{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:21:33Z","timestamp":1760059293993,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,6,4]],"date-time":"2025-06-04T00:00:00Z","timestamp":1748995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>Establishing a partnership necessitates agreeing on how to divide future profits or losses. We consider parties who wish to contract on the division of uncertain future profits. We propose to divide profits according to the Raiffa\u2013Kalai\u2013Smorodinsky (K-S) solution, which is the intersection point of the feasible region\u2019s boundary and the line connecting the disagreement and ideal points. It is the only function which satisfies invariance to linear transformations, symmetry, strong Pareto optimality, and monotonicity. We formulate the general problem of designing a contract which divides uncertain future profit between the partners and determines shares of each partner. We first focus on linear and, later, non-linear contracts between two partners, providing analytical and numerical solutions for various special cases in terms of the utility functions of the partners, their beliefs, and the disagreement point. We then generalize the analysis to any number of partners. We also consider a contract which is partially based on the parties\u2019 financial contribution to the partnership, which have a positive impact on profit. Finally, we address asymmetric K-S solutions. K-S solutions are seen to be a useful predictor of the outcome of negotiations, similar to Nash\u2019s bargaining solution.<\/jats:p>","DOI":"10.3390\/g16030029","type":"journal-article","created":{"date-parts":[[2025,6,4]],"date-time":"2025-06-04T11:14:12Z","timestamp":1749035652000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Raiffa\u2013Kalai\u2013Smorodinsky Solution as a Mechanism for Dividing the Uncertain Future Profit of a Partnership"],"prefix":"10.3390","volume":"16","author":[{"given":"Yigal","family":"Gerchak","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, Tel Aviv University, Tel-Aviv 69978, Israel"}]},{"given":"Eugene","family":"Khmelnitsky","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, Tel Aviv University, Tel-Aviv 69978, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/S0165-1765(01)00426-8","article-title":"An asymmetric Kalai-Smorodinsky solution","volume":"73","author":"Dubra","year":"2001","journal-title":"Economics Letters"},{"key":"ref_2","first-page":"429","article-title":"Performance measure congruity and diversity in multi-task principal\/agent relations","volume":"69","author":"Feltham","year":"1994","journal-title":"The Accounting Review"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5868","DOI":"10.1287\/mnsc.2021.4184","article-title":"Negotiations in competing supply chains: The Kalai-Smorodinsky bargaining solution","volume":"68","author":"Feng","year":"2022","journal-title":"Management Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s40070-019-00095-4","article-title":"Bargaining over shares of uncertain future profits","volume":"7","author":"Gerchak","year":"2019a","journal-title":"EURO Journal on Decision Processes"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0219198919400085","article-title":"Partnership\u2019s profit sharing: Linear and non-linear contracts","volume":"21","author":"Gerchak","year":"2019b","journal-title":"International Game Theory Review"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kilgour, D. M., and Eden, C. (2021). Sharing profit and risk in partnership. Handbook of group decision and negotiation, Springer.","DOI":"10.1007\/978-3-030-49629-6"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"513","DOI":"10.2307\/1914280","article-title":"Other solutions to Nash\u2019s bargaining problem","volume":"43","author":"Kalai","year":"1975","journal-title":"Econometrica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1007\/s00182-018-0611-4","article-title":"A generalization of the Egalitarian and the Kalai-Smorodinsky bargaining solutions","volume":"47","author":"Karos","year":"2018","journal-title":"International Journal of Game Theory"},{"key":"ref_9","first-page":"131","article-title":"Profit sharing and the role of professional partnerships","volume":"120","author":"Levin","year":"2005","journal-title":"The Quarterly Journal of Economics"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Muthoo, A. (1999). Bargaining theory with applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511607950"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"128","DOI":"10.2307\/1906951","article-title":"Two-person cooperative games","volume":"21","author":"Nash","year":"1953","journal-title":"Econometrica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2307\/1909607","article-title":"The theory of syndicates","volume":"36","author":"Wilson","year":"1968","journal-title":"Econometrica"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/16\/3\/29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:46:46Z","timestamp":1760032006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/16\/3\/29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,4]]},"references-count":12,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["g16030029"],"URL":"https:\/\/doi.org\/10.3390\/g16030029","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2025,6,4]]}}}