Abstract
We present a novel modeling approach for supervised teams, which can determine optimal incentives when individual team member contributions are unknown. Our approach is based on multiscale decision theory, which models the agents’ decision processes and their mutual influence. To estimate the initially unknown influence of team members on their supervisor’s success, we develop a linear approximation method that estimates model parameters from historic team performance data. In our analysis, we derive the optimal incentives the supervisor should offer to team members accounting for their varying skill levels. In addition, we identify the information and communication requirements between all agents such that the supervisor can calculate the optimal incentives, and such that team members can calculate their optimal effort responses. We illustrate our methods and the results through a systems engineering example.
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Alchian, A. A., & Demsetz, H. (1972). Production, information costs, and economic organization. The American Economic Review, 62(5), 777–795.
Baker, G., Gibbons, R., & Murphy, K. J. (1994). Subjective performance measures in optimal incentive contracts. The Quarterly Journal of Economics, 109(4), 1125–1156.
Baker, G. P. (1992). Incentive contracts and performance measurement. Journal of Political Economy, 100(3), 598–614.
Baker, G. P., Jensen, M. C., & Murphy, K. J. (1988). Compensation and incentives: Practice vs. theory. The Journal of Finance, 43(3), 593–616.
Bar-Yam, Y. (2004). Multiscale variety in complex systems. Complexity, 9(4), 37–45.
Beling, P. A. (2013). Multi-scale decision making: Challenges in engineering and environmental systems. Environment Systems and Decisions, 33(3), 323–325.
Bolton, P., & Dewatripont, M. (2005). Contract theory. Cambridge, MA: MIT Press.
Caillaud, B., Guesnerie, R., & Rey, P. (1992). Noisy observation in adverse selection models. The Review of Economic Studies, 59(3), 595–615.
Che, Y. K., & Yoo, S. W. (2001). Optimal incentives for teams. American Economic Review, 91(3), 525–541.
Coase, R. H. (1937). The nature of the firm. Economica, 4(16), 386–405.
Dolgov, D., & Durfee, E. (2004). Graphical models in local, asymmetric multi-agent Markov decision processes. In Proceedings of the third international joint conference on autonomous agents and multiagent systems (Vol. 2, pp. 956–963). IEEE Computer Society.
Fudenberg, D., & Tirole, J. (1990). Moral hazard and renegotiation in agency contracts. Econometrica: Journal of the Econometric Society, 58, 1279–1319.
Gibbons, R. (1998). Incentives in organizations. Journal of Economic Perspectives, 12(4), 115–132.
Green, J. R., & Stokey, N. L. (1983). A comparison of tournaments and contracts. Journal of Political Economy, 91(3), 349–364.
Grossman, S. J., & Hart, O. D. (1983). An analysis of the principal-agent problem. Econometrica: Journal of the Econometric Society, 51(1), 7–45.
Groves, T. (1973). Incentives in teams. Econometrica: Journal of the Econometric Society, 41(4), 617–631.
Henry, A., & Wernz, C. (2010). Optimal incentives in three-level agent systems. In Proceedings of the 2010 industrial engineering research conference, Cancun, Mexico (pp. 1–6).
Henry, A., & Wernz, C. (2015). A multiscale decision theory analysis for revenue sharing in three-stage supply chains. Annals of Operations Research, 226(1), 277–300.
Hölmstrom, B. (1979). Moral hazard and observability. The Bell Journal of Economics, 10(1), 74–91.
Holmstrom, B. (1982). Moral hazard in teams. The Bell Journal of Economics, 13(2), 324–340.
Holmstrom, B., & Milgrom, P. (1991). Multitask principal-agent analyses: Incentive contracts, asset ownership, and job design. The Journal of Law, Economics, & Organization, 7, 24.
Hong, S., Wernz, C., & Stillinger, J. D. (2016). Optimizing maintenance service contracts through mechanism design theory. Applied Mathematical Modelling, 40(21–22), 8849–8861.
Itoh, H. (1991). Incentives to help in multi-agent situations. Econometrica: Journal of the Econometric Society, 59, 611–636.
Itoh, H. (2001). Job design and incentives in hierarchies with team production. Hitotsubashi Journal of Commerce and Management, 36(1), 1–17.
Kendall, D.L., & Salas, E. (2004). Measuring team performance: Review of current methods and consideration of future needs. In J. Ness, V. Tepe, & D. Ritzer (Eds.), The science and simulation of human performance (pp. 307–326). Emerald Group Publishing Limited.
Laffont, J. J., & Martimort, D. (2009). The theory of incentives: The principal-agent model. Princeton, NJ: Princeton University Press.
Lazear, E. P., & Rosen, S. (1981). Rank-order tournaments as optimum labor contracts. Journal of Political Economy, 89(5), 841–864.
Levin, J. (2003). Relational incentive contracts. The American Economic Review, 93(3), 835–857.
Levinthal, D. (1988). A survey of agency models of organizations. Journal of Economic Behavior & Organization, 9(2), 153–185.
Marschak, J., & Radner, R. (1972). Economic theory of teams. New Haven, CT: Yale University Press.
Mesarovic, M. D., Macko, D., & Takahara, Y. (1970). Theory of hierarchical, multilevel, systems. New York, NY: Academic Press.
Mirrlees, J. A. (1999). The theory of moral hazard and unobservable behaviour: Part I. The Review of Economic Studies, 66(1), 3–21.
Mookherjee, D. (2006). Decentralization, hierarchies, and incentives: A mechanism design perspective. Journal of Economic Literature, 44(2), 367–390.
Nalebuff, B. J., & Stiglitz, J. E. (1983). Prizes and incentives: Towards a general theory of compensation and competition. The Bell Journal of Economics, 14(1), 21–43.
Prendergast, C. (1999). The provision of incentives in firms. Journal of Economic Literature, 37(1), 7–63.
Puterman, M. L. (2014). Markov decision processes: Discrete stochastic dynamic programming. Hoboken, NJ: Wiley.
Radner, R. (1991). Dynamic games in organization theory. Journal of Economic Behavior & Organization, 16(1), 217–260.
Radner, R. (1992). Hierarchy: The economics of managing. Journal of Economic Literature, 30(3), 1382–1415.
Rudin, W. (1964). Principles of mathematical analysis (Vol. 3). New York, NY: McGraw-Hill.
Sinuany-Stern, Z. (2014). Quadratic model for allocating operational budget in public and nonprofit organizations. Annals of Operations Research, 221(1), 357–376.
Thompson, R. G., & George, M. D. (1984). A stochastic investiment model for a survival conscious firm. Annals of Operations Research, 2(1), 157–182.
Wernz, C. (2008). Multiscale decision-making: Bridging temporal and organizational scales in hierarchical systems. Ph.D. thesis, University of Massachusetts, Amherst
Wernz, C. (2013). Multi-time-scale Markov decision processes for organizational decision-making. EURO Journal on Decision Processes, 1(3), 299–324.
Wernz, C., & Deshmukh, A. (2007a). Decision strategies and design of agent interactions in hierarchical manufacturing systems. Journal of Manufacturing Systems, 26(2), 135–143.
Wernz, C., & Deshmukh, A. (2007b). Managing hierarchies in a flat world. In Proceedings of the 2007 industrial engineering research conference, Nashville, TN (pp. 1266–1271).
Wernz, C., & Deshmukh, A. (2009). An incentive-based, multi-period decision model for hierarchical systems. In Proceedings of the 3rd annual conference of the Indian Subcontinent Decision Sciences Institute Region (ISDSI), Hyderabad, India (pp. 12–17).
Wernz, C., & Deshmukh, A. (2010a). Multi-time-scale decision making for strategic agent interactions. In Proceedings of the 2010 industrial engineering research conference, Cancun, Mexico (pp. 1–6).
Wernz, C., & Deshmukh, A. (2010b). Multiscale decision-making: Bridging organizational scales in systems with distributed decision-makers. European Journal of Operational Research, 202(3), 828–840.
Wernz, C., & Deshmukh, A. (2012). Unifying temporal and organizational scales in multiscale decision-making. European Journal of Operational Research, 223(3), 739–751.
Wernz, C., & Henry, A. (2009). Multi-level coordination and decision-making in service operations. Service Science, 1(4), 270–283.
Zhang, H., Wernz, C., & Slonim, A. D. (2016). Aligning incentives in health care: A multiscale decision theory approach. EURO Journal on Decision Processes, 4(3), 219–244.
Zhang, H., Wernz, C., & Hughes, D. R. (2018). Modeling and designing health care payment innovations for medical imaging. Health Care Management Science, 21(1), 37–51.
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This research was funded in part by the National Science Foundation (Grants 1549896 and 1762336), and the VCU Presidential Research Quest Fund.
Appendices
Appendix A
Using equation (12), the first order derivative of \(e^{\text {INF}x}_{*}(b^{\text {INF}x})\) is given by
Using Eqs. (12) and (24), the second order derivative of \(e^{\text {INF}x}_{*}(b^{\text {INF}x})\) is given by
Appendix B
Proof of Theorem 2
From Eq. (24), we know that \(\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x}) \big / \mathrm {d}b^{\text {INF}x} > 0\) for all x and for all \(b \in (0,1]^n\). In addition, if \( \displaystyle { \frac{ \partial ^2 }{\partial (e^{\text {INF}x})^2} \Big (\frac{\partial }{\partial e^{\text {INF}x}} R^{\mathrm {INF}x}(e,b) \Big ) \le 0 }\), then
for all x and for all e. From Eq. (25) it follows that \(\mathrm {d}^2e^{\text {INF}x}_{*}(b^{\text {INF}x}) \big / \mathrm {d}(b^{\text {INF}x})^2 < 0\) for all x and for all \(b \in (0,1]^n\). The second order partial derivatives of \(R^{\mathrm {SUP}}(b)\) are
and
for \(w \in \{1,\ldots ,n\}\) and \(w \ne x\). Since all the second order partial derivatives of \(R^{\mathrm {SUP}}(b)\) are negative, we know that the Hessian matrix of \(R^{\mathrm {SUP}}(b)\) is negative definite for all \(b \in (0,1]^n\). This implies that \(R^{\mathrm {SUP}}(b)\) is strictly concave for all \(b \in (0,1]^n\). \(\square \)
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Kulkarni, A.U., Wernz, C. Optimal incentives for teams: a multiscale decision theory approach. Ann Oper Res 288, 307–329 (2020). https://doi.org/10.1007/s10479-019-03478-7
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DOI: https://doi.org/10.1007/s10479-019-03478-7