Abstract
In this chapter we study a team dynamic problem in which a group of agents collaborate over time to complete a project. The project progresses at a rate that depends on the agents’ efforts, and it generates a payoff upon completion. First, we show that agents work harder the closer the project is to completion, and members of a larger team work harder than members of a smaller team—both individually and on aggregate—if and only if the project is sufficiently far from completion. Second, we analyze the problem faced by a manager who is the residual claimant of the project and she chooses the size of the team and the agents’ incentive contracts to maximize her discounted payoff. We show that the optimal symmetric contract compensates the agents only upon completion of the project. Finally, we endogenize the size of the project, where a bigger project is one that requires greater cumulative effort and generates a larger upon completion. We show that if the manager can commit to her optimal project size at the outset of the game, then she will choose a smaller project relative to the case without commitment. An implication of this result is that without commitment, the manager is better off delegating the decision rights over the project size to the agents.
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Notes
- 1.
In practice, the relevant employees are rewarded by a combination of flow payments (i.e., periodic salary) and compensation after completion of the project. The latter can take the form of bonus lump-sum payments, stock options (that are correlated to the profit generated by the project), and reputational benefits. In the base model, we assume (for tractability) that the agents are compensated only by a lump-sum upon completion of the project. Georgiadis (2015a) also considers the case in which, in addition to a lump-sum payment upon completion, they also receive a per unit of time compensation while the project is ongoing.
- 2.
The assumptions that efforts are perfect substitutes and the project progresses deterministically are made for tractability. Georgiadis et al. (2014) and Georgiadis (2015a) also examine the case in which efforts are complementary and the project progresses stochastically, and they show that all results continue to hold.
- 3.
We implicitly assume that the agents do not face a deadline to complete the project. This assumption is made (1) for simplicity, and (2) because deadlines are generally not renegotiation proof. As a result, if the project has not been completed by the deadline, the agents find it mutually beneficial to extend the deadline. For a treatment of deadlines, see Georgiadis (2015b).
- 4.
When progress is deterministic, as Georgiadis et al. (2014) show, the game also admits non-Markovian equilibria where at every moment t, each agent’s strategy is a function of the entire path of the project {q s } s ≤ t . We use the deterministic specification as a reduced form for a stochastic process, in which case as Georgiadis (2015a) conjectures, the agents’ payoffs from the best symmetric Public Perfect equilibrium are equal to the payoffs corresponding to the MPE.
- 5.
Because the agents’ rewards are independent of the completion time, the game is stationary, and so we can drop the subscript t.
- 6.
To simplify notation, because the equilibrium is symmetric and unique, the subscript i is dropped throughout the remainder of this article. Moreover, \([\cdot ]^{+} =\max \{ \cdot, 0\}\).
- 7.
So see why such an equilibrium exists, suppose that all agents except for i exert no effort at q 0 = 0. Then agent i finds it optimal to also exert no effort, because he is not willing to undertake the entire project single-handedly (since \(\left.C\right \vert _{n=1} \geq 0\)).
- 8.
This result follows by noting that \(C = Q -\sqrt{(2V /r) \cdot (2n - 1 )/n}\) decreases in n, while both \(a\left (Q\right )\) and \(na\left (Q\right )\) decrease in n.
- 9.
The manager’s contracting space is restricted. In principle, the optimal contract should condition each agent’s payoff on the path of q t (and hence on the completion time of the project). However, when the project progresses deterministically, the problem becomes trivial as efforts effectively become contractible, and in the stochastic case the problem is not tractable. For example, the contracting approach developed in Sannikov (2008) boils down a partial differential equation with at least variables (i.e., the state of the project q and the continuation value of each agent), which is intractable.
- 10.
This case raises the question of what happens to the agents’ beliefs off the equilibrium path if the manager does not complete the project at Q NC M. Suppose that the manager did not complete the project at Q NC M so that q > Q NC M. Clearly, \(Q,\tilde{Q}> Q_{NC}^{M}\), and it is straightforward to verify that \(\partial W(q; Q,\tilde{Q})/\partial Q <0\) for all \(q,Q,\tilde{Q}> Q_{NC}^{M}\), which implies that the manager would be better off had she completed the project at Q NC M irrespective of the agents’ beliefs.
- 11.
Conceptually, this commitment problem could be resolved by allowing β to be contingent on the project size. In particular, suppose that the manager can fix β, and let \(\hat{\beta }\left (Q\right )\) equal β if Q = Q FC M, and 1 otherwise. Then, her optimal project size is equal to Q FC M regardless of her commitment power because any other project size will yield her a net profit of 0. However, this implicitly assumes that Q FC M is contractible at q = 0, which is clearly not true without commitment. Therefore, we rule out this possibility by assuming that β is independent of Q.
- 12.
Letting \(Q^{M}(q) =\arg \max _{Q\geq q}\{W(q\,; Q,Q)\}\), one can show that Q M(q) increases in q.
References
Admati AR, Perry M (1991) Joint projects without commitment. Rev Econ Stud 58:259–276
Aghion P, Tirole J (1994) The management of innovation. Q J Econ 109:1185–1209
Aghion P, Tirole J (1997) Formal and real authority in organizations. J Polit Econ 105:1–29
Bowen TR, Georgiadis G, Lambert N (2015) Collective choice in dynamic public good provision: real versus formal authority. Working paper
Cvitanić J, Georgiadis G (2015) Achieving efficiency in dynamic contribution games. Working paper
Dessein W (2002) Authority and communication in organizations’. Rev Econ Stud 69:811–838
Dixit A (1999) The art of smooth pasting. Taylor & Francis, London
Ederer FP, Georgiadis G, Nunnari S (2015) Team size effects in dynamic contribution games: experimental evidence. Working paper
Fershtman C, Nitzan S (1991) Dynamic voluntary provision of public goods. Eur Econ Rev 35:1057–1067
Freixas X, Guesnerie R, Tirole J (1985) Planning under incomplete information and the ratchet effect. Rev Econ Stud 52:173–191
Georgiadis G (2015a) Projects and team dynamics. Rev Econ Stud 82(1):187–218
Georgiadis G (2015b) Deadlines and infrequent monitoring in the dynamic provision of public goods. Working paper
Georgiadis G, Lippman SA, Tang CS (2014) Project design with limited commitment and teams. RAND J Econ 45(3):598–623
Grossman SJ, Hart OD (1986) The cost and benefits of ownership: a theory of vertical and lateral integration. J Polit Econ 94:691–719
Harvard Business School (2004) Managing teams: forming a team that makes a difference. Harvard Business School Press, Boston
Ichniowski C, Shaw K (2003) Beyond incentive pay: insiders’ estimates of the value of complementary human resource management practices. J Econ Perspect 17(1):155–180
Kessing SG (2007) Strategic complementarity in the dynamic private provision of a discrete public good. J Public Econ Theory 9:699–710
Laffont J-J, Tirole J (1988) The dynamics of incentive contracts. Econometrica 56:1153–1175
Lakhani KR, Carlile PR (2012) Myelin repair foundation: accelerating drug discovery through collaboration. HBS Case No. 9-610-074, Harvard Business School, Boston
Lawler EE, Mohrman SA, Benson G (2001) Organizing for high performance: employee involvement, TQM, reengineering, and knowledge management in the fortune 1000. Jossey-Bass, San Francisco
Marx LM, Matthews SA (2000) Dynamic voluntary contribution to a public project. Rev Econ Stud 67:327–358
Olson M (1965) The logic of collective action: public goods and the theory of groups. Harvard University Press, Cambridge
Sannikov Y (2008) A continuous-time version of the principal-agent problem. Rev Econ Stud 75:957–984
Tirole J (1999) Incomplete contracts: where do we stand? Econometrica 67:741–781
Wired Magazine (2004) http://www.wired.com/gadgets/mac/news/2004/07/64286? Accessed 28 June 2015
Yildirim H (2006) Getting the ball rolling: voluntary contributions to a large scale public project. J Public Econ Theory 8:503–528
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Georgiadis, G., Tang, C.S. (2017). Project Contracting Strategies for Managing Team Dynamics. In: Ha, A., Tang, C. (eds) Handbook of Information Exchange in Supply Chain Management. Springer Series in Supply Chain Management, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-32441-8_5
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