A Primer on Cardinal versus Ordinal Tournaments

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. …