizes his expected total profit

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Maximizing Condition (4) with respect to [[beta].sub.R]. and then using Condition (3) yields:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Condition (5) indicates that the higher the total (i.e., common and idiosyncratic) uncertainty, the lower the power of incentives and effort, because uncertainty hampers the strength of the relationship between effort and output.

Given Condition (5), the principal's expected profit under the piece rate scheme is

(6) ET[[PI].sub.R] = n [a + 1/2 [a.sup.2]/a + r ([[sigma].sup.2.sub.[eta]] + [[sigma].sup.2.sub.[epsilon]]].

IV. THE CARDINAL TOURNAMENT

The cardinal tournament (T) is the payment scheme in which the compensation to each agent is determined by relative performance evaluation. Specifically the payment scheme is

[w.sub.i] = [b.sub.T] + [[beta].sub.T] ([x.sub.i] - [bar.x]) = [b.sub.T] + [[beta].sub.T] (n - 1/n [x.sub.i] - 1/n [summation over (j[not equal to]i)] [x.sub.j]]),

where [bar.x] is the average output obtained by all agents, and ([b.sub.T], [[beta].sub.T]) are the contractual parameters to be determined by the principal. (17) Note that under tournament the total wage bill is proportional to the base payment [b.sub.T], in particular, [summation][w.sub.i] = n[b.sub.T], which reflects a balanced budget for the performance part of pay. Thus, in contrast to the piece rate scheme, the principal's total payment to the agents and, hence, the expected payment per agent are independent of output. The agent's expected utility is

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The effort level that maximizes Condition (7) satisfies

(8) [e.sub.iT] = n - 1/n a[[beta].sub.T].

Further, the individual rationality constraint implies

(9) E[U.sub.T] = -1 [??] [b.sub.T] = 1/2 n - 1/n (n - 1/n a + r[[sigma].sup.2.sub.[epsilon]]) [[beta].sup.2.sub.T].

Then, given Conditions (8) and (9), the principal maximizes expected total profit

(10) ET[[PI].sub.T] = n [a + n - 1/n a[[beta].sub.T] - 1/2 n - 1/n x (n - 1/n a + r[[sigma].sup.2.sub.[epsilon]]) [[beta].sup.2.sub.T]].

Maximizing Conditions (10) with respect to [[beta].sub.T] and then using Condition (9) yields:

(11) [b.sub.T] = 1/2 [a.sup.2]/a + n/n-1 r[[sigma].sup.2.sub.[epsilon]], [[beta].sub.T] = a/n-1/n a + r[[sigma].sup.2.sub.[epsilon]].

Thus, [b.sub.T] > [b.sub.R] and [[beta].sub.T] > [[beta].sub.R]. The intuition behind the former condition is that the expected bonus under tournaments is zero, whereas under piece rates it is positive; hence, the base payment increases under tournaments to ensure agent participation. The rationale behind the latter inequality is that the removal of common uncertainty from the responsibility of the agents allows the principal to implement higher power incentives under tournaments. Note that, unlike Condition (5), Condition (11) indicates that common uncertainty does not affect incentives under tournaments. Given Condition (11), the principal's expected profit under tournament is

(12) ET[[PI].sub.T] = n [a + 1/2 [a.sup.2]/a + n/n-1 r[[sigma].sup.2.sub.[epsilon]]].

Comparing expected profit under piece rates and tournaments yields the following proposition.

PROPOSITION 1. (Marinakis and Tsoulouhas 2013) Under the modeling assumptions, with an arbitrarily large number of agents, cardinal tournaments are Pareto superior to piece rates if

[[sigma].sup.2.sub.[eta]] > 1/n-1 [[sigma].sup.2.sub.[epsilon]].

Proposition 1 is stating that tournaments are superior provided that the variance of the common shock is larger than only a fraction of the variance of the idiosyncratic shock, where the fraction decreases when the number of agents increases. A large number of agents strengthens the dominance of tournaments over piece rates because idiosyncratic shocks cancel out, which enables the principal to offer better insurance by filtering away common shocks from the responsibility of the agents through the average output obtained by them. Therefore, piece rate schemes are superior when the variance of the common shock is sufficiently small or the number of agents is sufficiently small.

This result is a reversal of what happens with risk-neutral agents (see Marinakis and Tsoulouhas 2012). The intuition is that the principal cannot charge risk-neutral agents a risk-premium for removing common uncertainty from their responsibility, and the contract induces the agents to exert the same effort [e.sub.i] = a they would exert under piece rates. (18) This result extends the Lazear and Rosen (1981) finding, that both piece rates and ordinal tournaments are equally efficient with risk-neutral agents, to cardinal tournaments.

V. THE ORDINAL TOURNAMENT

The ordinal tournament (O) is another payment scheme in which the compensation to each agent is determined by relative performance evaluation. This scheme rewards agents with pre-specified prizes [w.sub.n] > [w.sub.n-1] > [w.sub.n-2] > ... > [w.sub.1]. Let P([w.sub.l]) denote the probability that agent i wins reward [w.sub.l]. Under an ordinal tournament the agent's expected utility is:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The effort level that maximizes Condition (13) satisfies

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to determine the optimal effort, first, we need to determine the probabilities of winning. The probability that an agent i beats another agent), given realizations [[epsilon].sub.i] and [eta], is the probability that his output [x.sub.i] = a + [e.sub.i] + [eta] + [[epsilon].sub.i] is larger than the output of the other agent [x.sub.j] = a + [e.sub.j] + [eta] + [[epsilon].sub.j], that is,

p ([x.sub.i] [greater than or equal to] [x.sub.j]) = P ([[epsilon].sub.j] [less than or equal to] [[epsilon].sub.i] + [e.sub.i] - [e.sub.j]) = G ([[epsilon].sub.i] + [e.sub.i] - [e.sub.j]),

where G(*) is the distribution function of the idiosyncratic shock. Thus, the probability that i beats every other agent j, given realizations [[epsilon].sub.i] and [eta], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

hence, the prior probability that agent i wins the highest reward is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The probability that i beats every other agent j except one agent n. that is, the probability that n beats i who beats the rest of the agents, given realizations [[epsilon].sub.i], [[epsilon].sub.n], and [eta], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the prior probability that agent i wins the second highest reward is

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Therefore, the prior probability of i winning any reward [w.sub.l] is

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One possible simplification of the analysis is a clever shortcut provided by Lazear and Rosen (1981). (19) Specifically,

P ([x.sub.i] [greater than or equal to] [x.sub.j]) = P ([e.sub.i] + [[epsilon].sub.i] [greater than or equal to] [e.sub.j] + [[epsilon].sub.j]) = P ([[epsilon].sub.j] - [[epsilon].sub.i] [less than or equal to] [e.sub.i] - [e.sub.j]).

Next, define the random variable [xi] [equivalent to] [[epsilon].sub.j] - [[epsilon].sub.i], with cumulative distribution function F and density f, E([xi]) = 0 and E ([[xi].sup.2]) = 2[[sigma].sup.2.sub.[epsilon]] (because [[epsilon].sub.j] and [[epsilon].sub.i] are i.i.d.). Thus,

(15) P([x.sub.i] [greater than or equal to] [x.sub.j]) = F([e.sub.i] - [e.sub.j]).

Hence, the prior probability that agent i wins the highest reward is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The prior probability that agent i wins the second highest reward is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lastly, the prior probability of i winning any reward [w.sub.l] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Even though the Lazear and Rosen "shortcut" can simplify the analysis, it can easily be seen that determining the optimal effort in Condition (14) with an arbitrarily large (albeit finite) number of agents leads to a pretty messy analysis pretty quickly. Thus, the reader should have been convinced by now that there is a good reason why authors typically consider only two agents in the analysis of ordinal tournaments. This is exactly what we are doing next. At the end of this analysis, we show that the results generalize to an arbitrarily large number of agents.

Let P denote the prior probability that agent i wins. Hence, E([w.sub.i]) = P[w.sub.2] + (1 - P)[w.sub.1]. In the Nash equilibrium, that is, taking the effort of the other agent as given, Condition (14) implies

[e.sub.iO] = a([w.sub.2] - [w.sub.1]) [partial derivative]P/[partial derivative][e.sup.i] x [1 + r(E([w.sub.i]) - [[w.sub.2] + [w.sub.1]] /2)] ,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because agents are homogeneous, they exert the same effort [e.sub.i] = [e.sub.j] in equilibrium and, therefore, P = 1/2 so that E([w.sub.i]) = ([w.sub.2] + [w.sub.1])/2. Hence,

(16) [e.sub.iO] = a([w.sub.2] - [w.sub.1]) [[integra].sup.+[infinity].sub.-[infinity]] [(g ([[epsilon].sub.i])).sup.2] d[[epsilon].sub.i],

implying that effort increases in ability and in the prize spread [w.sub.2] - [w.sub.1]. Further intuition can be obtained through the Lazear and Rosen shortcut. By using Condition (15), Condition (16) can be rewritten as

(17) [e.sub.iO] = a([w.sub.2] - [w.sub.1]) f(0).

As [[epsilon].sub.i] ~ N (0, [[sigma].sup.2.sub.[epsilon]]), it follows that f(0) = f([e.sub.i] - [e.sub.j] = 0) = 1/2[[sigma].sub.[epsilon]] [square root of [pi]] (note a typing error on p. 847 in Lazear and Rosen 1981). Thus,

(18) [e.sub.iO] = a ([w.sub.2] - [w.sub.1])/2[[sigma].sub.[epsilon]] [square root of [pi]],

implying that, similar to cardinal tournaments, the larger the variance of idiosyncratic uncertainty the lower the effort because the more outcomes depend on luck the less effort the agent will exert. (20)

Given Condition (13), the individual rationality constraint implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(19) [??] [w.sup.2.sub.1] (-A) + [w.sub.1] [2[w.sub.2]A + 1/2) + [1/2 [w.sub.2] - [w.sup.2.sub.2]A) = 0,

where A [equivalent to] a[(f(0)).sup.2]/2 + r/2 1/4. Condition (19) is a quadratic in [w.sub.1] with two distinct solutions

[w.sub.1O] = -1/2 - 2[w.sub.2]A [+ or -] [square root of (1/4 + 4[w.sub.2]A)]/2(-A).

To provide maximum incentives and, hence, maximize the principal's profit, the optimal [w.sub.1] is the smallest root vv, so that the prize spread [w.sub.2] - [w.sub.1] is maximized. Thus,

(20) [w.sub.1O] = - -1/2 - 2[w.sub.2]A + [square root of (1/4 + 4[w.sub.2]A)]/2A.

Then, given Conditions (17) and (20), the principal maximizes expected total profit

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with respect to [w.sub.2]. Thus, the optimal [w.sub.2] satisfies

(22) [w.sub.2O] = [a.sup.2] [(f(0)).sup.2] + af(0)/2a (f(0)).sup.2] + r/2 = 1/2 [a.sup.2] + 2a[[sigma].sub.[epsilon]] [square root of [pi]]/a + [pi]r[[sigma].sup.2.sub.[epsilon]].

Condition (20) then implies

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to verify that the prize spread is positive, which ascertains the logical consistently of the solution process:

(24) [w.sub.2O] - [w.sub.1O] = 2a[[sigma].sub.[epsilon]] [square root of [pi]]/a + [pi]r[[sigma].sup.2.sub.[epsilon]] > 0.

Conditions (21), (22), and (23), given f(0) = 1/2[[sigma].sub.[epsilon]][square root of [pi]], imply that the principal's expected total profit under ordinal tournaments is

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By contrast, Condition (12) implies that the corresponding profit under cardinal tournaments is

(26) ET[[PI].sub.T] = 2a + [a.sup.2]/a 2r[[sigma].sup.2.sub.[epsilon]].

Comparing expected profit under the two schemes yields the following proposition.

PROPOSITION 2. Under the modeling assumptions, with two agents, cardinal tournaments are Pareto superior to ordinal tournaments regardless of common or idiosyncratic uncertainty.

Proof. Given Conditions (25) and (26), note that

ET[[PI].sub.T] > ET[[PI].sub.O] [??] [pi] > 2.

The latter statement is a correct statement, which completes the proof.

The result in Proposition 2 is pretty strong because it holds regardless of the magnitude of common or idiosyncratic uncertainty. To the best of our knowledge, this theoretical result has never been obtained in the literature which, if using cardinal tournaments, has been reciting the Holmstrom (1982) premise that ordinal tournaments are informationally inefficient by ignoring the agents' cardinal performance, and if using ordinal tournaments, has implicitly been assuming that cardinal performance data are relatively costly to obtain or ordinal tournaments should be optimal because they do not constrain the form of the contract. Our claim is that the real reason why cardinal tournaments are optimal is that they provide higher power incentives. To see this, note that Conditions (8) and (11) imply optimal effort

[e.sub.iT] = [a.sup.2]/a + 2r[[sigma].sup.2.sup.[epsilon]],

while Conditions (18) and (24) imply

[e.sub.iO] = [a.sup.2]/a + [pi]r[[sigma].sup.2.sub.[epsilon]],

that is, given that [pi] > 2, it follows that [e.sub.iT] > [e.sub.iO] or agents exert less effort under ordinal tournaments than under cardinal tournaments. The intuition behind the finding that cardinal tournaments provide higher-power incentives can be traced to the payment specifications under cardinal and ordinal tournaments. Figure 1 pictorially captures the intuition through a simple graph of [w.sub.i] against [x.sub.i], for an arbitrarily given output [x.sub.j] of another agent. As you can see in the graph, high performance is rewarded more under cardinal tournaments. (21)

[FIGURE 1 OMITTED]

Some comparative statics are in order. Given Conditions (25) and (26) it follows that

[partial derivative] (ET[[PI].sup.T] - ET[[PI].sub.O])/[partial derivative]a > 0,

that is, the more able agents inherently are, the more the principal benefits by relying on cardinal tournaments which provide the agents with higher power incentives. However, note that the value of the corresponding derivatives with respect to the risk-aversion rate r and the variance of idiosyncratic uncertainty [[sigma].sup.2.sub.[epsilon]] depend on the values of the parameters (i.e., they may be positive or negative).

Next, Condition (6) for piece rates implies

(27) ET[[PI].sub.R] = 2a + [a.sup.2]/a + r ([[sigma].sup.2.sub.[eta]] + [[sigma].sub.2.sup.[epsilon]].

Comparing expected profit under ordinal tournaments and piece rates then yields the following proposition.

PROPOSITION 3. Under the modeling assumptions, with two agents, ordinal tournaments are Pareto superior to piece rates if [[sigma].sup.2.sub.[eta]] > ([pi] - 1) [sigma].sup.2.sub.[epsilon]].

Proof. Given Conditions (25) and (27), the proof is straightforward by noting that ET[PI].sub.O] > ET[[PI].sub.R] [??] [[sigma].sup.2.sub.[eta]] > ([pi] - 1) [[sigma].sup.2.sub.[epsilon]].

Proposition 3 states that ordinal tournaments are superior to piece rates if common uncertainty is approximately at least twice as large as idiosyncratic. By contrast, as Proposition 1 indicates for n = 2, cardinal tournaments are superior to piece rates whenever common uncertainty is larger than idiosyncratic regardless of magnitude. Thus, common uncertainty must be relatively much more substantial for ordinal tournaments to dominate absolute performance evaluation via piece rates.

Even though the results above consider two agents, we conjecture that they hold more generally. To see this, observe that expected profit per agent under ordinal tournaments, E[[PI].sub.O] = E([x.sub.i]) - E([w.sub.i]), should be independent of the number of agents n, given the assumptions of agent homogeneity, prespecified prizes and a number of rewards that always equals the number of agents. (22) Agent homogeneity implies that effort is the same for each agent. However, effort should be independent of the number of agents. This is so because with homogeneous agents exposed to the same common shock and exerting the same effort, the agent ranking will be determined by sheer luck. Each agent always expects half the agents to beat him and half the agents to lose to him due to luck. Therefore, expected reward E([w.sub.i]) should always be the same, and so should effort and expected output E([x.sub.i]) be.

By contrast, under cardinal tournaments, agent homogeneity still implies that agents exert the same effort, but this effort increases with the number of agents, see Condition (8). This is so because performance relative to the sample average output [bar.x] matters, and according to the law of large numbers the sample average converges closer to the expected output with more agents. Given Conditions (25) and (12), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

However, the latter statement is a correct statement because it holds for n = 2, while [lim.sub.n[right arrow][infinity]] n/n-1 = 1. (23) Thus, we conjecture that cardinal tournaments are Pareto superior to ordinal tournaments even for an arbitrarily large number of agents, regardless of common or idiosyncratic uncertainty. The following proposition states and proves that our conjecture is correct and the result does generalize indeed.

PROPOSITION 4. Under the modeling assumptions, with an arbitrarily large number of agents, cardinal tournaments are Pareto superior to ordinal tournaments regardless of common or idiosyncratic uncertainty.

Proof. We prove the proposition by following the method of mathematical induction. Proposition 2 states that cardinal tournaments are Pareto superior to ordinal tournaments, regardless of common or idiosyncratic uncertainty, for n = 2, which is the minimum required n for relative performance evaluation to have bite. Suppose now that cardinal tournaments are Pareto superior to ordinal tournaments, regardless of common or idiosyncratic uncertainty, for an arbitrary but finite n > 2. Then, we need to prove that cardinal tournaments are Pareto superior to ordinal tournaments, regardless of common or idiosyncratic uncertainty, for n + 1.

Starting from n agents, if we add one more agent, expected profit under cardinal tournaments will change by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By contrast, expected profit under ordinal tournaments will change by

E([x.sub.i]) - E([w.sub.i]) = a + [e.sub.iO] - E([w.sub.iT]).

First note that starting from n agents receiving prespecified prizes, when one more agent is added, every other agent expects the new agent to win over him with probability 1/2 and to lose to him with probability 1/2 due only to the realization of the idiosyncratic shock e;, given that agents are homogeneous so that ability a is the same for all agents, and given that the common shock [eta] is the same. Thus, E([w.sub.i]) for n + 1 agents is identical to the expected reward per agent with n agents (even though one more reward will be added). In other words, each agent views himself as the average agent, and the average agent always expects half of the agents to beat him and half of the agents to lose to him, therefore there is no need for his incentive pay to adjust. Given that E([w.sub.i]) stays the same, [e.sub.iO] is identical to effort per agent with n agents and equal to [a.sup.2]/a+[pi]r[[sigma].sup.2.sub.[epsilon]]. How ever, as cardinal tournaments are assumed to be Pareto superior to ordinal tournaments, regardless of common or idiosyncratic uncertainty, for an arbitrary but finite n > 2, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But then, given that n+1/n < n/n-1, [for all] n, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, when moving from n to n + 1 agents, expected profit will increase more under cardinal than under ordinal tournaments.

Similarly, the result in Proposition 3 also generalizes to an arbitrarily large number of agents, that is, ordinal tournaments are always Pareto superior to piece rates provided that common uncertainty is relatively large. This is stated in the following proposition.

PROPOSITION 5. Under the modeling assumptions, with an arbitrarily large number of agents, ordinal tournaments are Pareto superior to piece rates if [[sigma].sup.2.sub.[eta]] > ([pi] - 1) [[sigma].sup.2.sub.[epsilon]].

Proof. Given Proposition 4, the proof is straightforward.

VI. CONCLUSION

Prior literature has contrasted the efficiency properties of either ordinal or cardinal tournaments against those of piece rates; however, a comparison of ordinal to cardinal tournaments on a theoretical level is, to the best of our knowledge, still missing. This paper fills in this gap in theory by developing a framework that can be used to analyze both cardinal and ordinal tournaments, as well as piece rates, and in doing so it provides a primer on tournaments and piece rates. The analysis aims at obtaining a Pareto ranking of cardinal versus ordinal tournaments.

Overall, and by combining the hybrid tournament case in Tsoulouhas (2010a), our analysis indicates that hybrid cardinal tournaments are Pareto superior to standard cardinal tournaments, which are Pareto superior to ordinal tournaments, which are Pareto superior to piece rates, if there is enough common uncertainty. The rationale is that the principal can induce the agents to exert more effort by subjecting them to some common uncertainty through hybrid cardinal tournaments. Agents, then, work harder in order to insure themselves against bad realizations of the common shocks. Further, by utilizing all the available information more efficiently, hybrid and standard cardinal tournaments allow the principal to implement higher power incentives than ordinal tournaments, which makes them superior even though they restrict the form of the contract more than ordinal tournaments. Thus, our analysis provides a theoretical justification for using cardinal instead of ordinal tournaments when cardinal performance data are available and agent performance is subject to common shocks.

doi: 10.1111/ecin.12168

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(1.) Note that this paper assumes that, under ordinal tournaments, all agents receive prespecified rewards and, under cardinal tournaments, the total prespecified reward depends on the number of agents. Size effects are analyzed by Eggert and Kolmar (2006), Lee (2007) and Fu and Lu (2009). Eggert and Kolmar examined contests where either the total rent depends on the number of individuals and agents engage in rent-seeking, or the total prize depends on the time invested in productive activities as well as on the number of agents. Lee extends the analysis to the case in which the unit cost of effort decreases with the number of agents. Fu and Lu show that in contests with pre-contest investment, the relationship between the number of contestants and equilibrium investment is non-monotonic.

(2.) The base payment ensures agent participation and the bonus provides incentives to perform. An agent receives a bonus if his performance is above that of his peers, and a penalty otherwise.

(3.) Obviously, the main result of this paper about the superiority of cardinal tournaments would not hold if individual output could only be observed at a relatively high cost. The existing tournament literature that focuses on ordinal tournaments builds on the implicit or explicit assumption that cardinal performance data are either not available or costly to obtain.

(4.) To some extent, the complexity of the theoretically optimal contract is due to the fact that contracts accommodate all possible events. Holmstrom and Milgrom (1987), however, have argued that schemes that adjust compensation to account for rare events may not provide correct incentives in ordinary high probability circumstances. Nevertheless, the optimal contract can be a simple linear scheme under specific assumptions (see Holmstrom 1979).

(5.) See Lazear (1986) and Gibbons (1987) on piece rates alone.

(6.) Also see Bull, Schotter, and Weigelt (1987), van Dijk, Sonnemans, and van Winden (2001), and Vandegrift, Yavas, and Brown (2007).

(7.) See Knoeber (1989). Knoeber and Thurman (1994, 995), Tsoulouhas (1999), Tsoulouhas and Vukina (1999, 2001), Levy and Vukina (2004), Wu and Roe (2005, 2006), Tsoulouhas and Marinakis (2007), Tsoulouhas (2010a, 2010b, 2012), Marinakis and Tsoulouhas (2012, 2013), and Knoeber and Tsoulouhas (2013). The experimental analysis in Carpenter, Matthews and Schirm (2010) lies between piece rates and cardinal tournaments in that participants receive a fixed bonus for winning the tournament in addition to a piece rate reward.

(8.) This interesting insight was offered to us by Charles Knoeber.

(9.) Since so far it has not been clear in the literature if ordinal or if cardinal tournaments are theoretically superior, a few authors have empirically investigated the question. Zheng and Vukina (2007) use historical data from broiler production and provide simulation results that are consistent with our theoretical findings. Further evidence is provided by Agranov and Tergiman (2013) who show that hybrid cardinal tournaments are superior to ordinal tournaments or to piece rates, from the principal's perspective, in a controlled laboratory experiment.

(10.) Agent heterogeneity has been examined in a number of recent papers. Konrad and Kovenock (2010) examine discriminating contests with stochastic contestant abilities. Tsoulouhas, Knoeber, and Agrawal (2007) consider CEO contests that are open to heterogeneous outsider contestants. Tsoulouhas and Marinakis (2007) analyze ex post agent heterogeneity. Instead, Riis (2010) allows for agents who are heterogeneous ex ante. Tsoulouhas (2012) shows that ex post sorting through relative performance evaluation reduces the scope for screening agents ex ante via a menu of offers when agents are not very heterogeneous and the principal utilizes a cardinal tournament. This is so because the principal becomes better informed ex post about agent types, via the realization of common uncertainty, and can effectively penalize or reward the agents ex post. Also see Tsoulouhas (2010b) and the symposium papers introduced therein, and Knoeber and Tsoulouhas (2013).

(11.) Thus, in this model there is one-sided moral hazard. The optimality of tournaments under two-sided moral hazard has been analyzed by Carmichael (1983) and Tsoulouhas (1999).

(12.) This is so because E [exp (-r[w.sub.i] + 1/2 r/a [e.sup.2.sub.i])] = exp [[mu] + [[sigma].sup.2]/2], when -r[w.sub.i] + 1/2 r/a [e.sup.2.sub.i] ~ N ([mu], [[sigma].sup.2]), which allows us to obtain a closed form solution for the expected utility.

(13.) By contrast, Tsoulouhas (1999) and Tsoulouhas and Vukina (1999) considered first-order Taylor approximations of the optimal nonlinear contract in order to approximate it by a linear tournament.

(14.) For hybrid tournaments, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(15.) Note that the analysis is directly applicable to any (negative) normalization other than -1.

(16.) Note that a negative [b.sub.R] is feasible because agents are not liquidity constrained. Tsoulouhas and Vukina (1999) and Marinakis and Tsoulouhas (2012, 2013) examine the optimality of piece rates and tournaments under bankruptcy, limited liability, or liquidity constraints for the agents or the principal.

(17.) Note that [w.sub.i] = [b.sub.T] + [[beta].sub.T] ([x.sub.i] - [bar.x]) = [b.sub.T] + [[beta].sub.T] n-1/n ([x.sub.i] - 1/n-1] [summation over (j[not equal to]i] [x.sub.j]). Thus, one can define [B.sub.T] = [[beta].sub.T] n- 1/n, so that [w.sub.i] = [b.sub.T] + [[beta].sub.T] ([x.sub.i] - 1/n-1] [summation over (j[not equal to]i] [x.sub.j]). In other words, [w.sub.i] = [b.sub.T] + [[beta].sub.T] ([x.sub.i] - [bar.x]) has the same properties as [w.sub.i] = [b.sub.T] + [[beta].sub.T] ([x.sub.i] - 1/n-1] [summation over (j[not equal to]i] [x.sub.j]) The latter is essentially Shleifer's "yardstick competition" form.

(18.) With piece rates, in particular, the optimum is the "selling the enterprise to the agent" solution, that is, a piece rate of 1 so that, in light of moral hazard, the agent assumes all the risk.

(19.) By contrast, Chan (1996) and Tsoulouhas, Knoeber and Agrawal (2007) used probability of success functions similar to those above.

(20.) Even if one used Condition (17) directly, an increase in the variance [[sigma].sup.2.sub.[epsilon]] is a mean-preserving spread for distribution f resulting in a lower value f(0) and a lower effort.

(21.) Note that whereas [b.sub.T] > [w.sub.1O], the relationship between [b.sub.T] and [w.sub.2O] depends on the coefficient of risk aversion and the variance of the indiosyncratic shock. The graph depicts the case when [w.sub.2O] > [b.sub.T], which occurs when the coefficient of risk-aversion and the standard deviation of the indiosyncratic shock are relatively small. If [b.sub.T] > [w.sub.2O], instead, then cardinal tournaments provide stronger incentives over a wider range of output. Also note that only [w.sub.2O] and [b.sub.T] are always positive.

(22.) The result would not hold if any of these assumptions were dropped (see Eggert and Kolmar 2006, Lee 2007, and Fu and Lu 2009 for such size effects, as well as the cardinal tournament case in this paper).

(23.) In other words, the sequence defined by n/n-1 is strictly monotonically decreasing.

THEOFANIS TSOULOUHAS, The paper has benefited from several useful conversations with Charles Knoeber. I am also thankful to two anonymous referees for very thoughtful comments. An earlier version of the paper was titled "A Primer on Tournaments." Tsoulouhas: Ernest & Julio Gallo Management Program, School of Social Sciences, Humanities & Arts, University of California, Merced, CA 95343. Phone (209) 228-4640, Fax (209) 228-4007, E-mail ftsoulouhas@ucmerced.edu