This paper studies optimal Public Private Partnerships contract between a public entity and a consortium, with the possibility for the public to stop the contract. The public (“she”) pays a continuous rent to the consortium (“he”), while the latter gives a response characterized by his effort. Usually, the public cannot observe the effort done by the consortium, in addition, the law of the Agent’s effort which affects the distribution of the total output is to be chosen over a set of probabilities, leading to a Principal-Agent problem with moral hazard and Knightian uncertainty characterized by $\kappa$-ignorance. Our problem is formulated as a Stackelberg leadership model between the public and the consortium. This problem is solved in three steps. The first one consists on characterizing the worst case, then we derive the Agent’s best response. In the third step, we solve the Principal problem by maximizing the social value of the project minus the rent paid, which is a standard stochastic control problem. The public value function is characterized by the solution of an associated Hamilton–Jacobi–Bellman variational inequality. The public value function, the optimal effort and the optimal rent are computed numerically in a feedback form by using the Howard algorithm. We show numerically that each increase in the degree of Knightian uncertainty leads to increase the optimal effort, and decrease the value function.

本文研究了公共实体与财团之间的最优公私合作伙伴关系合同，以及公众终止合同的可能性。公众（“她”）向财团（“他”）持续支付租金，而财团则给出以自己的努力为特征的回应。通常，公众无法观察到财团所做的努力，此外，影响总产出分配的代理人努力规律是在一组概率上选择的，导致存在道德风险的委托代理问题和奈特不确定性的特点是$\kappa$-无知。我们的问题被表述为公众和联盟之间的斯塔克尔伯格领导模型。这个问题分三步解决。第一个是描述最坏情况的特征，然后我们得出代理的最佳响应。第三步，我们通过最大化项目的社会价值减去支付的租金来解决本金问题，这是一个标准的随机控制问题。公共价值函数的特征是相关的 Hamilton-Jacobi-Bellman 变分不等式的解。使用霍华德算法以反馈形式对公共价值函数、最优努力和最优租金进行数值计算。我们用数字表明，奈特不确定性程度的每次增加都会导致最优努力的增加，并减少价值函数。