A Note on the Multi-Agent Contracts in Continuous Time

RePEc: Research Papers in Economics, 2010

Dynamic Incentive Contracts under Parameter Uncertainty * We analyze a long-term contracting problem involving common uncertainty about a parameter capturing the productivity of the relationship, and featuring a hidden action for the agent. We develop an approach that works for any utility function when the parameter and noise are normally distributed and when the effort and noise affect output additively. We then analytically solve for the optimal contract when the agent has exponential utility. We find that the Pareto frontier shifts out as information about the agent's quality improves. In the standard spot-market setup, by contrast, when the parameter measures the agent's "quality", the Pareto frontier shifts inwards with better information. Commitment is therefore more valuable when quality is known more precisely. Incentives then are easier to provide because the agent has less room to manipulate the beliefs of the principal. Moreover, in contrast to results under one-period commitment, wage volatility declines as experience accumulates.

SSRN Electronic Journal, 2000

A recursive dynamic agency model is developed for situations where the state of nature follows a Markov process. The repeated agency model is a special case. It is found that the optimal e¤ort depends not only on current performance but also on past performances, and the disparity between current and past performances is a crucial determinant in the optimal contract. In a special case when both the principal and the agent are risk neutral, the …rst best is achieved by a semi-linear contract. In another special case of a repeated version, the …rst best is again achieved when the discount rate converges to zero. For the general model, a computing algorithm is developed, which can be implemented in MathCAD to …nd the solution numerically.

For a non-cooperative m-persons differential game, the value functions ofthe various players satisfy a system of Hamilton-Jacobi-Bellman equations.Nashequilibrium solutions in feedback form can be obtained by studying a related system of P.D.E's.A new approach, which is proposed in this paper allows one to construct the feedback optimal control   x   1  x, . . . ,  m  x and cost functions J i t, x 0 , i  1, . . . , m directly,i.e.,without any reference to the corresponding Hamilton-Jacobi-Bellman equations.

European Journal of Operational Research, 1999

SIAM Journal on Control and Optimization

We consider a general formulation of the random horizon principal-agent problem with a continuous payment and a lump-sum payment at termination. In the European version of the problem, the random horizon is chosen solely by the principal with no other possible action from the agent than exerting effort on the dynamics of the output process. We also consider the American version of the contract, where the agent can also quit by optimally choosing the termination time of the contract. Our main result reduces such non-zero-sum stochastic differential games to appropriate stochastic control problems which may be solved by standard methods of stochastic control theory. This reduction is obtained by following the Sannikov [22] approach, further developed in . We first introduce an appropriate class of contracts for which the agent's optimal effort is immediately characterized by the standard verification argument in stochastic control theory. We then show that this class of contracts is dense in an appropriate sense, so that the optimization over this restricted family of contracts represents no loss of generality. The result is obtained by using the recent well-posedness result of random horizon second-order backward SDE in .

International Series in Operations Research & Management Science, 2014

We consider a wide class of dynamic problems characterized by multiple, non-cooperative agents operating under a general control rule. Since each agent follows its own objective function and these functions are interdependent, control efforts made by each agent may affect the performance of the other agents and thus affect the overall performance of the system. We show that recently developed dynamic linear reward/penalty schemes can be generalized to provide coordination of the multiple agents in a broad-spectrum dynamic environment. When the reward scheme is applied, the agents are induced to choose the system-wide optimal solution even though they operate in a decentralized decision-making environment.

Annals of Operations Research

We consider a variational problem modelling the evolution with time of two probability measures representing the subjective beliefs of a couple of agents engaged in a continuous-time bargaining pricing scheme with the goal of finding a unique price for a contingent claim in a continuous-time financial market. This optimization problem is coupled with two finite dimensional portfolio optimization problems, one for each agent involved in the bargaining scheme. Under mild conditions, we prove that the optimization problem under consideration here admits a unique solution, yielding a unique price for the contingent claim. We thank the four anonymous referees for their useful comments and suggestions. N.

American Economic Journal: Microeconomics, 2015

We propose a finite-horizon continuous-time framework for coalitional bargaining, in which players can make offers at random discrete times. In our model: (i) expected payoffs in Markov perfect equilibrium (MPE) are unique, generating sharp predictions and facilitating comparative statics; and (ii) MPE are the only subgame perfect Nash equilibria (SPNE) that can be approximated by SPNE of nearby discrete-time bargaining models. We investigate the limit MPE payoffs as the time horizon goes to infinity and players get infinitely patient. In convex games, we establish that the set of these limit payoffs achievable by varying recognition rates is exactly the core of the characteristic function. (JEL C78)

Eprint Arxiv 0806 0240, 2008

We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an $R^d$-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered.

arXiv: Optimization and Control, 2020

We consider a general formulation of the random horizon Principal-Agent problem with a continuous payment and a lump-sum payment at termination. In the European version of the problem, the random horizon is chosen solely by the principal with no other possible action from the agent than exerting effort on the dynamics of the output process. We also consider the American version of the contract, which covers the seminal Sannikov's model, where the agent can also quit by optimally choosing the termination time of the contract. Our main result reduces such non-zero-sum stochastic differential games to appropriate stochastic control problems which may be solved by standard methods of stochastic control theory. This reduction is obtained by following Sannikov's approach, further developed by Cvitanic, Possamai, and Touzi. We first introduce an appropriate class of contracts for which the agent's optimal effort is immediately characterized by the standard verification argument...

Journal of Mathematical Sciences, 2008

We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an R d-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered.

For a non-cooperative m-persons differential game, the value functions ofthe various players satisfy a system of Hamilton-Jacobi-Bellman equations.Nashequilibrium solutions in feedback form can be obtained by studying a related system of P.D.E's.A new approach, which is proposed in this this paper

Automatica, 1985

In this paper we consider a general class of stochastic incentive decision problems in which the leader has access to the control value of the follower and to private as well as common information on the unknown state of nature. The follower's cost function depends on a finite number of parameters whose values are not known accurately by the leader, and in spite of this parametric uncertainty the leader seeks a policy which would induce the desired behavior on the follower. We obtain such policies for the leader, which are smooth, induce the desired behavior at the nominal values of these parameters, and furthermore make the follower's optimal reaction either minimally sensitive or totally insensitive to variations in the values of these parameters from the nominals. The general solution is determined by some orthogonality relations in some appropriately constructed (probability) measure spaces, and leads to particularly simple incentive policies. The features presented here are intrinsic to stochastic decision problems and have no counterparts in deterministic incentive problems. t

We consider multi-agent optimization problems in which one agent is a leader and the others are followers. The leader is that agent who can declare his choice of control first. We explore the concept of control structures with incentives that the leader may wish to implement in order to induce the followers to choose their control vectors in such a way that the leader's objective function is globally optimized. Such a structure is useful in modeling future military multilevel command and control systems in intelligent hostile environments.

Journal of Optimization Theory and Applications, 2012

We study optimal stochastic control problems under model uncertainty. We rewrite such problems as (zero-sum) stochastic differential games of forward-backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the non-zero sum games (finding conditions for Nash equilibria). We then apply these results to study optimal portfolio and consumption problems under model uncertainty. We combine the optimality conditions given by the stochastic maximum principles with Malliavin calculus to obtain a set of equations which determine the optimal strategies.