Income Pools for Superstar Markets
Bibliographic record
Abstract
“Superstar” markets, where a small portion of individuals earn disproportionately high incomes, are common in fields like entrepreneurship, sports, and entertainment. Participants in these markets face significant income uncertainty, which can deter entry or prompt early exit. To address this difficulty, we propose income pools: contracts where individuals agree to share a portion of future earnings with pool members if a specific salary milestone is achieved. To date, hundreds of income pool contracts have been signed in practice. This paper develops the first mathematical model to analyze such contracts, focusing on stability (i.e., pools where no agents leave or join). When constrained to creating a single pool, we show that a stable pool always exists with specific structural properties. Generally, stronger agents tend to be more “collaborative” and favor larger pools, whereas weaker agents are more “selfish” and prefer smaller pools. Next, we analyze pools that adhere to a maximum size (mirroring current practice) and show that stability persists. These pools typically require more weaker agents than strong ones and we find an interesting “Pareto dominance” result, whereby all agents in a stable pool prefer a particular unique stable pool. Finally, we study general partition structures, prove that a stable partition always exists under certain conditions, and provide an algorithm to construct such a partition. We conclude with a case study on data from 2,000 professional baseball players to demonstrate a 20%–30% increase in social welfare if players join income pools, under varying contract parameters. This paper was accepted by Victor Martinez de Albeniz, operations management. Funding: T. C. Y. Chan received funding support from the Natural Sciences and Engineering Research Council of Canada. N. Chen has no funding sources to report. C. Fernandes received funding support from the Natural Sciences and Engineering Research Council of Canada and the TD Management Data and Analytics Lab. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2024.07480 .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.019 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".