The Distribution of Prizes in a Match-Play Tournament with Single Eliminations

This paper begins to study the reward-incentive structure in sequential knock-out or elimination tournaments with matched, pairwise comparisons among players at each stage. The prize structure required to elicit constant expected quality of play in all matches throughout the tournament is characterized for competition among equally talented (or perfectly handicapped), players.The incentive maintaining prize structure is shown to concentrate' extra weight on the top ranking prize, a phenomenon observed in most tournaments. More can be said. Prizes that maintain performance incentives at all stages award a constant increment for each match won up to the last stage; and an amount greater than this for the player who wins the final match. Players' incentives to perform in early rounds are propelled by the probability of achieving higher ranks and surviving to later stages where rewards are larger. These continuation options are played out in the final match, so it is only the difference between winning and losing prizes in the finals that controls incentives there. Many athletic tournaments are structured in the manner analyzed here,but the general framework ultimately may have application to certain career games as well. More generally, a tournament structure may he viewed as a statistical, experimental design problem.The prize structure interacts with the design in providing incentives for the best players to survive to the finals and win the top prizes.

Prizes and Incentives in Elimination Tournaments

The role of rewards for maintaining performance incentives in multistage, sequential games of survival is studied. The sequential structure is a statistical design-of-experiments for selecting and ranking contestants. It promotes survival of the fittest and saves sampling costs by early elimination of weaker contenders. Analysis begins with the case where competitors' talents are common knowledge and is extended to cases where talents are unknown. It is shown that extra weight must be placed on top ranking prizes to maintain performance incentives of survivors at all stages of the game. The extra weight at the top induces competitors to aspire to higher goals independent of past achievements. In career games workers have many rungs in the hierarchical ladder to aspire to in the early stages of their careers, and this plays an important role in maintaining their enthusiasm for continuing. But the further one has climbed, the fewer the rungs left to attain. If top prizes are not large enough, those who have succeeded in attaining higher ranks rest on their laurels and slack off in their attempts to climb higher. Elevating the top prizes makes the ladder appear longer for higher ranking contestants, and in the limit makes it appear of unbounded length: no matter how far one has climbed, it looks as if there is always the same length to go. Concentrating prize money on the top ranks eliminates the no-tomorrow aspects of competition in the final stages.

Labor Quality, the Demand for Skill, and Market Selection

This paper investigates some alternative definitions of labor for productivity and demand analysis. The paper is organized as follows: Section II considers the organization of work activities in a simple fixed coefficient technology in the presence of comparative advantage among various classes of workers. Assuming that the number of independent productive activities exceeds the number of comparative advantage classes, an application of the envelope theorem shows the derivation from first principles of a neoclassical production function with input dimension (the number of workers of each type) smaller than the engineering technology(the number of activities). This is the basic result illustrating that occupational classifications depend on both the technology and the distribution of skills (factor supplies) in the working population, a fact that may be relevant to international and other cross-sectional differences in productivity and the demand for labor. The situation is reversed in section III, which treats the case where the number of worker classifications exceeds the number of production activities. In this case the micro-technology cannot be reduced below the basic set of work activities one starts with, and within these categories labor can be aggregated according to efficiency units. However, the nature of factor endowments in economies of this sort is rather different than in the neoclassical model, and leads to an output transformation function that has all the neoclassical properties. This result is reminiscent of an example of Houthakker (also, see Sato) who also obtained smooth neoclassical behavioral functions from underlying distributional phenomena. Section IV examines the characteristics-factor approach to labor aggregation and relates it to the results in section III, noting an inherent difficulty arising from selectivity of various ability groups of workers among work activities due to comparative advantage. In effect, the existence of rent destroys the possibility of simple linear aggregation. Finally, section V indicates some problems with applying the theory of marriage directly to labor demand. These issues become most interesting when there are incomplete markets that limit the gains from fully exploiting comparative advantage, due to transactions costs. The results are limited, but some examples show that any predictions concerning positive or negative assortive matching of workers depends not only on the correlation of talents among members of the work force, but also on the nature of technology and the distribution of demands for various outputs.

Transactions Costs and Internal Labor Markets

The concept of transactions costs used by Coase in "The Nature of the Firm" is applied to the internal labor market of an organization. Under joint production it is shown that the number of transaction-specific prices necessary to decentralize labor allocations rises geometrically with the size of the work force. Complexity of calculation and costs of implementation constrains the possibilities for internal decentralization through a price mechanism and substitutes a more authoritarian system of allocation instead. These same issues of complexity and implementation costs limit the usefulness of agency theory as a conceptual framework for this problem. The analysis suggests that an internal labor market must be viewed in a more comprehensive framework of a personnel management system.

Labor Markets in Professional Sports

Many interesting elements of supply and demand are starkly observable in professional athletics. Understanding institutional arrangements, competitive balance and labor-management relations requires a basic understanding of sports labor markets and the struggle for control of those markets between interest groups. In this paper we treat historical and contemporary labor issues in North America and Europe, from reserve rules and free agency, high levels of player pay and work stoppages, to the distribution of playing talents across teams. We discuss the relationship between personal productivity and pay; relative versus absolute demand; competitive and cooperative interactions across firms (teams); factor substitutions; player mobility and the Coase theorem. We briefly consider how property rights affect supply, athletic talent, arms races and restrictions on competition. The problem of (excess) incentives to compete leading to externalities and inefficiencies are noted throughout the paper. Restrictive agreements such as reverse-order drafts, payroll caps and revenue sharing may constrain these forces, but they also redistribute rents from players to owners. All of these schemes, in one way or another, punish success. The European approach -- promotion of better-performing teams and relegation of those with the poorest records -- punishes failure. It remains an interesting economic question as to which system is better.

The New Economics of Teachers and Education

Rapidly growing costs of elementary and secondary education are studied in the context of the rising value of women's time. The three-fold increase in direct costs of education per student in the past three decades was caused by increasing demand and utilization of teacher and staff inputs, attributable to growing market opportunities of women and changes in the structure of families. Substitution of purchased teacher and staff inputs for own household time in the total production of children's education and maturation is a predictable economic response to these forces. On the supply side, the 'flexibility option,' that female teachers who take temporary leaves to raise children do not suffer subsequent wage loss upon reentry, is shown to be an important attraction of the teaching profession to women. Other college educated women suffer reentry wage losses of 10 percent per year of leave. The estimated value of flexibility in teaching is 5 percent of life-cycle earnings and will fall as labor force interruptions of women for child-rearing become less frequent. Both supply and demand considerations suggest that the direct costs of education per student will continue to increase in the future, independent of political and other organization reforms of schools.